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Theorem minmar1val 20870
 Description: Third substitution for the definition 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
minmar1val ((𝑀𝐵𝐾𝑁𝐿𝑁) → (𝐾(𝑄𝑀)𝐿) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗))))
Distinct variable groups:   𝑖,𝑁,𝑗   𝑅,𝑖,𝑗   𝑖,𝑀,𝑗   𝑖,𝐾,𝑗   𝑖,𝐿,𝑗
Allowed substitution hints:   𝐴(𝑖,𝑗)   𝐵(𝑖,𝑗)   𝑄(𝑖,𝑗)   1 (𝑖,𝑗)   0 (𝑖,𝑗)

Proof of Theorem minmar1val
Dummy variables 𝑘 𝑙 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 minmar1fval.a . . . 4 𝐴 = (𝑁 Mat 𝑅)
2 minmar1fval.b . . . 4 𝐵 = (Base‘𝐴)
3 minmar1fval.q . . . 4 𝑄 = (𝑁 minMatR1 𝑅)
4 minmar1fval.o . . . 4 1 = (1r𝑅)
5 minmar1fval.z . . . 4 0 = (0g𝑅)
61, 2, 3, 4, 5minmar1val0 20869 . . 3 (𝑀𝐵 → (𝑄𝑀) = (𝑘𝑁, 𝑙𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 1 , 0 ), (𝑖𝑀𝑗)))))
763ad2ant1 1124 . 2 ((𝑀𝐵𝐾𝑁𝐿𝑁) → (𝑄𝑀) = (𝑘𝑁, 𝑙𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 1 , 0 ), (𝑖𝑀𝑗)))))
8 simp2 1128 . . 3 ((𝑀𝐵𝐾𝑁𝐿𝑁) → 𝐾𝑁)
9 simpl3 1203 . . 3 (((𝑀𝐵𝐾𝑁𝐿𝑁) ∧ 𝑘 = 𝐾) → 𝐿𝑁)
101, 2matrcl 20633 . . . . . . . 8 (𝑀𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
1110simpld 490 . . . . . . 7 (𝑀𝐵𝑁 ∈ Fin)
1211, 11jca 507 . . . . . 6 (𝑀𝐵 → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
13123ad2ant1 1124 . . . . 5 ((𝑀𝐵𝐾𝑁𝐿𝑁) → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
1413adantr 474 . . . 4 (((𝑀𝐵𝐾𝑁𝐿𝑁) ∧ (𝑘 = 𝐾𝑙 = 𝐿)) → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
15 mpt2exga 7528 . . . 4 ((𝑁 ∈ Fin ∧ 𝑁 ∈ Fin) → (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 1 , 0 ), (𝑖𝑀𝑗))) ∈ V)
1614, 15syl 17 . . 3 (((𝑀𝐵𝐾𝑁𝐿𝑁) ∧ (𝑘 = 𝐾𝑙 = 𝐿)) → (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 1 , 0 ), (𝑖𝑀𝑗))) ∈ V)
17 eqeq2 2789 . . . . . . 7 (𝑘 = 𝐾 → (𝑖 = 𝑘𝑖 = 𝐾))
1817adantr 474 . . . . . 6 ((𝑘 = 𝐾𝑙 = 𝐿) → (𝑖 = 𝑘𝑖 = 𝐾))
19 eqeq2 2789 . . . . . . . 8 (𝑙 = 𝐿 → (𝑗 = 𝑙𝑗 = 𝐿))
2019ifbid 4329 . . . . . . 7 (𝑙 = 𝐿 → if(𝑗 = 𝑙, 1 , 0 ) = if(𝑗 = 𝐿, 1 , 0 ))
2120adantl 475 . . . . . 6 ((𝑘 = 𝐾𝑙 = 𝐿) → if(𝑗 = 𝑙, 1 , 0 ) = if(𝑗 = 𝐿, 1 , 0 ))
2218, 21ifbieq1d 4330 . . . . 5 ((𝑘 = 𝐾𝑙 = 𝐿) → if(𝑖 = 𝑘, if(𝑗 = 𝑙, 1 , 0 ), (𝑖𝑀𝑗)) = if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗)))
2322mpt2eq3dv 7000 . . . 4 ((𝑘 = 𝐾𝑙 = 𝐿) → (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 1 , 0 ), (𝑖𝑀𝑗))) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗))))
2423adantl 475 . . 3 (((𝑀𝐵𝐾𝑁𝐿𝑁) ∧ (𝑘 = 𝐾𝑙 = 𝐿)) → (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 1 , 0 ), (𝑖𝑀𝑗))) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗))))
258, 9, 16, 24ovmpt2dv2 7073 . 2 ((𝑀𝐵𝐾𝑁𝐿𝑁) → ((𝑄𝑀) = (𝑘𝑁, 𝑙𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 1 , 0 ), (𝑖𝑀𝑗)))) → (𝐾(𝑄𝑀)𝐿) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗)))))
267, 25mpd 15 1 ((𝑀𝐵𝐾𝑁𝐿𝑁) → (𝐾(𝑄𝑀)𝐿) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   ∧ w3a 1071   = wceq 1601   ∈ wcel 2107  Vcvv 3398  ifcif 4307  ‘cfv 6137  (class class class)co 6924   ↦ cmpt2 6926  Fincfn 8243  Basecbs 16266  0gc0g 16497  1rcur 18899   Mat cmat 20628   minMatR1 cminmar1 20855 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5008  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-1st 7447  df-2nd 7448  df-slot 16270  df-base 16272  df-mat 20629  df-minmar1 20857 This theorem is referenced by:  minmar1eval  20871  maducoevalmin1  20875  smadiadet  20893  smadiadetglem2  20895
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