Theorem isassa 15002

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		fveq2 5695	. . . 4 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊))
2		fveq2 5695	. . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
3		fveq2 5695	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → ( ·𝑠 ‘𝑤) = ( ·𝑠 ‘𝑊))
4		fveq2 5695	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (.r‘𝑤) = (.r‘𝑊))
5	4	sbceq1d 3056	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → ([(.r‘𝑤) / 𝑡](((𝑟𝑠𝑥)𝑡𝑦) = (𝑟𝑠(𝑥𝑡𝑦)) ∧ (𝑥𝑡(𝑟𝑠𝑦)) = (𝑟𝑠(𝑥𝑡𝑦))) ↔ [(.r‘𝑊) / 𝑡](((𝑟𝑠𝑥)𝑡𝑦) = (𝑟𝑠(𝑥𝑡𝑦)) ∧ (𝑥𝑡(𝑟𝑠𝑦)) = (𝑟𝑠(𝑥𝑡𝑦)))))
6	3, 5	sbceqbid 3058	. . . . . . 7 ⊢ (𝑤 = 𝑊 → ([( ·𝑠 ‘𝑤) / 𝑠][(.r‘𝑤) / 𝑡](((𝑟𝑠𝑥)𝑡𝑦) = (𝑟𝑠(𝑥𝑡𝑦)) ∧ (𝑥𝑡(𝑟𝑠𝑦)) = (𝑟𝑠(𝑥𝑡𝑦))) ↔ [( ·𝑠 ‘𝑊) / 𝑠][(.r‘𝑊) / 𝑡](((𝑟𝑠𝑥)𝑡𝑦) = (𝑟𝑠(𝑥𝑡𝑦)) ∧ (𝑥𝑡(𝑟𝑠𝑦)) = (𝑟𝑠(𝑥𝑡𝑦)))))
7	2, 6	raleqbidv 2765	. . . . . 6 ⊢ (𝑤 = 𝑊 → (∀𝑦 ∈ (Base‘𝑤)[( ·𝑠 ‘𝑤) / 𝑠][(.r‘𝑤) / 𝑡](((𝑟𝑠𝑥)𝑡𝑦) = (𝑟𝑠(𝑥𝑡𝑦)) ∧ (𝑥𝑡(𝑟𝑠𝑦)) = (𝑟𝑠(𝑥𝑡𝑦))) ↔ ∀𝑦 ∈ (Base‘𝑊)[( ·𝑠 ‘𝑊) / 𝑠][(.r‘𝑊) / 𝑡](((𝑟𝑠𝑥)𝑡𝑦) = (𝑟𝑠(𝑥𝑡𝑦)) ∧ (𝑥𝑡(𝑟𝑠𝑦)) = (𝑟𝑠(𝑥𝑡𝑦)))))
8	2, 7	raleqbidv 2765	. . . . 5 ⊢ (𝑤 = 𝑊 → (∀𝑥 ∈ (Base‘𝑤)∀𝑦 ∈ (Base‘𝑤)[( ·𝑠 ‘𝑤) / 𝑠][(.r‘𝑤) / 𝑡](((𝑟𝑠𝑥)𝑡𝑦) = (𝑟𝑠(𝑥𝑡𝑦)) ∧ (𝑥𝑡(𝑟𝑠𝑦)) = (𝑟𝑠(𝑥𝑡𝑦))) ↔ ∀𝑥 ∈ (Base‘𝑊)∀𝑦 ∈ (Base‘𝑊)[( ·𝑠 ‘𝑊) / 𝑠][(.r‘𝑊) / 𝑡](((𝑟𝑠𝑥)𝑡𝑦) = (𝑟𝑠(𝑥𝑡𝑦)) ∧ (𝑥𝑡(𝑟𝑠𝑦)) = (𝑟𝑠(𝑥𝑡𝑦)))))
9	8	ralbidv 2550	. . . 4 ⊢ (𝑤 = 𝑊 → (∀𝑟 ∈ (Base‘𝑓)∀𝑥 ∈ (Base‘𝑤)∀𝑦 ∈ (Base‘𝑤)[( ·𝑠 ‘𝑤) / 𝑠][(.r‘𝑤) / 𝑡](((𝑟𝑠𝑥)𝑡𝑦) = (𝑟𝑠(𝑥𝑡𝑦)) ∧ (𝑥𝑡(𝑟𝑠𝑦)) = (𝑟𝑠(𝑥𝑡𝑦))) ↔ ∀𝑟 ∈ (Base‘𝑓)∀𝑥 ∈ (Base‘𝑊)∀𝑦 ∈ (Base‘𝑊)[( ·𝑠 ‘𝑊) / 𝑠][(.r‘𝑊) / 𝑡](((𝑟𝑠𝑥)𝑡𝑦) = (𝑟𝑠(𝑥𝑡𝑦)) ∧ (𝑥𝑡(𝑟𝑠𝑦)) = (𝑟𝑠(𝑥𝑡𝑦)))))
10	1, 9	sbceqbid 3058	. . 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))) ↔ [(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)))))
11		df-assa 14999	. . 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)))}
