Theorem madufval 22620

Description: First substitution for the adjunct (cofactor) matrix. (Contributed by SO, 11-Jul-2018.)

Hypotheses

Ref	Expression
madufval.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
madufval.d	⊢ 𝐷 = (𝑁 maDet 𝑅)
madufval.j	⊢ 𝐽 = (𝑁 maAdju 𝑅)
madufval.b	⊢ 𝐵 = (Base‘𝐴)
madufval.o	⊢ 1 = (1r‘𝑅)
madufval.z	⊢ 0 = (0g‘𝑅)

Assertion

Ref	Expression
madufval	⊢ 𝐽 = (𝑚 ∈ 𝐵 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑚𝑙))))))

Distinct variable groups:   𝑚,𝑁,𝑖,𝑗,𝑘,𝑙   𝑅,𝑚,𝑖,𝑗,𝑘,𝑙   𝐵,𝑚

Allowed substitution hints:   𝐴(𝑖,𝑗,𝑘,𝑚,𝑙)   𝐵(𝑖,𝑗,𝑘,𝑙)   𝐷(𝑖,𝑗,𝑘,𝑚,𝑙)   1 (𝑖,𝑗,𝑘,𝑚,𝑙)   𝐽(𝑖,𝑗,𝑘,𝑚,𝑙)   0 (𝑖,𝑗,𝑘,𝑚,𝑙)

