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Theorem maducoeval 20667
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 20666 . . 3 (𝑀𝐵 → (𝐽𝑀) = (𝑖𝑁, 𝑗𝑁 ↦ (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑀𝑙))))))
873ad2ant1 1128 . 2 ((𝑀𝐵𝐼𝑁𝐻𝑁) → (𝐽𝑀) = (𝑖𝑁, 𝑗𝑁 ↦ (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑀𝑙))))))
9 simp1r 1241 . . . . . . 7 (((𝑖 = 𝐼𝑗 = 𝐻) ∧ 𝑘𝑁𝑙𝑁) → 𝑗 = 𝐻)
109eqeq2d 2770 . . . . . 6 (((𝑖 = 𝐼𝑗 = 𝐻) ∧ 𝑘𝑁𝑙𝑁) → (𝑘 = 𝑗𝑘 = 𝐻))
11 simp1l 1240 . . . . . . . 8 (((𝑖 = 𝐼𝑗 = 𝐻) ∧ 𝑘𝑁𝑙𝑁) → 𝑖 = 𝐼)
1211eqeq2d 2770 . . . . . . 7 (((𝑖 = 𝐼𝑗 = 𝐻) ∧ 𝑘𝑁𝑙𝑁) → (𝑙 = 𝑖𝑙 = 𝐼))
1312ifbid 4252 . . . . . 6 (((𝑖 = 𝐼𝑗 = 𝐻) ∧ 𝑘𝑁𝑙𝑁) → if(𝑙 = 𝑖, 1 , 0 ) = if(𝑙 = 𝐼, 1 , 0 ))
1410, 13ifbieq1d 4253 . . . . 5 (((𝑖 = 𝐼𝑗 = 𝐻) ∧ 𝑘𝑁𝑙𝑁) → if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑀𝑙)) = if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), (𝑘𝑀𝑙)))
1514mpt2eq3dva 6885 . . . 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 6357 . . 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 473 . 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 1132 . 2 ((𝑀𝐵𝐼𝑁𝐻𝑁) → 𝐼𝑁)
19 simp3 1133 . 2 ((𝑀𝐵𝐼𝑁𝐻𝑁) → 𝐻𝑁)
20 fvexd 6365 . 2 ((𝑀𝐵𝐼𝑁𝐻𝑁) → (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), (𝑘𝑀𝑙)))) ∈ V)
218, 17, 18, 19, 20ovmpt2d 6954 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 383  w3a 1072   = wceq 1632  wcel 2139  Vcvv 3340  ifcif 4230  cfv 6049  (class class class)co 6814  cmpt2 6816  Basecbs 16079  0gc0g 16322  1rcur 18721   Mat cmat 20435   maDet cmdat 20612   maAdju cmadu 20660
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  ax-un 7115
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-pw 4304  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 6817  df-oprab 6818  df-mpt2 6819  df-1st 7334  df-2nd 7335  df-slot 16083  df-base 16085  df-mat 20436  df-madu 20662
This theorem is referenced by:  maducoeval2  20668  madugsum  20671  maducoevalmin1  20680
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