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Theorem madufval 22567
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 𝐡 = (Baseβ€˜π΄)
madufval.o 1 = (1rβ€˜π‘…)
madufval.z 0 = (0gβ€˜π‘…)
Assertion
Ref Expression
madufval 𝐽 = (π‘š ∈ 𝐡 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (π·β€˜(π‘˜ ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(π‘˜ = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (π‘˜π‘šπ‘™))))))
Distinct variable groups:   π‘š,𝑁,𝑖,𝑗,π‘˜,𝑙   𝑅,π‘š,𝑖,𝑗,π‘˜,𝑙   𝐡,π‘š
Allowed substitution hints:   𝐴(𝑖,𝑗,π‘˜,π‘š,𝑙)   𝐡(𝑖,𝑗,π‘˜,𝑙)   𝐷(𝑖,𝑗,π‘˜,π‘š,𝑙)   1 (𝑖,𝑗,π‘˜,π‘š,𝑙)   𝐽(𝑖,𝑗,π‘˜,π‘š,𝑙)   0 (𝑖,𝑗,π‘˜,π‘š,𝑙)
Proof of Theorem madufval
Dummy variables 𝑛 π‘Ÿ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 madufval.j . 2 𝐽 = (𝑁 maAdju 𝑅)
2 fvoveq1 7449 . . . . . 6 (𝑛 = 𝑁 β†’ (Baseβ€˜(𝑛 Mat π‘Ÿ)) = (Baseβ€˜(𝑁 Mat π‘Ÿ)))
3 id 22 . . . . . . 7 (𝑛 = 𝑁 β†’ 𝑛 = 𝑁)
4 oveq1 7433 . . . . . . . 8 (𝑛 = 𝑁 β†’ (𝑛 maDet π‘Ÿ) = (𝑁 maDet π‘Ÿ))
5 eqidd 2729 . . . . . . . . 9 (𝑛 = 𝑁 β†’ if(π‘˜ = 𝑗, if(𝑙 = 𝑖, (1rβ€˜π‘Ÿ), (0gβ€˜π‘Ÿ)), (π‘˜π‘šπ‘™)) = if(π‘˜ = 𝑗, if(𝑙 = 𝑖, (1rβ€˜π‘Ÿ), (0gβ€˜π‘Ÿ)), (π‘˜π‘šπ‘™)))
63, 3, 5mpoeq123dv 7502 . . . . . . . 8 (𝑛 = 𝑁 β†’ (π‘˜ ∈ 𝑛, 𝑙 ∈ 𝑛 ↦ if(π‘˜ = 𝑗, if(𝑙 = 𝑖, (1rβ€˜π‘Ÿ), (0gβ€˜π‘Ÿ)), (π‘˜π‘šπ‘™))) = (π‘˜ ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(π‘˜ = 𝑗, if(𝑙 = 𝑖, (1rβ€˜π‘Ÿ), (0gβ€˜π‘Ÿ)), (π‘˜π‘šπ‘™))))
74, 6fveq12d 6909 . . . . . . 7 (𝑛 = 𝑁 β†’ ((𝑛 maDet π‘Ÿ)β€˜(π‘˜ ∈ 𝑛, 𝑙 ∈ 𝑛 ↦ if(π‘˜ = 𝑗, if(𝑙 = 𝑖, (1rβ€˜π‘Ÿ), (0gβ€˜π‘Ÿ)), (π‘˜π‘šπ‘™)))) = ((𝑁 maDet π‘Ÿ)β€˜(π‘˜ ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(π‘˜ = 𝑗, if(𝑙 = 𝑖, (1rβ€˜π‘Ÿ), (0gβ€˜π‘Ÿ)), (π‘˜π‘šπ‘™)))))
83, 3, 7mpoeq123dv 7502 . . . . . 6 (𝑛 = 𝑁 β†’ (𝑖 ∈ 𝑛, 𝑗 ∈ 𝑛 ↦ ((𝑛 maDet π‘Ÿ)β€˜(π‘˜ ∈ 𝑛, 𝑙 ∈ 𝑛 ↦ if(π‘˜ = 𝑗, if(𝑙 = 𝑖, (1rβ€˜π‘Ÿ), (0gβ€˜π‘Ÿ)), (π‘˜π‘šπ‘™))))) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑁 maDet π‘Ÿ)β€˜(π‘˜ ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(π‘˜ = 𝑗, if(𝑙 = 𝑖, (1rβ€˜π‘Ÿ), (0gβ€˜π‘Ÿ)), (π‘˜π‘šπ‘™))))))
92, 8mpteq12dv 5243 . . . . 5 (𝑛 = 𝑁 β†’ (π‘š ∈ (Baseβ€˜(𝑛 Mat π‘Ÿ)) ↦ (𝑖 ∈ 𝑛, 𝑗 ∈ 𝑛 ↦ ((𝑛 maDet π‘Ÿ)β€˜(π‘˜ ∈ 𝑛, 𝑙 ∈ 𝑛 ↦ if(π‘˜ = 𝑗, if(𝑙 = 𝑖, (1rβ€˜π‘Ÿ), (0gβ€˜π‘Ÿ)), (π‘˜π‘šπ‘™)))))) = (π‘š ∈ (Baseβ€˜(𝑁 Mat π‘Ÿ)) ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑁 maDet π‘Ÿ)β€˜(π‘˜ ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(π‘˜ = 𝑗, if(𝑙 = 𝑖, (1rβ€˜π‘Ÿ), (0gβ€˜π‘Ÿ)), (π‘˜π‘šπ‘™)))))))
