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Theorem mat2pmatvalel 22614
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 𝐡 = (Baseβ€˜π΄)
mat2pmatfval.p 𝑃 = (Poly1β€˜π‘…)
mat2pmatfval.s 𝑆 = (algScβ€˜π‘ƒ)
Assertion
Ref Expression
mat2pmatvalel (((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐡) ∧ (𝑋 ∈ 𝑁 ∧ π‘Œ ∈ 𝑁)) β†’ (𝑋(π‘‡β€˜π‘€)π‘Œ) = (π‘†β€˜(π‘‹π‘€π‘Œ)))

Proof of Theorem mat2pmatvalel
Dummy variables π‘₯ 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 𝐡 = (Baseβ€˜π΄)
4 mat2pmatfval.p . . . 4 𝑃 = (Poly1β€˜π‘…)
5 mat2pmatfval.s . . . 4 𝑆 = (algScβ€˜π‘ƒ)
61, 2, 3, 4, 5mat2pmatval 22613 . . 3 ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐡) β†’ (π‘‡β€˜π‘€) = (π‘₯ ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (π‘†β€˜(π‘₯𝑀𝑦))))
76adantr 480 . 2 (((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐡) ∧ (𝑋 ∈ 𝑁 ∧ π‘Œ ∈ 𝑁)) β†’ (π‘‡β€˜π‘€) = (π‘₯ ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (π‘†β€˜(π‘₯𝑀𝑦))))
8 oveq12 7423 . . . 4 ((π‘₯ = 𝑋 ∧ 𝑦 = π‘Œ) β†’ (π‘₯𝑀𝑦) = (π‘‹π‘€π‘Œ))
98fveq2d 6895 . . 3 ((π‘₯ = 𝑋 ∧ 𝑦 = π‘Œ) β†’ (π‘†β€˜(π‘₯𝑀𝑦)) = (π‘†β€˜(π‘‹π‘€π‘Œ)))
109adantl 481 . 2 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐡) ∧ (𝑋 ∈ 𝑁 ∧ π‘Œ ∈ 𝑁)) ∧ (π‘₯ = 𝑋 ∧ 𝑦 = π‘Œ)) β†’ (π‘†β€˜(π‘₯𝑀𝑦)) = (π‘†β€˜(π‘‹π‘€π‘Œ)))
11 simprl 770 . 2 (((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐡) ∧ (𝑋 ∈ 𝑁 ∧ π‘Œ ∈ 𝑁)) β†’ 𝑋 ∈ 𝑁)
12 simprr 772 . 2 (((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐡) ∧ (𝑋 ∈ 𝑁 ∧ π‘Œ ∈ 𝑁)) β†’ π‘Œ ∈ 𝑁)
13 fvexd 6906 . 2 (((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐡) ∧ (𝑋 ∈ 𝑁 ∧ π‘Œ ∈ 𝑁)) β†’ (π‘†β€˜(π‘‹π‘€π‘Œ)) ∈ V)
147, 10, 11, 12, 13ovmpod 7567 1 (((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐡) ∧ (𝑋 ∈ 𝑁 ∧ π‘Œ ∈ 𝑁)) β†’ (𝑋(π‘‡β€˜π‘€)π‘Œ) = (π‘†β€˜(π‘‹π‘€π‘Œ)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099  Vcvv 3469  β€˜cfv 6542  (class class class)co 7414   ∈ cmpo 7416  Fincfn 8955  Basecbs 17171  algSccascl 21773  Poly1cpl1 22083   Mat cmat 22294   matToPolyMat cmat2pmat 22593
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1st 7987  df-2nd 7988  df-mat2pmat 22596
This theorem is referenced by:  mat2pmatf1  22618  mat2pmat1  22621  mat2pmatlin  22624  m2cpm  22630  m2cpminvid  22642  monmatcollpw  22668  chpmat1dlem  22724  chpdmatlem2  22728  chpdmatlem3  22729
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