Theorem assaass 22146

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 22145	. 2 ⊢ ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ 𝐵 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → (((𝐴 · 𝑋) × 𝑌) = (𝐴 · (𝑋 × 𝑌)) ∧ (𝑋 × (𝐴 · 𝑌)) = (𝐴 · (𝑋 × 𝑌))))
7	6	simpld 500	1 ⊢ ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ 𝐵 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → ((𝐴 · 𝑋) × 𝑌) = (𝐴 · (𝑋 × 𝑌)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ‘cfv 6531  (class class class)co 7412  Basecbs 17367  ._rcmulr 17409  Scalarcsca 17411   ·_𝑠 cvsca 17412  AssAlgcasa 22138

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-nul 5260

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rab 3414  df-v 3453  df-sbc 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-iota 6487  df-fv 6539  df-ov 7415  df-assa 22141

This theorem is used by:  assa2ass  22151  assa2ass2  22152  issubassa3  22154  asclmul1  22174  assamulgscmlem2  22188  mplmon2mul  22358  matinv  22972  assaassd  34069