10 oveq2 7434 . . . . . . 7 (π‘Ÿ = 𝑅 β†’ (𝑁 Mat π‘Ÿ) = (𝑁 Mat 𝑅))
1110fveq2d 6906 . . . . . 6 (π‘Ÿ = 𝑅 β†’ (Baseβ€˜(𝑁 Mat π‘Ÿ)) = (Baseβ€˜(𝑁 Mat 𝑅)))
12 oveq2 7434 . . . . . . . 8 (π‘Ÿ = 𝑅 β†’ (𝑁 maDet π‘Ÿ) = (𝑁 maDet 𝑅))
13 fveq2 6902 . . . . . . . . . . 11 (π‘Ÿ = 𝑅 β†’ (1rβ€˜π‘Ÿ) = (1rβ€˜π‘…))
14 fveq2 6902 . . . . . . . . . . 11 (π‘Ÿ = 𝑅 β†’ (0gβ€˜π‘Ÿ) = (0gβ€˜π‘…))
1513, 14ifeq12d 4553 . . . . . . . . . 10 (π‘Ÿ = 𝑅 β†’ if(𝑙 = 𝑖, (1rβ€˜π‘Ÿ), (0gβ€˜π‘Ÿ)) = if(𝑙 = 𝑖, (1rβ€˜π‘…), (0gβ€˜π‘…)))
1615ifeq1d 4551 . . . . . . . . 9 (π‘Ÿ = 𝑅 β†’ if(π‘˜ = 𝑗, if(𝑙 = 𝑖, (1rβ€˜π‘Ÿ), (0gβ€˜π‘Ÿ)), (π‘˜π‘šπ‘™)) = if(π‘˜ = 𝑗, if(𝑙 = 𝑖, (1rβ€˜π‘…), (0gβ€˜π‘…)), (π‘˜π‘šπ‘™)))
1716mpoeq3dv 7506 . . . . . . . 8 (π‘Ÿ = 𝑅 β†’ (π‘˜ ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(π‘˜ = 𝑗, if(𝑙 = 𝑖, (1rβ€˜π‘Ÿ), (0gβ€˜π‘Ÿ)), (π‘˜π‘šπ‘™))) = (π‘˜ ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(π‘˜ = 𝑗, if(𝑙 = 𝑖, (1rβ€˜π‘…), (0gβ€˜π‘…)), (π‘˜π‘šπ‘™))))
1812, 17fveq12d 6909 . . . . . . 7 (π‘Ÿ = 𝑅 β†’ ((𝑁 maDet π‘Ÿ)β€˜(π‘˜ ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(π‘˜ = 𝑗, if(𝑙 = 𝑖, (1rβ€˜π‘Ÿ), (0gβ€˜π‘Ÿ)), (π‘˜π‘šπ‘™)))) = ((𝑁 maDet 𝑅)β€˜(π‘˜ ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(π‘˜ = 𝑗, if(𝑙 = 𝑖, (1rβ€˜π‘…), (0gβ€˜π‘…)), (π‘˜π‘šπ‘™)))))
1918mpoeq3dv 7506 . . . . . 6 (π‘Ÿ = 𝑅 β†’ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑁 maDet π‘Ÿ)β€˜(π‘˜ ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(π‘˜ = 𝑗, if(𝑙 = 𝑖, (1rβ€˜π‘Ÿ), (0gβ€˜π‘Ÿ)), (π‘˜π‘šπ‘™))))) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑁 maDet 𝑅)β€˜(π‘˜ ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(π‘˜ = 𝑗, if(𝑙 = 𝑖, (1rβ€˜π‘…), (0gβ€˜π‘…)), (π‘˜π‘šπ‘™))))))
2011, 19mpteq12dv 5243 . . . . 5 (π‘Ÿ = 𝑅 β†’ (π‘š ∈ (Baseβ€˜(𝑁 Mat π‘Ÿ)) ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑁 maDet π‘Ÿ)β€˜(π‘˜ ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(π‘˜ = 𝑗, if(𝑙 = 𝑖, (1rβ€˜π‘Ÿ), (0gβ€˜π‘Ÿ)), (π‘˜π‘šπ‘™)))))) = (π‘š ∈ (Baseβ€˜(𝑁 Mat 𝑅)) ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑁 maDet 𝑅)β€˜(π‘˜ ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(π‘˜ = 𝑗, if(𝑙 = 𝑖, (1rβ€˜π‘…), (0gβ€˜π‘…)), (π‘˜π‘šπ‘™)))))))
21 df-madu 22564 . . . . 5 maAdju = (𝑛 ∈ V, π‘Ÿ ∈ V ↦ (π‘š ∈ (Baseβ€˜(𝑛 Mat π‘Ÿ)) ↦ (𝑖 ∈ 𝑛, 𝑗 ∈ 𝑛 ↦ ((𝑛 maDet π‘Ÿ)β€˜(π‘˜ ∈ 𝑛, 𝑙 ∈ 𝑛 ↦ if(π‘˜ = 𝑗, if(𝑙 = 𝑖, (1rβ€˜π‘Ÿ), (0gβ€˜π‘Ÿ)), (π‘˜π‘šπ‘™)))))))
