Theorem assalem 22007

Description: The properties 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
assalem	⊢ ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ 𝐵 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → (((𝐴 · 𝑋) × 𝑌) = (𝐴 · (𝑋 × 𝑌)) ∧ (𝑋 × (𝐴 · 𝑌)) = (𝐴 · (𝑋 × 𝑌))))

Proof of Theorem assalem

Dummy variables 𝑥 𝑟 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isassa.v	. . . 4 ⊢ 𝑉 = (Base‘𝑊)
2		isassa.f	. . . 4 ⊢ 𝐹 = (Scalar‘𝑊)
3		isassa.b	. . . 4 ⊢ 𝐵 = (Base‘𝐹)
4		isassa.s	. . . 4 ⊢ · = ( ·𝑠 ‘𝑊)
5		isassa.t	. . . 4 ⊢ × = (.r‘𝑊)
6	1, 2, 3, 4, 5	isassa 22006	. . 3 ⊢ (𝑊 ∈ AssAlg ↔ ((𝑊 ∈ LMod ∧ 𝑊 ∈ Ring) ∧ ∀𝑟 ∈ 𝐵 ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)) ∧ (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦)))))
7	6	simprbi 502	. 2 ⊢ (𝑊 ∈ AssAlg → ∀𝑟 ∈ 𝐵 ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)) ∧ (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦))))
8		oveq1 7417	. . . . . 6 ⊢ (𝑟 = 𝐴 → (𝑟 · 𝑥) = (𝐴 · 𝑥))
9	8	oveq1d 7425	. . . . 5 ⊢ (𝑟 = 𝐴 → ((𝑟 · 𝑥) × 𝑦) = ((𝐴 · 𝑥) × 𝑦))
10		oveq1 7417	. . . . 5 ⊢ (𝑟 = 𝐴 → (𝑟 · (𝑥 × 𝑦)) = (𝐴 · (𝑥 × 𝑦)))
11	9, 10	eqeq12d 2779	. . . 4 ⊢ (𝑟 = 𝐴 → (((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)) ↔ ((𝐴 · 𝑥) × 𝑦) = (𝐴 · (𝑥 × 𝑦))))
12		oveq1 7417	. . . . . 6 ⊢ (𝑟 = 𝐴 → (𝑟 · 𝑦) = (𝐴 · 𝑦))
13	12	oveq2d 7426	. . . . 5 ⊢ (𝑟 = 𝐴 → (𝑥 × (𝑟 · 𝑦)) = (𝑥 × (𝐴 · 𝑦)))
14	13, 10	eqeq12d 2779	. . . 4 ⊢ (𝑟 = 𝐴 → ((𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦)) ↔ (𝑥 × (𝐴 · 𝑦)) = (𝐴 · (𝑥 × 𝑦))))
15	11, 14	anbi12d 643	. . 3 ⊢ (𝑟 = 𝐴 → ((((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)) ∧ (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦))) ↔ (((𝐴 · 𝑥) × 𝑦) = (𝐴 · (𝑥 × 𝑦)) ∧ (𝑥 × (𝐴 · 𝑦)) = (𝐴 · (𝑥 × 𝑦)))))
16		oveq2 7418	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝐴 · 𝑥) = (𝐴 · 𝑋))
17	16	oveq1d 7425	. . . . 5 ⊢ (𝑥 = 𝑋 → ((𝐴 · 𝑥) × 𝑦) = ((𝐴 · 𝑋) × 𝑦))
18		oveq1 7417	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑥 × 𝑦) = (𝑋 × 𝑦))
19	18	oveq2d 7426	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝐴 · (𝑥 × 𝑦)) = (𝐴 · (𝑋 × 𝑦)))
20	17, 19	eqeq12d 2779	. . . 4 ⊢ (𝑥 = 𝑋 → (((𝐴 · 𝑥) × 𝑦) = (𝐴 · (𝑥 × 𝑦)) ↔ ((𝐴 · 𝑋) × 𝑦) = (𝐴 · (𝑋 × 𝑦))))
