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Theorem mat2pmatval 22069
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 22068 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝑇 = (𝑚𝐵 ↦ (𝑥𝑁, 𝑦𝑁 ↦ (𝑆‘(𝑥𝑚𝑦)))))
763adant3 1132 . 2 ((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝐵) → 𝑇 = (𝑚𝐵 ↦ (𝑥𝑁, 𝑦𝑁 ↦ (𝑆‘(𝑥𝑚𝑦)))))
8 oveq 7360 . . . . 5 (𝑚 = 𝑀 → (𝑥𝑚𝑦) = (𝑥𝑀𝑦))
98fveq2d 6844 . . . 4 (𝑚 = 𝑀 → (𝑆‘(𝑥𝑚𝑦)) = (𝑆‘(𝑥𝑀𝑦)))
109mpoeq3dv 7433 . . 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 8007 . . 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 6953 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 3444  cmpt 5187  cfv 6494  (class class class)co 7354  cmpo 7356  Fincfn 8880  Basecbs 17080  algSccascl 21254  Poly1cpl1 21544   Mat cmat 21750   matToPolyMat cmat2pmat 22049
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 5241  ax-sep 5255  ax-nul 5262  ax-pow 5319  ax-pr 5383  ax-un 7669
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 2888  df-ne 2943  df-ral 3064  df-rex 3073  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3739  df-csb 3855  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-iun 4955  df-br 5105  df-opab 5167  df-mpt 5188  df-id 5530  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6446  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-ov 7357  df-oprab 7358  df-mpo 7359  df-1st 7918  df-2nd 7919  df-mat2pmat 22052
This theorem is referenced by:  mat2pmatvalel  22070  mat2pmatbas  22071  mat2pmatghm  22075  mat2pmatmul  22076  d0mat2pmat  22083  d1mat2pmat  22084  m2cpminvid2  22100  pmatcollpwlem  22125  pmatcollpwscmatlem2  22135
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