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Theorem mat2pmatval 22073
Description: The result of a 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
mat2pmatval ((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝐵) → (𝑇𝑀) = (𝑥𝑁, 𝑦𝑁 ↦ (𝑆‘(𝑥𝑀𝑦))))
Distinct variable groups:   𝑥,𝑦,𝑁   𝑥,𝑅,𝑦   𝑥,𝑀,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)   𝑃(𝑥,𝑦)   𝑆(𝑥,𝑦)   𝑇(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem mat2pmatval
Dummy variable 𝑚 is distinct from all other variables.
StepHypRef Expression
1 mat2pmatfval.t . . . 4 𝑇 = (𝑁 matToPolyMat 𝑅)
2 mat2pmatfval.a . . . 4 𝐴 = (𝑁 Mat 𝑅)
3 mat2pmatfval.b . . . 4 𝐵 = (Base‘𝐴)
4 mat2pmatfval.p . . . 4 𝑃 = (Poly1𝑅)
5 mat2pmatfval.s . . . 4 𝑆 = (algSc‘𝑃)
61, 2, 3, 4, 5mat2pmatfval 22072 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝑇 = (𝑚𝐵 ↦ (𝑥𝑁, 𝑦𝑁 ↦ (𝑆‘(𝑥𝑚𝑦)))))
763adant3 1132 . 2 ((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝐵) → 𝑇 = (𝑚𝐵 ↦ (𝑥𝑁, 𝑦𝑁 ↦ (𝑆‘(𝑥𝑚𝑦)))))
8 oveq 7363 . . . . 5 (𝑚 = 𝑀 → (𝑥𝑚𝑦) = (𝑥𝑀𝑦))
98fveq2d 6846 . . . 4 (𝑚 = 𝑀 → (𝑆‘(𝑥𝑚𝑦)) = (𝑆‘(𝑥𝑀𝑦)))
109mpoeq3dv 7436 . . 3 (𝑚 = 𝑀 → (𝑥𝑁, 𝑦𝑁 ↦ (𝑆‘(𝑥𝑚𝑦))) = (𝑥𝑁, 𝑦𝑁 ↦ (𝑆‘(𝑥𝑀𝑦))))
1110adantl 482 . 2 (((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝐵) ∧ 𝑚 = 𝑀) → (𝑥𝑁, 𝑦𝑁 ↦ (𝑆‘(𝑥𝑚𝑦))) = (𝑥𝑁, 𝑦𝑁 ↦ (𝑆‘(𝑥𝑀𝑦))))
12 simp3 1138 . 2 ((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝐵) → 𝑀𝐵)
13 simp1 1136 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝐵) → 𝑁 ∈ Fin)
14 mpoexga 8010 . . 3 ((𝑁 ∈ Fin ∧ 𝑁 ∈ Fin) → (𝑥𝑁, 𝑦𝑁 ↦ (𝑆‘(𝑥𝑀𝑦))) ∈ V)
1513, 13, 14syl2anc 584 . 2 ((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝐵) → (𝑥𝑁, 𝑦𝑁 ↦ (𝑆‘(𝑥𝑀𝑦))) ∈ V)
167, 11, 12, 15fvmptd 6955 1 ((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝐵) → (𝑇𝑀) = (𝑥𝑁, 𝑦𝑁 ↦ (𝑆‘(𝑥𝑀𝑦))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1087   = wceq 1541  wcel 2106  Vcvv 3445  cmpt 5188  cfv 6496  (class class class)co 7357  cmpo 7359  Fincfn 8883  Basecbs 17083  algSccascl 21258  Poly1cpl1 21548   Mat cmat 21754   matToPolyMat cmat2pmat 22053
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7921  df-2nd 7922  df-mat2pmat 22056
This theorem is referenced by:  mat2pmatvalel  22074  mat2pmatbas  22075  mat2pmatghm  22079  mat2pmatmul  22080  d0mat2pmat  22087  d1mat2pmat  22088  m2cpminvid2  22104  pmatcollpwlem  22129  pmatcollpwscmatlem2  22139
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