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Theorem maducoevalmin1 21907
Description: The coefficients of an adjunct (matrix of cofactors) expressed as determinants of the minor matrices (alternative definition) of the original matrix. (Contributed by AV, 31-Dec-2018.)
Hypotheses
Ref Expression
maducoevalmin1.a 𝐴 = (𝑁 Mat 𝑅)
maducoevalmin1.b 𝐵 = (Base‘𝐴)
maducoevalmin1.d 𝐷 = (𝑁 maDet 𝑅)
maducoevalmin1.j 𝐽 = (𝑁 maAdju 𝑅)
Assertion
Ref Expression
maducoevalmin1 ((𝑀𝐵𝐼𝑁𝐻𝑁) → (𝐼(𝐽𝑀)𝐻) = (𝐷‘(𝐻((𝑁 minMatR1 𝑅)‘𝑀)𝐼)))

Proof of Theorem maducoevalmin1
Dummy variables 𝑖 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 maducoevalmin1.a . . 3 𝐴 = (𝑁 Mat 𝑅)
2 maducoevalmin1.d . . 3 𝐷 = (𝑁 maDet 𝑅)
3 maducoevalmin1.j . . 3 𝐽 = (𝑁 maAdju 𝑅)
4 maducoevalmin1.b . . 3 𝐵 = (Base‘𝐴)
5 eqid 2736 . . 3 (1r𝑅) = (1r𝑅)
6 eqid 2736 . . 3 (0g𝑅) = (0g𝑅)
71, 2, 3, 4, 5, 6maducoeval 21894 . 2 ((𝑀𝐵𝐼𝑁𝐻𝑁) → (𝐼(𝐽𝑀)𝐻) = (𝐷‘(𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐻, if(𝑗 = 𝐼, (1r𝑅), (0g𝑅)), (𝑖𝑀𝑗)))))
8 eqid 2736 . . . . . 6 (𝑁 minMatR1 𝑅) = (𝑁 minMatR1 𝑅)
91, 4, 8, 5, 6minmar1val 21903 . . . . 5 ((𝑀𝐵𝐻𝑁𝐼𝑁) → (𝐻((𝑁 minMatR1 𝑅)‘𝑀)𝐼) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐻, if(𝑗 = 𝐼, (1r𝑅), (0g𝑅)), (𝑖𝑀𝑗))))
1093com23 1125 . . . 4 ((𝑀𝐵𝐼𝑁𝐻𝑁) → (𝐻((𝑁 minMatR1 𝑅)‘𝑀)𝐼) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐻, if(𝑗 = 𝐼, (1r𝑅), (0g𝑅)), (𝑖𝑀𝑗))))
1110eqcomd 2742 . . 3 ((𝑀𝐵𝐼𝑁𝐻𝑁) → (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐻, if(𝑗 = 𝐼, (1r𝑅), (0g𝑅)), (𝑖𝑀𝑗))) = (𝐻((𝑁 minMatR1 𝑅)‘𝑀)𝐼))
1211fveq2d 6829 . 2 ((𝑀𝐵𝐼𝑁𝐻𝑁) → (𝐷‘(𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐻, if(𝑗 = 𝐼, (1r𝑅), (0g𝑅)), (𝑖𝑀𝑗)))) = (𝐷‘(𝐻((𝑁 minMatR1 𝑅)‘𝑀)𝐼)))
137, 12eqtrd 2776 1 ((𝑀𝐵𝐼𝑁𝐻𝑁) → (𝐼(𝐽𝑀)𝐻) = (𝐷‘(𝐻((𝑁 minMatR1 𝑅)‘𝑀)𝐼)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1086   = wceq 1540  wcel 2105  ifcif 4473  cfv 6479  (class class class)co 7337  cmpo 7339  Basecbs 17009  0gc0g 17247  1rcur 19832   Mat cmat 21660   maDet cmdat 21839   maAdju cmadu 21887   minMatR1 cminmar1 21888
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5229  ax-sep 5243  ax-nul 5250  ax-pow 5308  ax-pr 5372  ax-un 7650  ax-cnex 11028  ax-1cn 11030  ax-addcl 11032
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3917  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5176  df-tr 5210  df-id 5518  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5575  df-we 5577  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-pred 6238  df-ord 6305  df-on 6306  df-lim 6307  df-suc 6308  df-iota 6431  df-fun 6481  df-fn 6482  df-f 6483  df-f1 6484  df-fo 6485  df-f1o 6486  df-fv 6487  df-ov 7340  df-oprab 7341  df-mpo 7342  df-om 7781  df-1st 7899  df-2nd 7900  df-frecs 8167  df-wrecs 8198  df-recs 8272  df-rdg 8311  df-nn 12075  df-slot 16980  df-ndx 16992  df-base 17010  df-mat 21661  df-madu 21889  df-minmar1 21890
This theorem is referenced by:  madjusmdetlem1  32075
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