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Theorem cpm2mfval 22082
Description: Value of the inverse matrix transformation. (Contributed by AV, 14-Dec-2019.)
Hypotheses
Ref Expression
cpm2mfval.i 𝐼 = (𝑁 cPolyMatToMat 𝑅)
cpm2mfval.s 𝑆 = (𝑁 ConstPolyMat 𝑅)
Assertion
Ref Expression
cpm2mfval ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝐼 = (𝑚𝑆 ↦ (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))))
Distinct variable groups:   𝑚,𝑁,𝑥,𝑦   𝑅,𝑚,𝑥,𝑦   𝑆,𝑚
Allowed substitution hints:   𝑆(𝑥,𝑦)   𝐼(𝑥,𝑦,𝑚)   𝑉(𝑥,𝑦,𝑚)

Proof of Theorem cpm2mfval
Dummy variables 𝑛 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cpm2mfval.i . 2 𝐼 = (𝑁 cPolyMatToMat 𝑅)
2 df-cpmat2mat 22041 . . . 4 cPolyMatToMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ (𝑚 ∈ (𝑛 ConstPolyMat 𝑟) ↦ (𝑥𝑛, 𝑦𝑛 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))))
32a1i 11 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → cPolyMatToMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ (𝑚 ∈ (𝑛 ConstPolyMat 𝑟) ↦ (𝑥𝑛, 𝑦𝑛 ↦ ((coe1‘(𝑥𝑚𝑦))‘0)))))
4 oveq12 7362 . . . . . 6 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑛 ConstPolyMat 𝑟) = (𝑁 ConstPolyMat 𝑅))
5 cpm2mfval.s . . . . . 6 𝑆 = (𝑁 ConstPolyMat 𝑅)
64, 5eqtr4di 2794 . . . . 5 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑛 ConstPolyMat 𝑟) = 𝑆)
7 simpl 483 . . . . . 6 ((𝑛 = 𝑁𝑟 = 𝑅) → 𝑛 = 𝑁)
8 eqidd 2737 . . . . . 6 ((𝑛 = 𝑁𝑟 = 𝑅) → ((coe1‘(𝑥𝑚𝑦))‘0) = ((coe1‘(𝑥𝑚𝑦))‘0))
97, 7, 8mpoeq123dv 7428 . . . . 5 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑥𝑛, 𝑦𝑛 ↦ ((coe1‘(𝑥𝑚𝑦))‘0)) = (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑚𝑦))‘0)))
106, 9mpteq12dv 5194 . . . 4 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑚 ∈ (𝑛 ConstPolyMat 𝑟) ↦ (𝑥𝑛, 𝑦𝑛 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))) = (𝑚𝑆 ↦ (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))))
1110adantl 482 . . 3 (((𝑁 ∈ Fin ∧ 𝑅𝑉) ∧ (𝑛 = 𝑁𝑟 = 𝑅)) → (𝑚 ∈ (𝑛 ConstPolyMat 𝑟) ↦ (𝑥𝑛, 𝑦𝑛 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))) = (𝑚𝑆 ↦ (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))))
12 simpl 483 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝑁 ∈ Fin)
13 elex 3461 . . . 4 (𝑅𝑉𝑅 ∈ V)
1413adantl 482 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝑅 ∈ V)
155ovexi 7387 . . . 4 𝑆 ∈ V
16 mptexg 7167 . . . 4 (𝑆 ∈ V → (𝑚𝑆 ↦ (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))) ∈ V)
1715, 16mp1i 13 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (𝑚𝑆 ↦ (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))) ∈ V)
183, 11, 12, 14, 17ovmpod 7503 . 2 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (𝑁 cPolyMatToMat 𝑅) = (𝑚𝑆 ↦ (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))))
191, 18eqtrid 2788 1 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝐼 = (𝑚𝑆 ↦ (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  Vcvv 3443  cmpt 5186  cfv 6493  (class class class)co 7353  cmpo 7355  Fincfn 8879  0cc0 11047  coe1cco1 21533   ConstPolyMat ccpmat 22036   cPolyMatToMat ccpmat2mat 22038
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5240  ax-sep 5254  ax-nul 5261  ax-pr 5382
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7356  df-oprab 7357  df-mpo 7358  df-cpmat2mat 22041
This theorem is referenced by:  cpm2mval  22083  cpm2mf  22085  m2cpmfo  22089  cayleyhamiltonALT  22224
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