Theorem assaass 20820

Description: Left-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
assaass	⊢ ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ 𝐵 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → ((𝐴 · 𝑋) × 𝑌) = (𝐴 · (𝑋 × 𝑌)))

Proof of Theorem assaass

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 20819	. 2 ⊢ ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ 𝐵 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → (((𝐴 · 𝑋) × 𝑌) = (𝐴 · (𝑋 × 𝑌)) ∧ (𝑋 × (𝐴 · 𝑌)) = (𝐴 · (𝑋 × 𝑌))))
7	6	simpld 498	1 ⊢ ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ 𝐵 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → ((𝐴 · 𝑋) × 𝑌) = (𝐴 · (𝑋 × 𝑌)))

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1089   = wceq 1543   ∈ wcel 2110  ‘cfv 6380  (class class class)co 7213  Basecbs 16760  ._rcmulr 16803  Scalarcsca 16805   ·_𝑠 cvsca 16806  AssAlgcasa 20812

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-nul 5199

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-sbc 3695  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-iota 6338  df-fv 6388  df-ov 7216  df-assa 20815

This theorem is referenced by:  assa2ass  20825  issubassa3  20827  asclmul1  20845  ascldimulOLD  20848  assamulgscmlem2  20860  mplmon2mul  21027  matinv  21574