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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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