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Theorem cpm2mfval 20774
 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 20733 . . . 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 6805 . . . . . 6 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑛 ConstPolyMat 𝑟) = (𝑁 ConstPolyMat 𝑅))
5 cpm2mfval.s . . . . . 6 𝑆 = (𝑁 ConstPolyMat 𝑅)
64, 5syl6eqr 2823 . . . . 5 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑛 ConstPolyMat 𝑟) = 𝑆)
7 simpl 468 . . . . . 6 ((𝑛 = 𝑁𝑟 = 𝑅) → 𝑛 = 𝑁)
8 eqidd 2772 . . . . . 6 ((𝑛 = 𝑁𝑟 = 𝑅) → ((coe1‘(𝑥𝑚𝑦))‘0) = ((coe1‘(𝑥𝑚𝑦))‘0))
97, 7, 8mpt2eq123dv 6868 . . . . 5 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑥𝑛, 𝑦𝑛 ↦ ((coe1‘(𝑥𝑚𝑦))‘0)) = (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑚𝑦))‘0)))
106, 9mpteq12dv 4868 . . . 4 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑚 ∈ (𝑛 ConstPolyMat 𝑟) ↦ (𝑥𝑛, 𝑦𝑛 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))) = (𝑚𝑆 ↦ (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))))
1110adantl 467 . . 3 (((𝑁 ∈ Fin ∧ 𝑅𝑉) ∧ (𝑛 = 𝑁𝑟 = 𝑅)) → (𝑚 ∈ (𝑛 ConstPolyMat 𝑟) ↦ (𝑥𝑛, 𝑦𝑛 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))) = (𝑚𝑆 ↦ (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))))
12 simpl 468 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝑁 ∈ Fin)
13 elex 3364 . . . 4 (𝑅𝑉𝑅 ∈ V)
1413adantl 467 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝑅 ∈ V)
15 ovex 6827 . . . . 5 (𝑁 ConstPolyMat 𝑅) ∈ V
165, 15eqeltri 2846 . . . 4 𝑆 ∈ V
17 mptexg 6631 . . . 4 (𝑆 ∈ V → (𝑚𝑆 ↦ (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))) ∈ V)
1816, 17mp1i 13 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (𝑚𝑆 ↦ (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))) ∈ V)
193, 11, 12, 14, 18ovmpt2d 6939 . 2 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (𝑁 cPolyMatToMat 𝑅) = (𝑚𝑆 ↦ (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))))
201, 19syl5eq 2817 1 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝐼 = (𝑚𝑆 ↦ (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 382   = wceq 1631   ∈ wcel 2145  Vcvv 3351   ↦ cmpt 4864  ‘cfv 6030  (class class class)co 6796   ↦ cmpt2 6798  Fincfn 8113  0cc0 10142  coe1cco1 19763   ConstPolyMat ccpmat 20728   cPolyMatToMat ccpmat2mat 20730 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pr 5035 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-cpmat2mat 20733 This theorem is referenced by:  cpm2mval  20775  cpm2mf  20777  m2cpmfo  20781  cayleyhamiltonALT  20916
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