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Theorem mat2pmatfval 20728
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 20712 . . . 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 6820 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑛 Mat 𝑟) = (𝑁 Mat 𝑅))
54fveq2d 6354 . . . . . 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 6353 . . . . . . 7 (Base‘𝐴) = (Base‘(𝑁 Mat 𝑅))
96, 8eqtr2i 2781 . . . . . 6 (Base‘(𝑁 Mat 𝑅)) = 𝐵
105, 9syl6eq 2808 . . . . 5 ((𝑛 = 𝑁𝑟 = 𝑅) → (Base‘(𝑛 Mat 𝑟)) = 𝐵)
11 simpl 474 . . . . . 6 ((𝑛 = 𝑁𝑟 = 𝑅) → 𝑛 = 𝑁)
12 fveq2 6350 . . . . . . . . . 10 (𝑟 = 𝑅 → (Poly1𝑟) = (Poly1𝑅))
1312fveq2d 6354 . . . . . . . . 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 6353 . . . . . . . . 9 (algSc‘𝑃) = (algSc‘(Poly1𝑅))
1815, 17eqtr2i 2781 . . . . . . . 8 (algSc‘(Poly1𝑅)) = 𝑆
1914, 18syl6eq 2808 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → (algSc‘(Poly1𝑟)) = 𝑆)
2019fveq1d 6352 . . . . . 6 ((𝑛 = 𝑁𝑟 = 𝑅) → ((algSc‘(Poly1𝑟))‘(𝑥𝑚𝑦)) = (𝑆‘(𝑥𝑚𝑦)))
2111, 11, 20mpt2eq123dv 6880 . . . . 5 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑥𝑛, 𝑦𝑛 ↦ ((algSc‘(Poly1𝑟))‘(𝑥𝑚𝑦))) = (𝑥𝑁, 𝑦𝑁 ↦ (𝑆‘(𝑥𝑚𝑦))))
2210, 21mpteq12dv 4883 . . . 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 3350 . . . 4 (𝑅𝑉𝑅 ∈ V)
2625adantl 473 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝑅 ∈ V)
27 fvex 6360 . . . . 5 (Base‘𝐴) ∈ V
286, 27eqeltri 2833 . . . 4 𝐵 ∈ V
29 mptexg 6646 . . . 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 6951 . 2 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (𝑁 matToPolyMat 𝑅) = (𝑚𝐵 ↦ (𝑥𝑁, 𝑦𝑁 ↦ (𝑆‘(𝑥𝑚𝑦)))))
321, 31syl5eq 2804 1 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝑇 = (𝑚𝐵 ↦ (𝑥𝑁, 𝑦𝑁 ↦ (𝑆‘(𝑥𝑚𝑦)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1630  wcel 2137  Vcvv 3338  cmpt 4879  cfv 6047  (class class class)co 6811  cmpt2 6813  Fincfn 8119  Basecbs 16057  algSccascl 19511  Poly1cpl1 19747   Mat cmat 20413   matToPolyMat cmat2pmat 20709
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-rep 4921  ax-sep 4931  ax-nul 4939  ax-pr 5053
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ne 2931  df-ral 3053  df-rex 3054  df-reu 3055  df-rab 3057  df-v 3340  df-sbc 3575  df-csb 3673  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-nul 4057  df-if 4229  df-sn 4320  df-pr 4322  df-op 4326  df-uni 4587  df-iun 4672  df-br 4803  df-opab 4863  df-mpt 4880  df-id 5172  df-xp 5270  df-rel 5271  df-cnv 5272  df-co 5273  df-dm 5274  df-rn 5275  df-res 5276  df-ima 5277  df-iota 6010  df-fun 6049  df-fn 6050  df-f 6051  df-f1 6052  df-fo 6053  df-f1o 6054  df-fv 6055  df-ov 6814  df-oprab 6815  df-mpt2 6816  df-mat2pmat 20712
This theorem is referenced by:  mat2pmatval  20729  mat2pmatf  20733  m2cpmf  20747
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