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 Description: An entry of the adjunct (cofactor) matrix. (Contributed by SO, 11-Jul-2018.)
Hypotheses
Ref Expression
madufval.a 𝐴 = (𝑁 Mat 𝑅)
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)

Dummy variables 𝑖 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 madufval.a . . . 4 𝐴 = (𝑁 Mat 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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