Theorem isassa 20545

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
isassa	⊢ (𝑊 ∈ AssAlg ↔ ((𝑊 ∈ LMod ∧ 𝑊 ∈ Ring ∧ 𝐹 ∈ CRing) ∧ ∀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 6660	. . . 4 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) ∈ V)
2		fveq2 6645	. . . . 5 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊))
3		isassa.f	. . . . 5 ⊢ 𝐹 = (Scalar‘𝑊)
4	2, 3	eqtr4di 2851	. . . 4 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = 𝐹)
5		simpr 488	. . . . . 6 ⊢ ((𝑤 = 𝑊 ∧ 𝑓 = 𝐹) → 𝑓 = 𝐹)
6	5	eleq1d 2874	. . . . 5 ⊢ ((𝑤 = 𝑊 ∧ 𝑓 = 𝐹) → (𝑓 ∈ CRing ↔ 𝐹 ∈ CRing))
7	5	fveq2d 6649	. . . . . . 7 ⊢ ((𝑤 = 𝑊 ∧ 𝑓 = 𝐹) → (Base‘𝑓) = (Base‘𝐹))
8		isassa.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐹)
9	7, 8	eqtr4di 2851	. . . . . 6 ⊢ ((𝑤 = 𝑊 ∧ 𝑓 = 𝐹) → (Base‘𝑓) = 𝐵)
10		fveq2 6645	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
11		isassa.v	. . . . . . . . 9 ⊢ 𝑉 = (Base‘𝑊)
12	10, 11	eqtr4di 2851	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝑉)
13		fvexd 6660	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → ( ·𝑠 ‘𝑤) ∈ V)
14		fvexd 6660	. . . . . . . . . . 11 ⊢ ((𝑤 = 𝑊 ∧ 𝑠 = ( ·𝑠 ‘𝑤)) → (.r‘𝑤) ∈ V)
15		simpr 488	. . . . . . . . . . . . . . 15 ⊢ (((𝑤 = 𝑊 ∧ 𝑠 = ( ·𝑠 ‘𝑤)) ∧ 𝑡 = (.r‘𝑤)) → 𝑡 = (.r‘𝑤))
16		fveq2 6645	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = 𝑊 → (.r‘𝑤) = (.r‘𝑊))
17	16	ad2antrr 725	. . . . . . . . . . . . . . . 16 ⊢ (((𝑤 = 𝑊 ∧ 𝑠 = ( ·𝑠 ‘𝑤)) ∧ 𝑡 = (.r‘𝑤)) → (.r‘𝑤) = (.r‘𝑊))
18		isassa.t	. . . . . . . . . . . . . . . 16 ⊢ × = (.r‘𝑊)
19	17, 18	eqtr4di 2851	. . . . . . . . . . . . . . 15 ⊢ (((𝑤 = 𝑊 ∧ 𝑠 = ( ·𝑠 ‘𝑤)) ∧ 𝑡 = (.r‘𝑤)) → (.r‘𝑤) = × )
20	15, 19	eqtrd 2833	. . . . . . . . . . . . . 14 ⊢ (((𝑤 = 𝑊 ∧ 𝑠 = ( ·𝑠 ‘𝑤)) ∧ 𝑡 = (.r‘𝑤)) → 𝑡 = × )
21		simplr 768	. . . . . . . . . . . . . . . 16 ⊢ (((𝑤 = 𝑊 ∧ 𝑠 = ( ·𝑠 ‘𝑤)) ∧ 𝑡 = (.r‘𝑤)) → 𝑠 = ( ·𝑠 ‘𝑤))
22		fveq2 6645	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = 𝑊 → ( ·𝑠 ‘𝑤) = ( ·𝑠 ‘𝑊))
