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Theorem cpmidpmatlem2 22784

Description: Lemma 2 for cpmidpmat 22786. (Contributed by AV, 14-Nov-2019.) (Proof shortened by AV, 7-Dec-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 = (1_r‘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 = (1_r‘𝐴)
cpmidgsumm2pm.m	⊢ ∗ = ( ·_𝑠 ‘𝐴)
cpmidgsumm2pm.t	⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)
cpmidpmat.g	⊢ 𝐺 = (𝑘 ∈ ℕ₀ ↦ (((coe₁‘𝐾)‘𝑘) ∗ 𝑂))

**Assertion**
Ref	Expression
cpmidpmatlem2	⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝐺 ∈ (𝐵 ↑_m ℕ₀))

Distinct variable groups: 𝐵,𝑘 𝑘,𝐻 𝑘,𝑁 𝑃,𝑘 𝑅,𝑘 𝑘,𝑌 𝑘,𝐾 𝑘,𝑂 ∗ ,𝑘 𝑘,𝑀 Allowed substitution hints: 𝐴(𝑘) 𝐶(𝑘) 𝑇(𝑘) · (𝑘) 𝑈(𝑘) 1 (𝑘) ↑ (𝑘) 𝐺(𝑘) 𝑋(𝑘)

**Proof of Theorem cpmidpmatlem2**
Step	Hyp	Ref	Expression
1		simpl1 1192	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑘 ∈ ℕ₀) → 𝑁 ∈ Fin)
2		crngring 20161	. . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
3	2	3ad2ant2 1134	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑅 ∈ Ring)
4	3	adantr 480	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑘 ∈ ℕ₀) → 𝑅 ∈ Ring)
5		cpmidgsum.k	. . . . . 6 ⊢ 𝐾 = (𝐶‘𝑀)
6		cpmidgsum.c	. . . . . . 7 ⊢ 𝐶 = (𝑁 CharPlyMat 𝑅)
7		cpmidgsum.a	. . . . . . 7 ⊢ 𝐴 = (𝑁 Mat 𝑅)
8		cpmidgsum.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐴)
9		cpmidgsum.p	. . . . . . 7 ⊢ 𝑃 = (Poly₁‘𝑅)
10		eqid 2731	. . . . . . 7 ⊢ (Base‘𝑃) = (Base‘𝑃)
11	6, 7, 8, 9, 10	chpmatply1 22745	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐶‘𝑀) ∈ (Base‘𝑃))
12	5, 11	eqeltrid 2835	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝐾 ∈ (Base‘𝑃))
13		eqid 2731	. . . . . 6 ⊢ (coe₁‘𝐾) = (coe₁‘𝐾)
14		eqid 2731	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
15	13, 10, 9, 14	coe1fvalcl 22123	. . . . 5 ⊢ ((𝐾 ∈ (Base‘𝑃) ∧ 𝑘 ∈ ℕ₀) → ((coe₁‘𝐾)‘𝑘) ∈ (Base‘𝑅))
16	12, 15	sylan 580	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑘 ∈ ℕ₀) → ((coe₁‘𝐾)‘𝑘) ∈ (Base‘𝑅))
17	2	anim2i 617	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
18	7	matring 22356	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
19		cpmidgsumm2pm.o	. . . . . . . 8 ⊢ 𝑂 = (1_r‘𝐴)
20	8, 19	ringidcl 20181	. . . . . . 7 ⊢ (𝐴 ∈ Ring → 𝑂 ∈ 𝐵)
21	17, 18, 20	3syl 18	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑂 ∈ 𝐵)
22	21	3adant3 1132	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑂 ∈ 𝐵)
23	22	adantr 480	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑘 ∈ ℕ₀) → 𝑂 ∈ 𝐵)
24		cpmidgsumm2pm.m	. . . . 5 ⊢ ∗ = ( ·_𝑠 ‘𝐴)
25	14, 7, 8, 24	matvscl 22344	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (((coe₁‘𝐾)‘𝑘) ∈ (Base‘𝑅) ∧ 𝑂 ∈ 𝐵)) → (((coe₁‘𝐾)‘𝑘) ∗ 𝑂) ∈ 𝐵)
26	1, 4, 16, 23, 25	syl22anc 838	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑘 ∈ ℕ₀) → (((coe₁‘𝐾)‘𝑘) ∗ 𝑂) ∈ 𝐵)
27		cpmidpmat.g	. . 3 ⊢ 𝐺 = (𝑘 ∈ ℕ₀ ↦ (((coe₁‘𝐾)‘𝑘) ∗ 𝑂))
28	26, 27	fmptd 7047	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝐺:ℕ₀⟶𝐵)
29	8	fvexi 6836	. . . 4 ⊢ 𝐵 ∈ V
30		nn0ex 12384	. . . 4 ⊢ ℕ₀ ∈ V
31	29, 30	pm3.2i 470	. . 3 ⊢ (𝐵 ∈ V ∧ ℕ₀ ∈ V)
32		elmapg 8763	. . 3 ⊢ ((𝐵 ∈ V ∧ ℕ₀ ∈ V) → (𝐺 ∈ (𝐵 ↑_m ℕ₀) ↔ 𝐺:ℕ₀⟶𝐵))
33	31, 32	mp1i 13	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐺 ∈ (𝐵 ↑_m ℕ₀) ↔ 𝐺:ℕ₀⟶𝐵))
34	28, 33	mpbird 257	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝐺 ∈ (𝐵 ↑_m ℕ₀))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2111 Vcvv 3436 ↦ cmpt 5172 ⟶wf 6477 ‘cfv 6481 (class class class)co 7346 ↑_m cmap 8750 Fincfn 8869 ℕ₀cn0 12378 Basecbs 17117 ·_𝑠 cvsca 17162 ._gcmg 18977 mulGrpcmgp 20056 1_rcur 20097 Ringcrg 20149 CRingccrg 20150 algSccascl 21787 var₁cv1 22086 Poly₁cpl1 22087 coe₁cco1 22088 Mat cmat 22320 matToPolyMat cmat2pmat 22617 CharPlyMat cchpmat 22739