12	10, 11	elrab2 2985	. 2 ⊢ (𝑊 ∈ 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)))))
13		scaslid 13507	. . . . . 6 ⊢ (Scalar = Slot (Scalar‘ndx) ∧ (Scalar‘ndx) ∈ ℕ)
14	13	slotex 13379	. . . . 5 ⊢ (𝑊 ∈ (LMod ∩ Ring) → (Scalar‘𝑊) ∈ V)
15		fveq2 5695	. . . . . . 7 ⊢ (𝑓 = (Scalar‘𝑊) → (Base‘𝑓) = (Base‘(Scalar‘𝑊)))
16	15	raleqdv 2755	. . . . . 6 ⊢ (𝑓 = (Scalar‘𝑊) → (∀𝑟 ∈ (Base‘𝑓)∀𝑥 ∈ (Base‘𝑊)∀𝑦 ∈ (Base‘𝑊)[( ·𝑠 ‘𝑊) / 𝑠][(.r‘𝑊) / 𝑡](((𝑟𝑠𝑥)𝑡𝑦) = (𝑟𝑠(𝑥𝑡𝑦)) ∧ (𝑥𝑡(𝑟𝑠𝑦)) = (𝑟𝑠(𝑥𝑡𝑦))) ↔ ∀𝑟 ∈ (Base‘(Scalar‘𝑊))∀𝑥 ∈ (Base‘𝑊)∀𝑦 ∈ (Base‘𝑊)[( ·𝑠 ‘𝑊) / 𝑠][(.r‘𝑊) / 𝑡](((𝑟𝑠𝑥)𝑡𝑦) = (𝑟𝑠(𝑥𝑡𝑦)) ∧ (𝑥𝑡(𝑟𝑠𝑦)) = (𝑟𝑠(𝑥𝑡𝑦)))))
17	16	sbcieg 3084	. . . . 5 ⊢ ((Scalar‘𝑊) ∈ V → ([(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 ∈ (Base‘(Scalar‘𝑊))∀𝑥 ∈ (Base‘𝑊)∀𝑦 ∈ (Base‘𝑊)[( ·𝑠 ‘𝑊) / 𝑠][(.r‘𝑊) / 𝑡](((𝑟𝑠𝑥)𝑡𝑦) = (𝑟𝑠(𝑥𝑡𝑦)) ∧ (𝑥𝑡(𝑟𝑠𝑦)) = (𝑟𝑠(𝑥𝑡𝑦)))))
18	14, 17	syl 14	. . . 4 ⊢ (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))) ↔ ∀𝑟 ∈ (Base‘(Scalar‘𝑊))∀𝑥 ∈ (Base‘𝑊)∀𝑦 ∈ (Base‘𝑊)[( ·𝑠 ‘𝑊) / 𝑠][(.r‘𝑊) / 𝑡](((𝑟𝑠𝑥)𝑡𝑦) = (𝑟𝑠(𝑥𝑡𝑦)) ∧ (𝑥𝑡(𝑟𝑠𝑦)) = (𝑟𝑠(𝑥𝑡𝑦)))))
19		isassa.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐹)
20		isassa.f	. . . . . . . 8 ⊢ 𝐹 = (Scalar‘𝑊)
21	20	fveq2i 5698	. . . . . . 7 ⊢ (Base‘𝐹) = (Base‘(Scalar‘𝑊))
22	19, 21	eqtr2i 2260	. . . . . 6 ⊢ (Base‘(Scalar‘𝑊)) = 𝐵
23	22	a1i 9	. . . . 5 ⊢ (𝑊 ∈ (LMod ∩ Ring) → (Base‘(Scalar‘𝑊)) = 𝐵)
24		isassa.v	. . . . . . . 8 ⊢ 𝑉 = (Base‘𝑊)
25	24	eqcomi 2242	. . . . . . 7 ⊢ (Base‘𝑊) = 𝑉
26	25	a1i 9	. . . . . 6 ⊢ (𝑊 ∈ (LMod ∩ Ring) → (Base‘𝑊) = 𝑉)
27		vscaslid 13517	. . . . . . . . . 10 ⊢ ( ·𝑠 = Slot ( ·𝑠 ‘ndx) ∧ ( ·𝑠 ‘ndx) ∈ ℕ)
28	27	slotex 13379	. . . . . . . . 9 ⊢ (𝑊 ∈ (LMod ∩ Ring) → ( ·𝑠 ‘𝑊) ∈ V)
29		isassa.s	. . . . . . . . . . . . . . . . 17 ⊢ · = ( ·𝑠 ‘𝑊)
