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Theorem submafval 20790
Description: First 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
submafval 𝑄 = (𝑚𝐵 ↦ (𝑘𝑁, 𝑙𝑁 ↦ (𝑖 ∈ (𝑁 ∖ {𝑘}), 𝑗 ∈ (𝑁 ∖ {𝑙}) ↦ (𝑖𝑚𝑗))))
Distinct variable groups:   𝐵,𝑚   𝑖,𝑁,𝑗,𝑘,𝑙,𝑚   𝑅,𝑖,𝑗,𝑘,𝑙,𝑚
Allowed substitution hints:   𝐴(𝑖,𝑗,𝑘,𝑚,𝑙)   𝐵(𝑖,𝑗,𝑘,𝑙)   𝑄(𝑖,𝑗,𝑘,𝑚,𝑙)

Proof of Theorem submafval
Dummy variables 𝑛 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 submafval.q . 2 𝑄 = (𝑁 subMat 𝑅)
2 oveq12 6931 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑛 Mat 𝑟) = (𝑁 Mat 𝑅))
3 submafval.a . . . . . . . 8 𝐴 = (𝑁 Mat 𝑅)
42, 3syl6eqr 2832 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑛 Mat 𝑟) = 𝐴)
54fveq2d 6450 . . . . . 6 ((𝑛 = 𝑁𝑟 = 𝑅) → (Base‘(𝑛 Mat 𝑟)) = (Base‘𝐴))
6 submafval.b . . . . . 6 𝐵 = (Base‘𝐴)
75, 6syl6eqr 2832 . . . . 5 ((𝑛 = 𝑁𝑟 = 𝑅) → (Base‘(𝑛 Mat 𝑟)) = 𝐵)
8 simpl 476 . . . . . 6 ((𝑛 = 𝑁𝑟 = 𝑅) → 𝑛 = 𝑁)
9 difeq1 3944 . . . . . . . 8 (𝑛 = 𝑁 → (𝑛 ∖ {𝑘}) = (𝑁 ∖ {𝑘}))
109adantr 474 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑛 ∖ {𝑘}) = (𝑁 ∖ {𝑘}))
11 difeq1 3944 . . . . . . . 8 (𝑛 = 𝑁 → (𝑛 ∖ {𝑙}) = (𝑁 ∖ {𝑙}))
1211adantr 474 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑛 ∖ {𝑙}) = (𝑁 ∖ {𝑙}))
13 eqidd 2779 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑖𝑚𝑗) = (𝑖𝑚𝑗))
1410, 12, 13mpt2eq123dv 6994 . . . . . 6 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑖 ∈ (𝑛 ∖ {𝑘}), 𝑗 ∈ (𝑛 ∖ {𝑙}) ↦ (𝑖𝑚𝑗)) = (𝑖 ∈ (𝑁 ∖ {𝑘}), 𝑗 ∈ (𝑁 ∖ {𝑙}) ↦ (𝑖𝑚𝑗)))
158, 8, 14mpt2eq123dv 6994 . . . . 5 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑘𝑛, 𝑙𝑛 ↦ (𝑖 ∈ (𝑛 ∖ {𝑘}), 𝑗 ∈ (𝑛 ∖ {𝑙}) ↦ (𝑖𝑚𝑗))) = (𝑘𝑁, 𝑙𝑁 ↦ (𝑖 ∈ (𝑁 ∖ {𝑘}), 𝑗 ∈ (𝑁 ∖ {𝑙}) ↦ (𝑖𝑚𝑗))))
167, 15mpteq12dv 4969 . . . 4 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑘𝑛, 𝑙𝑛 ↦ (𝑖 ∈ (𝑛 ∖ {𝑘}), 𝑗 ∈ (𝑛 ∖ {𝑙}) ↦ (𝑖𝑚𝑗)))) = (𝑚𝐵 ↦ (𝑘𝑁, 𝑙𝑁 ↦ (𝑖 ∈ (𝑁 ∖ {𝑘}), 𝑗 ∈ (𝑁 ∖ {𝑙}) ↦ (𝑖𝑚𝑗)))))
17 df-subma 20788 . . . 4 subMat = (𝑛 ∈ V, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑘𝑛, 𝑙𝑛 ↦ (𝑖 ∈ (𝑛 ∖ {𝑘}), 𝑗 ∈ (𝑛 ∖ {𝑙}) ↦ (𝑖𝑚𝑗)))))
186fvexi 6460 . . . . 5 𝐵 ∈ V
