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Theorem maducoeval 20212
Description: An entry of 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
maducoeval ((𝑀𝐵𝐼𝑁𝐻𝑁) → (𝐼(𝐽𝑀)𝐻) = (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), (𝑘𝑀𝑙)))))
Distinct variable groups:   𝑘,𝑁,𝑙   𝑅,𝑘,𝑙   𝑘,𝑀,𝑙   𝑘,𝐼,𝑙   𝑘,𝐻,𝑙
Allowed substitution hints:   𝐴(𝑘,𝑙)   𝐵(𝑘,𝑙)   𝐷(𝑘,𝑙)   1 (𝑘,𝑙)   𝐽(𝑘,𝑙)   0 (𝑘,𝑙)

Proof of Theorem maducoeval
Dummy variables 𝑖 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 madufval.a . . . 4 𝐴 = (𝑁 Mat 𝑅)
2 madufval.d . . . 4 𝐷 = (𝑁 maDet 𝑅)
3 madufval.j . . . 4 𝐽 = (𝑁 maAdju 𝑅)
4 madufval.b . . . 4 𝐵 = (Base‘𝐴)
5 madufval.o . . . 4 1 = (1r𝑅)
6 madufval.z . . . 4 0 = (0g𝑅)
71, 2, 3, 4, 5, 6maduval 20211 . . 3 (𝑀𝐵 → (𝐽𝑀) = (𝑖𝑁, 𝑗𝑁 ↦ (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑀𝑙))))))
873ad2ant1 1074 . 2 ((𝑀𝐵𝐼𝑁𝐻𝑁) → (𝐽𝑀) = (𝑖𝑁, 𝑗𝑁 ↦ (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑀𝑙))))))
9 simp1r 1078 . . . . . . 7 (((𝑖 = 𝐼𝑗 = 𝐻) ∧ 𝑘𝑁𝑙𝑁) → 𝑗 = 𝐻)
109eqeq2d 2619 . . . . . 6 (((𝑖 = 𝐼𝑗 = 𝐻) ∧ 𝑘𝑁𝑙𝑁) → (𝑘 = 𝑗𝑘 = 𝐻))
11 simp1l 1077 . . . . . . . 8 (((𝑖 = 𝐼𝑗 = 𝐻) ∧ 𝑘𝑁𝑙𝑁) → 𝑖 = 𝐼)
1211eqeq2d 2619 . . . . . . 7 (((𝑖 = 𝐼𝑗 = 𝐻) ∧ 𝑘𝑁𝑙𝑁) → (𝑙 = 𝑖𝑙 = 𝐼))
1312ifbid 4057 . . . . . 6 (((𝑖 = 𝐼𝑗 = 𝐻) ∧ 𝑘𝑁𝑙𝑁) → if(𝑙 = 𝑖, 1 , 0 ) = if(𝑙 = 𝐼, 1 , 0 ))
1410, 13ifbieq1d 4058 . . . . 5 (((𝑖 = 𝐼𝑗 = 𝐻) ∧ 𝑘𝑁𝑙𝑁) → if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑀𝑙)) = if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), (𝑘𝑀𝑙)))
1514mpt2eq3dva 6595 . . . 4 ((𝑖 = 𝐼𝑗 = 𝐻) → (𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑀𝑙))) = (𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), (𝑘𝑀𝑙))))
1615fveq2d 6092 . . 3 ((𝑖 = 𝐼𝑗 = 𝐻) → (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑀𝑙)))) = (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), (𝑘𝑀𝑙)))))
1716adantl 480 . 2 (((𝑀𝐵𝐼𝑁𝐻𝑁) ∧ (𝑖 = 𝐼𝑗 = 𝐻)) → (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑀𝑙)))) = (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), (𝑘𝑀𝑙)))))
18 simp2 1054 . 2 ((𝑀𝐵𝐼𝑁𝐻𝑁) → 𝐼𝑁)
19 simp3 1055 . 2 ((𝑀𝐵𝐼𝑁𝐻𝑁) → 𝐻𝑁)
20 fvex 6098 . . 3 (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), (𝑘𝑀𝑙)))) ∈ V
2120a1i 11 . 2 ((𝑀𝐵𝐼𝑁𝐻𝑁) → (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), (𝑘𝑀𝑙)))) ∈ V)
228, 17, 18, 19, 21ovmpt2d 6664 1 ((𝑀𝐵𝐼𝑁𝐻𝑁) → (𝐼(𝐽𝑀)𝐻) = (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), (𝑘𝑀𝑙)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1030   = wceq 1474  wcel 1976  Vcvv 3172  ifcif 4035  cfv 5790  (class class class)co 6527  cmpt2 6529  Basecbs 15644  0gc0g 15872  1rcur 18273   Mat cmat 19980   maDet cmdat 20157   maAdju cmadu 20205
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-1st 7037  df-2nd 7038  df-slot 15648  df-base 15649  df-mat 19981  df-madu 20207
This theorem is referenced by:  maducoeval2  20213  madugsum  20216  maducoevalmin1  20225
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