Theorem isassa 21781

Description: The properties of an associative algebra. (Contributed by Mario Carneiro, 29-Dec-2014.) (Revised by SN, 2-Mar-2025.)

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
isassa	⊢ (𝑊 ∈ AssAlg ↔ ((𝑊 ∈ LMod ∧ 𝑊 ∈ Ring) ∧ ∀𝑟 ∈ 𝐵 ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)) ∧ (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦)))))

Distinct variable groups:   𝑥,𝑟,𝑦   𝐵,𝑟   𝐹,𝑟   𝑉,𝑟,𝑥,𝑦   · ,𝑟,𝑥,𝑦   × ,𝑟,𝑥,𝑦   𝑊,𝑟,𝑥,𝑦

Allowed substitution hints:   𝐵(𝑥,𝑦)   𝐹(𝑥,𝑦)

Proof of Theorem isassa

Dummy variables 𝑓 𝑤 𝑠 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvexd 6841	. . . 4 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) ∈ V)
2		fveq2 6826	. . . . 5 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊))
3		isassa.f	. . . . 5 ⊢ 𝐹 = (Scalar‘𝑊)
4	2, 3	eqtr4di 2782	. . . 4 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = 𝐹)
5		fveq2 6826	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (Base‘𝑓) = (Base‘𝐹))
6		isassa.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐹)
7	5, 6	eqtr4di 2782	. . . . . 6 ⊢ (𝑓 = 𝐹 → (Base‘𝑓) = 𝐵)
8	7	adantl 481	. . . . 5 ⊢ ((𝑤 = 𝑊 ∧ 𝑓 = 𝐹) → (Base‘𝑓) = 𝐵)
9		fveq2 6826	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
10		isassa.v	. . . . . . . 8 ⊢ 𝑉 = (Base‘𝑊)
11	9, 10	eqtr4di 2782	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝑉)
12		isassa.s	. . . . . . . . 9 ⊢ · = ( ·𝑠 ‘𝑊)
13		isassa.t	. . . . . . . . 9 ⊢ × = (.r‘𝑊)
14		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝑠 = · ∧ 𝑡 = × ) → 𝑡 = × )
15		simpl 482	. . . . . . . . . . . . 13 ⊢ ((𝑠 = · ∧ 𝑡 = × ) → 𝑠 = · )
16	15	oveqd 7370	. . . . . . . . . . . 12 ⊢ ((𝑠 = · ∧ 𝑡 = × ) → (𝑟𝑠𝑥) = (𝑟 · 𝑥))
17		eqidd 2730	. . . . . . . . . . . 12 ⊢ ((𝑠 = · ∧ 𝑡 = × ) → 𝑦 = 𝑦)
18	14, 16, 17	oveq123d 7374	. . . . . . . . . . 11 ⊢ ((𝑠 = · ∧ 𝑡 = × ) → ((𝑟𝑠𝑥)𝑡𝑦) = ((𝑟 · 𝑥) × 𝑦))
19		eqidd 2730	. . . . . . . . . . . 12 ⊢ ((𝑠 = · ∧ 𝑡 = × ) → 𝑟 = 𝑟)
20	14	oveqd 7370	. . . . . . . . . . . 12 ⊢ ((𝑠 = · ∧ 𝑡 = × ) → (𝑥𝑡𝑦) = (𝑥 × 𝑦))
21	15, 19, 20	oveq123d 7374	. . . . . . . . . . 11 ⊢ ((𝑠 = · ∧ 𝑡 = × ) → (𝑟𝑠(𝑥𝑡𝑦)) = (𝑟 · (𝑥 × 𝑦)))
22	18, 21	eqeq12d 2745	. . . . . . . . . 10 ⊢ ((𝑠 = · ∧ 𝑡 = × ) → (((𝑟𝑠𝑥)𝑡𝑦) = (𝑟𝑠(𝑥𝑡𝑦)) ↔ ((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦))))
23		eqidd 2730	. . . . . . . . . . . 12 ⊢ ((𝑠 = · ∧ 𝑡 = × ) → 𝑥 = 𝑥)
24	15	oveqd 7370	. . . . . . . . . . . 12 ⊢ ((𝑠 = · ∧ 𝑡 = × ) → (𝑟𝑠𝑦) = (𝑟 · 𝑦))
25	14, 23, 24	oveq123d 7374	. . . . . . . . . . 11 ⊢ ((𝑠 = · ∧ 𝑡 = × ) → (𝑥𝑡(𝑟𝑠𝑦)) = (𝑥 × (𝑟 · 𝑦)))
26	25, 21	eqeq12d 2745	. . . . . . . . . 10 ⊢ ((𝑠 = · ∧ 𝑡 = × ) → ((𝑥𝑡(𝑟𝑠𝑦)) = (𝑟𝑠(𝑥𝑡𝑦)) ↔ (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦))))
27	22, 26	anbi12d 632	. . . . . . . . 9 ⊢ ((𝑠 = · ∧ 𝑡 = × ) → ((((𝑟𝑠𝑥)𝑡𝑦) = (𝑟𝑠(𝑥𝑡𝑦)) ∧ (𝑥𝑡(𝑟𝑠𝑦)) = (𝑟𝑠(𝑥𝑡𝑦))) ↔ (((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)) ∧ (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦)))))
28	12, 13, 27	sbcie2s 17090	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → ([( ·𝑠 ‘𝑤) / 𝑠][(.r‘𝑤) / 𝑡](((𝑟𝑠𝑥)𝑡𝑦) = (𝑟𝑠(𝑥𝑡𝑦)) ∧ (𝑥𝑡(𝑟𝑠𝑦)) = (𝑟𝑠(𝑥𝑡𝑦))) ↔ (((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)) ∧ (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦)))))
