Theorem assaassr 19366

Description: Right-associative property of an associative algebra. (Contributed by Mario Carneiro, 29-Dec-2014.)

Hypotheses

Ref	Expression
isassa.v	⊢ 𝑉 = (Base‘𝑊)
isassa.f	⊢ 𝐹 = (Scalar‘𝑊)
isassa.b	⊢ 𝐵 = (Base‘𝐹)
isassa.s	⊢ · = ( ·𝑠 ‘𝑊)
isassa.t	⊢ × = (.r‘𝑊)

Assertion

Ref	Expression
assaassr	⊢ ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ 𝐵 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → (𝑋 × (𝐴 · 𝑌)) = (𝐴 · (𝑋 × 𝑌)))

Proof of Theorem assaassr

Step	Hyp	Ref	Expression
1		isassa.v	. . 3 ⊢ 𝑉 = (Base‘𝑊)
2		isassa.f	. . 3 ⊢ 𝐹 = (Scalar‘𝑊)
3		isassa.b	. . 3 ⊢ 𝐵 = (Base‘𝐹)
4		isassa.s	. . 3 ⊢ · = ( ·𝑠 ‘𝑊)
5		isassa.t	. . 3 ⊢ × = (.r‘𝑊)
6	1, 2, 3, 4, 5	assalem 19364	. 2 ⊢ ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ 𝐵 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → (((𝐴 · 𝑋) × 𝑌) = (𝐴 · (𝑋 × 𝑌)) ∧ (𝑋 × (𝐴 · 𝑌)) = (𝐴 · (𝑋 × 𝑌))))
7	6	simprd 478	1 ⊢ ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ 𝐵 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → (𝑋 × (𝐴 · 𝑌)) = (𝐴 · (𝑋 × 𝑌)))

Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1054   = wceq 1523   ∈ wcel 2030  ‘cfv 5926  (class class class)co 6690  Basecbs 15904  ._rcmulr 15989  Scalarcsca 15991   ·_𝑠 cvsca 15992  AssAlgcasa 19357

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-nul 4822

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-iota 5889  df-fv 5934  df-ov 6693  df-assa 19360

This theorem is referenced by:  assa2ass  19370  issubassa  19372  asclmul2  19388  asclrhm  19390  assamulgscmlem2  19397  mplmon2mul  19549  matinv  20531  cpmadugsumlemC  20728