30	29	eqeq2i 2249	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 = · ↔ 𝑠 = ( ·𝑠 ‘𝑊))
31	30	biimpri 133	. . . . . . . . . . . . . . 15 ⊢ (𝑠 = ( ·𝑠 ‘𝑊) → 𝑠 = · )
32	31	oveqd 6102	. . . . . . . . . . . . . 14 ⊢ (𝑠 = ( ·𝑠 ‘𝑊) → (𝑟𝑠𝑥) = (𝑟 · 𝑥))
33	32	oveq1d 6100	. . . . . . . . . . . . 13 ⊢ (𝑠 = ( ·𝑠 ‘𝑊) → ((𝑟𝑠𝑥)𝑡𝑦) = ((𝑟 · 𝑥)𝑡𝑦))
34	31	oveqd 6102	. . . . . . . . . . . . 13 ⊢ (𝑠 = ( ·𝑠 ‘𝑊) → (𝑟𝑠(𝑥𝑡𝑦)) = (𝑟 · (𝑥𝑡𝑦)))
35	33, 34	eqeq12d 2253	. . . . . . . . . . . 12 ⊢ (𝑠 = ( ·𝑠 ‘𝑊) → (((𝑟𝑠𝑥)𝑡𝑦) = (𝑟𝑠(𝑥𝑡𝑦)) ↔ ((𝑟 · 𝑥)𝑡𝑦) = (𝑟 · (𝑥𝑡𝑦))))
36	31	oveqd 6102	. . . . . . . . . . . . . 14 ⊢ (𝑠 = ( ·𝑠 ‘𝑊) → (𝑟𝑠𝑦) = (𝑟 · 𝑦))
37	36	oveq2d 6101	. . . . . . . . . . . . 13 ⊢ (𝑠 = ( ·𝑠 ‘𝑊) → (𝑥𝑡(𝑟𝑠𝑦)) = (𝑥𝑡(𝑟 · 𝑦)))
38	37, 34	eqeq12d 2253	. . . . . . . . . . . 12 ⊢ (𝑠 = ( ·𝑠 ‘𝑊) → ((𝑥𝑡(𝑟𝑠𝑦)) = (𝑟𝑠(𝑥𝑡𝑦)) ↔ (𝑥𝑡(𝑟 · 𝑦)) = (𝑟 · (𝑥𝑡𝑦))))
39	35, 38	anbi12d 477	. . . . . . . . . . 11 ⊢ (𝑠 = ( ·𝑠 ‘𝑊) → ((((𝑟𝑠𝑥)𝑡𝑦) = (𝑟𝑠(𝑥𝑡𝑦)) ∧ (𝑥𝑡(𝑟𝑠𝑦)) = (𝑟𝑠(𝑥𝑡𝑦))) ↔ (((𝑟 · 𝑥)𝑡𝑦) = (𝑟 · (𝑥𝑡𝑦)) ∧ (𝑥𝑡(𝑟 · 𝑦)) = (𝑟 · (𝑥𝑡𝑦)))))
40	39	sbcbidv 3110	. . . . . . . . . 10 ⊢ (𝑠 = ( ·𝑠 ‘𝑊) → ([(.r‘𝑊) / 𝑡](((𝑟𝑠𝑥)𝑡𝑦) = (𝑟𝑠(𝑥𝑡𝑦)) ∧ (𝑥𝑡(𝑟𝑠𝑦)) = (𝑟𝑠(𝑥𝑡𝑦))) ↔ [(.r‘𝑊) / 𝑡](((𝑟 · 𝑥)𝑡𝑦) = (𝑟 · (𝑥𝑡𝑦)) ∧ (𝑥𝑡(𝑟 · 𝑦)) = (𝑟 · (𝑥𝑡𝑦)))))
41	40	sbcieg 3084	. . . . . . . . 9 ⊢ (( ·𝑠 ‘𝑊) ∈ V → ([( ·𝑠 ‘𝑊) / 𝑠][(.r‘𝑊) / 𝑡](((𝑟𝑠𝑥)𝑡𝑦) = (𝑟𝑠(𝑥𝑡𝑦)) ∧ (𝑥𝑡(𝑟𝑠𝑦)) = (𝑟𝑠(𝑥𝑡𝑦))) ↔ [(.r‘𝑊) / 𝑡](((𝑟 · 𝑥)𝑡𝑦) = (𝑟 · (𝑥𝑡𝑦)) ∧ (𝑥𝑡(𝑟 · 𝑦)) = (𝑟 · (𝑥𝑡𝑦)))))
42	28, 41	syl 14	. . . . . . . 8 ⊢ (𝑊 ∈ (LMod ∩ Ring) → ([( ·𝑠 ‘𝑊) / 𝑠][(.r‘𝑊) / 𝑡](((𝑟𝑠𝑥)𝑡𝑦) = (𝑟𝑠(𝑥𝑡𝑦)) ∧ (𝑥𝑡(𝑟𝑠𝑦)) = (𝑟𝑠(𝑥𝑡𝑦))) ↔ [(.r‘𝑊) / 𝑡](((𝑟 · 𝑥)𝑡𝑦) = (𝑟 · (𝑥𝑡𝑦)) ∧ (𝑥𝑡(𝑟 · 𝑦)) = (𝑟 · (𝑥𝑡𝑦)))))
43		mulrslid 13486	. . . . . . . . . 10 ⊢ (.r = Slot (.r‘ndx) ∧ (.r‘ndx) ∈ ℕ)
44	43	slotex 13379	. . . . . . . . 9 ⊢ (𝑊 ∈ (LMod ∩ Ring) → (.r‘𝑊) ∈ V)
45		isassa.t	. . . . . . . . . . . . . . 15 ⊢ × = (.r‘𝑊)
