text: # Metamath Proof Explorer
  
  ## Theorem fmul01lt1
  
  Description: Given a finite multiplication of values betweeen 0 and 1, a value E larger than any multiplicand, is larger than the whole multiplication. (Contributed by Glauco Siliprandi, 20-Apr-2017)
  
  Ref Expression
  Hypotheses fmul01lt1.1 ${⊢}\underset{_}{Ⅎ}{i}\phantom{\rule{.4em}{0ex}}{B}$
  fmul01lt1.2 ${⊢}Ⅎ{i}\phantom{\rule{.4em}{0ex}}{\phi }$
  fmul01lt1.3 ${⊢}\underset{_}{Ⅎ}{j}\phantom{\rule{.4em}{0ex}}{A}$
  fmul01lt1.4 ${⊢}{A}=seq1\left(×,{B}\right)$
  fmul01lt1.5 ${⊢}{\phi }\to {M}\in ℕ$
  fmul01lt1.6 ${⊢}{\phi }\to {B}:\left(1\dots {M}\right)⟶ℝ$
  fmul01lt1.7 ${⊢}\left({\phi }\wedge {i}\in \left(1\dots {M}\right)\right)\to 0\le {B}\left({i}\right)$
  fmul01lt1.8 ${⊢}\left({\phi }\wedge {i}\in \left(1\dots {M}\right)\right)\to {B}\left({i}\right)\le 1$
  fmul01lt1.9 ${⊢}{\phi }\to {E}\in {ℝ}^{+}$
  fmul01lt1.10 ${⊢}{\phi }\to \exists {j}\in \left(1\dots {M}\right)\phantom{\rule{.4em}{0ex}}{B}\left({j}\right)<{E}$
  Assertion fmul01lt1 ${⊢}{\phi }\to {A}\left({M}\right)<{E}$
  
