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Theorem madufval 20645
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 = (1r𝑅)
madufval.z 0 = (0g𝑅)
Assertion
Ref Expression
madufval 𝐽 = (𝑚𝐵 ↦ (𝑖𝑁, 𝑗𝑁 ↦ (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ 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.
StepHypRef Expression
1 madufval.j . 2 𝐽 = (𝑁 maAdju 𝑅)
2 oveq1 6820 . . . . . . 7 (𝑛 = 𝑁 → (𝑛 Mat 𝑟) = (𝑁 Mat 𝑟))
32fveq2d 6356 . . . . . 6 (𝑛 = 𝑁 → (Base‘(𝑛 Mat 𝑟)) = (Base‘(𝑁 Mat 𝑟)))
4 id 22 . . . . . . 7 (𝑛 = 𝑁𝑛 = 𝑁)
5 oveq1 6820 . . . . . . . 8 (𝑛 = 𝑁 → (𝑛 maDet 𝑟) = (𝑁 maDet 𝑟))
6 eqidd 2761 . . . . . . . . 9 (𝑛 = 𝑁 → if(𝑘 = 𝑗, if(𝑙 = 𝑖, (1r𝑟), (0g𝑟)), (𝑘𝑚𝑙)) = if(𝑘 = 𝑗, if(𝑙 = 𝑖, (1r𝑟), (0g𝑟)), (𝑘𝑚𝑙)))
74, 4, 6mpt2eq123dv 6882 . . . . . . . 8 (𝑛 = 𝑁 → (𝑘𝑛, 𝑙𝑛 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, (1r𝑟), (0g𝑟)), (𝑘𝑚𝑙))) = (𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, (1r𝑟), (0g𝑟)), (𝑘𝑚𝑙))))
85, 7fveq12d 6358 . . . . . . 7 (𝑛 = 𝑁 → ((𝑛 maDet 𝑟)‘(𝑘𝑛, 𝑙𝑛 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, (1r𝑟), (0g𝑟)), (𝑘𝑚𝑙)))) = ((𝑁 maDet 𝑟)‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, (1r𝑟), (0g𝑟)), (𝑘𝑚𝑙)))))
94, 4, 8mpt2eq123dv 6882 . . . . . 6 (𝑛 = 𝑁 → (𝑖𝑛, 𝑗𝑛 ↦ ((𝑛 maDet 𝑟)‘(𝑘𝑛, 𝑙𝑛 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, (1r𝑟), (0g𝑟)), (𝑘𝑚𝑙))))) = (𝑖𝑁, 𝑗𝑁 ↦ ((𝑁 maDet 𝑟)‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, (1r𝑟), (0g𝑟)), (𝑘𝑚𝑙))))))
103, 9mpteq12dv 4885 . . . . 5 (𝑛 = 𝑁 → (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑖𝑛, 𝑗𝑛 ↦ ((𝑛 maDet 𝑟)‘(𝑘𝑛, 𝑙𝑛 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, (1r𝑟), (0g𝑟)), (𝑘𝑚𝑙)))))) = (𝑚 ∈ (Base‘(𝑁 Mat 𝑟)) ↦ (𝑖𝑁, 𝑗𝑁 ↦ ((𝑁 maDet 𝑟)‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, (1r𝑟), (0g𝑟)), (𝑘𝑚𝑙)))))))
11 oveq2 6821 . . . . . . 7 (𝑟 = 𝑅 → (𝑁 Mat 𝑟) = (𝑁 Mat 𝑅))
1211fveq2d 6356 . . . . . 6 (𝑟 = 𝑅 → (Base‘(𝑁 Mat 𝑟)) = (Base‘(𝑁 Mat 𝑅)))
13 oveq2 6821 . . . . . . . 8 (𝑟 = 𝑅 → (𝑁 maDet 𝑟) = (𝑁 maDet 𝑅))
14 fveq2 6352 . . . . . . . . . . 11 (𝑟 = 𝑅 → (1r𝑟) = (1r𝑅))
15 fveq2 6352 . . . . . . . . . . 11 (𝑟 = 𝑅 → (0g𝑟) = (0g𝑅))
1614, 15ifeq12d 4250 . . . . . . . . . 10 (𝑟 = 𝑅 → if(𝑙 = 𝑖, (1r𝑟), (0g𝑟)) = if(𝑙 = 𝑖, (1r𝑅), (0g𝑅)))
1716ifeq1d 4248 . . . . . . . . 9 (𝑟 = 𝑅 → if(𝑘 = 𝑗, if(𝑙 = 𝑖, (1r𝑟), (0g𝑟)), (𝑘𝑚𝑙)) = if(𝑘 = 𝑗, if(𝑙 = 𝑖, (1r𝑅), (0g𝑅)), (𝑘𝑚𝑙)))
1817mpt2eq3dv 6886 . . . . . . . 8 (𝑟 = 𝑅 → (𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, (1r𝑟), (0g𝑟)), (𝑘𝑚𝑙))) = (𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, (1r𝑅), (0g𝑅)), (𝑘𝑚𝑙))))
1913, 18fveq12d 6358 . . . . . . 7 (𝑟 = 𝑅 → ((𝑁 maDet 𝑟)‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, (1r𝑟), (0g𝑟)), (𝑘𝑚𝑙)))) = ((𝑁 maDet 𝑅)‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, (1r𝑅), (0g𝑅)), (𝑘𝑚𝑙)))))
