Theorem assaassd 33585

Description: Left-associative property of an associative algebra, deduction version. (Contributed by Thierry Arnoux, 15-Feb-2026.)

Hypotheses

Ref	Expression
assaassd.1	⊢ 𝑉 = (Base‘𝑊)
assaassd.2	⊢ 𝐹 = (Scalar‘𝑊)
assaassd.3	⊢ 𝐵 = (Base‘𝐹)
assaassd.4	⊢ · = ( ·𝑠 ‘𝑊)
assaassd.5	⊢ × = (.r‘𝑊)
assaassd.6	⊢ (𝜑 → 𝑊 ∈ AssAlg)
assaassd.7	⊢ (𝜑 → 𝐴 ∈ 𝐵)
assaassd.8	⊢ (𝜑 → 𝑋 ∈ 𝑉)
assaassd.9	⊢ (𝜑 → 𝑌 ∈ 𝑉)

Assertion

Ref	Expression
assaassd	⊢ (𝜑 → ((𝐴 · 𝑋) × 𝑌) = (𝐴 · (𝑋 × 𝑌)))

Proof of Theorem assaassd

Step	Hyp	Ref	Expression
1		assaassd.6	. 2 ⊢ (𝜑 → 𝑊 ∈ AssAlg)
2		assaassd.7	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
3		assaassd.8	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
4		assaassd.9	. 2 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
5		assaassd.1	. . 3 ⊢ 𝑉 = (Base‘𝑊)
6		assaassd.2	. . 3 ⊢ 𝐹 = (Scalar‘𝑊)
7		assaassd.3	. . 3 ⊢ 𝐵 = (Base‘𝐹)
8		assaassd.4	. . 3 ⊢ · = ( ·𝑠 ‘𝑊)
9		assaassd.5	. . 3 ⊢ × = (.r‘𝑊)
10	5, 6, 7, 8, 9	assaass 21811	. 2 ⊢ ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ 𝐵 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → ((𝐴 · 𝑋) × 𝑌) = (𝐴 · (𝑋 × 𝑌)))
11	1, 2, 3, 4, 10	syl13anc 1374	1 ⊢ (𝜑 → ((𝐴 · 𝑋) × 𝑌) = (𝐴 · (𝑋 × 𝑌)))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113  ‘cfv 6490  (class class class)co 7356  Basecbs 17134  ._rcmulr 17176  Scalarcsca 17178   ·_𝑠 cvsca 17179  AssAlgcasa 21803

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2706  ax-nul 5249

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-ne 2931  df-ral 3050  df-rab 3398  df-v 3440  df-sbc 3739  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-iota 6446  df-fv 6498  df-ov 7359  df-assa 21806

This theorem is referenced by:  vietalem  33684