Proof of Theorem madufval

Dummy variables 𝑛 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		madufval.j	. 2 ⊢ 𝐽 = (𝑁 maAdju 𝑅)
2		fvoveq1 7379	. . . . . 6 ⊢ (𝑛 = 𝑁 → (Base‘(𝑛 Mat 𝑟)) = (Base‘(𝑁 Mat 𝑟)))
3		id 22	. . . . . . 7 ⊢ (𝑛 = 𝑁 → 𝑛 = 𝑁)
4		oveq1 7363	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (𝑛 maDet 𝑟) = (𝑁 maDet 𝑟))
5		eqidd 2740	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → if(𝑘 = 𝑗, if(𝑙 = 𝑖, (1r‘𝑟), (0g‘𝑟)), (𝑘𝑚𝑙)) = if(𝑘 = 𝑗, if(𝑙 = 𝑖, (1r‘𝑟), (0g‘𝑟)), (𝑘𝑚𝑙)))
6	3, 3, 5	mpoeq123dv 7431	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (𝑘 ∈ 𝑛, 𝑙 ∈ 𝑛 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, (1r‘𝑟), (0g‘𝑟)), (𝑘𝑚𝑙))) = (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, (1r‘𝑟), (0g‘𝑟)), (𝑘𝑚𝑙))))
7	4, 6	fveq12d 6834	. . . . . . 7 ⊢ (𝑛 = 𝑁 → ((𝑛 maDet 𝑟)‘(𝑘 ∈ 𝑛, 𝑙 ∈ 𝑛 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, (1r‘𝑟), (0g‘𝑟)), (𝑘𝑚𝑙)))) = ((𝑁 maDet 𝑟)‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, (1r‘𝑟), (0g‘𝑟)), (𝑘𝑚𝑙)))))
8	3, 3, 7	mpoeq123dv 7431	. . . . . 6 ⊢ (𝑛 = 𝑁 → (𝑖 ∈ 𝑛, 𝑗 ∈ 𝑛 ↦ ((𝑛 maDet 𝑟)‘(𝑘 ∈ 𝑛, 𝑙 ∈ 𝑛 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, (1r‘𝑟), (0g‘𝑟)), (𝑘𝑚𝑙))))) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑁 maDet 𝑟)‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, (1r‘𝑟), (0g‘𝑟)), (𝑘𝑚𝑙))))))
9	2, 8	mpteq12dv 5159	. . . . 5 ⊢ (𝑛 = 𝑁 → (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑖 ∈ 𝑛, 𝑗 ∈ 𝑛 ↦ ((𝑛 maDet 𝑟)‘(𝑘 ∈ 𝑛, 𝑙 ∈ 𝑛 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, (1r‘𝑟), (0g‘𝑟)), (𝑘𝑚𝑙)))))) = (𝑚 ∈ (Base‘(𝑁 Mat 𝑟)) ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑁 maDet 𝑟)‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, (1r‘𝑟), (0g‘𝑟)), (𝑘𝑚𝑙)))))))
10		oveq2 7364	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (𝑁 Mat 𝑟) = (𝑁 Mat 𝑅))
11	10	fveq2d 6831	. . . . . 6 ⊢ (𝑟 = 𝑅 → (Base‘(𝑁 Mat 𝑟)) = (Base‘(𝑁 Mat 𝑅)))
12		oveq2 7364	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (𝑁 maDet 𝑟) = (𝑁 maDet 𝑅))
13		fveq2 6827	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (1r‘𝑟) = (1r‘𝑅))
14		fveq2 6827	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = (0g‘𝑅))
15	13, 14	ifeq12d 4476	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → if(𝑙 = 𝑖, (1r‘𝑟), (0g‘𝑟)) = if(𝑙 = 𝑖, (1r‘𝑅), (0g‘𝑅)))
16	15	ifeq1d 4474	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → if(𝑘 = 𝑗, if(𝑙 = 𝑖, (1r‘𝑟), (0g‘𝑟)), (𝑘𝑚𝑙)) = if(𝑘 = 𝑗, if(𝑙 = 𝑖, (1r‘𝑅), (0g‘𝑅)), (𝑘𝑚𝑙)))
17	16	mpoeq3dv 7435	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, (1r‘𝑟), (0g‘𝑟)), (𝑘𝑚𝑙))) = (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, (1r‘𝑅), (0g‘𝑅)), (𝑘𝑚𝑙))))
18	12, 17	fveq12d 6834	. . . . . . 7 ⊢ (𝑟 = 𝑅 → ((𝑁 maDet 𝑟)‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, (1r‘𝑟), (0g‘𝑟)), (𝑘𝑚𝑙)))) = ((𝑁 maDet 𝑅)‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, (1r‘𝑅), (0g‘𝑅)), (𝑘𝑚𝑙)))))
19	18	mpoeq3dv 7435	. . . . . 6 ⊢ (𝑟 = 𝑅 → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑁 maDet 𝑟)‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, (1r‘𝑟), (0g‘𝑟)), (𝑘𝑚𝑙))))) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑁 maDet 𝑅)‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, (1r‘𝑅), (0g‘𝑅)), (𝑘𝑚𝑙))))))
20	11, 19	mpteq12dv 5159	. . . . 5 ⊢ (𝑟 = 𝑅 → (𝑚 ∈ (Base‘(𝑁 Mat 𝑟)) ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑁 maDet 𝑟)‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, (1r‘𝑟), (0g‘𝑟)), (𝑘𝑚𝑙)))))) = (𝑚 ∈ (Base‘(𝑁 Mat 𝑅)) ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑁 maDet 𝑅)‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, (1r‘𝑅), (0g‘𝑅)), (𝑘𝑚𝑙)))))))
21		df-madu 22617	. . . . 5 ⊢ maAdju = (𝑛 ∈ V, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑖 ∈ 𝑛, 𝑗 ∈ 𝑛 ↦ ((𝑛 maDet 𝑟)‘(𝑘 ∈ 𝑛, 𝑙 ∈ 𝑛 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, (1r‘𝑟), (0g‘𝑟)), (𝑘𝑚𝑙)))))))
22		fvex 6840	. . . . . 6 ⊢ (Base‘(𝑁 Mat 𝑅)) ∈ V
23	22	mptex 7167	. . . . 5 ⊢ (𝑚 ∈ (Base‘(𝑁 Mat 𝑅)) ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑁 maDet 𝑅)‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, (1r‘𝑅), (0g‘𝑅)), (𝑘𝑚𝑙)))))) ∈ V
24	9, 20, 21, 23	ovmpo 7516	. . . 4 ⊢ ((𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 maAdju 𝑅) = (𝑚 ∈ (Base‘(𝑁 Mat 𝑅)) ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑁 maDet 𝑅)‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, (1r‘𝑅), (0g‘𝑅)), (𝑘𝑚𝑙)))))))
25		madufval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐴)
26		madufval.a	. . . . . . 7 ⊢ 𝐴 = (𝑁 Mat 𝑅)