23	22	ad2antrr 725	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑤 = 𝑊 ∧ 𝑠 = ( ·𝑠 ‘𝑤)) ∧ 𝑡 = (.r‘𝑤)) → ( ·𝑠 ‘𝑤) = ( ·𝑠 ‘𝑊))
24		isassa.s	. . . . . . . . . . . . . . . . 17 ⊢ · = ( ·𝑠 ‘𝑊)
25	23, 24	eqtr4di 2851	. . . . . . . . . . . . . . . 16 ⊢ (((𝑤 = 𝑊 ∧ 𝑠 = ( ·𝑠 ‘𝑤)) ∧ 𝑡 = (.r‘𝑤)) → ( ·𝑠 ‘𝑤) = · )
26	21, 25	eqtrd 2833	. . . . . . . . . . . . . . 15 ⊢ (((𝑤 = 𝑊 ∧ 𝑠 = ( ·𝑠 ‘𝑤)) ∧ 𝑡 = (.r‘𝑤)) → 𝑠 = · )
27	26	oveqd 7152	. . . . . . . . . . . . . 14 ⊢ (((𝑤 = 𝑊 ∧ 𝑠 = ( ·𝑠 ‘𝑤)) ∧ 𝑡 = (.r‘𝑤)) → (𝑟𝑠𝑥) = (𝑟 · 𝑥))
28		eqidd 2799	. . . . . . . . . . . . . 14 ⊢ (((𝑤 = 𝑊 ∧ 𝑠 = ( ·𝑠 ‘𝑤)) ∧ 𝑡 = (.r‘𝑤)) → 𝑦 = 𝑦)
29	20, 27, 28	oveq123d 7156	. . . . . . . . . . . . 13 ⊢ (((𝑤 = 𝑊 ∧ 𝑠 = ( ·𝑠 ‘𝑤)) ∧ 𝑡 = (.r‘𝑤)) → ((𝑟𝑠𝑥)𝑡𝑦) = ((𝑟 · 𝑥) × 𝑦))
30		eqidd 2799	. . . . . . . . . . . . . 14 ⊢ (((𝑤 = 𝑊 ∧ 𝑠 = ( ·𝑠 ‘𝑤)) ∧ 𝑡 = (.r‘𝑤)) → 𝑟 = 𝑟)
31	20	oveqd 7152	. . . . . . . . . . . . . 14 ⊢ (((𝑤 = 𝑊 ∧ 𝑠 = ( ·𝑠 ‘𝑤)) ∧ 𝑡 = (.r‘𝑤)) → (𝑥𝑡𝑦) = (𝑥 × 𝑦))
32	26, 30, 31	oveq123d 7156	. . . . . . . . . . . . 13 ⊢ (((𝑤 = 𝑊 ∧ 𝑠 = ( ·𝑠 ‘𝑤)) ∧ 𝑡 = (.r‘𝑤)) → (𝑟𝑠(𝑥𝑡𝑦)) = (𝑟 · (𝑥 × 𝑦)))
33	29, 32	eqeq12d 2814	. . . . . . . . . . . 12 ⊢ (((𝑤 = 𝑊 ∧ 𝑠 = ( ·𝑠 ‘𝑤)) ∧ 𝑡 = (.r‘𝑤)) → (((𝑟𝑠𝑥)𝑡𝑦) = (𝑟𝑠(𝑥𝑡𝑦)) ↔ ((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦))))
34		eqidd 2799	. . . . . . . . . . . . . 14 ⊢ (((𝑤 = 𝑊 ∧ 𝑠 = ( ·𝑠 ‘𝑤)) ∧ 𝑡 = (.r‘𝑤)) → 𝑥 = 𝑥)
35	26	oveqd 7152	. . . . . . . . . . . . . 14 ⊢ (((𝑤 = 𝑊 ∧ 𝑠 = ( ·𝑠 ‘𝑤)) ∧ 𝑡 = (.r‘𝑤)) → (𝑟𝑠𝑦) = (𝑟 · 𝑦))
36	20, 34, 35	oveq123d 7156	. . . . . . . . . . . . 13 ⊢ (((𝑤 = 𝑊 ∧ 𝑠 = ( ·𝑠 ‘𝑤)) ∧ 𝑡 = (.r‘𝑤)) → (𝑥𝑡(𝑟𝑠𝑦)) = (𝑥 × (𝑟 · 𝑦)))
37	36, 32	eqeq12d 2814	. . . . . . . . . . . 12 ⊢ (((𝑤 = 𝑊 ∧ 𝑠 = ( ·𝑠 ‘𝑤)) ∧ 𝑡 = (.r‘𝑤)) → ((𝑥𝑡(𝑟𝑠𝑦)) = (𝑟𝑠(𝑥𝑡𝑦)) ↔ (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦))))
38	33, 37	anbi12d 633	. . . . . . . . . . 11 ⊢ (((𝑤 = 𝑊 ∧ 𝑠 = ( ·𝑠 ‘𝑤)) ∧ 𝑡 = (.r‘𝑤)) → ((((𝑟𝑠𝑥)𝑡𝑦) = (𝑟𝑠(𝑥𝑡𝑦)) ∧ (𝑥𝑡(𝑟𝑠𝑦)) = (𝑟𝑠(𝑥𝑡𝑦))) ↔ (((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)) ∧ (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦)))))
