Theorem isassa 19315

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 6203	. . . 4 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) ∈ V)
2		fveq2 6191	. . . . 5 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊))
3		isassa.f	. . . . 5 ⊢ 𝐹 = (Scalar‘𝑊)
4	2, 3	syl6eqr 2674	. . . 4 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = 𝐹)
5		simpr 477	. . . . . 6 ⊢ ((𝑤 = 𝑊 ∧ 𝑓 = 𝐹) → 𝑓 = 𝐹)
6	5	eleq1d 2686	. . . . 5 ⊢ ((𝑤 = 𝑊 ∧ 𝑓 = 𝐹) → (𝑓 ∈ CRing ↔ 𝐹 ∈ CRing))
7	5	fveq2d 6195	. . . . . . 7 ⊢ ((𝑤 = 𝑊 ∧ 𝑓 = 𝐹) → (Base‘𝑓) = (Base‘𝐹))
8		isassa.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐹)
9	7, 8	syl6eqr 2674	. . . . . 6 ⊢ ((𝑤 = 𝑊 ∧ 𝑓 = 𝐹) → (Base‘𝑓) = 𝐵)
10		fveq2 6191	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
11		isassa.v	. . . . . . . . 9 ⊢ 𝑉 = (Base‘𝑊)
12	10, 11	syl6eqr 2674	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝑉)
13		fvexd 6203	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → ( ·𝑠 ‘𝑤) ∈ V)
14		fvexd 6203	. . . . . . . . . . 11 ⊢ ((𝑤 = 𝑊 ∧ 𝑠 = ( ·𝑠 ‘𝑤)) → (.r‘𝑤) ∈ V)
15		simpr 477	. . . . . . . . . . . . . . 15 ⊢ (((𝑤 = 𝑊 ∧ 𝑠 = ( ·𝑠 ‘𝑤)) ∧ 𝑡 = (.r‘𝑤)) → 𝑡 = (.r‘𝑤))
16		fveq2 6191	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = 𝑊 → (.r‘𝑤) = (.r‘𝑊))
17	16	ad2antrr 762	. . . . . . . . . . . . . . . 16 ⊢ (((𝑤 = 𝑊 ∧ 𝑠 = ( ·𝑠 ‘𝑤)) ∧ 𝑡 = (.r‘𝑤)) → (.r‘𝑤) = (.r‘𝑊))
18		isassa.t	. . . . . . . . . . . . . . . 16 ⊢ × = (.r‘𝑊)
19	17, 18	syl6eqr 2674	. . . . . . . . . . . . . . 15 ⊢ (((𝑤 = 𝑊 ∧ 𝑠 = ( ·𝑠 ‘𝑤)) ∧ 𝑡 = (.r‘𝑤)) → (.r‘𝑤) = × )
20	15, 19	eqtrd 2656	. . . . . . . . . . . . . 14 ⊢ (((𝑤 = 𝑊 ∧ 𝑠 = ( ·𝑠 ‘𝑤)) ∧ 𝑡 = (.r‘𝑤)) → 𝑡 = × )
21		simplr 792	. . . . . . . . . . . . . . . 16 ⊢ (((𝑤 = 𝑊 ∧ 𝑠 = ( ·𝑠 ‘𝑤)) ∧ 𝑡 = (.r‘𝑤)) → 𝑠 = ( ·𝑠 ‘𝑤))
22		fveq2 6191	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = 𝑊 → ( ·𝑠 ‘𝑤) = ( ·𝑠 ‘𝑊))
23	22	ad2antrr 762	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑤 = 𝑊 ∧ 𝑠 = ( ·𝑠 ‘𝑤)) ∧ 𝑡 = (.r‘𝑤)) → ( ·𝑠 ‘𝑤) = ( ·𝑠 ‘𝑊))
24		isassa.s	. . . . . . . . . . . . . . . . 17 ⊢ · = ( ·𝑠 ‘𝑊)
25	23, 24	syl6eqr 2674	. . . . . . . . . . . . . . . 16 ⊢ (((𝑤 = 𝑊 ∧ 𝑠 = ( ·𝑠 ‘𝑤)) ∧ 𝑡 = (.r‘𝑤)) → ( ·𝑠 ‘𝑤) = · )
26	21, 25	eqtrd 2656	. . . . . . . . . . . . . . 15 ⊢ (((𝑤 = 𝑊 ∧ 𝑠 = ( ·𝑠 ‘𝑤)) ∧ 𝑡 = (.r‘𝑤)) → 𝑠 = · )
27	26	oveqd 6667	. . . . . . . . . . . . . 14 ⊢ (((𝑤 = 𝑊 ∧ 𝑠 = ( ·𝑠 ‘𝑤)) ∧ 𝑡 = (.r‘𝑤)) → (𝑟𝑠𝑥) = (𝑟 · 𝑥))
28		eqidd 2623	. . . . . . . . . . . . . 14 ⊢ (((𝑤 = 𝑊 ∧ 𝑠 = ( ·𝑠 ‘𝑤)) ∧ 𝑡 = (.r‘𝑤)) → 𝑦 = 𝑦)
29	20, 27, 28	oveq123d 6671	. . . . . . . . . . . . 13 ⊢ (((𝑤 = 𝑊 ∧ 𝑠 = ( ·𝑠 ‘𝑤)) ∧ 𝑡 = (.r‘𝑤)) → ((𝑟𝑠𝑥)𝑡𝑦) = ((𝑟 · 𝑥) × 𝑦))
30		eqidd 2623	. . . . . . . . . . . . . 14 ⊢ (((𝑤 = 𝑊 ∧ 𝑠 = ( ·𝑠 ‘𝑤)) ∧ 𝑡 = (.r‘𝑤)) → 𝑟 = 𝑟)
