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Theorem pm2mpf1lem 20599

Description: Lemma for pm2mpf1 20604. (Contributed by AV, 14-Oct-2019.) (Revised by AV, 4-Dec-2019.)

**Hypotheses**
Ref	Expression
pm2mpf1lem.p	⊢ 𝑃 = (Poly₁‘𝑅)
pm2mpf1lem.c	⊢ 𝐶 = (𝑁 Mat 𝑃)
pm2mpf1lem.b	⊢ 𝐵 = (Base‘𝐶)
pm2mpf1lem.m	⊢ ∗ = ( ·_𝑠 ‘𝑄)
pm2mpf1lem.e	⊢ ↑ = (._g‘(mulGrp‘𝑄))
pm2mpf1lem.x	⊢ 𝑋 = (var₁‘𝐴)
pm2mpf1lem.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
pm2mpf1lem.q	⊢ 𝑄 = (Poly₁‘𝐴)

**Assertion**
Ref	Expression
pm2mpf1lem	⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑈 ∈ 𝐵 ∧ 𝐾 ∈ ℕ₀)) → ((coe₁‘(𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑈 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋)))))‘𝐾) = (𝑈 decompPMat 𝐾))

Distinct variable groups: 𝐴,𝑘 𝐵,𝑘 𝑘,𝐾 𝑘,𝑁 𝑄,𝑘 𝑅,𝑘 𝑈,𝑘 ∗ ,𝑘 ↑ ,𝑘 Allowed substitution hints: 𝐶(𝑘) 𝑃(𝑘) 𝑋(𝑘)

