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Theorem maducoevalmin1 20219
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 2609 . . 3 (1r𝑅) = (1r𝑅)
6 eqid 2609 . . 3 (0g𝑅) = (0g𝑅)
71, 2, 3, 4, 5, 6maducoeval 20206 . 2 ((𝑀𝐵𝐼𝑁𝐻𝑁) → (𝐼(𝐽𝑀)𝐻) = (𝐷‘(𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐻, if(𝑗 = 𝐼, (1r𝑅), (0g𝑅)), (𝑖𝑀𝑗)))))
8 eqid 2609 . . . . . 6 (𝑁 minMatR1 𝑅) = (𝑁 minMatR1 𝑅)
91, 4, 8, 5, 6minmar1val 20215 . . . . 5 ((𝑀𝐵𝐻𝑁𝐼𝑁) → (𝐻((𝑁 minMatR1 𝑅)‘𝑀)𝐼) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐻, if(𝑗 = 𝐼, (1r𝑅), (0g𝑅)), (𝑖𝑀𝑗))))
1093com23 1262 . . . 4 ((𝑀𝐵𝐼𝑁𝐻𝑁) → (𝐻((𝑁 minMatR1 𝑅)‘𝑀)𝐼) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐻, if(𝑗 = 𝐼, (1r𝑅), (0g𝑅)), (𝑖𝑀𝑗))))
1110eqcomd 2615 . . 3 ((𝑀𝐵𝐼𝑁𝐻𝑁) → (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐻, if(𝑗 = 𝐼, (1r𝑅), (0g𝑅)), (𝑖𝑀𝑗))) = (𝐻((𝑁 minMatR1 𝑅)‘𝑀)𝐼))
1211fveq2d 6092 . 2 ((𝑀𝐵𝐼𝑁𝐻𝑁) → (𝐷‘(𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐻, if(𝑗 = 𝐼, (1r𝑅), (0g𝑅)), (𝑖𝑀𝑗)))) = (𝐷‘(𝐻((𝑁 minMatR1 𝑅)‘𝑀)𝐼)))
137, 12eqtrd 2643 1 ((𝑀𝐵𝐼𝑁𝐻𝑁) → (𝐼(𝐽𝑀)𝐻) = (𝐷‘(𝐻((𝑁 minMatR1 𝑅)‘𝑀)𝐼)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1030   = wceq 1474  wcel 1976  ifcif 4035  cfv 5790  (class class class)co 6527  cmpt2 6529  Basecbs 15641  0gc0g 15869  1rcur 18270   Mat cmat 19974   maDet cmdat 20151   maAdju cmadu 20199   minMatR1 cminmar1 20200
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 6824
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 7036  df-2nd 7037  df-slot 15645  df-base 15646  df-mat 19975  df-madu 20201  df-minmar1 20202
This theorem is referenced by:  madjusmdetlem1  29027
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