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Theorem mply1topmatval 22750
Description: A polynomial over matrices transformed into a polynomial matrix. 𝐼 is the inverse function of the transformation 𝑇 of polynomial matrices into polynomials over matrices: (𝑇‘(𝐼𝑂)) = 𝑂) (see mp2pm2mp 22757). (Contributed by AV, 6-Oct-2019.)
Hypotheses
Ref Expression
mply1topmat.a 𝐴 = (𝑁 Mat 𝑅)
mply1topmat.q 𝑄 = (Poly1𝐴)
mply1topmat.l 𝐿 = (Base‘𝑄)
mply1topmat.p 𝑃 = (Poly1𝑅)
mply1topmat.m · = ( ·𝑠𝑃)
mply1topmat.e 𝐸 = (.g‘(mulGrp‘𝑃))
mply1topmat.y 𝑌 = (var1𝑅)
mply1topmat.i 𝐼 = (𝑝𝐿 ↦ (𝑖𝑁, 𝑗𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌))))))
Assertion
Ref Expression
mply1topmatval ((𝑁𝑉𝑂𝐿) → (𝐼𝑂) = (𝑖𝑁, 𝑗𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1𝑂)‘𝑘)𝑗) · (𝑘𝐸𝑌))))))
Distinct variable groups:   𝑖,𝑁,𝑗,𝑝   𝐸,𝑝   𝐿,𝑝   𝑃,𝑝   𝑉,𝑝   𝑌,𝑝   𝑖,𝑂,𝑗,𝑘,𝑝   · ,𝑘,𝑝
Allowed substitution hints:   𝐴(𝑖,𝑗,𝑘,𝑝)   𝑃(𝑖,𝑗,𝑘)   𝑄(𝑖,𝑗,𝑘,𝑝)   𝑅(𝑖,𝑗,𝑘,𝑝)   · (𝑖,𝑗)   𝐸(𝑖,𝑗,𝑘)   𝐼(𝑖,𝑗,𝑘,𝑝)   𝐿(𝑖,𝑗,𝑘)   𝑁(𝑘)   𝑉(𝑖,𝑗,𝑘)   𝑌(𝑖,𝑗,𝑘)

Proof of Theorem mply1topmatval
StepHypRef Expression
1 mply1topmat.i . 2 𝐼 = (𝑝𝐿 ↦ (𝑖𝑁, 𝑗𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌))))))
2 fveq2 6896 . . . . . . . 8 (𝑝 = 𝑂 → (coe1𝑝) = (coe1𝑂))
32fveq1d 6898 . . . . . . 7 (𝑝 = 𝑂 → ((coe1𝑝)‘𝑘) = ((coe1𝑂)‘𝑘))
43oveqd 7436 . . . . . 6 (𝑝 = 𝑂 → (𝑖((coe1𝑝)‘𝑘)𝑗) = (𝑖((coe1𝑂)‘𝑘)𝑗))
54oveq1d 7434 . . . . 5 (𝑝 = 𝑂 → ((𝑖((coe1𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌)) = ((𝑖((coe1𝑂)‘𝑘)𝑗) · (𝑘𝐸𝑌)))
65mpteq2dv 5251 . . . 4 (𝑝 = 𝑂 → (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌))) = (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1𝑂)‘𝑘)𝑗) · (𝑘𝐸𝑌))))
76oveq2d 7435 . . 3 (𝑝 = 𝑂 → (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌)))) = (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1𝑂)‘𝑘)𝑗) · (𝑘𝐸𝑌)))))
87mpoeq3dv 7499 . 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 simpr 483 . 2 ((𝑁𝑉𝑂𝐿) → 𝑂𝐿)
10 simpl 481 . . 3 ((𝑁𝑉𝑂𝐿) → 𝑁𝑉)
11 mpoexga 8082 . . 3 ((𝑁𝑉𝑁𝑉) → (𝑖𝑁, 𝑗𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1𝑂)‘𝑘)𝑗) · (𝑘𝐸𝑌))))) ∈ V)
1210, 11syldan 589 . 2 ((𝑁𝑉𝑂𝐿) → (𝑖𝑁, 𝑗𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1𝑂)‘𝑘)𝑗) · (𝑘𝐸𝑌))))) ∈ V)
131, 8, 9, 12fvmptd3 7027 1 ((𝑁𝑉𝑂𝐿) → (𝐼𝑂) = (𝑖𝑁, 𝑗𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1𝑂)‘𝑘)𝑗) · (𝑘𝐸𝑌))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  Vcvv 3461  cmpt 5232  cfv 6549  (class class class)co 7419  cmpo 7421  0cn0 12505  Basecbs 17183   ·𝑠 cvsca 17240   Σg cgsu 17425  .gcmg 19031  mulGrpcmgp 20086  var1cv1 22118  Poly1cpl1 22119  coe1cco1 22120   Mat cmat 22351
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-1st 7994  df-2nd 7995
This theorem is referenced by:  mply1topmatcl  22751
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