  ### Proof
  
  Step Hyp Ref Expression
  1 fmul01lt1.1 ${⊢}\underset{_}{Ⅎ}{i}\phantom{\rule{.4em}{0ex}}{B}$
  2 fmul01lt1.2 ${⊢}Ⅎ{i}\phantom{\rule{.4em}{0ex}}{\phi }$
  3 fmul01lt1.3 ${⊢}\underset{_}{Ⅎ}{j}\phantom{\rule{.4em}{0ex}}{A}$
  4 fmul01lt1.4 ${⊢}{A}=seq1\left(×,{B}\right)$
  5 fmul01lt1.5 ${⊢}{\phi }\to {M}\in ℕ$
  6 fmul01lt1.6 ${⊢}{\phi }\to {B}:\left(1\dots {M}\right)⟶ℝ$
  7 fmul01lt1.7 ${⊢}\left({\phi }\wedge {i}\in \left(1\dots {M}\right)\right)\to 0\le {B}\left({i}\right)$
  8 fmul01lt1.8 ${⊢}\left({\phi }\wedge {i}\in \left(1\dots {M}\right)\right)\to {B}\left({i}\right)\le 1$
  9 fmul01lt1.9 ${⊢}{\phi }\to {E}\in {ℝ}^{+}$
  10 fmul01lt1.10 ${⊢}{\phi }\to \exists {j}\in \left(1\dots {M}\right)\phantom{\rule{.4em}{0ex}}{B}\left({j}\right)<{E}$
  11 nfv ${⊢}Ⅎ{j}\phantom{\rule{.4em}{0ex}}{\phi }$
  12 nfcv ${⊢}\underset{_}{Ⅎ}{j}\phantom{\rule{.4em}{0ex}}{M}$
  13 3 12 nffv ${⊢}\underset{_}{Ⅎ}{j}\phantom{\rule{.4em}{0ex}}{A}\left({M}\right)$
  14 nfcv ${⊢}\underset{_}{Ⅎ}{j}\phantom{\rule{.4em}{0ex}}<$
  15 nfcv ${⊢}\underset{_}{Ⅎ}{j}\phantom{\rule{.4em}{0ex}}{E}$
  16 13 14 15 nfbr ${⊢}Ⅎ{j}\phantom{\rule{.4em}{0ex}}{A}\left({M}\right)<{E}$
  17 nfv ${⊢}Ⅎ{i}\phantom{\rule{.4em}{0ex}}{j}\in \left(1\dots {M}\right)$
  18 nfcv ${⊢}\underset{_}{Ⅎ}{i}\phantom{\rule{.4em}{0ex}}{j}$
  19 1 18 nffv ${⊢}\underset{_}{Ⅎ}{i}\phantom{\rule{.4em}{0ex}}{B}\left({j}\right)$
  20 nfcv ${⊢}\underset{_}{Ⅎ}{i}\phantom{\rule{.4em}{0ex}}<$
  21 nfcv ${⊢}\underset{_}{Ⅎ}{i}\phantom{\rule{.4em}{0ex}}{E}$
  22 19 20 21 nfbr ${⊢}Ⅎ{i}\phantom{\rule{.4em}{0ex}}{B}\left({j}\right)<{E}$
  23 2 17 22 nf3an ${⊢}Ⅎ{i}\phantom{\rule{.4em}{0ex}}\left({\phi }\wedge {j}\in \left(1\dots {M}\right)\wedge {B}\left({j}\right)<{E}\right)$
  24 1zzd ${⊢}\left({\phi }\wedge {j}\in \left(1\dots {M}\right)\wedge {B}\left({j}\right)<{E}\right)\to 1\in ℤ$
  25 elnnuz ${⊢}{M}\in ℕ↔{M}\in {ℤ}_{\ge 1}$
  26 5 25 sylib ${⊢}{\phi }\to {M}\in {ℤ}_{\ge 1}$
  27 26 3ad2ant1 ${⊢}\left({\phi }\wedge {j}\in \left(1\dots {M}\right)\wedge {B}\left({j}\right)<{E}\right)\to {M}\in {ℤ}_{\ge 1}$
  28 6 ffvelrnda ${⊢}\left({\phi }\wedge {i}\in \left(1\dots {M}\right)\right)\to {B}\left({i}\right)\in ℝ$
  29 28 3ad2antl1 ${⊢}\left(\left({\phi }\wedge {j}\in \left(1\dots {M}\right)\wedge {B}\left({j}\right)<{E}\right)\wedge {i}\in \left(1\dots {M}\right)\right)\to {B}\left({i}\right)\in ℝ$
  30 7 3ad2antl1 ${⊢}\left(\left({\phi }\wedge {j}\in \left(1\dots {M}\right)\wedge {B}\left({j}\right)<{E}\right)\wedge {i}\in \left(1\dots {M}\right)\right)\to 0\le {B}\left({i}\right)$
  31 8 3ad2antl1 ${⊢}\left(\left({\phi }\wedge {j}\in \left(1\dots {M}\right)\wedge {B}\left({j}\right)<{E}\right)\wedge {i}\in \left(1\dots {M}\right)\right)\to {B}\left({i}\right)\le 1$
  32 9 3ad2ant1 ${⊢}\left({\phi }\wedge {j}\in \left(1\dots {M}\right)\wedge {B}\left({j}\right)<{E}\right)\to {E}\in {ℝ}^{+}$
  33 simp2 ${⊢}\left({\phi }\wedge {j}\in \left(1\dots {M}\right)\wedge {B}\left({j}\right)<{E}\right)\to {j}\in \left(1\dots {M}\right)$
  34 simp3 ${⊢}\left({\phi }\wedge {j}\in \left(1\dots {M}\right)\wedge {B}\left({j}\right)<{E}\right)\to {B}\left({j}\right)<{E}$
  35 1 23 4 24 27 29 30 31 32 33 34 fmul01lt1lem2 ${⊢}\left({\phi }\wedge {j}\in \left(1\dots {M}\right)\wedge {B}\left({j}\right)<{E}\right)\to {A}\left({M}\right)<{E}$
  36 35 3exp ${⊢}{\phi }\to \left({j}\in \left(1\dots {M}\right)\to \left({B}\left({j}\right)<{E}\to {A}\left({M}\right)<{E}\right)\right)$
  37 11 16 36 rexlimd ${⊢}{\phi }\to \left(\exists {j}\in \left(1\dots {M}\right)\phantom{\rule{.4em}{0ex}}{B}\left({j}\right)<{E}\to {A}\left({M}\right)<{E}\right)$
  38 10 37 mpd ${⊢}{\phi }\to {A}\left({M}\right)<{E}$
explanation: The provided text is a detailed mathematical theorem and proof from the Metamath Proof Explorer. It describes a theorem (fmul01lt1) that asserts a property of finite multiplications of values between 0 and 1. The theorem states that if there is a value \( E \) larger than any multiplicand, then \( E \) is larger than the whole multiplication. The text includes hypotheses, an assertion, and a step-by-step proof.
  
