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Theorem mat2pmatvalel 22731
Description: A (matrix) element of 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
mat2pmatvalel (((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝐵) ∧ (𝑋𝑁𝑌𝑁)) → (𝑋(𝑇𝑀)𝑌) = (𝑆‘(𝑋𝑀𝑌)))

Proof of Theorem mat2pmatvalel
Dummy variables 𝑥 𝑦 are mutually distinct and 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, 5mat2pmatval 22730 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝐵) → (𝑇𝑀) = (𝑥𝑁, 𝑦𝑁 ↦ (𝑆‘(𝑥𝑀𝑦))))
76adantr 480 . 2 (((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝐵) ∧ (𝑋𝑁𝑌𝑁)) → (𝑇𝑀) = (𝑥𝑁, 𝑦𝑁 ↦ (𝑆‘(𝑥𝑀𝑦))))
8 oveq12 7440 . . . 4 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝑥𝑀𝑦) = (𝑋𝑀𝑌))
98fveq2d 6910 . . 3 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝑆‘(𝑥𝑀𝑦)) = (𝑆‘(𝑋𝑀𝑌)))
109adantl 481 . 2 ((((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝐵) ∧ (𝑋𝑁𝑌𝑁)) ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝑆‘(𝑥𝑀𝑦)) = (𝑆‘(𝑋𝑀𝑌)))
11 simprl 771 . 2 (((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝐵) ∧ (𝑋𝑁𝑌𝑁)) → 𝑋𝑁)
12 simprr 773 . 2 (((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝐵) ∧ (𝑋𝑁𝑌𝑁)) → 𝑌𝑁)
13 fvexd 6921 . 2 (((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝐵) ∧ (𝑋𝑁𝑌𝑁)) → (𝑆‘(𝑋𝑀𝑌)) ∈ V)
147, 10, 11, 12, 13ovmpod 7585 1 (((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝐵) ∧ (𝑋𝑁𝑌𝑁)) → (𝑋(𝑇𝑀)𝑌) = (𝑆‘(𝑋𝑀𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  Vcvv 3480  cfv 6561  (class class class)co 7431  cmpo 7433  Fincfn 8985  Basecbs 17247  algSccascl 21872  Poly1cpl1 22178   Mat cmat 22411   matToPolyMat cmat2pmat 22710
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-mat2pmat 22713
This theorem is referenced by:  mat2pmatf1  22735  mat2pmat1  22738  mat2pmatlin  22741  m2cpm  22747  m2cpminvid  22759  monmatcollpw  22785  chpmat1dlem  22841  chpdmatlem2  22845  chpdmatlem3  22846
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