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Theorem maduval 21389
Description: Second 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
maduval (𝑀𝐵 → (𝐽𝑀) = (𝑖𝑁, 𝑗𝑁 ↦ (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑀𝑙))))))
Distinct variable groups:   𝑖,𝑁,𝑗,𝑘,𝑙   𝑅,𝑖,𝑗,𝑘,𝑙   𝑖,𝑀,𝑗,𝑘,𝑙
Allowed substitution hints:   𝐴(𝑖,𝑗,𝑘,𝑙)   𝐵(𝑖,𝑗,𝑘,𝑙)   𝐷(𝑖,𝑗,𝑘,𝑙)   1 (𝑖,𝑗,𝑘,𝑙)   𝐽(𝑖,𝑗,𝑘,𝑙)   0 (𝑖,𝑗,𝑘,𝑙)

Proof of Theorem maduval
Dummy variable 𝑚 is distinct from all other variables.
StepHypRef Expression
1 madufval.a . . . . 5 𝐴 = (𝑁 Mat 𝑅)
2 madufval.b . . . . 5 𝐵 = (Base‘𝐴)
31, 2matrcl 21163 . . . 4 (𝑀𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
43simpld 498 . . 3 (𝑀𝐵𝑁 ∈ Fin)
5 mpoexga 7801 . . 3 ((𝑁 ∈ Fin ∧ 𝑁 ∈ Fin) → (𝑖𝑁, 𝑗𝑁 ↦ (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑀𝑙))))) ∈ V)
64, 4, 5syl2anc 587 . 2 (𝑀𝐵 → (𝑖𝑁, 𝑗𝑁 ↦ (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑀𝑙))))) ∈ V)
7 oveq 7176 . . . . . . . 8 (𝑚 = 𝑀 → (𝑘𝑚𝑙) = (𝑘𝑀𝑙))
87ifeq2d 4434 . . . . . . 7 (𝑚 = 𝑀 → if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑚𝑙)) = if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑀𝑙)))
98mpoeq3dv 7247 . . . . . 6 (𝑚 = 𝑀 → (𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑚𝑙))) = (𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑀𝑙))))
1093ad2ant1 1134 . . . . 5 ((𝑚 = 𝑀𝑖𝑁𝑗𝑁) → (𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑚𝑙))) = (𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑀𝑙))))
1110fveq2d 6678 . . . 4 ((𝑚 = 𝑀𝑖𝑁𝑗𝑁) → (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑚𝑙)))) = (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑀𝑙)))))
1211mpoeq3dva 7245 . . 3 (𝑚 = 𝑀 → (𝑖𝑁, 𝑗𝑁 ↦ (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑚𝑙))))) = (𝑖𝑁, 𝑗𝑁 ↦ (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑀𝑙))))))
13 madufval.d . . . 4 𝐷 = (𝑁 maDet 𝑅)
14 madufval.j . . . 4 𝐽 = (𝑁 maAdju 𝑅)
15 madufval.o . . . 4 1 = (1r𝑅)
16 madufval.z . . . 4 0 = (0g𝑅)
171, 13, 14, 2, 15, 16madufval 21388 . . 3 𝐽 = (𝑚𝐵 ↦ (𝑖𝑁, 𝑗𝑁 ↦ (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑚𝑙))))))
1812, 17fvmptg 6773 . 2 ((𝑀𝐵 ∧ (𝑖𝑁, 𝑗𝑁 ↦ (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑀𝑙))))) ∈ V) → (𝐽𝑀) = (𝑖𝑁, 𝑗𝑁 ↦ (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑀𝑙))))))
196, 18mpdan 687 1 (𝑀𝐵 → (𝐽𝑀) = (𝑖𝑁, 𝑗𝑁 ↦ (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑀𝑙))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1088   = wceq 1542  wcel 2114  Vcvv 3398  ifcif 4414  cfv 6339  (class class class)co 7170  cmpo 7172  Fincfn 8555  Basecbs 16586  0gc0g 16816  1rcur 19370   Mat cmat 21158   maDet cmdat 21335   maAdju cmadu 21383
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-ov 7173  df-oprab 7174  df-mpo 7175  df-1st 7714  df-2nd 7715  df-slot 16590  df-base 16592  df-mat 21159  df-madu 21385
This theorem is referenced by:  maducoeval  21390
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