Theorem assaassd 33790

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 21977	. 2 ⊢ ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ 𝐵 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → ((𝐴 · 𝑋) × 𝑌) = (𝐴 · (𝑋 × 𝑌)))
11	1, 2, 3, 4, 10	syl13anc 1397	1 ⊢ (𝜑 → ((𝐴 · 𝑋) × 𝑌) = (𝐴 · (𝑋 × 𝑌)))

Syntax hints:   → wi 4   = wceq 1567   ∈ wcel 2149  ‘cfv 6537  (class class class)co 7411  Basecbs 17269  ._rcmulr 17311  Scalarcsca 17313   ·_𝑠 cvsca 17314  AssAlgcasa 21969

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-nul 5271

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-ral 3086  df-rab 3424  df-v 3465  df-sbc 3754  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-iota 6493  df-fv 6545  df-ov 7414  df-assa 21972

This theorem is referenced by:  vietalem  33914