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Theorem cpm2mvalel 20758
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 20757 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝑆) → (𝐼𝑀) = (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑀𝑦))‘0)))
43adantr 472 . 2 (((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝑆) ∧ (𝑋𝑁𝑌𝑁)) → (𝐼𝑀) = (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑀𝑦))‘0)))
5 oveq12 6822 . . . . 5 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝑥𝑀𝑦) = (𝑋𝑀𝑌))
65fveq2d 6356 . . . 4 ((𝑥 = 𝑋𝑦 = 𝑌) → (coe1‘(𝑥𝑀𝑦)) = (coe1‘(𝑋𝑀𝑌)))
76fveq1d 6354 . . 3 ((𝑥 = 𝑋𝑦 = 𝑌) → ((coe1‘(𝑥𝑀𝑦))‘0) = ((coe1‘(𝑋𝑀𝑌))‘0))
87adantl 473 . 2 ((((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝑆) ∧ (𝑋𝑁𝑌𝑁)) ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → ((coe1‘(𝑥𝑀𝑦))‘0) = ((coe1‘(𝑋𝑀𝑌))‘0))
9 simprl 811 . 2 (((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝑆) ∧ (𝑋𝑁𝑌𝑁)) → 𝑋𝑁)
10 simprr 813 . 2 (((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝑆) ∧ (𝑋𝑁𝑌𝑁)) → 𝑌𝑁)
11 fvexd 6364 . 2 (((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝑆) ∧ (𝑋𝑁𝑌𝑁)) → ((coe1‘(𝑋𝑀𝑌))‘0) ∈ V)
124, 8, 9, 10, 11ovmpt2d 6953 1 (((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝑆) ∧ (𝑋𝑁𝑌𝑁)) → (𝑋(𝐼𝑀)𝑌) = ((coe1‘(𝑋𝑀𝑌))‘0))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1072   = wceq 1632  wcel 2139  Vcvv 3340  cfv 6049  (class class class)co 6813  cmpt2 6815  Fincfn 8121  0cc0 10128  coe1cco1 19750   ConstPolyMat ccpmat 20710   cPolyMatToMat ccpmat2mat 20712
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-1st 7333  df-2nd 7334  df-cpmat2mat 20715
This theorem is referenced by: (None)
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