Theorem madufval 22572

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 7378	. . . . . 6 ⊢ (𝑛 = 𝑁 → (Base‘(𝑛 Mat 𝑟)) = (Base‘(𝑁 Mat 𝑟)))
3		id 22	. . . . . . 7 ⊢ (𝑛 = 𝑁 → 𝑛 = 𝑁)
4		oveq1 7362	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (𝑛 maDet 𝑟) = (𝑁 maDet 𝑟))
5		eqidd 2734	. . . . . . . . 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 7430	. . . . . . . 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 6838	. . . . . . 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 7430	. . . . . 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 5182	. . . . 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 7363	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (𝑁 Mat 𝑟) = (𝑁 Mat 𝑅))
11	10	fveq2d 6835	. . . . . 6 ⊢ (𝑟 = 𝑅 → (Base‘(𝑁 Mat 𝑟)) = (Base‘(𝑁 Mat 𝑅)))
12		oveq2 7363	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (𝑁 maDet 𝑟) = (𝑁 maDet 𝑅))
13		fveq2 6831	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (1r‘𝑟) = (1r‘𝑅))
14		fveq2 6831	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = (0g‘𝑅))
15	13, 14	ifeq12d 4498	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → if(𝑙 = 𝑖, (1r‘𝑟), (0g‘𝑟)) = if(𝑙 = 𝑖, (1r‘𝑅), (0g‘𝑅)))
16	15	ifeq1d 4496	. . . . . . . . 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 7434	. . . . . . . 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 6838	. . . . . . 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 7434	. . . . . 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 5182	. . . . 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 22569	. . . . 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 6844	. . . . . 6 ⊢ (Base‘(𝑁 Mat 𝑅)) ∈ V
23	22	mptex 7166	. . . . 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 7515	. . . 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 6834	. . . . . 6 ⊢ (Base‘𝐴) = (Base‘(𝑁 Mat 𝑅))
28	25, 27	eqtri 2756	. . . . 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 4498	. . . . . . . . . 10 ⊢ ((𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) → if(𝑙 = 𝑖, 1 , 0 ) = if(𝑙 = 𝑖, (1r‘𝑅), (0g‘𝑅)))
35	34	ifeq1d 4496	. . . . . . . . 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 7433	. . . . . . . 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 6837	. . . . . . 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 7433	. . . . 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 5192	. . . 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 2786	. . 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 7489	. . . . 5 ⊢ Rel dom maAdju
43	42	ovprc 7393	. . . 4 ⊢ (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 maAdju 𝑅) = ∅)
44		df-mat 22343	. . . . . . . . . . 11 ⊢ Mat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ ((𝑟 freeLMod (𝑛 × 𝑛)) sSet ⟨(.r‘ndx), (𝑟 maMul ⟨𝑛, 𝑛, 𝑛⟩)⟩))
45	44	reldmmpo 7489	. . . . . . . . . 10 ⊢ Rel dom Mat
46	45	ovprc 7393	. . . . . . . . 9 ⊢ (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 Mat 𝑅) = ∅)
47	26, 46	eqtrid 2780	. . . . . . . 8 ⊢ (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → 𝐴 = ∅)
48	47	fveq2d 6835	. . . . . . 7 ⊢ (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (Base‘𝐴) = (Base‘∅))
49		base0 17132	. . . . . . 7 ⊢ ∅ = (Base‘∅)
50	48, 25, 49	3eqtr4g 2793	. . . . . 6 ⊢ (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → 𝐵 = ∅)
51	50	mpteq1d 5185	. . . . 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 6631	. . . . 5 ⊢ (𝑚 ∈ ∅ ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑚𝑙)))))) = ∅
53	51, 52	eqtrdi 2784	. . . 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 2771	. . 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 182	. 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 2756	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 395   = wceq 1541   ∈ wcel 2113  Vcvv 3437  ∅c0 4282  ifcif 4476  ⟨cop 4583  ⟨cotp 4585   ↦ cmpt 5176   × cxp 5619  ‘cfv 6489  (class class class)co 7355   ∈ cmpo 7357  Fincfn 8879   sSet csts 17081  ndxcnx 17111  Basecbs 17127  ._rcmulr 17169  0_gc0g 17350  1_rcur 20107   freeLMod cfrlm 21692   maMul cmmul 22325   Mat cmat 22342   maDet cmdat 22519   maAdju cmadu 22567

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677  ax-cnex 11073  ax-1cn 11075  ax-addcl 11077

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-ov 7358  df-oprab 7359  df-mpo 7360  df-om 7806  df-2nd 7931  df-frecs 8220  df-wrecs 8251  df-recs 8300  df-rdg 8338  df-nn 12137  df-slot 17100  df-ndx 17112  df-base 17128  df-mat 22343  df-madu 22569

This theorem is referenced by:  maduval  22573  maduf  22576