21		oveq1 7417	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥 × (𝐴 · 𝑦)) = (𝑋 × (𝐴 · 𝑦)))
22	21, 19	eqeq12d 2779	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝑥 × (𝐴 · 𝑦)) = (𝐴 · (𝑥 × 𝑦)) ↔ (𝑋 × (𝐴 · 𝑦)) = (𝐴 · (𝑋 × 𝑦))))
23	20, 22	anbi12d 643	. . 3 ⊢ (𝑥 = 𝑋 → ((((𝐴 · 𝑥) × 𝑦) = (𝐴 · (𝑥 × 𝑦)) ∧ (𝑥 × (𝐴 · 𝑦)) = (𝐴 · (𝑥 × 𝑦))) ↔ (((𝐴 · 𝑋) × 𝑦) = (𝐴 · (𝑋 × 𝑦)) ∧ (𝑋 × (𝐴 · 𝑦)) = (𝐴 · (𝑋 × 𝑦)))))
24		oveq2 7418	. . . . 5 ⊢ (𝑦 = 𝑌 → ((𝐴 · 𝑋) × 𝑦) = ((𝐴 · 𝑋) × 𝑌))
25		oveq2 7418	. . . . . 6 ⊢ (𝑦 = 𝑌 → (𝑋 × 𝑦) = (𝑋 × 𝑌))
26	25	oveq2d 7426	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝐴 · (𝑋 × 𝑦)) = (𝐴 · (𝑋 × 𝑌)))
27	24, 26	eqeq12d 2779	. . . 4 ⊢ (𝑦 = 𝑌 → (((𝐴 · 𝑋) × 𝑦) = (𝐴 · (𝑋 × 𝑦)) ↔ ((𝐴 · 𝑋) × 𝑌) = (𝐴 · (𝑋 × 𝑌))))
28		oveq2 7418	. . . . . 6 ⊢ (𝑦 = 𝑌 → (𝐴 · 𝑦) = (𝐴 · 𝑌))
29	28	oveq2d 7426	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝑋 × (𝐴 · 𝑦)) = (𝑋 × (𝐴 · 𝑌)))
30	29, 26	eqeq12d 2779	. . . 4 ⊢ (𝑦 = 𝑌 → ((𝑋 × (𝐴 · 𝑦)) = (𝐴 · (𝑋 × 𝑦)) ↔ (𝑋 × (𝐴 · 𝑌)) = (𝐴 · (𝑋 × 𝑌))))
31	27, 30	anbi12d 643	. . 3 ⊢ (𝑦 = 𝑌 → ((((𝐴 · 𝑋) × 𝑦) = (𝐴 · (𝑋 × 𝑦)) ∧ (𝑋 × (𝐴 · 𝑦)) = (𝐴 · (𝑋 × 𝑦))) ↔ (((𝐴 · 𝑋) × 𝑌) = (𝐴 · (𝑋 × 𝑌)) ∧ (𝑋 × (𝐴 · 𝑌)) = (𝐴 · (𝑋 × 𝑌)))))
32	15, 23, 31	rspc3v 3597	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (∀𝑟 ∈ 𝐵 ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)) ∧ (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦))) → (((𝐴 · 𝑋) × 𝑌) = (𝐴 · (𝑋 × 𝑌)) ∧ (𝑋 × (𝐴 · 𝑌)) = (𝐴 · (𝑋 × 𝑌)))))
33	7, 32	mpan9 515	1 ⊢ ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ 𝐵 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → (((𝐴 · 𝑋) × 𝑌) = (𝐴 · (𝑋 × 𝑌)) ∧ (𝑋 × (𝐴 · 𝑌)) = (𝐴 · (𝑋 × 𝑌))))

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  ∀wral 3079  ‘cfv 6536  (class class class)co 7410  Basecbs 17264  ._rcmulr 17306  Scalarcsca 17308   ·_𝑠 cvsca 17309  Ringcrg 20310  LModclmod 20981  AssAlgcasa 22000

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-nul 5269

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rab 3417  df-v 3457  df-sbc 3745  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-iota 6492  df-fv 6544  df-ov 7413  df-assa 22003

This theorem is referenced by:  assaass  22008  assaassr  22009