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Theorem madufval 21970
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 7376 . . . . . 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 7360 . . . . . . . 8 (𝑛 = 𝑁 β†’ (𝑛 maDet π‘Ÿ) = (𝑁 maDet π‘Ÿ))
5 eqidd 2737 . . . . . . . . 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 7428 . . . . . . . 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 6846 . . . . . . 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 7428 . . . . . 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 5194 . . . . 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 7361 . . . . . . 7 (π‘Ÿ = 𝑅 β†’ (𝑁 Mat π‘Ÿ) = (𝑁 Mat 𝑅))
1110fveq2d 6843 . . . . . 6 (π‘Ÿ = 𝑅 β†’ (Baseβ€˜(𝑁 Mat π‘Ÿ)) = (Baseβ€˜(𝑁 Mat 𝑅)))
12 oveq2 7361 . . . . . . . 8 (π‘Ÿ = 𝑅 β†’ (𝑁 maDet π‘Ÿ) = (𝑁 maDet 𝑅))
13 fveq2 6839 . . . . . . . . . . 11 (π‘Ÿ = 𝑅 β†’ (1rβ€˜π‘Ÿ) = (1rβ€˜π‘…))
14 fveq2 6839 . . . . . . . . . . 11 (π‘Ÿ = 𝑅 β†’ (0gβ€˜π‘Ÿ) = (0gβ€˜π‘…))
1513, 14ifeq12d 4505 . . . . . . . . . 10 (π‘Ÿ = 𝑅 β†’ if(𝑙 = 𝑖, (1rβ€˜π‘Ÿ), (0gβ€˜π‘Ÿ)) = if(𝑙 = 𝑖, (1rβ€˜π‘…), (0gβ€˜π‘…)))
1615ifeq1d 4503 . . . . . . . . 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 7432 . . . . . . . 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 6846 . . . . . . 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 7432 . . . . . 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 5194 . . . . 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 21967 . . . . 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 6852 . . . . . 6 (Baseβ€˜(𝑁 Mat 𝑅)) ∈ V
2322mptex 7169 . . . . 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 7511 . . . 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 6842 . . . . . 6 (Baseβ€˜π΄) = (Baseβ€˜(𝑁 Mat 𝑅))
2825, 27eqtri 2764 . . . . 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 4505 . . . . . . . . . 10 ((π‘˜ ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) β†’ if(𝑙 = 𝑖, 1 , 0 ) = if(𝑙 = 𝑖, (1rβ€˜π‘…), (0gβ€˜π‘…)))
3534ifeq1d 4503 . . . . . . . . 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 7431 . . . . . . . 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 6845 . . . . . . 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 7431 . . . . 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 5209 . . . 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 2794 . . 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 7486 . . . . 5 Rel dom maAdju
4342ovprc 7391 . . . 4 (Β¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) β†’ (𝑁 maAdju 𝑅) = βˆ…)
44 df-mat 21739 . . . . . . . . . . 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 7486 . . . . . . . . . 10 Rel dom Mat
4645ovprc 7391 . . . . . . . . 9 (Β¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) β†’ (𝑁 Mat 𝑅) = βˆ…)
4726, 46eqtrid 2788 . . . . . . . 8 (Β¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) β†’ 𝐴 = βˆ…)
4847fveq2d 6843 . . . . . . 7 (Β¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) β†’ (Baseβ€˜π΄) = (Baseβ€˜βˆ…))
49 base0 17080 . . . . . . 7 βˆ… = (Baseβ€˜βˆ…)
5048, 25, 493eqtr4g 2801 . . . . . 6 (Β¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) β†’ 𝐡 = βˆ…)
5150mpteq1d 5198 . . . . 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 6640 . . . . 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 2792 . . . 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 2779 . . 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 2764 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 396   = wceq 1541   ∈ wcel 2106  Vcvv 3443  βˆ…c0 4280  ifcif 4484  βŸ¨cop 4590  βŸ¨cotp 4592   ↦ cmpt 5186   Γ— cxp 5629  β€˜cfv 6493  (class class class)co 7353   ∈ cmpo 7355  Fincfn 8879   sSet csts 17027  ndxcnx 17057  Basecbs 17075  .rcmulr 17126  0gc0g 17313  1rcur 19904   freeLMod cfrlm 21137   maMul cmmul 21716   Mat cmat 21738   maDet cmdat 21917   maAdju cmadu 21965
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5240  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7668  ax-cnex 11103  ax-1cn 11105  ax-addcl 11107
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-tr 5221  df-id 5529  df-eprel 5535  df-po 5543  df-so 5544  df-fr 5586  df-we 5588  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6251  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7356  df-oprab 7357  df-mpo 7358  df-om 7799  df-2nd 7918  df-frecs 8208  df-wrecs 8239  df-recs 8313  df-rdg 8352  df-nn 12150  df-slot 17046  df-ndx 17058  df-base 17076  df-mat 21739  df-madu 21967
This theorem is referenced by:  maduval  21971  maduf  21974
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