22 fvex 6915 . . . . . 6 (Baseβ€˜(𝑁 Mat 𝑅)) ∈ V
2322mptex 7241 . . . . 5 (π‘š ∈ (Baseβ€˜(𝑁 Mat 𝑅)) ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑁 maDet 𝑅)β€˜(π‘˜ ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(π‘˜ = 𝑗, if(𝑙 = 𝑖, (1rβ€˜π‘…), (0gβ€˜π‘…)), (π‘˜π‘šπ‘™)))))) ∈ V
249, 20, 21, 23ovmpo 7588 . . . 4 ((𝑁 ∈ V ∧ 𝑅 ∈ V) β†’ (𝑁 maAdju 𝑅) = (π‘š ∈ (Baseβ€˜(𝑁 Mat 𝑅)) ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑁 maDet 𝑅)β€˜(π‘˜ ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(π‘˜ = 𝑗, if(𝑙 = 𝑖, (1rβ€˜π‘…), (0gβ€˜π‘…)), (π‘˜π‘šπ‘™)))))))
25 madufval.b . . . . . 6 𝐡 = (Baseβ€˜π΄)
26 madufval.a . . . . . . 7 𝐴 = (𝑁 Mat 𝑅)
2726fveq2i 6905 . . . . . 6 (Baseβ€˜π΄) = (Baseβ€˜(𝑁 Mat 𝑅))
2825, 27eqtri 2756 . . . . 5 𝐡 = (Baseβ€˜(𝑁 Mat 𝑅))
29 madufval.d . . . . . . . 8 𝐷 = (𝑁 maDet 𝑅)
30 madufval.o . . . . . . . . . . . 12 1 = (1rβ€˜π‘…)
3130a1i 11 . . . . . . . . . . 11 ((π‘˜ ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) β†’ 1 = (1rβ€˜π‘…))
32 madufval.z . . . . . . . . . . . 12 0 = (0gβ€˜π‘…)
3332a1i 11 . . . . . . . . . . 11 ((π‘˜ ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) β†’ 0 = (0gβ€˜π‘…))
3431, 33ifeq12d 4553 . . . . . . . . . 10 ((π‘˜ ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) β†’ if(𝑙 = 𝑖, 1 , 0 ) = if(𝑙 = 𝑖, (1rβ€˜π‘…), (0gβ€˜π‘…)))
3534ifeq1d 4551 . . . . . . . . 9 ((π‘˜ ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) β†’ if(π‘˜ = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (π‘˜π‘šπ‘™)) = if(π‘˜ = 𝑗, if(𝑙 = 𝑖, (1rβ€˜π‘…), (0gβ€˜π‘…)), (π‘˜π‘šπ‘™)))
3635mpoeq3ia 7505 . . . . . . . 8 (π‘˜ ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(π‘˜ = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (π‘˜π‘šπ‘™))) = (π‘˜ ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(π‘˜ = 𝑗, if(𝑙 = 𝑖, (1rβ€˜π‘…), (0gβ€˜π‘…)), (π‘˜π‘šπ‘™)))
3729, 36fveq12i 6908 . . . . . . 7 (π·β€˜(π‘˜ ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(π‘˜ = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (π‘˜π‘šπ‘™)))) = ((𝑁 maDet 𝑅)β€˜(π‘˜ ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(π‘˜ = 𝑗, if(𝑙 = 𝑖, (1rβ€˜π‘…), (0gβ€˜π‘…)), (π‘˜π‘šπ‘™))))
3837a1i 11 . . . . . 6 ((𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) β†’ (π·β€˜(π‘˜ ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(π‘˜ = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (π‘˜π‘šπ‘™)))) = ((𝑁 maDet 𝑅)β€˜(π‘˜ ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(π‘˜ = 𝑗, if(𝑙 = 𝑖, (1rβ€˜π‘…), (0gβ€˜π‘…)), (π‘˜π‘šπ‘™)))))
