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Theorem maducoevalmin1 22567
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 2728 . . 3 (1r𝑅) = (1r𝑅)
6 eqid 2728 . . 3 (0g𝑅) = (0g𝑅)
71, 2, 3, 4, 5, 6maducoeval 22554 . 2 ((𝑀𝐵𝐼𝑁𝐻𝑁) → (𝐼(𝐽𝑀)𝐻) = (𝐷‘(𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐻, if(𝑗 = 𝐼, (1r𝑅), (0g𝑅)), (𝑖𝑀𝑗)))))
8 eqid 2728 . . . . . 6 (𝑁 minMatR1 𝑅) = (𝑁 minMatR1 𝑅)
91, 4, 8, 5, 6minmar1val 22563 . . . . 5 ((𝑀𝐵𝐻𝑁𝐼𝑁) → (𝐻((𝑁 minMatR1 𝑅)‘𝑀)𝐼) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐻, if(𝑗 = 𝐼, (1r𝑅), (0g𝑅)), (𝑖𝑀𝑗))))
1093com23 1124 . . . 4 ((𝑀𝐵𝐼𝑁𝐻𝑁) → (𝐻((𝑁 minMatR1 𝑅)‘𝑀)𝐼) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐻, if(𝑗 = 𝐼, (1r𝑅), (0g𝑅)), (𝑖𝑀𝑗))))
1110eqcomd 2734 . . 3 ((𝑀𝐵𝐼𝑁𝐻𝑁) → (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐻, if(𝑗 = 𝐼, (1r𝑅), (0g𝑅)), (𝑖𝑀𝑗))) = (𝐻((𝑁 minMatR1 𝑅)‘𝑀)𝐼))
1211fveq2d 6901 . 2 ((𝑀𝐵𝐼𝑁𝐻𝑁) → (𝐷‘(𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐻, if(𝑗 = 𝐼, (1r𝑅), (0g𝑅)), (𝑖𝑀𝑗)))) = (𝐷‘(𝐻((𝑁 minMatR1 𝑅)‘𝑀)𝐼)))
137, 12eqtrd 2768 1 ((𝑀𝐵𝐼𝑁𝐻𝑁) → (𝐼(𝐽𝑀)𝐻) = (𝐷‘(𝐻((𝑁 minMatR1 𝑅)‘𝑀)𝐼)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1085   = wceq 1534  wcel 2099  ifcif 4529  cfv 6548  (class class class)co 7420  cmpo 7422  Basecbs 17180  0gc0g 17421  1rcur 20121   Mat cmat 22320   maDet cmdat 22499   maAdju cmadu 22547   minMatR1 cminmar1 22548
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740  ax-cnex 11195  ax-1cn 11197  ax-addcl 11199
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6305  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7871  df-1st 7993  df-2nd 7994  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-nn 12244  df-slot 17151  df-ndx 17163  df-base 17181  df-mat 22321  df-madu 22549  df-minmar1 22550
This theorem is referenced by:  madjusmdetlem1  33428
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