27	26	fveq2i 6830	. . . . . 6 ⊢ (Base‘𝐴) = (Base‘(𝑁 Mat 𝑅))
28	25, 27	eqtri 2762	. . . . 5 ⊢ 𝐵 = (Base‘(𝑁 Mat 𝑅))
29		madufval.d	. . . . . . . 8 ⊢ 𝐷 = (𝑁 maDet 𝑅)
30		madufval.o	. . . . . . . . . . . 12 ⊢ 1 = (1r‘𝑅)
31	30	a1i 11	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) → 1 = (1r‘𝑅))
32		madufval.z	. . . . . . . . . . . 12 ⊢ 0 = (0g‘𝑅)
33	32	a1i 11	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) → 0 = (0g‘𝑅))
34	31, 33	ifeq12d 4476	. . . . . . . . . 10 ⊢ ((𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) → if(𝑙 = 𝑖, 1 , 0 ) = if(𝑙 = 𝑖, (1r‘𝑅), (0g‘𝑅)))
35	34	ifeq1d 4474	. . . . . . . . 9 ⊢ ((𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) → if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑚𝑙)) = if(𝑘 = 𝑗, if(𝑙 = 𝑖, (1r‘𝑅), (0g‘𝑅)), (𝑘𝑚𝑙)))
36	35	mpoeq3ia 7434	. . . . . . . 8 ⊢ (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑚𝑙))) = (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, (1r‘𝑅), (0g‘𝑅)), (𝑘𝑚𝑙)))
37	29, 36	fveq12i 6833	. . . . . . 7 ⊢ (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑚𝑙)))) = ((𝑁 maDet 𝑅)‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, (1r‘𝑅), (0g‘𝑅)), (𝑘𝑚𝑙))))
38	37	a1i 11	. . . . . 6 ⊢ ((𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑚𝑙)))) = ((𝑁 maDet 𝑅)‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, (1r‘𝑅), (0g‘𝑅)), (𝑘𝑚𝑙)))))
39	38	mpoeq3ia 7434	. . . . 5 ⊢ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑚𝑙))))) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑁 maDet 𝑅)‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, (1r‘𝑅), (0g‘𝑅)), (𝑘𝑚𝑙)))))
40	28, 39	mpteq12i 5169	. . . 4 ⊢ (𝑚 ∈ 𝐵 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑚𝑙)))))) = (𝑚 ∈ (Base‘(𝑁 Mat 𝑅)) ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑁 maDet 𝑅)‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, (1r‘𝑅), (0g‘𝑅)), (𝑘𝑚𝑙))))))
41	24, 40	eqtr4di 2792	. . 3 ⊢ ((𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 maAdju 𝑅) = (𝑚 ∈ 𝐵 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑚𝑙)))))))
42	21	reldmmpo 7490	. . . . 5 ⊢ Rel dom maAdju
43	42	ovprc 7394	. . . 4 ⊢ (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 maAdju 𝑅) = ∅)
44		df-mat 22391	. . . . . . . . . . 11 ⊢ Mat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ ((𝑟 freeLMod (𝑛 × 𝑛)) sSet ⟨(.r‘ndx), (𝑟 maMul ⟨𝑛, 𝑛, 𝑛⟩)⟩))
45	44	reldmmpo 7490	. . . . . . . . . 10 ⊢ Rel dom Mat
46	45	ovprc 7394	. . . . . . . . 9 ⊢ (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 Mat 𝑅) = ∅)
47	26, 46	eqtrid 2786	. . . . . . . 8 ⊢ (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → 𝐴 = ∅)
48	47	fveq2d 6831	. . . . . . 7 ⊢ (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (Base‘𝐴) = (Base‘∅))
49		base0 17175	. . . . . . 7 ⊢ ∅ = (Base‘∅)
50	48, 25, 49	3eqtr4g 2799	. . . . . 6 ⊢ (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → 𝐵 = ∅)
51	50	mpteq1d 5162	. . . . 5 ⊢ (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑚 ∈ 𝐵 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑚𝑙)))))) = (𝑚 ∈ ∅ ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑚𝑙)))))))
52		mpt0 6627	. . . . 5 ⊢ (𝑚 ∈ ∅ ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑚𝑙)))))) = ∅
53	51, 52	eqtrdi 2790	. . . 4 ⊢ (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑚 ∈ 𝐵 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑚𝑙)))))) = ∅)
54	43, 53	eqtr4d 2777	. . 3 ⊢ (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 maAdju 𝑅) = (𝑚 ∈ 𝐵 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑚𝑙)))))))
55	41, 54	pm2.61i 183	. 2 ⊢ (𝑁 maAdju 𝑅) = (𝑚 ∈ 𝐵 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑚𝑙))))))
56	1, 55	eqtri 2762	1 ⊢ 𝐽 = (𝑚 ∈ 𝐵 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑚𝑙))))))

Syntax hints:  ¬ wn 3   ∧ wa 396   = wceq 1547   ∈ wcel 2119  Vcvv 3431  ∅c0 4261  ifcif 4454  ⟨cop 4561  ⟨cotp 4563   ↦ cmpt 5153   × cxp 5616  ‘cfv 6485  (class class class)co 7356   ∈ cmpo 7358  Fincfn 8883   sSet csts 17124  ndxcnx 17154  Basecbs 17170  ._rcmulr 17212  0_gc0g 17393  1_rcur 20153   freeLMod cfrlm 21721   maMul cmmul 22373   Mat cmat 22390   maDet cmdat 22567   maAdju cmadu 22615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-cnex 11085  ax-1cn 11087  ax-addcl 11089

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-nn 12166  df-slot 17143  df-ndx 17155  df-base 17171  df-mat 22391  df-madu 22617

This theorem is referenced by:  maduval  22621  maduf  22624