Theorem mply1topmatval 23122

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 23129). (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 6885	. . . . . . . 8 ⊢ (𝑝 = 𝑂 → (coe1‘𝑝) = (coe1‘𝑂))
3	2	fveq1d 6887	. . . . . . 7 ⊢ (𝑝 = 𝑂 → ((coe1‘𝑝)‘𝑘) = ((coe1‘𝑂)‘𝑘))
4	3	oveqd 7437	. . . . . 6 ⊢ (𝑝 = 𝑂 → (𝑖((coe1‘𝑝)‘𝑘)𝑗) = (𝑖((coe1‘𝑂)‘𝑘)𝑗))
5	4	oveq1d 7435	. . . . 5 ⊢ (𝑝 = 𝑂 → ((𝑖((coe1‘𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌)) = ((𝑖((coe1‘𝑂)‘𝑘)𝑗) · (𝑘𝐸𝑌)))
6	5	mpteq2dv 5198	. . . 4 ⊢ (𝑝 = 𝑂 → (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1‘𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌))) = (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1‘𝑂)‘𝑘)𝑗) · (𝑘𝐸𝑌))))
7	6	oveq2d 7436	. . 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 7499	. 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 490	. 2 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑂 ∈ 𝐿) → 𝑂 ∈ 𝐿)
10		simpl 488	. . 3 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑂 ∈ 𝐿) → 𝑁 ∈ 𝑉)
11		mpoexga 8090	. . 3 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1‘𝑂)‘𝑘)𝑗) · (𝑘𝐸𝑌))))) ∈ V)
12	10, 11	syldan 603	. 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 7017	1 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑂 ∈ 𝐿) → (𝐼‘𝑂) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1‘𝑂)‘𝑘)𝑗) · (𝑘𝐸𝑌))))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  Vcvv 3450   ↦ cmpt 5185  ‘cfv 6538  (class class class)co 7420   ∈ cmpo 7422  ℕ₀cn0 12606  Basecbs 17387   ·_𝑠 cvsca 17432   Σ_g cgsu 17611  ._gcmg 19277  mulGrpcmgp 20360  var₁cv1 22494  Poly₁cpl1 22495  coe₁cco1 22496   Mat cmat 22722

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002

This theorem is used by:  mply1topmatcl  23123