Theorem mply1topmatval 21412

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 21419). (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 6670	. . . . . . . 8 ⊢ (𝑝 = 𝑂 → (coe1‘𝑝) = (coe1‘𝑂))
3	2	fveq1d 6672	. . . . . . 7 ⊢ (𝑝 = 𝑂 → ((coe1‘𝑝)‘𝑘) = ((coe1‘𝑂)‘𝑘))
4	3	oveqd 7173	. . . . . 6 ⊢ (𝑝 = 𝑂 → (𝑖((coe1‘𝑝)‘𝑘)𝑗) = (𝑖((coe1‘𝑂)‘𝑘)𝑗))
5	4	oveq1d 7171	. . . . 5 ⊢ (𝑝 = 𝑂 → ((𝑖((coe1‘𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌)) = ((𝑖((coe1‘𝑂)‘𝑘)𝑗) · (𝑘𝐸𝑌)))
6	5	mpteq2dv 5162	. . . 4 ⊢ (𝑝 = 𝑂 → (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1‘𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌))) = (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1‘𝑂)‘𝑘)𝑗) · (𝑘𝐸𝑌))))
7	6	oveq2d 7172	. . 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 7233	. 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 487	. 2 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑂 ∈ 𝐿) → 𝑂 ∈ 𝐿)
10		simpl 485	. . 3 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑂 ∈ 𝐿) → 𝑁 ∈ 𝑉)
11		mpoexga 7775	. . 3 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1‘𝑂)‘𝑘)𝑗) · (𝑘𝐸𝑌))))) ∈ V)
12	10, 11	syldan 593	. 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 6791	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 398   = wceq 1537   ∈ wcel 2114  Vcvv 3494   ↦ cmpt 5146  ‘cfv 6355  (class class class)co 7156   ∈ cmpo 7158  ℕ₀cn0 11898  Basecbs 16483   ·_𝑠 cvsca 16569   Σ_g cgsu 16714  ._gcmg 18224  mulGrpcmgp 19239  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 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-ov 7159  df-oprab 7160  df-mpo 7161  df-1st 7689  df-2nd 7690

This theorem is referenced by:  mply1topmatcl  21413