39	14, 38	sbcied 3762	. . . . . . . . . 10 ⊢ ((𝑤 = 𝑊 ∧ 𝑠 = ( ·𝑠 ‘𝑤)) → ([(.r‘𝑤) / 𝑡](((𝑟𝑠𝑥)𝑡𝑦) = (𝑟𝑠(𝑥𝑡𝑦)) ∧ (𝑥𝑡(𝑟𝑠𝑦)) = (𝑟𝑠(𝑥𝑡𝑦))) ↔ (((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)) ∧ (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦)))))
40	13, 39	sbcied 3762	. . . . . . . . 9 ⊢ (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)))))
41	12, 40	raleqbidv 3354	. . . . . . . 8 ⊢ (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)))))
42	12, 41	raleqbidv 3354	. . . . . . 7 ⊢ (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)))))
43	42	adantr 484	. . . . . 6 ⊢ ((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)))))
44	9, 43	raleqbidv 3354	. . . . 5 ⊢ ((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)))))
45	6, 44	anbi12d 633	. . . 4 ⊢ ((𝑤 = 𝑊 ∧ 𝑓 = 𝐹) → ((𝑓 ∈ CRing ∧ ∀𝑟 ∈ (Base‘𝑓)∀𝑥 ∈ (Base‘𝑤)∀𝑦 ∈ (Base‘𝑤)[( ·𝑠 ‘𝑤) / 𝑠][(.r‘𝑤) / 𝑡](((𝑟𝑠𝑥)𝑡𝑦) = (𝑟𝑠(𝑥𝑡𝑦)) ∧ (𝑥𝑡(𝑟𝑠𝑦)) = (𝑟𝑠(𝑥𝑡𝑦)))) ↔ (𝐹 ∈ CRing ∧ ∀𝑟 ∈ 𝐵 ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)) ∧ (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦))))))
46	1, 4, 45	sbcied2 3763	. . 3 ⊢ (𝑤 = 𝑊 → ([(Scalar‘𝑤) / 𝑓](𝑓 ∈ CRing ∧ ∀𝑟 ∈ (Base‘𝑓)∀𝑥 ∈ (Base‘𝑤)∀𝑦 ∈ (Base‘𝑤)[( ·𝑠 ‘𝑤) / 𝑠][(.r‘𝑤) / 𝑡](((𝑟𝑠𝑥)𝑡𝑦) = (𝑟𝑠(𝑥𝑡𝑦)) ∧ (𝑥𝑡(𝑟𝑠𝑦)) = (𝑟𝑠(𝑥𝑡𝑦)))) ↔ (𝐹 ∈ CRing ∧ ∀𝑟 ∈ 𝐵 ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)) ∧ (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦))))))
47		df-assa 20542	. . 3 ⊢ AssAlg = {𝑤 ∈ (LMod ∩ Ring) ∣ [(Scalar‘𝑤) / 𝑓](𝑓 ∈ CRing ∧ ∀𝑟 ∈ (Base‘𝑓)∀𝑥 ∈ (Base‘𝑤)∀𝑦 ∈ (Base‘𝑤)[( ·𝑠 ‘𝑤) / 𝑠][(.r‘𝑤) / 𝑡](((𝑟𝑠𝑥)𝑡𝑦) = (𝑟𝑠(𝑥𝑡𝑦)) ∧ (𝑥𝑡(𝑟𝑠𝑦)) = (𝑟𝑠(𝑥𝑡𝑦))))}
48	46, 47	elrab2 3631	. 2 ⊢ (𝑊 ∈ AssAlg ↔ (𝑊 ∈ (LMod ∩ Ring) ∧ (𝐹 ∈ CRing ∧ ∀𝑟 ∈ 𝐵 ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)) ∧ (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦))))))
