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Theorem madufval 22659
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 fvoveq1 7454 . . . . . 6 (𝑛 = 𝑁 → (Base‘(𝑛 Mat 𝑟)) = (Base‘(𝑁 Mat 𝑟)))
3 id 22 . . . . . . 7 (𝑛 = 𝑁𝑛 = 𝑁)
4 oveq1 7438 . . . . . . . 8 (𝑛 = 𝑁 → (𝑛 maDet 𝑟) = (𝑁 maDet 𝑟))
5 eqidd 2736 . . . . . . . . 9 (𝑛 = 𝑁 → if(𝑘 = 𝑗, if(𝑙 = 𝑖, (1r𝑟), (0g𝑟)), (𝑘𝑚𝑙)) = if(𝑘 = 𝑗, if(𝑙 = 𝑖, (1r𝑟), (0g𝑟)), (𝑘𝑚𝑙)))
63, 3, 5mpoeq123dv 7508 . . . . . . . 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))))
74, 6fveq12d 6914 . . . . . . 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)))))
83, 3, 7mpoeq123dv 7508 . . . . . 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))))))
92, 8mpteq12dv 5239 . . . . 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)))))))
10 oveq2 7439 . . . . . . 7 (𝑟 = 𝑅 → (𝑁 Mat 𝑟) = (𝑁 Mat 𝑅))
1110fveq2d 6911 . . . . . 6 (𝑟 = 𝑅 → (Base‘(𝑁 Mat 𝑟)) = (Base‘(𝑁 Mat 𝑅)))
12 oveq2 7439 . . . . . . . 8 (𝑟 = 𝑅 → (𝑁 maDet 𝑟) = (𝑁 maDet 𝑅))
13 fveq2 6907 . . . . . . . . . . 11 (𝑟 = 𝑅 → (1r𝑟) = (1r𝑅))
14 fveq2 6907 . . . . . . . . . . 11 (𝑟 = 𝑅 → (0g𝑟) = (0g𝑅))
1513, 14ifeq12d 4552 . . . . . . . . . 10 (𝑟 = 𝑅 → if(𝑙 = 𝑖, (1r𝑟), (0g𝑟)) = if(𝑙 = 𝑖, (1r𝑅), (0g𝑅)))
1615ifeq1d 4550 . . . . . . . . 9 (𝑟 = 𝑅 → if(𝑘 = 𝑗, if(𝑙 = 𝑖, (1r𝑟), (0g𝑟)), (𝑘𝑚𝑙)) = if(𝑘 = 𝑗, if(𝑙 = 𝑖, (1r𝑅), (0g𝑅)), (𝑘𝑚𝑙)))
1716mpoeq3dv 7512 . . . . . . . 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))))
1812, 17fveq12d 6914 . . . . . . 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)))))
1918mpoeq3dv 7512 . . . . . 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))))))
2011, 19mpteq12dv 5239 . . . . 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)))))))
21 df-madu 22656 . . . . 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)))))))
22 fvex 6920 . . . . . 6 (Base‘(𝑁 Mat 𝑅)) ∈ V
2322mptex 7243 . . . . 5 (𝑚 ∈ (Base‘(𝑁 Mat 𝑅)) ↦ (𝑖𝑁, 𝑗𝑁 ↦ ((𝑁 maDet 𝑅)‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, (1r𝑅), (0g𝑅)), (𝑘𝑚𝑙)))))) ∈ V
249, 20, 21, 23ovmpo 7593 . . . 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)))))))
25 madufval.b . . . . . 6 𝐵 = (Base‘𝐴)
26 madufval.a . . . . . . 7 𝐴 = (𝑁 Mat 𝑅)
2726fveq2i 6910 . . . . . 6 (Base‘𝐴) = (Base‘(𝑁 Mat 𝑅))
2825, 27eqtri 2763 . . . . 5 𝐵 = (Base‘(𝑁 Mat 𝑅))
29 madufval.d . . . . . . . 8 𝐷 = (𝑁 maDet 𝑅)
30 madufval.o . . . . . . . . . . . 12 1 = (1r𝑅)
3130a1i 11 . . . . . . . . . . 11 ((𝑘𝑁𝑙𝑁) → 1 = (1r𝑅))
32 madufval.z . . . . . . . . . . . 12 0 = (0g𝑅)
3332a1i 11 . . . . . . . . . . 11 ((𝑘𝑁𝑙𝑁) → 0 = (0g𝑅))
3431, 33ifeq12d 4552 . . . . . . . . . 10 ((𝑘𝑁𝑙𝑁) → if(𝑙 = 𝑖, 1 , 0 ) = if(𝑙 = 𝑖, (1r𝑅), (0g𝑅)))
3534ifeq1d 4550 . . . . . . . . 9 ((𝑘𝑁𝑙𝑁) → if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑚𝑙)) = if(𝑘 = 𝑗, if(𝑙 = 𝑖, (1r𝑅), (0g𝑅)), (𝑘𝑚𝑙)))
