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Theorem cpmidpmatlem2 22020

Description: Lemma 2 for cpmidpmat 22022. (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 1190	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑘 ∈ ℕ₀) → 𝑁 ∈ Fin)
2		crngring 19795	. . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
3	2	3ad2ant2 1133	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑅 ∈ Ring)
4	3	adantr 481	. . . 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 21981	. . . . . 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 21383	. . . . 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 21592	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
19		cpmidgsumm2pm.o	. . . . . . . 8 ⊢ 𝑂 = (1_r‘𝐴)
20	8, 19	ringidcl 19807	. . . . . . 7 ⊢ (𝐴 ∈ Ring → 𝑂 ∈ 𝐵)
21	17, 18, 20	3syl 18	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑂 ∈ 𝐵)
22	21	3adant3 1131	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑂 ∈ 𝐵)
23	22	adantr 481	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑘 ∈ ℕ₀) → 𝑂 ∈ 𝐵)
24		cpmidgsumm2pm.m	. . . . 5 ⊢ ∗ = ( ·_𝑠 ‘𝐴)
25	14, 7, 8, 24	matvscl 21580	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (((coe₁‘𝐾)‘𝑘) ∈ (Base‘𝑅) ∧ 𝑂 ∈ 𝐵)) → (((coe₁‘𝐾)‘𝑘) ∗ 𝑂) ∈ 𝐵)
26	1, 4, 16, 23, 25	syl22anc 836	. . 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 6988	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝐺:ℕ₀⟶𝐵)
29	8	fvexi 6788	. . . 4 ⊢ 𝐵 ∈ V
30		nn0ex 12239	. . . 4 ⊢ ℕ₀ ∈ V
31	29, 30	pm3.2i 471	. . 3 ⊢ (𝐵 ∈ V ∧ ℕ₀ ∈ V)
32		elmapg 8628	. . 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 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 Vcvv 3432 ↦ cmpt 5157 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 ↑_m cmap 8615 Fincfn 8733 ℕ₀cn0 12233 Basecbs 16912 ·_𝑠 cvsca 16966 ._gcmg 18700 mulGrpcmgp 19720 1_rcur 19737 Ringcrg 19783 CRingccrg 19784 algSccascl 21059 var₁cv1 21347 Poly₁cpl1 21348 coe₁cco1 21349 Mat cmat 21554 matToPolyMat cmat2pmat 21853 CharPlyMat cchpmat 21975

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-addf 10950 ax-mulf 10951

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-xor 1507 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-ot 4570 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-ofr 7534 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7978 df-tpos 8042 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-2o 8298 df-er 8498 df-map 8617 df-pm 8618 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fsupp 9129 df-sup 9201 df-oi 9269 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-xnn0 12306 df-z 12320 df-dec 12438 df-uz 12583 df-rp 12731 df-fz 13240 df-fzo 13383 df-seq 13722 df-exp 13783 df-hash 14045 df-word 14218 df-lsw 14266 df-concat 14274 df-s1 14301 df-substr 14354 df-pfx 14384 df-splice 14463 df-reverse 14472 df-s2 14561 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-starv 16977 df-sca 16978 df-vsca 16979 df-ip 16980 df-tset 16981 df-ple 16982 df-ds 16984 df-unif 16985 df-hom 16986 df-cco 16987 df-0g 17152 df-gsum 17153 df-prds 17158 df-pws 17160 df-mre 17295 df-mrc 17296 df-acs 17298 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-mhm 18430 df-submnd 18431 df-efmnd 18508 df-grp 18580 df-minusg 18581 df-sbg 18582 df-mulg 18701 df-subg 18752 df-ghm 18832 df-gim 18875 df-cntz 18923 df-oppg 18950 df-symg 18975 df-pmtr 19050 df-psgn 19099 df-cmn 19388 df-abl 19389 df-mgp 19721 df-ur 19738 df-ring 19785 df-cring 19786 df-oppr 19862 df-dvdsr 19883 df-unit 19884 df-invr 19914 df-dvr 19925 df-rnghom 19959 df-drng 19993 df-subrg 20022 df-lmod 20125 df-lss 20194 df-sra 20434 df-rgmod 20435 df-cnfld 20598 df-zring 20671 df-zrh 20705 df-dsmm 20939 df-frlm 20954 df-ascl 21062 df-psr 21112 df-mvr 21113 df-mpl 21114 df-opsr 21116 df-psr1 21351 df-vr1 21352 df-ply1 21353 df-coe1 21354 df-mamu 21533 df-mat 21555 df-mdet 21734 df-mat2pmat 21856 df-chpmat 21976

This theorem is referenced by: cpmidpmat 22022

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