**Proof of Theorem pm2mpf1lem**
Step	Hyp	Ref	Expression
1		pm2mpf1lem.q	. . 3 ⊢ 𝑄 = (Poly₁‘𝐴)
2		eqid 2622	. . 3 ⊢ (Base‘𝑄) = (Base‘𝑄)
3		pm2mpf1lem.x	. . 3 ⊢ 𝑋 = (var₁‘𝐴)
4		pm2mpf1lem.e	. . 3 ⊢ ↑ = (._g‘(mulGrp‘𝑄))
5		pm2mpf1lem.a	. . . . 5 ⊢ 𝐴 = (𝑁 Mat 𝑅)
6	5	matring 20249	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
7	6	adantr 481	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑈 ∈ 𝐵 ∧ 𝐾 ∈ ℕ₀)) → 𝐴 ∈ Ring)
8		eqid 2622	. . 3 ⊢ (Base‘𝐴) = (Base‘𝐴)
9		pm2mpf1lem.m	. . 3 ⊢ ∗ = ( ·_𝑠 ‘𝑄)
10		eqid 2622	. . 3 ⊢ (0_g‘𝐴) = (0_g‘𝐴)
11		simpllr 799	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑈 ∈ 𝐵 ∧ 𝐾 ∈ ℕ₀)) ∧ 𝑘 ∈ ℕ₀) → 𝑅 ∈ Ring)
12		simplrl 800	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑈 ∈ 𝐵 ∧ 𝐾 ∈ ℕ₀)) ∧ 𝑘 ∈ ℕ₀) → 𝑈 ∈ 𝐵)
13		simpr 477	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑈 ∈ 𝐵 ∧ 𝐾 ∈ ℕ₀)) ∧ 𝑘 ∈ ℕ₀) → 𝑘 ∈ ℕ₀)
14		pm2mpf1lem.p	. . . . . 6 ⊢ 𝑃 = (Poly₁‘𝑅)
15		pm2mpf1lem.c	. . . . . 6 ⊢ 𝐶 = (𝑁 Mat 𝑃)
16		pm2mpf1lem.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐶)
17	14, 15, 16, 5, 8	decpmatcl 20572	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐵 ∧ 𝑘 ∈ ℕ₀) → (𝑈 decompPMat 𝑘) ∈ (Base‘𝐴))
18	11, 12, 13, 17	syl3anc 1326	. . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑈 ∈ 𝐵 ∧ 𝐾 ∈ ℕ₀)) ∧ 𝑘 ∈ ℕ₀) → (𝑈 decompPMat 𝑘) ∈ (Base‘𝐴))
19	18	ralrimiva 2966	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑈 ∈ 𝐵 ∧ 𝐾 ∈ ℕ₀)) → ∀𝑘 ∈ ℕ₀ (𝑈 decompPMat 𝑘) ∈ (Base‘𝐴))
20	14, 15, 16, 5, 10	decpmatfsupp 20574	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐵) → (𝑘 ∈ ℕ₀ ↦ (𝑈 decompPMat 𝑘)) finSupp (0_g‘𝐴))
21	20	ad2ant2lr 784	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑈 ∈ 𝐵 ∧ 𝐾 ∈ ℕ₀)) → (𝑘 ∈ ℕ₀ ↦ (𝑈 decompPMat 𝑘)) finSupp (0_g‘𝐴))
22		simprr 796	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑈 ∈ 𝐵 ∧ 𝐾 ∈ ℕ₀)) → 𝐾 ∈ ℕ₀)
23	1, 2, 3, 4, 7, 8, 9, 10, 19, 21, 22	gsummoncoe1 19674	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑈 ∈ 𝐵 ∧ 𝐾 ∈ ℕ₀)) → ((coe₁‘(𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑈 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋)))))‘𝐾) = ⦋𝐾 / 𝑘⦌(𝑈 decompPMat 𝑘))
24		csbov2g 6691	. . 3 ⊢ (𝐾 ∈ ℕ₀ → ⦋𝐾 / 𝑘⦌(𝑈 decompPMat 𝑘) = (𝑈 decompPMat ⦋𝐾 / 𝑘⦌𝑘))
25	24	ad2antll 765	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑈 ∈ 𝐵 ∧ 𝐾 ∈ ℕ₀)) → ⦋𝐾 / 𝑘⦌(𝑈 decompPMat 𝑘) = (𝑈 decompPMat ⦋𝐾 / 𝑘⦌𝑘))
26		csbvarg 4003	. . . 4 ⊢ (𝐾 ∈ ℕ₀ → ⦋𝐾 / 𝑘⦌𝑘 = 𝐾)
27	26	ad2antll 765	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑈 ∈ 𝐵 ∧ 𝐾 ∈ ℕ₀)) → ⦋𝐾 / 𝑘⦌𝑘 = 𝐾)
28	27	oveq2d 6666	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑈 ∈ 𝐵 ∧ 𝐾 ∈ ℕ₀)) → (𝑈 decompPMat ⦋𝐾 / 𝑘⦌𝑘) = (𝑈 decompPMat 𝐾))
29	23, 25, 28	3eqtrd 2660	1 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑈 ∈ 𝐵 ∧ 𝐾 ∈ ℕ₀)) → ((coe₁‘(𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑈 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋)))))‘𝐾) = (𝑈 decompPMat 𝐾))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ⦋csb 3533 class class class wbr 4653 ↦ cmpt 4729 ‘cfv 5888 (class class class)co 6650 Fincfn 7955 finSupp cfsupp 8275 ℕ₀cn0 11292 Basecbs 15857 ·_𝑠 cvsca 15945 0_gc0g 16100 Σ_g cgsu 16101 ._gcmg 17540 mulGrpcmgp 18489 Ringcrg 18547 var₁cv1 19546 Poly₁cpl1 19547 coe₁cco1 19548 Mat cmat 20213 decompPMat cdecpmat 20567

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-ot 4186 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-ofr 6898 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-sup 8348 df-oi 8415 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-fz 12327 df-fzo 12466 df-seq 12802 df-hash 13118 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-hom 15966 df-cco 15967 df-0g 16102 df-gsum 16103 df-prds 16108 df-pws 16110 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-mhm 17335 df-submnd 17336 df-grp 17425 df-minusg 17426 df-sbg 17427 df-mulg 17541 df-subg 17591 df-ghm 17658 df-cntz 17750 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-ring 18549 df-subrg 18778 df-lmod 18865 df-lss 18933 df-sra 19172 df-rgmod 19173 df-psr 19356 df-mvr 19357 df-mpl 19358 df-opsr 19360 df-psr1 19550 df-vr1 19551 df-ply1 19552 df-coe1 19553 df-dsmm 20076 df-frlm 20091 df-mamu 20190 df-mat 20214 df-decpmat 20568

This theorem is referenced by: pm2mpf1 20604

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