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 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)
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 2801 . . 3 (1r𝑅) = (1r𝑅)
6 eqid 2801 . . 3 (0g𝑅) = (0g𝑅)
71, 2, 3, 4, 5, 6maducoeval 21248 . 2 ((𝑀𝐵𝐼𝑁𝐻𝑁) → (𝐼(𝐽𝑀)𝐻) = (𝐷‘(𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐻, if(𝑗 = 𝐼, (1r𝑅), (0g𝑅)), (𝑖𝑀𝑗)))))
8 eqid 2801 . . . . . 6 (𝑁 minMatR1 𝑅) = (𝑁 minMatR1 𝑅)
91, 4, 8, 5, 6minmar1val 21257 . . . . 5 ((𝑀𝐵𝐻𝑁𝐼𝑁) → (𝐻((𝑁 minMatR1 𝑅)‘𝑀)𝐼) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐻, if(𝑗 = 𝐼, (1r𝑅), (0g𝑅)), (𝑖𝑀𝑗))))
1093com23 1123 . . . 4 ((𝑀𝐵𝐼𝑁𝐻𝑁) → (𝐻((𝑁 minMatR1 𝑅)‘𝑀)𝐼) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐻, if(𝑗 = 𝐼, (1r𝑅), (0g𝑅)), (𝑖𝑀𝑗))))
1110eqcomd 2807 . . 3 ((𝑀𝐵𝐼𝑁𝐻𝑁) → (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐻, if(𝑗 = 𝐼, (1r𝑅), (0g𝑅)), (𝑖𝑀𝑗))) = (𝐻((𝑁 minMatR1 𝑅)‘𝑀)𝐼))
1211fveq2d 6653 . 2 ((𝑀𝐵𝐼𝑁𝐻𝑁) → (𝐷‘(𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐻, if(𝑗 = 𝐼, (1r𝑅), (0g𝑅)), (𝑖𝑀𝑗)))) = (𝐷‘(𝐻((𝑁 minMatR1 𝑅)‘𝑀)𝐼)))
137, 12eqtrd 2836 1 ((𝑀𝐵𝐼𝑁𝐻𝑁) → (𝐼(𝐽𝑀)𝐻) = (𝐷‘(𝐻((𝑁 minMatR1 𝑅)‘𝑀)𝐼)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1084   = wceq 1538   ∈ wcel 2112  ifcif 4428  ‘cfv 6328  (class class class)co 7139   ∈ cmpo 7141  Basecbs 16479  0gc0g 16709  1rcur 19248   Mat cmat 21016   maDet cmdat 21193   maAdju cmadu 21241   minMatR1 cminmar1 21242 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-1st 7675  df-2nd 7676  df-slot 16483  df-base 16485  df-mat 21017  df-madu 21243  df-minmar1 21244 This theorem is referenced by:  madjusmdetlem1  31184
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