Theorem mply1topmatval 21953

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 21960). (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 6774	. . . . . . . 8 ⊢ (𝑝 = 𝑂 → (coe1‘𝑝) = (coe1‘𝑂))
3	2	fveq1d 6776	. . . . . . 7 ⊢ (𝑝 = 𝑂 → ((coe1‘𝑝)‘𝑘) = ((coe1‘𝑂)‘𝑘))
4	3	oveqd 7292	. . . . . 6 ⊢ (𝑝 = 𝑂 → (𝑖((coe1‘𝑝)‘𝑘)𝑗) = (𝑖((coe1‘𝑂)‘𝑘)𝑗))
5	4	oveq1d 7290	. . . . 5 ⊢ (𝑝 = 𝑂 → ((𝑖((coe1‘𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌)) = ((𝑖((coe1‘𝑂)‘𝑘)𝑗) · (𝑘𝐸𝑌)))
6	5	mpteq2dv 5176	. . . 4 ⊢ (𝑝 = 𝑂 → (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1‘𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌))) = (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1‘𝑂)‘𝑘)𝑗) · (𝑘𝐸𝑌))))
7	6	oveq2d 7291	. . 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 7354	. 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 7918	. . 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 6898	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 1539   ∈ wcel 2106  Vcvv 3432   ↦ cmpt 5157  ‘cfv 6433  (class class class)co 7275   ∈ cmpo 7277  ℕ₀cn0 12233  Basecbs 16912   ·_𝑠 cvsca 16966   Σ_g cgsu 17151  ._gcmg 18700  mulGrpcmgp 19720  var₁cv1 21347  Poly₁cpl1 21348  coe₁cco1 21349   Mat cmat 21554

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832

This theorem is referenced by:  mply1topmatcl  21954