  The document does not contain a question or an answer to a question. It is purely a formal mathematical theorem and its proof.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: # Metamath Proof Explorer
  
  ## Theorem fmul01lt1
  
  Description: Given a finite multiplication of values betweeen 0 and 1, a value E larger than any multiplicand, is larger than the whole multiplication. (Contributed by Glauco Siliprandi, 20-Apr-2017)
  
  Ref Expression
  Hypotheses fmul01lt1.1 ${⊢}\underset{_}{Ⅎ}{i}\phantom{\rule{.4em}{0ex}}{B}$
  fmul01lt1.2 ${⊢}Ⅎ{i}\phantom{\rule{.4em}{0ex}}{\phi }$
  fmul01lt1.3 ${⊢}\underset{_}{Ⅎ}{j}\phantom{\rule{.4em}{0ex}}{A}$
  fmul01lt1.4 ${⊢}{A}=seq1\left(×,{B}\right)$
  fmul01lt1.5 ${⊢}{\phi }\to {M}\in ℕ$
  fmul01lt1.6 ${⊢}{\phi }\to {B}:\left(1\dots {M}\right)⟶ℝ$
  fmul01lt1.7 ${⊢}\left({\phi }\wedge {i}\in \left(1\dots {M}\right)\right)\to 0\le {B}\left({i}\right)$
  fmul01lt1.8 ${⊢}\left({\phi }\wedge {i}\in \left(1\dots {M}\right)\right)\to {B}\left({i}\right)\le 1$
  fmul01lt1.9 ${⊢}{\phi }\to {E}\in {ℝ}^{+}$
  fmul01lt1.10 ${⊢}{\phi }\to \exists {j}\in \left(1\dots {M}\right)\phantom{\rule{.4em}{0ex}}{B}\left({j}\right)<{E}$
  Assertion fmul01lt1 ${⊢}{\phi }\to {A}\left({M}\right)<{E}$
  