29	11, 28	raleqbidv 3310	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (∀𝑦 ∈ (Base‘𝑤)[( ·𝑠 ‘𝑤) / 𝑠][(.r‘𝑤) / 𝑡](((𝑟𝑠𝑥)𝑡𝑦) = (𝑟𝑠(𝑥𝑡𝑦)) ∧ (𝑥𝑡(𝑟𝑠𝑦)) = (𝑟𝑠(𝑥𝑡𝑦))) ↔ ∀𝑦 ∈ 𝑉 (((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)) ∧ (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦)))))
30	11, 29	raleqbidv 3310	. . . . . 6 ⊢ (𝑤 = 𝑊 → (∀𝑥 ∈ (Base‘𝑤)∀𝑦 ∈ (Base‘𝑤)[( ·𝑠 ‘𝑤) / 𝑠][(.r‘𝑤) / 𝑡](((𝑟𝑠𝑥)𝑡𝑦) = (𝑟𝑠(𝑥𝑡𝑦)) ∧ (𝑥𝑡(𝑟𝑠𝑦)) = (𝑟𝑠(𝑥𝑡𝑦))) ↔ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)) ∧ (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦)))))
31	30	adantr 480	. . . . 5 ⊢ ((𝑤 = 𝑊 ∧ 𝑓 = 𝐹) → (∀𝑥 ∈ (Base‘𝑤)∀𝑦 ∈ (Base‘𝑤)[( ·𝑠 ‘𝑤) / 𝑠][(.r‘𝑤) / 𝑡](((𝑟𝑠𝑥)𝑡𝑦) = (𝑟𝑠(𝑥𝑡𝑦)) ∧ (𝑥𝑡(𝑟𝑠𝑦)) = (𝑟𝑠(𝑥𝑡𝑦))) ↔ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)) ∧ (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦)))))
32	8, 31	raleqbidv 3310	. . . 4 ⊢ ((𝑤 = 𝑊 ∧ 𝑓 = 𝐹) → (∀𝑟 ∈ (Base‘𝑓)∀𝑥 ∈ (Base‘𝑤)∀𝑦 ∈ (Base‘𝑤)[( ·𝑠 ‘𝑤) / 𝑠][(.r‘𝑤) / 𝑡](((𝑟𝑠𝑥)𝑡𝑦) = (𝑟𝑠(𝑥𝑡𝑦)) ∧ (𝑥𝑡(𝑟𝑠𝑦)) = (𝑟𝑠(𝑥𝑡𝑦))) ↔ ∀𝑟 ∈ 𝐵 ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)) ∧ (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦)))))
33	1, 4, 32	sbcied2 3789	. . 3 ⊢ (𝑤 = 𝑊 → ([(Scalar‘𝑤) / 𝑓]∀𝑟 ∈ (Base‘𝑓)∀𝑥 ∈ (Base‘𝑤)∀𝑦 ∈ (Base‘𝑤)[( ·𝑠 ‘𝑤) / 𝑠][(.r‘𝑤) / 𝑡](((𝑟𝑠𝑥)𝑡𝑦) = (𝑟𝑠(𝑥𝑡𝑦)) ∧ (𝑥𝑡(𝑟𝑠𝑦)) = (𝑟𝑠(𝑥𝑡𝑦))) ↔ ∀𝑟 ∈ 𝐵 ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)) ∧ (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦)))))
34		df-assa 21778	. . 3 ⊢ AssAlg = {𝑤 ∈ (LMod ∩ Ring) ∣ [(Scalar‘𝑤) / 𝑓]∀𝑟 ∈ (Base‘𝑓)∀𝑥 ∈ (Base‘𝑤)∀𝑦 ∈ (Base‘𝑤)[( ·𝑠 ‘𝑤) / 𝑠][(.r‘𝑤) / 𝑡](((𝑟𝑠𝑥)𝑡𝑦) = (𝑟𝑠(𝑥𝑡𝑦)) ∧ (𝑥𝑡(𝑟𝑠𝑦)) = (𝑟𝑠(𝑥𝑡𝑦)))}
35	33, 34	elrab2 3653	. 2 ⊢ (𝑊 ∈ AssAlg ↔ (𝑊 ∈ (LMod ∩ Ring) ∧ ∀𝑟 ∈ 𝐵 ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)) ∧ (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦)))))
36		elin 3921	. . 3 ⊢ (𝑊 ∈ (LMod ∩ Ring) ↔ (𝑊 ∈ LMod ∧ 𝑊 ∈ Ring))
37	36	anbi1i 624	. 2 ⊢ ((𝑊 ∈ (LMod ∩ Ring) ∧ ∀𝑟 ∈ 𝐵 ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)) ∧ (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦)))) ↔ ((𝑊 ∈ LMod ∧ 𝑊 ∈ Ring) ∧ ∀𝑟 ∈ 𝐵 ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)) ∧ (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦)))))
38	35, 37	bitri 275	1 ⊢ (𝑊 ∈ AssAlg ↔ ((𝑊 ∈ LMod ∧ 𝑊 ∈ Ring) ∧ ∀𝑟 ∈ 𝐵 ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)) ∧ (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦)))))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  Vcvv 3438  [wsbc 3744   ∩ cin 3904  ‘cfv 6486  (class class class)co 7353  Basecbs 17138  ._rcmulr 17180  Scalarcsca 17182   ·_𝑠 cvsca 17183  Ringcrg 20136  LModclmod 20781  AssAlgcasa 21775

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-nul 5248

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rab 3397  df-v 3440  df-sbc 3745  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-iota 6442  df-fv 6494  df-ov 7356  df-assa 21778

This theorem is referenced by:  assalem  21782  assalmod  21785  assaring  21786  isassad  21790  assapropd  21797