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Theorem cpm2mval 20778
 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 20777 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝐼 = (𝑚𝑆 ↦ (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))))
433adant3 1127 . 2 ((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝑆) → 𝐼 = (𝑚𝑆 ↦ (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))))
5 oveq 6821 . . . . . 6 (𝑚 = 𝑀 → (𝑥𝑚𝑦) = (𝑥𝑀𝑦))
65fveq2d 6358 . . . . 5 (𝑚 = 𝑀 → (coe1‘(𝑥𝑚𝑦)) = (coe1‘(𝑥𝑀𝑦)))
76fveq1d 6356 . . . 4 (𝑚 = 𝑀 → ((coe1‘(𝑥𝑚𝑦))‘0) = ((coe1‘(𝑥𝑀𝑦))‘0))
87mpt2eq3dv 6888 . . 3 (𝑚 = 𝑀 → (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑚𝑦))‘0)) = (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑀𝑦))‘0)))
98adantl 473 . 2 (((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝑆) ∧ 𝑚 = 𝑀) → (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑚𝑦))‘0)) = (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑀𝑦))‘0)))
10 simp3 1133 . 2 ((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝑆) → 𝑀𝑆)
11 simp1 1131 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝑆) → 𝑁 ∈ Fin)
12 mpt2exga 7416 . . 3 ((𝑁 ∈ Fin ∧ 𝑁 ∈ Fin) → (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑀𝑦))‘0)) ∈ V)
1311, 11, 12syl2anc 696 . 2 ((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝑆) → (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑀𝑦))‘0)) ∈ V)
144, 9, 10, 13fvmptd 6452 1 ((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝑆) → (𝐼𝑀) = (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑀𝑦))‘0)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1072   = wceq 1632   ∈ wcel 2140  Vcvv 3341   ↦ cmpt 4882  ‘cfv 6050  (class class class)co 6815   ↦ cmpt2 6817  Fincfn 8124  0cc0 10149  coe1cco1 19771   ConstPolyMat ccpmat 20731   cPolyMatToMat ccpmat2mat 20733 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-rep 4924  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056  ax-un 7116 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-ral 3056  df-rex 3057  df-reu 3058  df-rab 3060  df-v 3343  df-sbc 3578  df-csb 3676  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-pw 4305  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-iun 4675  df-br 4806  df-opab 4866  df-mpt 4883  df-id 5175  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280  df-iota 6013  df-fun 6052  df-fn 6053  df-f 6054  df-f1 6055  df-fo 6056  df-f1o 6057  df-fv 6058  df-ov 6818  df-oprab 6819  df-mpt2 6820  df-1st 7335  df-2nd 7336  df-cpmat2mat 20736 This theorem is referenced by:  cpm2mvalel  20779  m2cpminvid  20781  m2cpminvid2  20783
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