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Theorem cpm2mvalel 22182
Description: A (matrix) element of the result of an inverse matrix transformation. (Contributed by AV, 14-Dec-2019.)
Hypotheses
Ref Expression
cpm2mfval.i 𝐼 = (𝑁 cPolyMatToMat 𝑅)
cpm2mfval.s 𝑆 = (𝑁 ConstPolyMat 𝑅)
Assertion
Ref Expression
cpm2mvalel (((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝑆) ∧ (𝑋𝑁𝑌𝑁)) → (𝑋(𝐼𝑀)𝑌) = ((coe1‘(𝑋𝑀𝑌))‘0))

Proof of Theorem cpm2mvalel
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cpm2mfval.i . . . 4 𝐼 = (𝑁 cPolyMatToMat 𝑅)
2 cpm2mfval.s . . . 4 𝑆 = (𝑁 ConstPolyMat 𝑅)
31, 2cpm2mval 22181 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝑆) → (𝐼𝑀) = (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑀𝑦))‘0)))
43adantr 481 . 2 (((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝑆) ∧ (𝑋𝑁𝑌𝑁)) → (𝐼𝑀) = (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑀𝑦))‘0)))
5 oveq12 7402 . . . . 5 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝑥𝑀𝑦) = (𝑋𝑀𝑌))
65fveq2d 6882 . . . 4 ((𝑥 = 𝑋𝑦 = 𝑌) → (coe1‘(𝑥𝑀𝑦)) = (coe1‘(𝑋𝑀𝑌)))
76fveq1d 6880 . . 3 ((𝑥 = 𝑋𝑦 = 𝑌) → ((coe1‘(𝑥𝑀𝑦))‘0) = ((coe1‘(𝑋𝑀𝑌))‘0))
87adantl 482 . 2 ((((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝑆) ∧ (𝑋𝑁𝑌𝑁)) ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → ((coe1‘(𝑥𝑀𝑦))‘0) = ((coe1‘(𝑋𝑀𝑌))‘0))
9 simprl 769 . 2 (((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝑆) ∧ (𝑋𝑁𝑌𝑁)) → 𝑋𝑁)
10 simprr 771 . 2 (((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝑆) ∧ (𝑋𝑁𝑌𝑁)) → 𝑌𝑁)
11 fvexd 6893 . 2 (((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝑆) ∧ (𝑋𝑁𝑌𝑁)) → ((coe1‘(𝑋𝑀𝑌))‘0) ∈ V)
124, 8, 9, 10, 11ovmpod 7543 1 (((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝑆) ∧ (𝑋𝑁𝑌𝑁)) → (𝑋(𝐼𝑀)𝑌) = ((coe1‘(𝑋𝑀𝑌))‘0))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087   = wceq 1541  wcel 2106  Vcvv 3473  cfv 6532  (class class class)co 7393  cmpo 7395  Fincfn 8922  0cc0 11092  coe1cco1 21631   ConstPolyMat ccpmat 22134   cPolyMatToMat ccpmat2mat 22136
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 2702  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708
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 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-f1 6537  df-fo 6538  df-f1o 6539  df-fv 6540  df-ov 7396  df-oprab 7397  df-mpo 7398  df-1st 7957  df-2nd 7958  df-cpmat2mat 22139
This theorem is referenced by: (None)
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