Theorem assaassd 34087

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 22166	. 2 ⊢ ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ 𝐵 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → ((𝐴 · 𝑋) × 𝑌) = (𝐴 · (𝑋 × 𝑌)))
11	1, 2, 3, 4, 10	syl13anc 1399	1 ⊢ (𝜑 → ((𝐴 · 𝑋) × 𝑌) = (𝐴 · (𝑋 × 𝑌)))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  ‘cfv 6538  (class class class)co 7420  Basecbs 17387  ._rcmulr 17429  Scalarcsca 17431   ·_𝑠 cvsca 17432  AssAlgcasa 22158

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 6494  df-fv 6546  df-ov 7423  df-assa 22161

This theorem is used by:  vietalem  34211