cpmidpmatlem2 - Metamath Proof Explorer

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Theorem cpmidpmatlem2 23194

Description: Lemma 2 for cpmidpmat 23196. (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 1210	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑘 ∈ ℕ₀) → 𝑁 ∈ Fin)
2		crngring 20477	. . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
3	2	3ad2ant2 1152	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑅 ∈ Ring)
4	3	adantr 486	. . . 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 2760	. . . . . . 7 ⊢ (Base‘𝑃) = (Base‘𝑃)
11	6, 7, 8, 9, 10	chpmatply1 23155	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐶‘𝑀) ∈ (Base‘𝑃))
12	5, 11	eqeltrid 2864	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝐾 ∈ (Base‘𝑃))
13		eqid 2760	. . . . . 6 ⊢ (coe₁‘𝐾) = (coe₁‘𝐾)
14		eqid 2760	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
15	13, 10, 9, 14	coe1fvalcl 22535	. . . . 5 ⊢ ((𝐾 ∈ (Base‘𝑃) ∧ 𝑘 ∈ ℕ₀) → ((coe₁‘𝐾)‘𝑘) ∈ (Base‘𝑅))
16	12, 15	sylan 592	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑘 ∈ ℕ₀) → ((coe₁‘𝐾)‘𝑘) ∈ (Base‘𝑅))
17	2	anim2i 629	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
18	7	matring 22763	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
19		cpmidgsumm2pm.o	. . . . . . . 8 ⊢ 𝑂 = (1_r‘𝐴)
20	8, 19	ringidcl 20499	. . . . . . 7 ⊢ (𝐴 ∈ Ring → 𝑂 ∈ 𝐵)
21	17, 18, 20	3syl 19	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑂 ∈ 𝐵)
22	21	3adant3 1150	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑂 ∈ 𝐵)
23	22	adantr 486	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑘 ∈ ℕ₀) → 𝑂 ∈ 𝐵)
24		cpmidgsumm2pm.m	. . . . 5 ⊢ ∗ = ( ·_𝑠 ‘𝐴)
25	14, 7, 8, 24	matvscl 22751	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (((coe₁‘𝐾)‘𝑘) ∈ (Base‘𝑅) ∧ 𝑂 ∈ 𝐵)) → (((coe₁‘𝐾)‘𝑘) ∗ 𝑂) ∈ 𝐵)
26	1, 4, 16, 23, 25	syl22anc 852	. . 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 7114	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝐺:ℕ₀⟶𝐵)
29	8	fvexi 6899	. . . 4 ⊢ 𝐵 ∈ V
30		nn0ex 12615	. . . 4 ⊢ ℕ₀ ∈ V
31	29, 30	pm3.2i 476	. . 3 ⊢ (𝐵 ∈ V ∧ ℕ₀ ∈ V)
32		elmapg 8862	. . 3 ⊢ ((𝐵 ∈ V ∧ ℕ₀ ∈ V) → (𝐺 ∈ (𝐵 ↑_m ℕ₀) ↔ 𝐺:ℕ₀⟶𝐵))
33	31, 32	mp1i 14	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐺 ∈ (𝐵 ↑_m ℕ₀) ↔ 𝐺:ℕ₀⟶𝐵))
34	28, 33	mpbird 260	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝐺 ∈ (𝐵 ↑_m ℕ₀))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1569 ∈ wcel 2144 Vcvv 3450 ↦ cmpt 5185 ⟶wf 6534 ‘cfv 6538 (class class class)co 7420 ↑_m cmap 8850 Fincfn 8976 ℕ₀cn0 12609 Basecbs 17392 ·_𝑠 cvsca 17437 ._gcmg 19282 mulGrpcmgp 20365 1_rcur 20412 Ringcrg 20464 CRingccrg 20465 algSccascl 22165 var₁cv1 22499 Poly₁cpl1 22500 coe₁cco1 22501 Mat cmat 22727 matToPolyMat cmat2pmat 23027 CharPlyMat cchpmat 23149

