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Theorem maducoeval 22451
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 22450 . . 3 (𝑀𝐵 → (𝐽𝑀) = (𝑖𝑁, 𝑗𝑁 ↦ (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑀𝑙))))))
873ad2ant1 1130 . 2 ((𝑀𝐵𝐼𝑁𝐻𝑁) → (𝐽𝑀) = (𝑖𝑁, 𝑗𝑁 ↦ (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑀𝑙))))))
9 simp1r 1195 . . . . . . 7 (((𝑖 = 𝐼𝑗 = 𝐻) ∧ 𝑘𝑁𝑙𝑁) → 𝑗 = 𝐻)
109eqeq2d 2735 . . . . . 6 (((𝑖 = 𝐼𝑗 = 𝐻) ∧ 𝑘𝑁𝑙𝑁) → (𝑘 = 𝑗𝑘 = 𝐻))
11 simp1l 1194 . . . . . . . 8 (((𝑖 = 𝐼𝑗 = 𝐻) ∧ 𝑘𝑁𝑙𝑁) → 𝑖 = 𝐼)
1211eqeq2d 2735 . . . . . . 7 (((𝑖 = 𝐼𝑗 = 𝐻) ∧ 𝑘𝑁𝑙𝑁) → (𝑙 = 𝑖𝑙 = 𝐼))
1312ifbid 4543 . . . . . 6 (((𝑖 = 𝐼𝑗 = 𝐻) ∧ 𝑘𝑁𝑙𝑁) → if(𝑙 = 𝑖, 1 , 0 ) = if(𝑙 = 𝐼, 1 , 0 ))
1410, 13ifbieq1d 4544 . . . . 5 (((𝑖 = 𝐼𝑗 = 𝐻) ∧ 𝑘𝑁𝑙𝑁) → if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑀𝑙)) = if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), (𝑘𝑀𝑙)))
1514mpoeq3dva 7478 . . . 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 6885 . . 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 481 . 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 1134 . 2 ((𝑀𝐵𝐼𝑁𝐻𝑁) → 𝐼𝑁)
19 simp3 1135 . 2 ((𝑀𝐵𝐼𝑁𝐻𝑁) → 𝐻𝑁)
20 fvexd 6896 . 2 ((𝑀𝐵𝐼𝑁𝐻𝑁) → (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), (𝑘𝑀𝑙)))) ∈ V)
218, 17, 18, 19, 20ovmpod 7552 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 395  w3a 1084   = wceq 1533  wcel 2098  Vcvv 3466  ifcif 4520  cfv 6533  (class class class)co 7401  cmpo 7403  Basecbs 17140  0gc0g 17381  1rcur 20071   Mat cmat 22217   maDet cmdat 22396   maAdju cmadu 22444
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718  ax-cnex 11161  ax-1cn 11163  ax-addcl 11165
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7404  df-oprab 7405  df-mpo 7406  df-om 7849  df-1st 7968  df-2nd 7969  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-nn 12209  df-slot 17111  df-ndx 17123  df-base 17141  df-mat 22218  df-madu 22446
This theorem is referenced by:  maducoeval2  22452  madugsum  22455  maducoevalmin1  22464
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