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 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5217 ax-sep 5234 ax-nul 5244 ax-pow 5303 ax-pr 5370 ax-un 7668 ax-cnex 11059 ax-resscn 11060 ax-1cn 11061 ax-icn 11062 ax-addcl 11063 ax-addrcl 11064 ax-mulcl 11065 ax-mulrcl 11066 ax-mulcom 11067 ax-addass 11068 ax-mulass 11069 ax-distr 11070 ax-i2m1 11071 ax-1ne0 11072 ax-1rid 11073 ax-rnegex 11074 ax-rrecex 11075 ax-cnre 11076 ax-pre-lttri 11077 ax-pre-lttrn 11078 ax-pre-ltadd 11079 ax-pre-mulgt0 11080 ax-addf 11082 ax-mulf 11083

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-xor 1513 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4476 df-pw 4552 df-sn 4577 df-pr 4579 df-tp 4581 df-op 4583 df-ot 4585 df-uni 4860 df-int 4898 df-iun 4943 df-iin 4944 df-br 5092 df-opab 5154 df-mpt 5173 df-tr 5199 df-id 5511 df-eprel 5516 df-po 5524 df-so 5525 df-fr 5569 df-se 5570 df-we 5571 df-xp 5622 df-rel 5623 df-cnv 5624 df-co 5625 df-dm 5626 df-rn 5627 df-res 5628 df-ima 5629 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-isom 6490 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-of 7610 df-ofr 7611 df-om 7797 df-1st 7921 df-2nd 7922 df-supp 8091 df-tpos 8156 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-2o 8386 df-er 8622 df-map 8752 df-pm 8753 df-ixp 8822 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-fsupp 9246 df-sup 9326 df-oi 9396 df-card 9829 df-pnf 11145 df-mnf 11146 df-xr 11147 df-ltxr 11148 df-le 11149 df-sub 11343 df-neg 11344 df-div 11772 df-nn 12123 df-2 12185 df-3 12186 df-4 12187 df-5 12188 df-6 12189 df-7 12190 df-8 12191 df-9 12192 df-n0 12379 df-xnn0 12452 df-z 12466 df-dec 12586 df-uz 12730 df-rp 12888 df-fz 13405 df-fzo 13552 df-seq 13906 df-exp 13966 df-hash 14235 df-word 14418 df-lsw 14467 df-concat 14475 df-s1 14501 df-substr 14546 df-pfx 14576 df-splice 14654 df-reverse 14663 df-s2 14752 df-struct 17055 df-sets 17072 df-slot 17090 df-ndx 17102 df-base 17118 df-ress 17139 df-plusg 17171 df-mulr 17172 df-starv 17173 df-sca 17174 df-vsca 17175 df-ip 17176 df-tset 17177 df-ple 17178 df-ds 17180 df-unif 17181 df-hom 17182 df-cco 17183 df-0g 17342 df-gsum 17343 df-prds 17348 df-pws 17350 df-mre 17485 df-mrc 17486 df-acs 17488 df-mgm 18545 df-sgrp 18624 df-mnd 18640 df-mhm 18688 df-submnd 18689 df-efmnd 18774 df-grp 18846 df-minusg 18847 df-sbg 18848 df-mulg 18978 df-subg 19033 df-ghm 19123 df-gim 19169 df-cntz 19227 df-oppg 19256 df-symg 19280 df-pmtr 19352 df-psgn 19401 df-cmn 19692 df-abl 19693 df-mgp 20057 df-rng 20069 df-ur 20098 df-ring 20151 df-cring 20152 df-oppr 20253 df-dvdsr 20273 df-unit 20274 df-invr 20304 df-dvr 20317 df-rhm 20388 df-subrng 20459 df-subrg 20483 df-drng 20644 df-lmod 20793 df-lss 20863 df-sra 21105 df-rgmod 21106 df-cnfld 21290 df-zring 21382 df-zrh 21438 df-dsmm 21667 df-frlm 21682 df-ascl 21790 df-psr 21844 df-mvr 21845 df-mpl 21846 df-opsr 21848 df-psr1 22090 df-vr1 22091 df-ply1 22092 df-coe1 22093 df-mamu 22304 df-mat 22321 df-mdet 22498 df-mat2pmat 22620 df-chpmat 22740

This theorem is referenced by: cpmidpmat 22786

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