  ### Proof
  
  Step Hyp Ref Expression
  1 fmul01lt1.1 ${⊢}\underset{_}{Ⅎ}{i}\phantom{\rule{.4em}{0ex}}{B}$
  2 fmul01lt1.2 ${⊢}Ⅎ{i}\phantom{\rule{.4em}{0ex}}{\phi }$
  3 fmul01lt1.3 ${⊢}\underset{_}{Ⅎ}{j}\phantom{\rule{.4em}{0ex}}{A}$
  4 fmul01lt1.4 ${⊢}{A}=seq1\left(×,{B}\right)$
  5 fmul01lt1.5 ${⊢}{\phi }\to {M}\in ℕ$
  6 fmul01lt1.6 ${⊢}{\phi }\to {B}:\left(1\dots {M}\right)⟶ℝ$
  7 fmul01lt1.7 ${⊢}\left({\phi }\wedge {i}\in \left(1\dots {M}\right)\right)\to 0\le {B}\left({i}\right)$
  8 fmul01lt1.8 ${⊢}\left({\phi }\wedge {i}\in \left(1\dots {M}\right)\right)\to {B}\left({i}\right)\le 1$
  9 fmul01lt1.9 ${⊢}{\phi }\to {E}\in {ℝ}^{+}$
  10 fmul01lt1.10 ${⊢}{\phi }\to \exists {j}\in \left(1\dots {M}\right)\phantom{\rule{.4em}{0ex}}{B}\left({j}\right)<{E}$
  11 nfv ${⊢}Ⅎ{j}\phantom{\rule{.4em}{0ex}}{\phi }$
  12 nfcv ${⊢}\underset{_}{Ⅎ}{j}\phantom{\rule{.4em}{0ex}}{M}$
  13 3 12 nffv ${⊢}\underset{_}{Ⅎ}{j}\phantom{\rule{.4em}{0ex}}{A}\left({M}\right)$
  14 nfcv ${⊢}\underset{_}{Ⅎ}{j}\phantom{\rule{.4em}{0ex}}<$
  15 nfcv ${⊢}\underset{_}{Ⅎ}{j}\phantom{\rule{.4em}{0ex}}{E}$
  16 13 14 15 nfbr ${⊢}Ⅎ{j}\phantom{\rule{.4em}{0ex}}{A}\left({M}\right)<{E}$
  17 nfv ${⊢}Ⅎ{i}\phantom{\rule{.4em}{0ex}}{j}\in \left(1\dots {M}\right)$
  18 nfcv ${⊢}\underset{_}{Ⅎ}{i}\phantom{\rule{.4em}{0ex}}{j}$
  19 1 18 nffv ${⊢}\underset{_}{Ⅎ}{i}\phantom{\rule{.4em}{0ex}}{B}\left({j}\right)$
  20 nfcv ${⊢}\underset{_}{Ⅎ}{i}\phantom{\rule{.4em}{0ex}}<$
  21 nfcv ${⊢}\underset{_}{Ⅎ}{i}\phantom{\rule{.4em}{0ex}}{E}$
  22 19 20 21 nfbr ${⊢}Ⅎ{i}\phantom{\rule{.4em}{0ex}}{B}\left({j}\right)<{E}$
  23 2 17 22 nf3an ${⊢}Ⅎ{i}\phantom{\rule{.4em}{0ex}}\left({\phi }\wedge {j}\in \left(1\dots {M}\right)\wedge {B}\left({j}\right)<{E}\right)$
  24 1zzd ${⊢}\left({\phi }\wedge {j}\in \left(1\dots {M}\right)\wedge {B}\left({j}\right)<{E}\right)\to 1\in ℤ$
  25 elnnuz ${⊢}{M}\in ℕ↔{M}\in {ℤ}_{\ge 1}$
  26 5 25 sylib ${⊢}{\phi }\to {M}\in {ℤ}_{\ge 1}$
  27 26 3ad2ant1 ${⊢}\left({\phi }\wedge {j}\in \left(1\dots {M}\right)\wedge {B}\left({j}\right)<{E}\right)\to {M}\in {ℤ}_{\ge 1}$
  28 6 ffvelrnda ${⊢}\left({\phi }\wedge {i}\in \left(1\dots {M}\right)\right)\to {B}\left({i}\right)\in ℝ$
  29 28 3ad2antl1 ${⊢}\left(\left({\phi }\wedge {j}\in \left(1\dots {M}\right)\wedge {B}\left({j}\right)<{E}\right)\wedge {i}\in \left(1\dots {M}\right)\right)\to {B}\left({i}\right)\in ℝ$
  30 7 3ad2antl1 ${⊢}\left(\left({\phi }\wedge {j}\in \left(1\dots {M}\right)\wedge {B}\left({j}\right)<{E}\right)\wedge {i}\in \left(1\dots {M}\right)\right)\to 0\le {B}\left({i}\right)$
  31 8 3ad2antl1 ${⊢}\left(\left({\phi }\wedge {j}\in \left(1\dots {M}\right)\wedge {B}\left({j}\right)<{E}\right)\wedge {i}\in \left(1\dots {M}\right)\right)\to {B}\left({i}\right)\le 1$
  32 9 3ad2ant1 ${⊢}\left({\phi }\wedge {j}\in \left(1\dots {M}\right)\wedge {B}\left({j}\right)<{E}\right)\to {E}\in {ℝ}^{+}$
  33 simp2 ${⊢}\left({\phi }\wedge {j}\in \left(1\dots {M}\right)\wedge {B}\left({j}\right)<{E}\right)\to {j}\in \left(1\dots {M}\right)$
  34 simp3 ${⊢}\left({\phi }\wedge {j}\in \left(1\dots {M}\right)\wedge {B}\left({j}\right)<{E}\right)\to {B}\left({j}\right)<{E}$
  35 1 23 4 24 27 29 30 31 32 33 34 fmul01lt1lem2 ${⊢}\left({\phi }\wedge {j}\in \left(1\dots {M}\right)\wedge {B}\left({j}\right)<{E}\right)\to {A}\left({M}\right)<{E}$
  36 35 3exp ${⊢}{\phi }\to \left({j}\in \left(1\dots {M}\right)\to \left({B}\left({j}\right)<{E}\to {A}\left({M}\right)<{E}\right)\right)$
  37 11 16 36 rexlimd ${⊢}{\phi }\to \left(\exists {j}\in \left(1\dots {M}\right)\phantom{\rule{.4em}{0ex}}{B}\left({j}\right)<{E}\to {A}\left({M}\right)<{E}\right)$
  38 10 37 mpd ${⊢}{\phi }\to {A}\left({M}\right)<{E}$
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The provided text is a detailed mathematical theorem and proof from the Metamath Proof Explorer. It describes a theorem (fmul01lt1) that asserts a property of finite multiplications of values between 0 and 1. The theorem states that if there is a value \( E \) larger than any multiplicand, then \( E \) is larger than the whole multiplication. The text includes hypotheses, an assertion, and a step-by-step proof.
  
  The document does not contain a question or an answer to a question. It is purely a formal mathematical theorem and its proof.
  
  Label: no-question