49		anass 472	. 2 ⊢ (((𝑊 ∈ (LMod ∩ Ring) ∧ 𝐹 ∈ CRing) ∧ ∀𝑟 ∈ 𝐵 ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)) ∧ (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦)))) ↔ (𝑊 ∈ (LMod ∩ Ring) ∧ (𝐹 ∈ CRing ∧ ∀𝑟 ∈ 𝐵 ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)) ∧ (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦))))))
50		elin 3897	. . . . 5 ⊢ (𝑊 ∈ (LMod ∩ Ring) ↔ (𝑊 ∈ LMod ∧ 𝑊 ∈ Ring))
51	50	anbi1i 626	. . . 4 ⊢ ((𝑊 ∈ (LMod ∩ Ring) ∧ 𝐹 ∈ CRing) ↔ ((𝑊 ∈ LMod ∧ 𝑊 ∈ Ring) ∧ 𝐹 ∈ CRing))
52		df-3an 1086	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑊 ∈ Ring ∧ 𝐹 ∈ CRing) ↔ ((𝑊 ∈ LMod ∧ 𝑊 ∈ Ring) ∧ 𝐹 ∈ CRing))
53	51, 52	bitr4i 281	. . 3 ⊢ ((𝑊 ∈ (LMod ∩ Ring) ∧ 𝐹 ∈ CRing) ↔ (𝑊 ∈ LMod ∧ 𝑊 ∈ Ring ∧ 𝐹 ∈ CRing))
54	53	anbi1i 626	. 2 ⊢ (((𝑊 ∈ (LMod ∩ Ring) ∧ 𝐹 ∈ CRing) ∧ ∀𝑟 ∈ 𝐵 ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)) ∧ (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦)))) ↔ ((𝑊 ∈ LMod ∧ 𝑊 ∈ Ring ∧ 𝐹 ∈ CRing) ∧ ∀𝑟 ∈ 𝐵 ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)) ∧ (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦)))))
55	48, 49, 54	3bitr2i 302	1 ⊢ (𝑊 ∈ AssAlg ↔ ((𝑊 ∈ LMod ∧ 𝑊 ∈ Ring ∧ 𝐹 ∈ CRing) ∧ ∀𝑟 ∈ 𝐵 ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)) ∧ (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦)))))

Syntax hints:   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3106  Vcvv 3441  [wsbc 3720   ∩ cin 3880  ‘cfv 6324  (class class class)co 7135  Basecbs 16475  ._rcmulr 16558  Scalarcsca 16560   ·_𝑠 cvsca 16561  Ringcrg 19290  CRingccrg 19291  LModclmod 19627  AssAlgcasa 20539

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-nul 5174

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-iota 6283  df-fv 6332  df-ov 7138  df-assa 20542

This theorem is referenced by:  assalem  20546  assalmod  20549  assaring  20550  assasca  20551  isassad  20553  assapropd  20558