46	45	eqeq2i 2249	. . . . . . . . . . . . . 14 ⊢ (𝑡 = × ↔ 𝑡 = (.r‘𝑊))
47	46	biimpri 133	. . . . . . . . . . . . 13 ⊢ (𝑡 = (.r‘𝑊) → 𝑡 = × )
48	47	oveqd 6102	. . . . . . . . . . . 12 ⊢ (𝑡 = (.r‘𝑊) → ((𝑟 · 𝑥)𝑡𝑦) = ((𝑟 · 𝑥) × 𝑦))
49	47	oveqd 6102	. . . . . . . . . . . . 13 ⊢ (𝑡 = (.r‘𝑊) → (𝑥𝑡𝑦) = (𝑥 × 𝑦))
50	49	oveq2d 6101	. . . . . . . . . . . 12 ⊢ (𝑡 = (.r‘𝑊) → (𝑟 · (𝑥𝑡𝑦)) = (𝑟 · (𝑥 × 𝑦)))
51	48, 50	eqeq12d 2253	. . . . . . . . . . 11 ⊢ (𝑡 = (.r‘𝑊) → (((𝑟 · 𝑥)𝑡𝑦) = (𝑟 · (𝑥𝑡𝑦)) ↔ ((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦))))
52	47	oveqd 6102	. . . . . . . . . . . 12 ⊢ (𝑡 = (.r‘𝑊) → (𝑥𝑡(𝑟 · 𝑦)) = (𝑥 × (𝑟 · 𝑦)))
53	52, 50	eqeq12d 2253	. . . . . . . . . . 11 ⊢ (𝑡 = (.r‘𝑊) → ((𝑥𝑡(𝑟 · 𝑦)) = (𝑟 · (𝑥𝑡𝑦)) ↔ (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦))))
54	51, 53	anbi12d 477	. . . . . . . . . 10 ⊢ (𝑡 = (.r‘𝑊) → ((((𝑟 · 𝑥)𝑡𝑦) = (𝑟 · (𝑥𝑡𝑦)) ∧ (𝑥𝑡(𝑟 · 𝑦)) = (𝑟 · (𝑥𝑡𝑦))) ↔ (((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)) ∧ (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦)))))
55	54	sbcieg 3084	. . . . . . . . 9 ⊢ ((.r‘𝑊) ∈ V → ([(.r‘𝑊) / 𝑡](((𝑟 · 𝑥)𝑡𝑦) = (𝑟 · (𝑥𝑡𝑦)) ∧ (𝑥𝑡(𝑟 · 𝑦)) = (𝑟 · (𝑥𝑡𝑦))) ↔ (((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)) ∧ (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦)))))
56	44, 55	syl 14	. . . . . . . 8 ⊢ (𝑊 ∈ (LMod ∩ Ring) → ([(.r‘𝑊) / 𝑡](((𝑟 · 𝑥)𝑡𝑦) = (𝑟 · (𝑥𝑡𝑦)) ∧ (𝑥𝑡(𝑟 · 𝑦)) = (𝑟 · (𝑥𝑡𝑦))) ↔ (((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)) ∧ (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦)))))
57	42, 56	bitrd 188	. . . . . . 7 ⊢ (𝑊 ∈ (LMod ∩ Ring) → ([( ·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)))))
58	26, 57	raleqbidv 2765	. . . . . 6 ⊢ (𝑊 ∈ (LMod ∩ Ring) → (∀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)))))