3635mpoeq3ia 7511 . . . . . . . 8 (𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑚𝑙))) = (𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, (1r𝑅), (0g𝑅)), (𝑘𝑚𝑙)))
3729, 36fveq12i 6913 . . . . . . 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))))
3837a1i 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)))))
3938mpoeq3ia 7511 . . . . 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)))))
4028, 39mpteq12i 5254 . . . 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))))))
4124, 40eqtr4di 2793 . . 3 ((𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 maAdju 𝑅) = (𝑚𝐵 ↦ (𝑖𝑁, 𝑗𝑁 ↦ (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑚𝑙)))))))
4221reldmmpo 7567 . . . . 5 Rel dom maAdju
4342ovprc 7469 . . . 4 (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 maAdju 𝑅) = ∅)
44 df-mat 22428 . . . . . . . . . . 11 Mat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ ((𝑟 freeLMod (𝑛 × 𝑛)) sSet ⟨(.r‘ndx), (𝑟 maMul ⟨𝑛, 𝑛, 𝑛⟩)⟩))
4544reldmmpo 7567 . . . . . . . . . 10 Rel dom Mat
4645ovprc 7469 . . . . . . . . 9 (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 Mat 𝑅) = ∅)
4726, 46eqtrid 2787 . . . . . . . 8 (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → 𝐴 = ∅)
4847fveq2d 6911 . . . . . . 7 (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (Base‘𝐴) = (Base‘∅))
49 base0 17250 . . . . . . 7 ∅ = (Base‘∅)
5048, 25, 493eqtr4g 2800 . . . . . 6 (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → 𝐵 = ∅)
5150mpteq1d 5243 . . . . 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)))))))
52 mpt0 6711 . . . . 5 (𝑚 ∈ ∅ ↦ (𝑖𝑁, 𝑗𝑁 ↦ (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑚𝑙)))))) = ∅
5351, 52eqtrdi 2791 . . . 4 (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑚𝐵 ↦ (𝑖𝑁, 𝑗𝑁 ↦ (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑚𝑙)))))) = ∅)
5443, 53eqtr4d 2778 . . 3 (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 maAdju 𝑅) = (𝑚𝐵 ↦ (𝑖𝑁, 𝑗𝑁 ↦ (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑚𝑙)))))))
5541, 54pm2.61i 182 . 2 (𝑁 maAdju 𝑅) = (𝑚𝐵 ↦ (𝑖𝑁, 𝑗𝑁 ↦ (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑚𝑙))))))
561, 55eqtri 2763 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 395   = wceq 1537  wcel 2106  Vcvv 3478  c0 4339  ifcif 4531  cop 4637  cotp 4639  cmpt 5231   × cxp 5687  cfv 6563  (class class class)co 7431  cmpo 7433  Fincfn 8984   sSet csts 17197  ndxcnx 17227  Basecbs 17245  .rcmulr 17299  0gc0g 17486  1rcur 20199   freeLMod cfrlm 21784   maMul cmmul 22410   Mat cmat 22427   maDet cmdat 22606   maAdju cmadu 22654
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-1cn 11211  ax-addcl 11213
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-nn 12265  df-slot 17216  df-ndx 17228  df-base 17246  df-mat 22428  df-madu 22656
This theorem is referenced by:  maduval  22660  maduf  22663
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