1918mptex 6758 . . . 4 (𝑚𝐵 ↦ (𝑘𝑁, 𝑙𝑁 ↦ (𝑖 ∈ (𝑁 ∖ {𝑘}), 𝑗 ∈ (𝑁 ∖ {𝑙}) ↦ (𝑖𝑚𝑗)))) ∈ V
2016, 17, 19ovmpt2a 7068 . . 3 ((𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 subMat 𝑅) = (𝑚𝐵 ↦ (𝑘𝑁, 𝑙𝑁 ↦ (𝑖 ∈ (𝑁 ∖ {𝑘}), 𝑗 ∈ (𝑁 ∖ {𝑙}) ↦ (𝑖𝑚𝑗)))))
2117mpt2ndm0 7152 . . . . 5 (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 subMat 𝑅) = ∅)
22 mpt0 6267 . . . . 5 (𝑚 ∈ ∅ ↦ (𝑘𝑁, 𝑙𝑁 ↦ (𝑖 ∈ (𝑁 ∖ {𝑘}), 𝑗 ∈ (𝑁 ∖ {𝑙}) ↦ (𝑖𝑚𝑗)))) = ∅
2321, 22syl6eqr 2832 . . . 4 (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 subMat 𝑅) = (𝑚 ∈ ∅ ↦ (𝑘𝑁, 𝑙𝑁 ↦ (𝑖 ∈ (𝑁 ∖ {𝑘}), 𝑗 ∈ (𝑁 ∖ {𝑙}) ↦ (𝑖𝑚𝑗)))))
243fveq2i 6449 . . . . . . 7 (Base‘𝐴) = (Base‘(𝑁 Mat 𝑅))
256, 24eqtri 2802 . . . . . 6 𝐵 = (Base‘(𝑁 Mat 𝑅))
26 matbas0pc 20619 . . . . . 6 (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (Base‘(𝑁 Mat 𝑅)) = ∅)
2725, 26syl5eq 2826 . . . . 5 (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → 𝐵 = ∅)
2827mpteq1d 4973 . . . 4 (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑚𝐵 ↦ (𝑘𝑁, 𝑙𝑁 ↦ (𝑖 ∈ (𝑁 ∖ {𝑘}), 𝑗 ∈ (𝑁 ∖ {𝑙}) ↦ (𝑖𝑚𝑗)))) = (𝑚 ∈ ∅ ↦ (𝑘𝑁, 𝑙𝑁 ↦ (𝑖 ∈ (𝑁 ∖ {𝑘}), 𝑗 ∈ (𝑁 ∖ {𝑙}) ↦ (𝑖𝑚𝑗)))))
2923, 28eqtr4d 2817 . . 3 (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 subMat 𝑅) = (𝑚𝐵 ↦ (𝑘𝑁, 𝑙𝑁 ↦ (𝑖 ∈ (𝑁 ∖ {𝑘}), 𝑗 ∈ (𝑁 ∖ {𝑙}) ↦ (𝑖𝑚𝑗)))))
3020, 29pm2.61i 177 . 2 (𝑁 subMat 𝑅) = (𝑚𝐵 ↦ (𝑘𝑁, 𝑙𝑁 ↦ (𝑖 ∈ (𝑁 ∖ {𝑘}), 𝑗 ∈ (𝑁 ∖ {𝑙}) ↦ (𝑖𝑚𝑗))))
311, 30eqtri 2802 1 𝑄 = (𝑚𝐵 ↦ (𝑘𝑁, 𝑙𝑁 ↦ (𝑖 ∈ (𝑁 ∖ {𝑘}), 𝑗 ∈ (𝑁 ∖ {𝑙}) ↦ (𝑖𝑚𝑗))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 386   = wceq 1601  wcel 2107  Vcvv 3398  cdif 3789  c0 4141  {csn 4398  cmpt 4965  cfv 6135  (class class class)co 6922  cmpt2 6924  Basecbs 16255   Mat cmat 20617   subMat csubma 20787
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 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138
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-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-slot 16259  df-base 16261  df-mat 20618  df-subma 20788
This theorem is referenced by:  submaval0  20791
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