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Theorem submaval0 21183
Description: Second substitution for a submatrix. (Contributed by AV, 28-Dec-2018.)
Hypotheses
Ref Expression
submafval.a 𝐴 = (𝑁 Mat 𝑅)
submafval.q 𝑄 = (𝑁 subMat 𝑅)
submafval.b 𝐵 = (Base‘𝐴)
Assertion
Ref Expression
submaval0 (𝑀𝐵 → (𝑄𝑀) = (𝑘𝑁, 𝑙𝑁 ↦ (𝑖 ∈ (𝑁 ∖ {𝑘}), 𝑗 ∈ (𝑁 ∖ {𝑙}) ↦ (𝑖𝑀𝑗))))
Distinct variable groups:   𝑖,𝑁,𝑗,𝑘,𝑙   𝑅,𝑖,𝑗,𝑘,𝑙   𝑖,𝑀,𝑗,𝑘,𝑙
Allowed substitution hints:   𝐴(𝑖,𝑗,𝑘,𝑙)   𝐵(𝑖,𝑗,𝑘,𝑙)   𝑄(𝑖,𝑗,𝑘,𝑙)

Proof of Theorem submaval0
Dummy variable 𝑚 is distinct from all other variables.
StepHypRef Expression
1 submafval.a . . . . 5 𝐴 = (𝑁 Mat 𝑅)
2 submafval.b . . . . 5 𝐵 = (Base‘𝐴)
31, 2matrcl 21015 . . . 4 (𝑀𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
43simpld 498 . . 3 (𝑀𝐵𝑁 ∈ Fin)
5 mpoexga 7762 . . 3 ((𝑁 ∈ Fin ∧ 𝑁 ∈ Fin) → (𝑘𝑁, 𝑙𝑁 ↦ (𝑖 ∈ (𝑁 ∖ {𝑘}), 𝑗 ∈ (𝑁 ∖ {𝑙}) ↦ (𝑖𝑀𝑗))) ∈ V)
64, 4, 5syl2anc 587 . 2 (𝑀𝐵 → (𝑘𝑁, 𝑙𝑁 ↦ (𝑖 ∈ (𝑁 ∖ {𝑘}), 𝑗 ∈ (𝑁 ∖ {𝑙}) ↦ (𝑖𝑀𝑗))) ∈ V)
7 oveq 7146 . . . . 5 (𝑚 = 𝑀 → (𝑖𝑚𝑗) = (𝑖𝑀𝑗))
87mpoeq3dv 7217 . . . 4 (𝑚 = 𝑀 → (𝑖 ∈ (𝑁 ∖ {𝑘}), 𝑗 ∈ (𝑁 ∖ {𝑙}) ↦ (𝑖𝑚𝑗)) = (𝑖 ∈ (𝑁 ∖ {𝑘}), 𝑗 ∈ (𝑁 ∖ {𝑙}) ↦ (𝑖𝑀𝑗)))
98mpoeq3dv 7217 . . 3 (𝑚 = 𝑀 → (𝑘𝑁, 𝑙𝑁 ↦ (𝑖 ∈ (𝑁 ∖ {𝑘}), 𝑗 ∈ (𝑁 ∖ {𝑙}) ↦ (𝑖𝑚𝑗))) = (𝑘𝑁, 𝑙𝑁 ↦ (𝑖 ∈ (𝑁 ∖ {𝑘}), 𝑗 ∈ (𝑁 ∖ {𝑙}) ↦ (𝑖𝑀𝑗))))
10 submafval.q . . . 4 𝑄 = (𝑁 subMat 𝑅)
111, 10, 2submafval 21182 . . 3 𝑄 = (𝑚𝐵 ↦ (𝑘𝑁, 𝑙𝑁 ↦ (𝑖 ∈ (𝑁 ∖ {𝑘}), 𝑗 ∈ (𝑁 ∖ {𝑙}) ↦ (𝑖𝑚𝑗))))
129, 11fvmptg 6748 . 2 ((𝑀𝐵 ∧ (𝑘𝑁, 𝑙𝑁 ↦ (𝑖 ∈ (𝑁 ∖ {𝑘}), 𝑗 ∈ (𝑁 ∖ {𝑙}) ↦ (𝑖𝑀𝑗))) ∈ V) → (𝑄𝑀) = (𝑘𝑁, 𝑙𝑁 ↦ (𝑖 ∈ (𝑁 ∖ {𝑘}), 𝑗 ∈ (𝑁 ∖ {𝑙}) ↦ (𝑖𝑀𝑗))))
136, 12mpdan 686 1 (𝑀𝐵 → (𝑄𝑀) = (𝑘𝑁, 𝑙𝑁 ↦ (𝑖 ∈ (𝑁 ∖ {𝑘}), 𝑗 ∈ (𝑁 ∖ {𝑙}) ↦ (𝑖𝑀𝑗))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2114  Vcvv 3469  cdif 3905  {csn 4539  cfv 6334  (class class class)co 7140  cmpo 7142  Fincfn 8496  Basecbs 16474   Mat cmat 21010   subMat csubma 21179
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 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
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 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-ov 7143  df-oprab 7144  df-mpo 7145  df-1st 7675  df-2nd 7676  df-slot 16478  df-base 16480  df-mat 21011  df-subma 21180
This theorem is referenced by:  submaval  21184
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