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Theorem cpm2mval 21494
Description: The result of an inverse matrix transformation. (Contributed by AV, 12-Nov-2019.) (Revised by AV, 14-Dec-2019.)
Hypotheses
Ref Expression
cpm2mfval.i 𝐼 = (𝑁 cPolyMatToMat 𝑅)
cpm2mfval.s 𝑆 = (𝑁 ConstPolyMat 𝑅)
Assertion
Ref Expression
cpm2mval ((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝑆) → (𝐼𝑀) = (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑀𝑦))‘0)))
Distinct variable groups:   𝑥,𝑁,𝑦   𝑥,𝑅,𝑦   𝑥,𝑀,𝑦
Allowed substitution hints:   𝑆(𝑥,𝑦)   𝐼(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem cpm2mval
Dummy variable 𝑚 is distinct from all other variables.
StepHypRef Expression
1 cpm2mfval.i . . . 4 𝐼 = (𝑁 cPolyMatToMat 𝑅)
2 cpm2mfval.s . . . 4 𝑆 = (𝑁 ConstPolyMat 𝑅)
31, 2cpm2mfval 21493 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝐼 = (𝑚𝑆 ↦ (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))))
433adant3 1133 . 2 ((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝑆) → 𝐼 = (𝑚𝑆 ↦ (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))))
5 oveq 7170 . . . . . 6 (𝑚 = 𝑀 → (𝑥𝑚𝑦) = (𝑥𝑀𝑦))
65fveq2d 6672 . . . . 5 (𝑚 = 𝑀 → (coe1‘(𝑥𝑚𝑦)) = (coe1‘(𝑥𝑀𝑦)))
76fveq1d 6670 . . . 4 (𝑚 = 𝑀 → ((coe1‘(𝑥𝑚𝑦))‘0) = ((coe1‘(𝑥𝑀𝑦))‘0))
87mpoeq3dv 7241 . . 3 (𝑚 = 𝑀 → (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑚𝑦))‘0)) = (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑀𝑦))‘0)))
98adantl 485 . 2 (((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝑆) ∧ 𝑚 = 𝑀) → (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑚𝑦))‘0)) = (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑀𝑦))‘0)))
10 simp3 1139 . 2 ((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝑆) → 𝑀𝑆)
11 simp1 1137 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝑆) → 𝑁 ∈ Fin)
12 mpoexga 7794 . . 3 ((𝑁 ∈ Fin ∧ 𝑁 ∈ Fin) → (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑀𝑦))‘0)) ∈ V)
1311, 11, 12syl2anc 587 . 2 ((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝑆) → (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑀𝑦))‘0)) ∈ V)
144, 9, 10, 13fvmptd 6776 1 ((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝑆) → (𝐼𝑀) = (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑀𝑦))‘0)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1088   = wceq 1542  wcel 2113  Vcvv 3397  cmpt 5107  cfv 6333  (class class class)co 7164  cmpo 7166  Fincfn 8548  0cc0 10608  coe1cco1 20946   ConstPolyMat ccpmat 21447   cPolyMatToMat ccpmat2mat 21449
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 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-rep 5151  ax-sep 5164  ax-nul 5171  ax-pow 5229  ax-pr 5293  ax-un 7473
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3399  df-sbc 3680  df-csb 3789  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-nul 4210  df-if 4412  df-pw 4487  df-sn 4514  df-pr 4516  df-op 4520  df-uni 4794  df-iun 4880  df-br 5028  df-opab 5090  df-mpt 5108  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6291  df-fun 6335  df-fn 6336  df-f 6337  df-f1 6338  df-fo 6339  df-f1o 6340  df-fv 6341  df-ov 7167  df-oprab 7168  df-mpo 7169  df-1st 7707  df-2nd 7708  df-cpmat2mat 21452
This theorem is referenced by:  cpm2mvalel  21495  m2cpminvid  21497  m2cpminvid2  21499
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