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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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