31	20	oveqd 6667	. . . . . . . . . . . . . 14 ⊢ (((𝑤 = 𝑊 ∧ 𝑠 = ( ·𝑠 ‘𝑤)) ∧ 𝑡 = (.r‘𝑤)) → (𝑥𝑡𝑦) = (𝑥 × 𝑦))
32	26, 30, 31	oveq123d 6671	. . . . . . . . . . . . 13 ⊢ (((𝑤 = 𝑊 ∧ 𝑠 = ( ·𝑠 ‘𝑤)) ∧ 𝑡 = (.r‘𝑤)) → (𝑟𝑠(𝑥𝑡𝑦)) = (𝑟 · (𝑥 × 𝑦)))
33	29, 32	eqeq12d 2637	. . . . . . . . . . . 12 ⊢ (((𝑤 = 𝑊 ∧ 𝑠 = ( ·𝑠 ‘𝑤)) ∧ 𝑡 = (.r‘𝑤)) → (((𝑟𝑠𝑥)𝑡𝑦) = (𝑟𝑠(𝑥𝑡𝑦)) ↔ ((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦))))
34		eqidd 2623	. . . . . . . . . . . . . 14 ⊢ (((𝑤 = 𝑊 ∧ 𝑠 = ( ·𝑠 ‘𝑤)) ∧ 𝑡 = (.r‘𝑤)) → 𝑥 = 𝑥)
35	26	oveqd 6667	. . . . . . . . . . . . . 14 ⊢ (((𝑤 = 𝑊 ∧ 𝑠 = ( ·𝑠 ‘𝑤)) ∧ 𝑡 = (.r‘𝑤)) → (𝑟𝑠𝑦) = (𝑟 · 𝑦))
36	20, 34, 35	oveq123d 6671	. . . . . . . . . . . . 13 ⊢ (((𝑤 = 𝑊 ∧ 𝑠 = ( ·𝑠 ‘𝑤)) ∧ 𝑡 = (.r‘𝑤)) → (𝑥𝑡(𝑟𝑠𝑦)) = (𝑥 × (𝑟 · 𝑦)))
37	36, 32	eqeq12d 2637	. . . . . . . . . . . 12 ⊢ (((𝑤 = 𝑊 ∧ 𝑠 = ( ·𝑠 ‘𝑤)) ∧ 𝑡 = (.r‘𝑤)) → ((𝑥𝑡(𝑟𝑠𝑦)) = (𝑟𝑠(𝑥𝑡𝑦)) ↔ (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦))))
38	33, 37	anbi12d 747	. . . . . . . . . . 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 3472	. . . . . . . . . 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 3472	. . . . . . . . 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 3152	. . . . . . . 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 3152	. . . . . . 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 481	. . . . . 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 3152	. . . . 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 747	. . . 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 3473	. . 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 19312	. . 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 3366	. 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 681	. 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 3796	. . . . 5 ⊢ (𝑊 ∈ (LMod ∩ Ring) ↔ (𝑊 ∈ LMod ∧ 𝑊 ∈ Ring))
51	50	anbi1i 731	. . . 4 ⊢ ((𝑊 ∈ (LMod ∩ Ring) ∧ 𝐹 ∈ CRing) ↔ ((𝑊 ∈ LMod ∧ 𝑊 ∈ Ring) ∧ 𝐹 ∈ CRing))
52		df-3an 1039	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑊 ∈ Ring ∧ 𝐹 ∈ CRing) ↔ ((𝑊 ∈ LMod ∧ 𝑊 ∈ Ring) ∧ 𝐹 ∈ CRing))
53	51, 52	bitr4i 267	. . 3 ⊢ ((𝑊 ∈ (LMod ∩ Ring) ∧ 𝐹 ∈ CRing) ↔ (𝑊 ∈ LMod ∧ 𝑊 ∈ Ring ∧ 𝐹 ∈ CRing))
54	53	anbi1i 731	. 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 288	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 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912  Vcvv 3200  [wsbc 3435   ∩ cin 3573  ‘cfv 5888  (class class class)co 6650  Basecbs 15857  ._rcmulr 15942  Scalarcsca 15944   ·_𝑠 cvsca 15945  Ringcrg 18547  CRingccrg 18548  LModclmod 18863  AssAlgcasa 19309

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653  df-assa 19312

This theorem is referenced by:  assalem  19316  assalmod  19319  assaring  19320  assasca  19321  isassad  19323  assapropd  19327