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  Theorem grplactfval 18195
   Description: The left group action of element 𝐴 of group 𝐺. (Contributed by Paul Chapman, 18-Mar-2008.)
  Hypotheses
  Ref Expression
  grplact.1 𝐹 = (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔 + 𝑎)))
  grplact.2 𝑋 = (Base‘𝐺)
  Assertion
  Ref Expression
  grplactfval (𝐴𝑋 → (𝐹𝐴) = (𝑎𝑋 ↦ (𝐴 + 𝑎)))
  Distinct variable groups:   𝑔,𝑎,𝐴   𝐺,𝑎,𝑔   + ,𝑎,𝑔   𝑋,𝑎,𝑔
  Allowed substitution hints:   𝐹(𝑔,𝑎)
  
  Proof of Theorem grplactfval
  StepHypRef Expression
  1 oveq1 7142 . . 3 (𝑔 = 𝐴 → (𝑔 + 𝑎) = (𝐴 + 𝑎))
  21mpteq2dv 5126 . 2 (𝑔 = 𝐴 → (𝑎𝑋 ↦ (𝑔 + 𝑎)) = (𝑎𝑋 ↦ (𝐴 + 𝑎)))
  3 grplact.1 . 2 𝐹 = (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔 + 𝑎)))
  4 grplact.2 . 2 𝑋 = (Base‘𝐺)
  52, 3, 4mptfvmpt 6968 1 (𝐴𝑋 → (𝐹𝐴) = (𝑎𝑋 ↦ (𝐴 + 𝑎)))
   Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111   ↦ cmpt 5110  ‘cfv 6324  (class class class)co 7135  Basecbs 16477 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pr 5295 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138 This theorem is referenced by:  grplactval  18196  grplactcnv  18197  eqglact  18326  eqgen  18328  tgplacthmeo  22715  tgpconncompeqg  22724
   Copyright terms: Public domain W3C validator
explanation: The provided text is a formal proof from the Metamath Proof Explorer, a project that aims to develop a comprehensive database of mathematical proofs. The text describes a theorem (grplactfval) related to the left group action of an element of a group. It includes the theorem's hypotheses, assertion, and proof steps. The text does not contain a question or an answer to a question.
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distilabel_metadata.content: Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  grplactfval Structured version   Visualization version   GIF version
  
  Theorem grplactfval 18195
   Description: The left group action of element 𝐴 of group 𝐺. (Contributed by Paul Chapman, 18-Mar-2008.)
  Hypotheses
  Ref Expression
  grplact.1 𝐹 = (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔 + 𝑎)))
  grplact.2 𝑋 = (Base‘𝐺)
  Assertion
  Ref Expression
  grplactfval (𝐴𝑋 → (𝐹𝐴) = (𝑎𝑋 ↦ (𝐴 + 𝑎)))
  Distinct variable groups:   𝑔,𝑎,𝐴   𝐺,𝑎,𝑔   + ,𝑎,𝑔   𝑋,𝑎,𝑔
  Allowed substitution hints:   𝐹(𝑔,𝑎)
  
  Proof of Theorem grplactfval
  StepHypRef Expression
  1 oveq1 7142 . . 3 (𝑔 = 𝐴 → (𝑔 + 𝑎) = (𝐴 + 𝑎))
  21mpteq2dv 5126 . 2 (𝑔 = 𝐴 → (𝑎𝑋 ↦ (𝑔 + 𝑎)) = (𝑎𝑋 ↦ (𝐴 + 𝑎)))
  3 grplact.1 . 2 𝐹 = (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔 + 𝑎)))
  4 grplact.2 . 2 𝑋 = (Base‘𝐺)
  52, 3, 4mptfvmpt 6968 1 (𝐴𝑋 → (𝐹𝐴) = (𝑎𝑋 ↦ (𝐴 + 𝑎)))
   Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111   ↦ cmpt 5110  ‘cfv 6324  (class class class)co 7135  Basecbs 16477 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pr 5295 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138 This theorem is referenced by:  grplactval  18196  grplactcnv  18197  eqglact  18326  eqgen  18328  tgplacthmeo  22715  tgpconncompeqg  22724
   Copyright terms: Public domain W3C validator
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distilabel_metadata.raw_output_q_a_detector_0: The provided text is a formal proof from the Metamath Proof Explorer, a project that aims to develop a comprehensive database of mathematical proofs. The text describes a theorem (grplactfval) related to the left group action of an element of a group. It includes the theorem's hypotheses, assertion, and proof steps. The text does not contain a question or an answer to a question.
  
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