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7751 ax-cnex 11259 ax-resscn 11260 ax-1cn 11261 ax-icn 11262 ax-addcl 11263 ax-addrcl 11264 ax-mulcl 11265 ax-mulrcl 11266 ax-mulcom 11267 ax-addass 11268 ax-mulass 11269 ax-distr 11270 ax-i2m1 11271 ax-1ne0 11272 ax-1rid 11273 ax-rnegex 11274 ax-rrecex 11275 ax-cnre 11276 ax-pre-lttri 11277 ax-pre-lttrn 11278 ax-pre-ltadd 11279 ax-pre-mulgt0 11280 ax-addf 11282 ax-mulf 11283

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-xor 1541 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4835 df-ot 4858 df-uni 4867 df-int 4907 df-iun 4952 df-iin 4953 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-se 5604 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-isom 6547 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7693 df-ofr 7694 df-om 7878 df-1st 8001 df-2nd 8002 df-supp 8180 df-tpos 8246 df-frecs 8302 df-wrecs 8333 df-recs 8382 df-rdg 8421 df-1o 8479 df-2o 8480 df-er 8720 df-map 8852 df-pm 8853 df-ixp 8929 df-en 8977 df-dom 8978 df-sdom 8979 df-fin 8980 df-fsupp 9357 df-sup 9437 df-oi 9507 df-card 10023 df-pnf 11348 df-mnf 11349 df-xr 11350 df-ltxr 11351 df-le 11352 df-sub 11546 df-neg 11547 df-div 11977 df-nn 12339 df-2 12408 df-3 12409 df-4 12410 df-5 12411 df-6 12412 df-7 12413 df-8 12414 df-9 12415 df-n0 12610 df-xnn0 12683 df-z 12697 df-dec 12818 df-uz 12969 df-rp 13124 df-fz 13643 df-fzo 13792 df-seq 14148 df-exp 14208 df-hash 14478 df-word 14662 df-lsw 14711 df-concat 14719 df-s1 14746 df-substr 14793 df-pfx 14825 df-splice 14903 df-reverse 14912 df-s2 15003 df-struct 17330 df-sets 17347 df-slot 17365 df-ndx 17377 df-base 17393 df-ress 17414 df-plusg 17446 df-mulr 17447 df-starv 17448 df-sca 17449 df-vsca 17450 df-ip 17451 df-tset 17452 df-ple 17453 df-ds 17455 df-unif 17456 df-hom 17457 df-cco 17458 df-0g 17617 df-gsum 17618 df-prds 17623 df-pws 17625 df-mre 17761 df-mrc 17762 df-acs 17764 df-mgm 18821 df-sgrp 18913 df-mnd 18929 df-mhm 18983 df-submnd 18984 df-efmnd 19070 df-grp 19152 df-minusg 19153 df-sbg 19154 df-mulg 19283 df-subg 19338 df-ghm 19433 df-gim 19478 df-cntz 19536 df-oppg 19565 df-symg 19589 df-pmtr 19661 df-psgn 19710 df-cmn 20001 df-abl 20002 df-mgp 20366 df-rng 20380 df-ur 20413 df-ring 20466 df-cring 20467 df-oppr 20572 df-dvdsr 20592 df-unit 20593 df-invr 20623 df-dvr 20636 df-rhm 20707 df-subrng 20803 df-subrg 20827 df-drng 20987 df-lmod 21142 df-lss 21212 df-sra 21453 df-rgmod 21454 df-cnfld 21684 df-zring 21758 df-zrh 21814 df-dsmm 22043 df-frlm 22058 df-ascl 22168 df-psr 22222 df-mvr 22223 df-mpl 22224 df-opsr 22226 df-psr1 22503 df-vr1 22504 df-ply1 22505 df-coe1 22506 df-mamu 22711 df-mat 22728 df-mdet 22905 df-mat2pmat 23030 df-chpmat 23150

This theorem is used by: cpmidpmat 23196

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