Theorem mply1topmatval 22297

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‘(𝐼‘𝑂)) = 𝑂) (see mp2pm2mp 22304). (Contributed by AV, 6-Oct-2019.)

Hypotheses

Ref	Expression
mply1topmat.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
mply1topmat.q	⊢ 𝑄 = (Poly1‘𝐴)
mply1topmat.l	⊢ 𝐿 = (Base‘𝑄)
mply1topmat.p	⊢ 𝑃 = (Poly1‘𝑅)
mply1topmat.m	⊢ · = ( ·𝑠 ‘𝑃)
mply1topmat.e	⊢ 𝐸 = (.g‘(mulGrp‘𝑃))
mply1topmat.y	⊢ 𝑌 = (var1‘𝑅)
mply1topmat.i	⊢ 𝐼 = (𝑝 ∈ 𝐿 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1‘𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌))))))

Assertion

Ref	Expression
mply1topmatval	⊢ ((𝑁 ∈ 𝑉 ∧ 𝑂 ∈ 𝐿) → (𝐼‘𝑂) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1‘𝑂)‘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

Step	Hyp	Ref	Expression
1		mply1topmat.i	. 2 ⊢ 𝐼 = (𝑝 ∈ 𝐿 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1‘𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌))))))
2		fveq2 6888	. . . . . . . 8 ⊢ (𝑝 = 𝑂 → (coe1‘𝑝) = (coe1‘𝑂))
3	2	fveq1d 6890	. . . . . . 7 ⊢ (𝑝 = 𝑂 → ((coe1‘𝑝)‘𝑘) = ((coe1‘𝑂)‘𝑘))
4	3	oveqd 7422	. . . . . 6 ⊢ (𝑝 = 𝑂 → (𝑖((coe1‘𝑝)‘𝑘)𝑗) = (𝑖((coe1‘𝑂)‘𝑘)𝑗))
5	4	oveq1d 7420	. . . . 5 ⊢ (𝑝 = 𝑂 → ((𝑖((coe1‘𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌)) = ((𝑖((coe1‘𝑂)‘𝑘)𝑗) · (𝑘𝐸𝑌)))
6	5	mpteq2dv 5249	. . . 4 ⊢ (𝑝 = 𝑂 → (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1‘𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌))) = (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1‘𝑂)‘𝑘)𝑗) · (𝑘𝐸𝑌))))
7	6	oveq2d 7421	. . 3 ⊢ (𝑝 = 𝑂 → (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1‘𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌)))) = (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1‘𝑂)‘𝑘)𝑗) · (𝑘𝐸𝑌)))))
8	7	mpoeq3dv 7484	. 2 ⊢ (𝑝 = 𝑂 → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1‘𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌))))) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1‘𝑂)‘𝑘)𝑗) · (𝑘𝐸𝑌))))))
9		simpr 485	. 2 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑂 ∈ 𝐿) → 𝑂 ∈ 𝐿)
10		simpl 483	. . 3 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑂 ∈ 𝐿) → 𝑁 ∈ 𝑉)
11		mpoexga 8060	. . 3 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1‘𝑂)‘𝑘)𝑗) · (𝑘𝐸𝑌))))) ∈ V)
12	10, 11	syldan 591	. 2 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑂 ∈ 𝐿) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1‘𝑂)‘𝑘)𝑗) · (𝑘𝐸𝑌))))) ∈ V)
13	1, 8, 9, 12	fvmptd3 7018	1 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑂 ∈ 𝐿) → (𝐼‘𝑂) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1‘𝑂)‘𝑘)𝑗) · (𝑘𝐸𝑌))))))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  Vcvv 3474   ↦ cmpt 5230  ‘cfv 6540  (class class class)co 7405   ∈ cmpo 7407  ℕ₀cn0 12468  Basecbs 17140   ·_𝑠 cvsca 17197   Σ_g cgsu 17382  ._gcmg 18944  mulGrpcmgp 19981  var₁cv1 21691  Poly₁cpl1 21692  coe₁cco1 21693   Mat cmat 21898

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 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721

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 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7971  df-2nd 7972

This theorem is referenced by:  mply1topmatcl  22298