R : Type ?u.480281
A : Type ?u.480284
inst✝² : CommSemiring R
inst✝¹ : Semiring A
inst✝ : Algebra R A
p s : Submodule R A
h1 : 1 ∈ s
hmul : ∀ (x y : A), x ∈ s → y ∈ s → x * y ∈ s
r : R
⊢ ↑(algebraMap R A) r ∈
    { toSubmonoid := { toSubsemigroup := { carrier := ↑s, mul_mem' := hmul }, one_mem' := h1 },
            add_mem' := (_ : ∀ {a b : A}, a ∈ s → b ∈ s → a + b ∈ s),
            zero_mem' := (_ : 0 ∈ s) }.toSubmonoid.toSubsemigroup.carrier
R : Type ?u.480281
A : Type ?u.480284
inst✝² : CommSemiring R
inst✝¹ : Semiring A
inst✝ : Algebra R A
p s : Submodule R A
h1 : 1 ∈ s
hmul : ∀ (x y : A), x ∈ s → y ∈ s → x * y ∈ s
⊢ ∀ (r : R),
    ↑(algebraMap R A) r ∈
      { toSubmonoid := { toSubsemigroup := { carrier := ↑s, mul_mem' := hmul }, one_mem' := h1 },
              add_mem' := (_ : ∀ {a b : A}, a ∈ s → b ∈ s → a + b ∈ s),
              zero_mem' := (_ : 0 ∈ s) }.toSubmonoid.toSubsemigroup.carrier
intro r
R : Type ?u.480281
A : Type ?u.480284
inst✝² : CommSemiring R
inst✝¹ : Semiring A
inst✝ : Algebra R A
p s : Submodule R A
h1 : 1 ∈ s
hmul : ∀ (x y : A), x ∈ s → y ∈ s → x * y ∈ s
r : R
⊢ r • 1 ∈
    { toSubmonoid := { toSubsemigroup := { carrier := ↑s, mul_mem' := hmul }, one_mem' := h1 },
            add_mem' := (_ : ∀ {a b : A}, a ∈ s → b ∈ s → a + b ∈ s),
            zero_mem' := (_ : 0 ∈ s) }.toSubmonoid.toSubsemigroup.carrier
R : Type ?u.480281
A : Type ?u.480284
inst✝² : CommSemiring R
inst✝¹ : Semiring A
inst✝ : Algebra R A
p s : Submodule R A
h1 : 1 ∈ s
hmul : ∀ (x y : A), x ∈ s → y ∈ s → x * y ∈ s
r : R
⊢ ↑(algebraMap R A) r ∈
    { toSubmonoid := { toSubsemigroup := { carrier := ↑s, mul_mem' := hmul }, one_mem' := h1 },
            add_mem' := (_ : ∀ {a b : A}, a ∈ s → b ∈ s → a + b ∈ s),
            zero_mem' := (_ : 0 ∈ s) }.toSubmonoid.toSubsemigroup.carrier
rw [Algebra.algebraMap_eq_smul_one]
no goals
R : Type ?u.480281
A : Type ?u.480284
inst✝² : CommSemiring R
inst✝¹ : Semiring A
inst✝ : Algebra R A
p s : Submodule R A
h1 : 1 ∈ s
hmul : ∀ (x y : A), x ∈ s → y ∈ s → x * y ∈ s
r : R
⊢ r • 1 ∈
    { toSubmonoid := { toSubsemigroup := { carrier := ↑s, mul_mem' := hmul }, one_mem' := h1 },
            add_mem' := (_ : ∀ {a b : A}, a ∈ s → b ∈ s → a + b ∈ s),
            zero_mem' := (_ : 0 ∈ s) }.toSubmonoid.toSubsemigroup.carrier
apply s.smul_mem _ h1