3938mpoeq3ia 7505 . . . . 5 (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (π·β€˜(π‘˜ ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(π‘˜ = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (π‘˜π‘šπ‘™))))) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑁 maDet 𝑅)β€˜(π‘˜ ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(π‘˜ = 𝑗, if(𝑙 = 𝑖, (1rβ€˜π‘…), (0gβ€˜π‘…)), (π‘˜π‘šπ‘™)))))
4028, 39mpteq12i 5258 . . . 4 (π‘š ∈ 𝐡 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (π·β€˜(π‘˜ ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(π‘˜ = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (π‘˜π‘šπ‘™)))))) = (π‘š ∈ (Baseβ€˜(𝑁 Mat 𝑅)) ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑁 maDet 𝑅)β€˜(π‘˜ ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(π‘˜ = 𝑗, if(𝑙 = 𝑖, (1rβ€˜π‘…), (0gβ€˜π‘…)), (π‘˜π‘šπ‘™))))))
4124, 40eqtr4di 2786 . . 3 ((𝑁 ∈ V ∧ 𝑅 ∈ V) β†’ (𝑁 maAdju 𝑅) = (π‘š ∈ 𝐡 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (π·β€˜(π‘˜ ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(π‘˜ = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (π‘˜π‘šπ‘™)))))))
4221reldmmpo 7562 . . . . 5 Rel dom maAdju
4342ovprc 7464 . . . 4 (Β¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) β†’ (𝑁 maAdju 𝑅) = βˆ…)
44 df-mat 22336 . . . . . . . . . . 11 Mat = (𝑛 ∈ Fin, π‘Ÿ ∈ V ↦ ((π‘Ÿ freeLMod (𝑛 Γ— 𝑛)) sSet ⟨(.rβ€˜ndx), (π‘Ÿ maMul βŸ¨π‘›, 𝑛, π‘›βŸ©)⟩))
4544reldmmpo 7562 . . . . . . . . . 10 Rel dom Mat
4645ovprc 7464 . . . . . . . . 9 (Β¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) β†’ (𝑁 Mat 𝑅) = βˆ…)
4726, 46eqtrid 2780 . . . . . . . 8 (Β¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) β†’ 𝐴 = βˆ…)
4847fveq2d 6906 . . . . . . 7 (Β¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) β†’ (Baseβ€˜π΄) = (Baseβ€˜βˆ…))
49 base0 17194 . . . . . . 7 βˆ… = (Baseβ€˜βˆ…)
5048, 25, 493eqtr4g 2793 . . . . . 6 (Β¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) β†’ 𝐡 = βˆ…)
5150mpteq1d 5247 . . . . 5 (Β¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) β†’ (π‘š ∈ 𝐡 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (π·β€˜(π‘˜ ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(π‘˜ = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (π‘˜π‘šπ‘™)))))) = (π‘š ∈ βˆ… ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (π·β€˜(π‘˜ ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(π‘˜ = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (π‘˜π‘šπ‘™)))))))