2019mpt2eq3dv 6886 . . . . . 6 (𝑟 = 𝑅 → (𝑖𝑁, 𝑗𝑁 ↦ ((𝑁 maDet 𝑟)‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, (1r𝑟), (0g𝑟)), (𝑘𝑚𝑙))))) = (𝑖𝑁, 𝑗𝑁 ↦ ((𝑁 maDet 𝑅)‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, (1r𝑅), (0g𝑅)), (𝑘𝑚𝑙))))))
2112, 20mpteq12dv 4885 . . . . 5 (𝑟 = 𝑅 → (𝑚 ∈ (Base‘(𝑁 Mat 𝑟)) ↦ (𝑖𝑁, 𝑗𝑁 ↦ ((𝑁 maDet 𝑟)‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, (1r𝑟), (0g𝑟)), (𝑘𝑚𝑙)))))) = (𝑚 ∈ (Base‘(𝑁 Mat 𝑅)) ↦ (𝑖𝑁, 𝑗𝑁 ↦ ((𝑁 maDet 𝑅)‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, (1r𝑅), (0g𝑅)), (𝑘𝑚𝑙)))))))
22 df-madu 20642 . . . . 5 maAdju = (𝑛 ∈ V, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑖𝑛, 𝑗𝑛 ↦ ((𝑛 maDet 𝑟)‘(𝑘𝑛, 𝑙𝑛 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, (1r𝑟), (0g𝑟)), (𝑘𝑚𝑙)))))))
23 fvex 6362 . . . . . 6 (Base‘(𝑁 Mat 𝑅)) ∈ V
2423mptex 6650 . . . . 5 (𝑚 ∈ (Base‘(𝑁 Mat 𝑅)) ↦ (𝑖𝑁, 𝑗𝑁 ↦ ((𝑁 maDet 𝑅)‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, (1r𝑅), (0g𝑅)), (𝑘𝑚𝑙)))))) ∈ V
2510, 21, 22, 24ovmpt2 6961 . . . 4 ((𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 maAdju 𝑅) = (𝑚 ∈ (Base‘(𝑁 Mat 𝑅)) ↦ (𝑖𝑁, 𝑗𝑁 ↦ ((𝑁 maDet 𝑅)‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, (1r𝑅), (0g𝑅)), (𝑘𝑚𝑙)))))))
26 madufval.b . . . . . 6 𝐵 = (Base‘𝐴)
27 madufval.a . . . . . . 7 𝐴 = (𝑁 Mat 𝑅)
2827fveq2i 6355 . . . . . 6 (Base‘𝐴) = (Base‘(𝑁 Mat 𝑅))
2926, 28eqtri 2782 . . . . 5 𝐵 = (Base‘(𝑁 Mat 𝑅))
30 madufval.d . . . . . . . 8 𝐷 = (𝑁 maDet 𝑅)
31 madufval.o . . . . . . . . . . . 12 1 = (1r𝑅)
3231a1i 11 . . . . . . . . . . 11 ((𝑘𝑁𝑙𝑁) → 1 = (1r𝑅))
33 madufval.z . . . . . . . . . . . 12 0 = (0g𝑅)
3433a1i 11 . . . . . . . . . . 11 ((𝑘𝑁𝑙𝑁) → 0 = (0g𝑅))
3532, 34ifeq12d 4250 . . . . . . . . . 10 ((𝑘𝑁𝑙𝑁) → if(𝑙 = 𝑖, 1 , 0 ) = if(𝑙 = 𝑖, (1r𝑅), (0g𝑅)))
3635ifeq1d 4248 . . . . . . . . 9 ((𝑘𝑁𝑙𝑁) → if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑚𝑙)) = if(𝑘 = 𝑗, if(𝑙 = 𝑖, (1r𝑅), (0g𝑅)), (𝑘𝑚𝑙)))
3736mpt2eq3ia 6885 . . . . . . . 8 (𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑚𝑙))) = (𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, (1r𝑅), (0g𝑅)), (𝑘𝑚𝑙)))
3830, 37fveq12i 6357 . . . . . . 7 (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑚𝑙)))) = ((𝑁 maDet 𝑅)‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, (1r𝑅), (0g𝑅)), (𝑘𝑚𝑙))))
3938a1i 11 . . . . . 6 ((𝑖𝑁𝑗𝑁) → (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑚𝑙)))) = ((𝑁 maDet 𝑅)‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, (1r𝑅), (0g𝑅)), (𝑘𝑚𝑙)))))
4039mpt2eq3ia 6885 . . . . 5 (𝑖𝑁, 𝑗𝑁 ↦ (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑚𝑙))))) = (𝑖𝑁, 𝑗𝑁 ↦ ((𝑁 maDet 𝑅)‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, (1r𝑅), (0g𝑅)), (𝑘𝑚𝑙)))))