59	26, 58	raleqbidv 2765	. . . . 5 ⊢ (𝑊 ∈ (LMod ∩ Ring) → (∀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)))))
60	23, 59	raleqbidv 2765	. . . 4 ⊢ (𝑊 ∈ (LMod ∩ Ring) → (∀𝑟 ∈ (Base‘(Scalar‘𝑊))∀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)))))
61	18, 60	bitrd 188	. . 3 ⊢ (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))) ↔ ∀𝑟 ∈ 𝐵 ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)) ∧ (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦)))))
62	61	pm5.32i 458	. 2 ⊢ ((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)))) ↔ (𝑊 ∈ (LMod ∩ Ring) ∧ ∀𝑟 ∈ 𝐵 ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)) ∧ (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦)))))
63		elin 3412	. . 3 ⊢ (𝑊 ∈ (LMod ∩ Ring) ↔ (𝑊 ∈ LMod ∧ 𝑊 ∈ Ring))
64	63	anbi1i 462	. 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)))))
65	12, 62, 64	3bitri 206	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)))))

This proof depends on syntax axioms:   ∧ wa 104   ↔ wb 105   = wceq 1402   ∈ wcel 2209  ∀wral 2528  Vcvv 2821  [wsbc 3051   ∩ cin 3219  ‘cfv 5377  (class class class)co 6085  Basecbs 13352  ._rcmulr 13432  Scalarcsca 13434   ·_𝑠 cvsca 13435  Ringcrg 14300  LModclmod 14623  AssAlgcasa 14996

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-cnex 8270  ax-resscn 8271  ax-1re 8273  ax-addrcl 8276

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-iota 5337  df-fun 5379  df-fn 5380  df-fv 5385  df-ov 6088  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-5 9366  df-6 9367  df-ndx 13355  df-slot 13356  df-mulr 13445  df-sca 13447  df-vsca 13448  df-assa 14999

This theorem is used by:  assalem  15003  assalmod  15006  assaring  15007  isassad  15011  assapropd  15014