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Theorem cpmidpmatlem2 21928

Description: Lemma 2 for cpmidpmat 21930. (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 1189	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑘 ∈ ℕ₀) → 𝑁 ∈ Fin)
2		crngring 19710	. . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
3	2	3ad2ant2 1132	. . . . 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 2738	. . . . . . 7 ⊢ (Base‘𝑃) = (Base‘𝑃)
11	6, 7, 8, 9, 10	chpmatply1 21889	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐶‘𝑀) ∈ (Base‘𝑃))
12	5, 11	eqeltrid 2843	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝐾 ∈ (Base‘𝑃))
13		eqid 2738	. . . . . 6 ⊢ (coe₁‘𝐾) = (coe₁‘𝐾)
14		eqid 2738	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
15	13, 10, 9, 14	coe1fvalcl 21293	. . . . 5 ⊢ ((𝐾 ∈ (Base‘𝑃) ∧ 𝑘 ∈ ℕ₀) → ((coe₁‘𝐾)‘𝑘) ∈ (Base‘𝑅))
16	12, 15	sylan 579	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑘 ∈ ℕ₀) → ((coe₁‘𝐾)‘𝑘) ∈ (Base‘𝑅))
17	2	anim2i 616	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
18	7	matring 21500	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
19		cpmidgsumm2pm.o	. . . . . . . 8 ⊢ 𝑂 = (1_r‘𝐴)
20	8, 19	ringidcl 19722	. . . . . . 7 ⊢ (𝐴 ∈ Ring → 𝑂 ∈ 𝐵)
21	17, 18, 20	3syl 18	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑂 ∈ 𝐵)
22	21	3adant3 1130	. . . . 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 21488	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (((coe₁‘𝐾)‘𝑘) ∈ (Base‘𝑅) ∧ 𝑂 ∈ 𝐵)) → (((coe₁‘𝐾)‘𝑘) ∗ 𝑂) ∈ 𝐵)
26	1, 4, 16, 23, 25	syl22anc 835	. . 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 6970	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝐺:ℕ₀⟶𝐵)
29	8	fvexi 6770	. . . 4 ⊢ 𝐵 ∈ V
30		nn0ex 12169	. . . 4 ⊢ ℕ₀ ∈ V
31	29, 30	pm3.2i 470	. . 3 ⊢ (𝐵 ∈ V ∧ ℕ₀ ∈ V)
32		elmapg 8586	. . 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 256	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝐺 ∈ (𝐵 ↑_m ℕ₀))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1085 = wceq 1539 ∈ wcel 2108 Vcvv 3422 ↦ cmpt 5153 ⟶wf 6414 ‘cfv 6418 (class class class)co 7255 ↑_m cmap 8573 Fincfn 8691 ℕ₀cn0 12163 Basecbs 16840 ·_𝑠 cvsca 16892 ._gcmg 18615 mulGrpcmgp 19635 1_rcur 19652 Ringcrg 19698 CRingccrg 19699 algSccascl 20969 var₁cv1 21257 Poly₁cpl1 21258 coe₁cco1 21259 Mat cmat 21464 matToPolyMat cmat2pmat 21761 CharPlyMat cchpmat 21883

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-addf 10881 ax-mulf 10882

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-xor 1504 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-ot 4567 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-ofr 7512 df-om 7688 df-1st 7804 df-2nd 7805 df-supp 7949 df-tpos 8013 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-2o 8268 df-er 8456 df-map 8575 df-pm 8576 df-ixp 8644 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fsupp 9059 df-sup 9131 df-oi 9199 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-xnn0 12236 df-z 12250 df-dec 12367 df-uz 12512 df-rp 12660 df-fz 13169 df-fzo 13312 df-seq 13650 df-exp 13711 df-hash 13973 df-word 14146 df-lsw 14194 df-concat 14202 df-s1 14229 df-substr 14282 df-pfx 14312 df-splice 14391 df-reverse 14400 df-s2 14489 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-starv 16903 df-sca 16904 df-vsca 16905 df-ip 16906 df-tset 16907 df-ple 16908 df-ds 16910 df-unif 16911 df-hom 16912 df-cco 16913 df-0g 17069 df-gsum 17070 df-prds 17075 df-pws 17077 df-mre 17212 df-mrc 17213 df-acs 17215 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-mhm 18345 df-submnd 18346 df-efmnd 18423 df-grp 18495 df-minusg 18496 df-sbg 18497 df-mulg 18616 df-subg 18667 df-ghm 18747 df-gim 18790 df-cntz 18838 df-oppg 18865 df-symg 18890 df-pmtr 18965 df-psgn 19014 df-cmn 19303 df-abl 19304 df-mgp 19636 df-ur 19653 df-ring 19700 df-cring 19701 df-oppr 19777 df-dvdsr 19798 df-unit 19799 df-invr 19829 df-dvr 19840 df-rnghom 19874 df-drng 19908 df-subrg 19937 df-lmod 20040 df-lss 20109 df-sra 20349 df-rgmod 20350 df-cnfld 20511 df-zring 20583 df-zrh 20617 df-dsmm 20849 df-frlm 20864 df-ascl 20972 df-psr 21022 df-mvr 21023 df-mpl 21024 df-opsr 21026 df-psr1 21261 df-vr1 21262 df-ply1 21263 df-coe1 21264 df-mamu 21443 df-mat 21465 df-mdet 21642 df-mat2pmat 21764 df-chpmat 21884

This theorem is referenced by: cpmidpmat 21930

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