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Theorem minmar1eval 21794
Description: An entry of a matrix for a minor. (Contributed by AV, 31-Dec-2018.)
Hypotheses
Ref Expression
minmar1fval.a 𝐴 = (𝑁 Mat 𝑅)
minmar1fval.b 𝐵 = (Base‘𝐴)
minmar1fval.q 𝑄 = (𝑁 minMatR1 𝑅)
minmar1fval.o 1 = (1r𝑅)
minmar1fval.z 0 = (0g𝑅)
Assertion
Ref Expression
minmar1eval ((𝑀𝐵 ∧ (𝐾𝑁𝐿𝑁) ∧ (𝐼𝑁𝐽𝑁)) → (𝐼(𝐾(𝑄𝑀)𝐿)𝐽) = if(𝐼 = 𝐾, if(𝐽 = 𝐿, 1 , 0 ), (𝐼𝑀𝐽)))

Proof of Theorem minmar1eval
Dummy variables 𝑖 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 minmar1fval.a . . . . 5 𝐴 = (𝑁 Mat 𝑅)
2 minmar1fval.b . . . . 5 𝐵 = (Base‘𝐴)
3 minmar1fval.q . . . . 5 𝑄 = (𝑁 minMatR1 𝑅)
4 minmar1fval.o . . . . 5 1 = (1r𝑅)
5 minmar1fval.z . . . . 5 0 = (0g𝑅)
61, 2, 3, 4, 5minmar1val 21793 . . . 4 ((𝑀𝐵𝐾𝑁𝐿𝑁) → (𝐾(𝑄𝑀)𝐿) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗))))
763expb 1119 . . 3 ((𝑀𝐵 ∧ (𝐾𝑁𝐿𝑁)) → (𝐾(𝑄𝑀)𝐿) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗))))
873adant3 1131 . 2 ((𝑀𝐵 ∧ (𝐾𝑁𝐿𝑁) ∧ (𝐼𝑁𝐽𝑁)) → (𝐾(𝑄𝑀)𝐿) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗))))
9 simp3l 1200 . . 3 ((𝑀𝐵 ∧ (𝐾𝑁𝐿𝑁) ∧ (𝐼𝑁𝐽𝑁)) → 𝐼𝑁)
10 simpl3r 1228 . . 3 (((𝑀𝐵 ∧ (𝐾𝑁𝐿𝑁) ∧ (𝐼𝑁𝐽𝑁)) ∧ 𝑖 = 𝐼) → 𝐽𝑁)
114fvexi 6783 . . . . . 6 1 ∈ V
125fvexi 6783 . . . . . 6 0 ∈ V
1311, 12ifex 4515 . . . . 5 if(𝑗 = 𝐿, 1 , 0 ) ∈ V
14 ovex 7302 . . . . 5 (𝑖𝑀𝑗) ∈ V
1513, 14ifex 4515 . . . 4 if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗)) ∈ V
1615a1i 11 . . 3 (((𝑀𝐵 ∧ (𝐾𝑁𝐿𝑁) ∧ (𝐼𝑁𝐽𝑁)) ∧ (𝑖 = 𝐼𝑗 = 𝐽)) → if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗)) ∈ V)
17 eqeq1 2744 . . . . . 6 (𝑖 = 𝐼 → (𝑖 = 𝐾𝐼 = 𝐾))
1817adantr 481 . . . . 5 ((𝑖 = 𝐼𝑗 = 𝐽) → (𝑖 = 𝐾𝐼 = 𝐾))
19 eqeq1 2744 . . . . . . 7 (𝑗 = 𝐽 → (𝑗 = 𝐿𝐽 = 𝐿))
2019adantl 482 . . . . . 6 ((𝑖 = 𝐼𝑗 = 𝐽) → (𝑗 = 𝐿𝐽 = 𝐿))
2120ifbid 4488 . . . . 5 ((𝑖 = 𝐼𝑗 = 𝐽) → if(𝑗 = 𝐿, 1 , 0 ) = if(𝐽 = 𝐿, 1 , 0 ))
22 oveq12 7278 . . . . 5 ((𝑖 = 𝐼𝑗 = 𝐽) → (𝑖𝑀𝑗) = (𝐼𝑀𝐽))
2318, 21, 22ifbieq12d 4493 . . . 4 ((𝑖 = 𝐼𝑗 = 𝐽) → if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗)) = if(𝐼 = 𝐾, if(𝐽 = 𝐿, 1 , 0 ), (𝐼𝑀𝐽)))
2423adantl 482 . . 3 (((𝑀𝐵 ∧ (𝐾𝑁𝐿𝑁) ∧ (𝐼𝑁𝐽𝑁)) ∧ (𝑖 = 𝐼𝑗 = 𝐽)) → if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗)) = if(𝐼 = 𝐾, if(𝐽 = 𝐿, 1 , 0 ), (𝐼𝑀𝐽)))
259, 10, 16, 24ovmpodv2 7423 . 2 ((𝑀𝐵 ∧ (𝐾𝑁𝐿𝑁) ∧ (𝐼𝑁𝐽𝑁)) → ((𝐾(𝑄𝑀)𝐿) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗))) → (𝐼(𝐾(𝑄𝑀)𝐿)𝐽) = if(𝐼 = 𝐾, if(𝐽 = 𝐿, 1 , 0 ), (𝐼𝑀𝐽))))
268, 25mpd 15 1 ((𝑀𝐵 ∧ (𝐾𝑁𝐿𝑁) ∧ (𝐼𝑁𝐽𝑁)) → (𝐼(𝐾(𝑄𝑀)𝐿)𝐽) = if(𝐼 = 𝐾, if(𝐽 = 𝐿, 1 , 0 ), (𝐼𝑀𝐽)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1086   = wceq 1542  wcel 2110  Vcvv 3431  ifcif 4465  cfv 6431  (class class class)co 7269  cmpo 7271  Basecbs 16908  0gc0g 17146  1rcur 19733   Mat cmat 21550   minMatR1 cminmar1 21778
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-rep 5214  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7580  ax-cnex 10926  ax-1cn 10928  ax-addcl 10930
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-ral 3071  df-rex 3072  df-reu 3073  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-tr 5197  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6200  df-ord 6267  df-on 6268  df-lim 6269  df-suc 6270  df-iota 6389  df-fun 6433  df-fn 6434  df-f 6435  df-f1 6436  df-fo 6437  df-f1o 6438  df-fv 6439  df-ov 7272  df-oprab 7273  df-mpo 7274  df-om 7705  df-1st 7822  df-2nd 7823  df-frecs 8086  df-wrecs 8117  df-recs 8191  df-rdg 8230  df-nn 11972  df-slot 16879  df-ndx 16891  df-base 16909  df-mat 21551  df-minmar1 21780
This theorem is referenced by:  madjusmdetlem1  31771
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