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Theorem mp2pm2mplem1 21414
Description: Lemma 1 for mp2pm2mp 21419. (Contributed by AV, 9-Oct-2019.) (Revised by AV, 5-Dec-2019.)
Hypotheses
Ref Expression
mp2pm2mp.a 𝐴 = (𝑁 Mat 𝑅)
mp2pm2mp.q 𝑄 = (Poly1𝐴)
mp2pm2mp.l 𝐿 = (Base‘𝑄)
mp2pm2mp.m · = ( ·𝑠𝑃)
mp2pm2mp.e 𝐸 = (.g‘(mulGrp‘𝑃))
mp2pm2mp.y 𝑌 = (var1𝑅)
mp2pm2mp.i 𝐼 = (𝑝𝐿 ↦ (𝑖𝑁, 𝑗𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌))))))
Assertion
Ref Expression
mp2pm2mplem1 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) → (𝐼𝑂) = (𝑖𝑁, 𝑗𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1𝑂)‘𝑘)𝑗) · (𝑘𝐸𝑌))))))
Distinct variable groups:   𝐸,𝑝   𝐿,𝑝   𝑖,𝑁,𝑗,𝑝   𝑖,𝑂,𝑗,𝑝   𝑘,𝑂,𝑝   𝑃,𝑝   𝑅,𝑝   𝑌,𝑝   · ,𝑝
Allowed substitution hints:   𝐴(𝑖,𝑗,𝑘,𝑝)   𝑃(𝑖,𝑗,𝑘)   𝑄(𝑖,𝑗,𝑘,𝑝)   𝑅(𝑖,𝑗,𝑘)   · (𝑖,𝑗,𝑘)   𝐸(𝑖,𝑗,𝑘)   𝐼(𝑖,𝑗,𝑘,𝑝)   𝐿(𝑖,𝑗,𝑘)   𝑁(𝑘)   𝑌(𝑖,𝑗,𝑘)

Proof of Theorem mp2pm2mplem1
StepHypRef Expression
1 mp2pm2mp.i . 2 𝐼 = (𝑝𝐿 ↦ (𝑖𝑁, 𝑗𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌))))))
2 fveq2 6661 . . . . . . . 8 (𝑝 = 𝑂 → (coe1𝑝) = (coe1𝑂))
32fveq1d 6663 . . . . . . 7 (𝑝 = 𝑂 → ((coe1𝑝)‘𝑘) = ((coe1𝑂)‘𝑘))
43oveqd 7166 . . . . . 6 (𝑝 = 𝑂 → (𝑖((coe1𝑝)‘𝑘)𝑗) = (𝑖((coe1𝑂)‘𝑘)𝑗))
54oveq1d 7164 . . . . 5 (𝑝 = 𝑂 → ((𝑖((coe1𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌)) = ((𝑖((coe1𝑂)‘𝑘)𝑗) · (𝑘𝐸𝑌)))
65mpteq2dv 5148 . . . 4 (𝑝 = 𝑂 → (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌))) = (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1𝑂)‘𝑘)𝑗) · (𝑘𝐸𝑌))))
76oveq2d 7165 . . 3 (𝑝 = 𝑂 → (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌)))) = (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1𝑂)‘𝑘)𝑗) · (𝑘𝐸𝑌)))))
87mpoeq3dv 7226 . 2 (𝑝 = 𝑂 → (𝑖𝑁, 𝑗𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌))))) = (𝑖𝑁, 𝑗𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1𝑂)‘𝑘)𝑗) · (𝑘𝐸𝑌))))))
9 simp3 1135 . 2 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) → 𝑂𝐿)
10 simp1 1133 . . 3 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) → 𝑁 ∈ Fin)
11 mpoexga 7771 . . 3 ((𝑁 ∈ Fin ∧ 𝑁 ∈ Fin) → (𝑖𝑁, 𝑗𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1𝑂)‘𝑘)𝑗) · (𝑘𝐸𝑌))))) ∈ V)
1210, 10, 11syl2anc 587 . 2 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) → (𝑖𝑁, 𝑗𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1𝑂)‘𝑘)𝑗) · (𝑘𝐸𝑌))))) ∈ V)
131, 8, 9, 12fvmptd3 6782 1 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) → (𝐼𝑂) = (𝑖𝑁, 𝑗𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1𝑂)‘𝑘)𝑗) · (𝑘𝐸𝑌))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084   = wceq 1538  wcel 2115  Vcvv 3480  cmpt 5132  cfv 6343  (class class class)co 7149  cmpo 7151  Fincfn 8505  0cn0 11894  Basecbs 16483   ·𝑠 cvsca 16569   Σg cgsu 16714  .gcmg 18224  mulGrpcmgp 19239  Ringcrg 19297  var1cv1 20344  Poly1cpl1 20345  coe1cco1 20346   Mat cmat 21016
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-ov 7152  df-oprab 7153  df-mpo 7154  df-1st 7684  df-2nd 7685
This theorem is referenced by:  mp2pm2mplem3  21416
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