52 mpt0 6702 . . . . 5 (π‘š ∈ βˆ… ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (π·β€˜(π‘˜ ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(π‘˜ = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (π‘˜π‘šπ‘™)))))) = βˆ…
5351, 52eqtrdi 2784 . . . 4 (Β¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) β†’ (π‘š ∈ 𝐡 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (π·β€˜(π‘˜ ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(π‘˜ = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (π‘˜π‘šπ‘™)))))) = βˆ…)
5443, 53eqtr4d 2771 . . 3 (Β¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) β†’ (𝑁 maAdju 𝑅) = (π‘š ∈ 𝐡 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (π·β€˜(π‘˜ ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(π‘˜ = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (π‘˜π‘šπ‘™)))))))
5541, 54pm2.61i 182 . 2 (𝑁 maAdju 𝑅) = (π‘š ∈ 𝐡 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (π·β€˜(π‘˜ ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(π‘˜ = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (π‘˜π‘šπ‘™))))))
561, 55eqtri 2756 1 𝐽 = (π‘š ∈ 𝐡 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (π·β€˜(π‘˜ ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(π‘˜ = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (π‘˜π‘šπ‘™))))))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   ∧ wa 394   = wceq 1533   ∈ wcel 2098  Vcvv 3473  βˆ…c0 4326  ifcif 4532  βŸ¨cop 4638  βŸ¨cotp 4640   ↦ cmpt 5235   Γ— cxp 5680  β€˜cfv 6553  (class class class)co 7426   ∈ cmpo 7428  Fincfn 8972   sSet csts 17141  ndxcnx 17171  Basecbs 17189  .rcmulr 17243  0gc0g 17430  1rcur 20135   freeLMod cfrlm 21694   maMul cmmul 22318   Mat cmat 22335   maDet cmdat 22514   maAdju cmadu 22562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7748  ax-cnex 11204  ax-1cn 11206  ax-addcl 11208
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7879  df-2nd 8002  df-frecs 8295  df-wrecs 8326  df-recs 8400  df-rdg 8439  df-nn 12253  df-slot 17160  df-ndx 17172  df-base 17190  df-mat 22336  df-madu 22564
This theorem is referenced by:  maduval  22568  maduf  22571
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