Theorem cpmidpmatlem1 21472

Description: Lemma 1 for cpmidpmat 21475. (Contributed by AV, 13-Nov-2019.)

Hypotheses

Ref	Expression
cpmidgsum.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
cpmidgsum.b	⊢ 𝐵 = (Base‘𝐴)
cpmidgsum.p	⊢ 𝑃 = (Poly1‘𝑅)
cpmidgsum.y	⊢ 𝑌 = (𝑁 Mat 𝑃)
cpmidgsum.x	⊢ 𝑋 = (var1‘𝑅)
cpmidgsum.e	⊢ ↑ = (.g‘(mulGrp‘𝑃))
cpmidgsum.m	⊢ · = ( ·𝑠 ‘𝑌)
cpmidgsum.1	⊢ 1 = (1r‘𝑌)
cpmidgsum.u	⊢ 𝑈 = (algSc‘𝑃)
cpmidgsum.c	⊢ 𝐶 = (𝑁 CharPlyMat 𝑅)
cpmidgsum.k	⊢ 𝐾 = (𝐶‘𝑀)
cpmidgsum.h	⊢ 𝐻 = (𝐾 · 1 )
cpmidgsumm2pm.o	⊢ 𝑂 = (1r‘𝐴)
cpmidgsumm2pm.m	⊢ ∗ = ( ·𝑠 ‘𝐴)
cpmidgsumm2pm.t	⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)
cpmidpmat.g	⊢ 𝐺 = (𝑘 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑘) ∗ 𝑂))

Assertion

Ref	Expression
cpmidpmatlem1	⊢ (𝐿 ∈ ℕ0 → (𝐺‘𝐿) = (((coe1‘𝐾)‘𝐿) ∗ 𝑂))

Distinct variable groups:   𝐵,𝑘   𝑘,𝐻   𝑘,𝑁   𝑃,𝑘   𝑅,𝑘   𝑘,𝑌   𝑘,𝐾   𝑘,𝐿   𝑘,𝑂   ∗ ,𝑘

Allowed substitution hints:   𝐴(𝑘)   𝐶(𝑘)   𝑇(𝑘)   · (𝑘)   𝑈(𝑘)   1 (𝑘)   ↑ (𝑘)   𝐺(𝑘)   𝑀(𝑘)   𝑋(𝑘)

Proof of Theorem cpmidpmatlem1

Step	Hyp	Ref	Expression
1		fveq2 6664	. . 3 ⊢ (𝑘 = 𝐿 → ((coe1‘𝐾)‘𝑘) = ((coe1‘𝐾)‘𝐿))
2	1	oveq1d 7165	. 2 ⊢ (𝑘 = 𝐿 → (((coe1‘𝐾)‘𝑘) ∗ 𝑂) = (((coe1‘𝐾)‘𝐿) ∗ 𝑂))
3		cpmidpmat.g	. 2 ⊢ 𝐺 = (𝑘 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑘) ∗ 𝑂))
4		ovex 7183	. 2 ⊢ (((coe1‘𝐾)‘𝐿) ∗ 𝑂) ∈ V
5	2, 3, 4	fvmpt 6762	1 ⊢ (𝐿 ∈ ℕ0 → (𝐺‘𝐿) = (((coe1‘𝐾)‘𝐿) ∗ 𝑂))

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2110   ↦ cmpt 5138  ‘cfv 6349  (class class class)co 7150  ℕ₀cn0 11891  Basecbs 16477   ·_𝑠 cvsca 16563  ._gcmg 18218  mulGrpcmgp 19233  1_rcur 19245  algSccascl 20078  var₁cv1 20338  Poly₁cpl1 20339  coe₁cco1 20340   Mat cmat 21010   matToPolyMat cmat2pmat 21306   CharPlyMat cchpmat 21428

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-iota 6308  df-fun 6351  df-fv 6357  df-ov 7153

This theorem is referenced by:  cpmidpmat  21475