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Theorem madufval 22494
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 7428 . . . . . 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 7412 . . . . . . . 8 (𝑛 = 𝑁 β†’ (𝑛 maDet π‘Ÿ) = (𝑁 maDet π‘Ÿ))
5 eqidd 2727 . . . . . . . . 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 7480 . . . . . . . 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 6892 . . . . . . 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 7480 . . . . . 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 5232 . . . . 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 7413 . . . . . . 7 (π‘Ÿ = 𝑅 β†’ (𝑁 Mat π‘Ÿ) = (𝑁 Mat 𝑅))
1110fveq2d 6889 . . . . . 6 (π‘Ÿ = 𝑅 β†’ (Baseβ€˜(𝑁 Mat π‘Ÿ)) = (Baseβ€˜(𝑁 Mat 𝑅)))
12 oveq2 7413 . . . . . . . 8 (π‘Ÿ = 𝑅 β†’ (𝑁 maDet π‘Ÿ) = (𝑁 maDet 𝑅))
13 fveq2 6885 . . . . . . . . . . 11 (π‘Ÿ = 𝑅 β†’ (1rβ€˜π‘Ÿ) = (1rβ€˜π‘…))
14 fveq2 6885 . . . . . . . . . . 11 (π‘Ÿ = 𝑅 β†’ (0gβ€˜π‘Ÿ) = (0gβ€˜π‘…))
1513, 14ifeq12d 4544 . . . . . . . . . 10 (π‘Ÿ = 𝑅 β†’ if(𝑙 = 𝑖, (1rβ€˜π‘Ÿ), (0gβ€˜π‘Ÿ)) = if(𝑙 = 𝑖, (1rβ€˜π‘…), (0gβ€˜π‘…)))
1615ifeq1d 4542 . . . . . . . . 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 7484 . . . . . . . 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 6892 . . . . . . 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 7484 . . . . . 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 5232 . . . . 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 22491 . . . . 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 6898 . . . . . 6 (Baseβ€˜(𝑁 Mat 𝑅)) ∈ V
2322mptex 7220 . . . . 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 7564 . . . 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 6888 . . . . . 6 (Baseβ€˜π΄) = (Baseβ€˜(𝑁 Mat 𝑅))
2825, 27eqtri 2754 . . . . 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 4544 . . . . . . . . . 10 ((π‘˜ ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) β†’ if(𝑙 = 𝑖, 1 , 0 ) = if(𝑙 = 𝑖, (1rβ€˜π‘…), (0gβ€˜π‘…)))
3534ifeq1d 4542 . . . . . . . . 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 7483 . . . . . . . 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 6891 . . . . . . 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 7483 . . . . 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 5247 . . . 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 2784 . . 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 7539 . . . . 5 Rel dom maAdju
4342ovprc 7443 . . . 4 (Β¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) β†’ (𝑁 maAdju 𝑅) = βˆ…)
44 df-mat 22263 . . . . . . . . . . 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 7539 . . . . . . . . . 10 Rel dom Mat
4645ovprc 7443 . . . . . . . . 9 (Β¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) β†’ (𝑁 Mat 𝑅) = βˆ…)
4726, 46eqtrid 2778 . . . . . . . 8 (Β¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) β†’ 𝐴 = βˆ…)
4847fveq2d 6889 . . . . . . 7 (Β¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) β†’ (Baseβ€˜π΄) = (Baseβ€˜βˆ…))
49 base0 17158 . . . . . . 7 βˆ… = (Baseβ€˜βˆ…)
5048, 25, 493eqtr4g 2791 . . . . . 6 (Β¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) β†’ 𝐡 = βˆ…)
5150mpteq1d 5236 . . . . 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 6686 . . . . 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 2782 . . . 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 2769 . . 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 2754 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 395   = wceq 1533   ∈ wcel 2098  Vcvv 3468  βˆ…c0 4317  ifcif 4523  βŸ¨cop 4629  βŸ¨cotp 4631   ↦ cmpt 5224   Γ— cxp 5667  β€˜cfv 6537  (class class class)co 7405   ∈ cmpo 7407  Fincfn 8941   sSet csts 17105  ndxcnx 17135  Basecbs 17153  .rcmulr 17207  0gc0g 17394  1rcur 20086   freeLMod cfrlm 21641   maMul cmmul 22240   Mat cmat 22262   maDet cmdat 22441   maAdju cmadu 22489
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-cnex 11168  ax-1cn 11170  ax-addcl 11172
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7853  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-nn 12217  df-slot 17124  df-ndx 17136  df-base 17154  df-mat 22263  df-madu 22491
This theorem is referenced by:  maduval  22495  maduf  22498
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