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Theorem mat2pmatfval 20750
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 20734 . . . 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 6823 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑛 Mat 𝑟) = (𝑁 Mat 𝑅))
54fveq2d 6357 . . . . . 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 6356 . . . . . . 7 (Base‘𝐴) = (Base‘(𝑁 Mat 𝑅))
96, 8eqtr2i 2783 . . . . . 6 (Base‘(𝑁 Mat 𝑅)) = 𝐵
105, 9syl6eq 2810 . . . . 5 ((𝑛 = 𝑁𝑟 = 𝑅) → (Base‘(𝑛 Mat 𝑟)) = 𝐵)
11 simpl 474 . . . . . 6 ((𝑛 = 𝑁𝑟 = 𝑅) → 𝑛 = 𝑁)
12 fveq2 6353 . . . . . . . . . 10 (𝑟 = 𝑅 → (Poly1𝑟) = (Poly1𝑅))
1312fveq2d 6357 . . . . . . . . 9 (𝑟 = 𝑅 → (algSc‘(Poly1𝑟)) = (algSc‘(Poly1𝑅)))
1413adantl 473 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → (algSc‘(Poly1𝑟)) = (algSc‘(Poly1𝑅)))
15 mat2pmatfval.s . . . . . . . . 9 𝑆 = (algSc‘𝑃)
16 mat2pmatfval.p . . . . . . . . . 10 𝑃 = (Poly1𝑅)
1716fveq2i 6356 . . . . . . . . 9 (algSc‘𝑃) = (algSc‘(Poly1𝑅))
1815, 17eqtr2i 2783 . . . . . . . 8 (algSc‘(Poly1𝑅)) = 𝑆
1914, 18syl6eq 2810 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → (algSc‘(Poly1𝑟)) = 𝑆)
2019fveq1d 6355 . . . . . 6 ((𝑛 = 𝑁𝑟 = 𝑅) → ((algSc‘(Poly1𝑟))‘(𝑥𝑚𝑦)) = (𝑆‘(𝑥𝑚𝑦)))
2111, 11, 20mpt2eq123dv 6883 . . . . 5 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑥𝑛, 𝑦𝑛 ↦ ((algSc‘(Poly1𝑟))‘(𝑥𝑚𝑦))) = (𝑥𝑁, 𝑦𝑁 ↦ (𝑆‘(𝑥𝑚𝑦))))
2210, 21mpteq12dv 4885 . . . 4 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑥𝑛, 𝑦𝑛 ↦ ((algSc‘(Poly1𝑟))‘(𝑥𝑚𝑦)))) = (𝑚𝐵 ↦ (𝑥𝑁, 𝑦𝑁 ↦ (𝑆‘(𝑥𝑚𝑦)))))
2322adantl 473 . . 3 (((𝑁 ∈ Fin ∧ 𝑅𝑉) ∧ (𝑛 = 𝑁𝑟 = 𝑅)) → (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑥𝑛, 𝑦𝑛 ↦ ((algSc‘(Poly1𝑟))‘(𝑥𝑚𝑦)))) = (𝑚𝐵 ↦ (𝑥𝑁, 𝑦𝑁 ↦ (𝑆‘(𝑥𝑚𝑦)))))
24 simpl 474 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝑁 ∈ Fin)
25 elex 3352 . . . 4 (𝑅𝑉𝑅 ∈ V)
2625adantl 473 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝑅 ∈ V)
27 fvex 6363 . . . . 5 (Base‘𝐴) ∈ V
286, 27eqeltri 2835 . . . 4 𝐵 ∈ V
29 mptexg 6649 . . . 4 (𝐵 ∈ V → (𝑚𝐵 ↦ (𝑥𝑁, 𝑦𝑁 ↦ (𝑆‘(𝑥𝑚𝑦)))) ∈ V)
3028, 29mp1i 13 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (𝑚𝐵 ↦ (𝑥𝑁, 𝑦𝑁 ↦ (𝑆‘(𝑥𝑚𝑦)))) ∈ V)
313, 23, 24, 26, 30ovmpt2d 6954 . 2 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (𝑁 matToPolyMat 𝑅) = (𝑚𝐵 ↦ (𝑥𝑁, 𝑦𝑁 ↦ (𝑆‘(𝑥𝑚𝑦)))))
321, 31syl5eq 2806 1 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝑇 = (𝑚𝐵 ↦ (𝑥𝑁, 𝑦𝑁 ↦ (𝑆‘(𝑥𝑚𝑦)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  Vcvv 3340  cmpt 4881  cfv 6049  (class class class)co 6814  cmpt2 6816  Fincfn 8123  Basecbs 16079  algSccascl 19533  Poly1cpl1 19769   Mat cmat 20435   matToPolyMat cmat2pmat 20731
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 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pr 5055
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 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-mat2pmat 20734
This theorem is referenced by:  mat2pmatval  20751  mat2pmatf  20755  m2cpmf  20769
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