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Theorem cpm2mvalel 22252
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 22251 . . 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 7417 . . . . 5 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝑥𝑀𝑦) = (𝑋𝑀𝑌))
65fveq2d 6895 . . . 4 ((𝑥 = 𝑋𝑦 = 𝑌) → (coe1‘(𝑥𝑀𝑦)) = (coe1‘(𝑋𝑀𝑌)))
76fveq1d 6893 . . 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 6906 . 2 (((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝑆) ∧ (𝑋𝑁𝑌𝑁)) → ((coe1‘(𝑋𝑀𝑌))‘0) ∈ V)
124, 8, 9, 10, 11ovmpod 7559 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 3474  cfv 6543  (class class class)co 7408  cmpo 7410  Fincfn 8938  0cc0 11109  coe1cco1 21701   ConstPolyMat ccpmat 22204   cPolyMatToMat ccpmat2mat 22206
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 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-1st 7974  df-2nd 7975  df-cpmat2mat 22209
This theorem is referenced by: (None)
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