Theorem assaassrd 33649

Description: Right-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
assaassrd	⊢ (𝜑 → (𝑋 × (𝐴 · 𝑌)) = (𝐴 · (𝑋 × 𝑌)))

Proof of Theorem assaassrd

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	assaassr 21826	. 2 ⊢ ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ 𝐵 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → (𝑋 × (𝐴 · 𝑌)) = (𝐴 · (𝑋 × 𝑌)))
11	1, 2, 3, 4, 10	syl13anc 1375	1 ⊢ (𝜑 → (𝑋 × (𝐴 · 𝑌)) = (𝐴 · (𝑋 × 𝑌)))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  ‘cfv 6500  (class class class)co 7368  Basecbs 17148  ._rcmulr 17190  Scalarcsca 17192   ·_𝑠 cvsca 17193  AssAlgcasa 21817

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-nul 5253

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rab 3402  df-v 3444  df-sbc 3743  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-iota 6456  df-fv 6508  df-ov 7371  df-assa 21820

This theorem is referenced by: (None)