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Theorem mat2pmatfval 22729
Description: Value of the matrix transformation. (Contributed by AV, 31-Jul-2019.)
Hypotheses
Ref Expression
mat2pmatfval.t 𝑇 = (𝑁 matToPolyMat 𝑅)
mat2pmatfval.a 𝐴 = (𝑁 Mat 𝑅)
mat2pmatfval.b 𝐵 = (Base‘𝐴)
mat2pmatfval.p 𝑃 = (Poly1𝑅)
mat2pmatfval.s 𝑆 = (algSc‘𝑃)
Assertion
Ref Expression
mat2pmatfval ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝑇 = (𝑚𝐵 ↦ (𝑥𝑁, 𝑦𝑁 ↦ (𝑆‘(𝑥𝑚𝑦)))))
Distinct variable groups:   𝐵,𝑚   𝑥,𝑚,𝑦,𝑁   𝑅,𝑚,𝑥,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦,𝑚)   𝐵(𝑥,𝑦)   𝑃(𝑥,𝑦,𝑚)   𝑆(𝑥,𝑦,𝑚)   𝑇(𝑥,𝑦,𝑚)   𝑉(𝑥,𝑦,𝑚)

Proof of Theorem mat2pmatfval
Dummy variables 𝑛 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mat2pmatfval.t . 2 𝑇 = (𝑁 matToPolyMat 𝑅)
2 df-mat2pmat 22713 . . . 4 matToPolyMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑥𝑛, 𝑦𝑛 ↦ ((algSc‘(Poly1𝑟))‘(𝑥𝑚𝑦)))))
32a1i 11 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → matToPolyMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑥𝑛, 𝑦𝑛 ↦ ((algSc‘(Poly1𝑟))‘(𝑥𝑚𝑦))))))
4 oveq12 7440 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑛 Mat 𝑟) = (𝑁 Mat 𝑅))
54fveq2d 6910 . . . . . 6 ((𝑛 = 𝑁𝑟 = 𝑅) → (Base‘(𝑛 Mat 𝑟)) = (Base‘(𝑁 Mat 𝑅)))
6 mat2pmatfval.b . . . . . . 7 𝐵 = (Base‘𝐴)
7 mat2pmatfval.a . . . . . . . 8 𝐴 = (𝑁 Mat 𝑅)
87fveq2i 6909 . . . . . . 7 (Base‘𝐴) = (Base‘(𝑁 Mat 𝑅))
96, 8eqtr2i 2766 . . . . . 6 (Base‘(𝑁 Mat 𝑅)) = 𝐵
105, 9eqtrdi 2793 . . . . 5 ((𝑛 = 𝑁𝑟 = 𝑅) → (Base‘(𝑛 Mat 𝑟)) = 𝐵)
11 simpl 482 . . . . . 6 ((𝑛 = 𝑁𝑟 = 𝑅) → 𝑛 = 𝑁)
12 2fveq3 6911 . . . . . . . . 9 (𝑟 = 𝑅 → (algSc‘(Poly1𝑟)) = (algSc‘(Poly1𝑅)))
1312adantl 481 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → (algSc‘(Poly1𝑟)) = (algSc‘(Poly1𝑅)))
14 mat2pmatfval.s . . . . . . . . 9 𝑆 = (algSc‘𝑃)
15 mat2pmatfval.p . . . . . . . . . 10 𝑃 = (Poly1𝑅)
1615fveq2i 6909 . . . . . . . . 9 (algSc‘𝑃) = (algSc‘(Poly1𝑅))
1714, 16eqtr2i 2766 . . . . . . . 8 (algSc‘(Poly1𝑅)) = 𝑆
1813, 17eqtrdi 2793 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → (algSc‘(Poly1𝑟)) = 𝑆)
1918fveq1d 6908 . . . . . 6 ((𝑛 = 𝑁𝑟 = 𝑅) → ((algSc‘(Poly1𝑟))‘(𝑥𝑚𝑦)) = (𝑆‘(𝑥𝑚𝑦)))
2011, 11, 19mpoeq123dv 7508 . . . . 5 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑥𝑛, 𝑦𝑛 ↦ ((algSc‘(Poly1𝑟))‘(𝑥𝑚𝑦))) = (𝑥𝑁, 𝑦𝑁 ↦ (𝑆‘(𝑥𝑚𝑦))))
2110, 20mpteq12dv 5233 . . . 4 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑥𝑛, 𝑦𝑛 ↦ ((algSc‘(Poly1𝑟))‘(𝑥𝑚𝑦)))) = (𝑚𝐵 ↦ (𝑥𝑁, 𝑦𝑁 ↦ (𝑆‘(𝑥𝑚𝑦)))))
2221adantl 481 . . 3 (((𝑁 ∈ Fin ∧ 𝑅𝑉) ∧ (𝑛 = 𝑁𝑟 = 𝑅)) → (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑥𝑛, 𝑦𝑛 ↦ ((algSc‘(Poly1𝑟))‘(𝑥𝑚𝑦)))) = (𝑚𝐵 ↦ (𝑥𝑁, 𝑦𝑁 ↦ (𝑆‘(𝑥𝑚𝑦)))))
23 simpl 482 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝑁 ∈ Fin)
24 elex 3501 . . . 4 (𝑅𝑉𝑅 ∈ V)
2524adantl 481 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝑅 ∈ V)
266fvexi 6920 . . . 4 𝐵 ∈ V
27 mptexg 7241 . . . 4 (𝐵 ∈ V → (𝑚𝐵 ↦ (𝑥𝑁, 𝑦𝑁 ↦ (𝑆‘(𝑥𝑚𝑦)))) ∈ V)
2826, 27mp1i 13 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (𝑚𝐵 ↦ (𝑥𝑁, 𝑦𝑁 ↦ (𝑆‘(𝑥𝑚𝑦)))) ∈ V)
293, 22, 23, 25, 28ovmpod 7585 . 2 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (𝑁 matToPolyMat 𝑅) = (𝑚𝐵 ↦ (𝑥𝑁, 𝑦𝑁 ↦ (𝑆‘(𝑥𝑚𝑦)))))
301, 29eqtrid 2789 1 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝑇 = (𝑚𝐵 ↦ (𝑥𝑁, 𝑦𝑁 ↦ (𝑆‘(𝑥𝑚𝑦)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  Vcvv 3480  cmpt 5225  cfv 6561  (class class class)co 7431  cmpo 7433  Fincfn 8985  Basecbs 17247  algSccascl 21872  Poly1cpl1 22178   Mat cmat 22411   matToPolyMat cmat2pmat 22710
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-mat2pmat 22713
This theorem is referenced by:  mat2pmatval  22730  mat2pmatf  22734  m2cpmf  22748
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