4129, 40mpteq12i 4894 . . . 4 (𝑚𝐵 ↦ (𝑖𝑁, 𝑗𝑁 ↦ (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑚𝑙)))))) = (𝑚 ∈ (Base‘(𝑁 Mat 𝑅)) ↦ (𝑖𝑁, 𝑗𝑁 ↦ ((𝑁 maDet 𝑅)‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, (1r𝑅), (0g𝑅)), (𝑘𝑚𝑙))))))
4225, 41syl6eqr 2812 . . 3 ((𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 maAdju 𝑅) = (𝑚𝐵 ↦ (𝑖𝑁, 𝑗𝑁 ↦ (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑚𝑙)))))))
4322reldmmpt2 6936 . . . . 5 Rel dom maAdju
4443ovprc 6846 . . . 4 (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 maAdju 𝑅) = ∅)
45 df-mat 20416 . . . . . . . . . . 11 Mat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ ((𝑟 freeLMod (𝑛 × 𝑛)) sSet ⟨(.r‘ndx), (𝑟 maMul ⟨𝑛, 𝑛, 𝑛⟩)⟩))
4645reldmmpt2 6936 . . . . . . . . . 10 Rel dom Mat
4746ovprc 6846 . . . . . . . . 9 (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 Mat 𝑅) = ∅)
4827, 47syl5eq 2806 . . . . . . . 8 (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → 𝐴 = ∅)
4948fveq2d 6356 . . . . . . 7 (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (Base‘𝐴) = (Base‘∅))
50 base0 16114 . . . . . . 7 ∅ = (Base‘∅)
5149, 26, 503eqtr4g 2819 . . . . . 6 (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → 𝐵 = ∅)
5251mpteq1d 4890 . . . . 5 (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑚𝐵 ↦ (𝑖𝑁, 𝑗𝑁 ↦ (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑚𝑙)))))) = (𝑚 ∈ ∅ ↦ (𝑖𝑁, 𝑗𝑁 ↦ (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑚𝑙)))))))
53 mpt0 6182 . . . . 5 (𝑚 ∈ ∅ ↦ (𝑖𝑁, 𝑗𝑁 ↦ (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑚𝑙)))))) = ∅
5452, 53syl6eq 2810 . . . 4 (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑚𝐵 ↦ (𝑖𝑁, 𝑗𝑁 ↦ (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑚𝑙)))))) = ∅)
5544, 54eqtr4d 2797 . . 3 (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 maAdju 𝑅) = (𝑚𝐵 ↦ (𝑖𝑁, 𝑗𝑁 ↦ (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑚𝑙)))))))
5642, 55pm2.61i 176 . 2 (𝑁 maAdju 𝑅) = (𝑚𝐵 ↦ (𝑖𝑁, 𝑗𝑁 ↦ (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑚𝑙))))))
571, 56eqtri 2782 1 𝐽 = (𝑚𝐵 ↦ (𝑖𝑁, 𝑗𝑁 ↦ (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑚𝑙))))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 383   = wceq 1632  wcel 2139  Vcvv 3340  c0 4058  ifcif 4230  cop 4327  cotp 4329  cmpt 4881   × cxp 5264  cfv 6049  (class class class)co 6813  cmpt2 6815  Fincfn 8121  ndxcnx 16056   sSet csts 16057  Basecbs 16059  .rcmulr 16144  0gc0g 16302  1rcur 18701   freeLMod cfrlm 20292   maMul cmmul 20391   Mat cmat 20415   maDet cmdat 20592   maAdju cmadu 20640
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-slot 16063  df-base 16065  df-mat 20416  df-madu 20642
This theorem is referenced by:  maduval  20646  maduf  20649
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