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Theorem madufval 22359
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 7434 . . . . . 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 7418 . . . . . . . 8 (𝑛 = 𝑁 β†’ (𝑛 maDet π‘Ÿ) = (𝑁 maDet π‘Ÿ))
5 eqidd 2731 . . . . . . . . 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 7486 . . . . . . . 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 6897 . . . . . . 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 7486 . . . . . 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 5238 . . . . 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 7419 . . . . . . 7 (π‘Ÿ = 𝑅 β†’ (𝑁 Mat π‘Ÿ) = (𝑁 Mat 𝑅))
1110fveq2d 6894 . . . . . 6 (π‘Ÿ = 𝑅 β†’ (Baseβ€˜(𝑁 Mat π‘Ÿ)) = (Baseβ€˜(𝑁 Mat 𝑅)))
12 oveq2 7419 . . . . . . . 8 (π‘Ÿ = 𝑅 β†’ (𝑁 maDet π‘Ÿ) = (𝑁 maDet 𝑅))
13 fveq2 6890 . . . . . . . . . . 11 (π‘Ÿ = 𝑅 β†’ (1rβ€˜π‘Ÿ) = (1rβ€˜π‘…))
14 fveq2 6890 . . . . . . . . . . 11 (π‘Ÿ = 𝑅 β†’ (0gβ€˜π‘Ÿ) = (0gβ€˜π‘…))
1513, 14ifeq12d 4548 . . . . . . . . . 10 (π‘Ÿ = 𝑅 β†’ if(𝑙 = 𝑖, (1rβ€˜π‘Ÿ), (0gβ€˜π‘Ÿ)) = if(𝑙 = 𝑖, (1rβ€˜π‘…), (0gβ€˜π‘…)))
1615ifeq1d 4546 . . . . . . . . 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 7490 . . . . . . . 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 6897 . . . . . . 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 7490 . . . . . 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 5238 . . . . 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 22356 . . . . 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 6903 . . . . . 6 (Baseβ€˜(𝑁 Mat 𝑅)) ∈ V
2322mptex 7226 . . . . 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 7570 . . . 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 6893 . . . . . 6 (Baseβ€˜π΄) = (Baseβ€˜(𝑁 Mat 𝑅))
2825, 27eqtri 2758 . . . . 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 4548 . . . . . . . . . 10 ((π‘˜ ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) β†’ if(𝑙 = 𝑖, 1 , 0 ) = if(𝑙 = 𝑖, (1rβ€˜π‘…), (0gβ€˜π‘…)))
3534ifeq1d 4546 . . . . . . . . 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 7489 . . . . . . . 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 6896 . . . . . . 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 7489 . . . . 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 5253 . . . 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 2788 . . 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 7545 . . . . 5 Rel dom maAdju
4342ovprc 7449 . . . 4 (Β¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) β†’ (𝑁 maAdju 𝑅) = βˆ…)
44 df-mat 22128 . . . . . . . . . . 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 7545 . . . . . . . . . 10 Rel dom Mat
4645ovprc 7449 . . . . . . . . 9 (Β¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) β†’ (𝑁 Mat 𝑅) = βˆ…)
4726, 46eqtrid 2782 . . . . . . . 8 (Β¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) β†’ 𝐴 = βˆ…)
4847fveq2d 6894 . . . . . . 7 (Β¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) β†’ (Baseβ€˜π΄) = (Baseβ€˜βˆ…))
49 base0 17153 . . . . . . 7 βˆ… = (Baseβ€˜βˆ…)
5048, 25, 493eqtr4g 2795 . . . . . 6 (Β¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) β†’ 𝐡 = βˆ…)
5150mpteq1d 5242 . . . . 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 6691 . . . . 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 2786 . . . 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 2773 . . 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 2758 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 1539   ∈ wcel 2104  Vcvv 3472  βˆ…c0 4321  ifcif 4527  βŸ¨cop 4633  βŸ¨cotp 4635   ↦ cmpt 5230   Γ— cxp 5673  β€˜cfv 6542  (class class class)co 7411   ∈ cmpo 7413  Fincfn 8941   sSet csts 17100  ndxcnx 17130  Basecbs 17148  .rcmulr 17202  0gc0g 17389  1rcur 20075   freeLMod cfrlm 21520   maMul cmmul 22105   Mat cmat 22127   maDet cmdat 22306   maAdju cmadu 22354
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-1cn 11170  ax-addcl 11172
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-nn 12217  df-slot 17119  df-ndx 17131  df-base 17149  df-mat 22128  df-madu 22356
This theorem is referenced by:  maduval  22360  maduf  22363
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