cpmidpmatlem2 - Metamath Proof Explorer

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Theorem cpmidpmatlem2 23102

Description: Lemma 2 for cpmidpmat 23104. (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 20390	. . . . . 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 2762	. . . . . . 7 ⊢ (Base‘𝑃) = (Base‘𝑃)
11	6, 7, 8, 9, 10	chpmatply1 23063	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐶‘𝑀) ∈ (Base‘𝑃))
12	5, 11	eqeltrid 2866	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝐾 ∈ (Base‘𝑃))
13		eqid 2762	. . . . . 6 ⊢ (coe₁‘𝐾) = (coe₁‘𝐾)
14		eqid 2762	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
15	13, 10, 9, 14	coe1fvalcl 22443	. . . . 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 22671	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
19		cpmidgsumm2pm.o	. . . . . . . 8 ⊢ 𝑂 = (1_r‘𝐴)
20	8, 19	ringidcl 20412	. . . . . . 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 22659	. . . 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 7111	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝐺:ℕ₀⟶𝐵)
29	8	fvexi 6896	. . . 4 ⊢ 𝐵 ∈ V
30		nn0ex 12538	. . . 4 ⊢ ℕ₀ ∈ V
31	29, 30	pm3.2i 476	. . 3 ⊢ (𝐵 ∈ V ∧ ℕ₀ ∈ V)
32		elmapg 8842	. . 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 1570 ∈ wcel 2145 Vcvv 3453 ↦ cmpt 5190 ⟶wf 6533 ‘cfv 6537 (class class class)co 7417 ↑_m cmap 8830 Fincfn 8956 ℕ₀cn0 12532 Basecbs 17307 ·_𝑠 cvsca 17352 ._gcmg 19196 mulGrpcmgp 20279 1_rcur 20326 Ringcrg 20378 CRingccrg 20379 algSccascl 22073 var₁cv1 22407 Poly₁cpl1 22408 coe₁cco1 22409 Mat cmat 22635 matToPolyMat cmat2pmat 22935 CharPlyMat cchpmat 23057

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7740 ax-cnex 11184 ax-resscn 11185 ax-1cn 11186 ax-icn 11187 ax-addcl 11188 ax-addrcl 11189 ax-mulcl 11190 ax-mulrcl 11191 ax-mulcom 11192 ax-addass 11193 ax-mulass 11194 ax-distr 11195 ax-i2m1 11196 ax-1ne0 11197 ax-1rid 11198 ax-rnegex 11199 ax-rrecex 11200 ax-cnre 11201 ax-pre-lttri 11202 ax-pre-lttrn 11203 ax-pre-ltadd 11204 ax-pre-mulgt0 11205 ax-addf 11207 ax-mulf 11208

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-xor 1542 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-ot 4596 df-uni 4871 df-int 4911 df-iun 4956 df-iin 4957 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-se 5613 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7374 df-ov 7420 df-oprab 7421 df-mpo 7422 df-of 7682 df-ofr 7683 df-om 7867 df-1st 7990 df-2nd 7991 df-supp 8163 df-tpos 8228 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-2o 8460 df-er 8700 df-map 8832 df-pm 8833 df-ixp 8909 df-en 8957 df-dom 8958 df-sdom 8959 df-fin 8960 df-fsupp 9336 df-sup 9416 df-oi 9486 df-card 9948 df-pnf 11273 df-mnf 11274 df-xr 11275 df-ltxr 11276 df-le 11277 df-sub 11471 df-neg 11472 df-div 11900 df-nn 12262 df-2 12331 df-3 12332 df-4 12333 df-5 12334 df-6 12335 df-7 12336 df-8 12337 df-9 12338 df-n0 12533 df-xnn0 12606 df-z 12620 df-dec 12741 df-uz 12892 df-rp 13047 df-fz 13566 df-fzo 13714 df-seq 14070 df-exp 14130 df-hash 14399 df-word 14583 df-lsw 14632 df-concat 14640 df-s1 14667 df-substr 14713 df-pfx 14745 df-splice 14823 df-reverse 14832 df-s2 14923 df-struct 17245 df-sets 17262 df-slot 17280 df-ndx 17292 df-base 17308 df-ress 17329 df-plusg 17361 df-mulr 17362 df-starv 17363 df-sca 17364 df-vsca 17365 df-ip 17366 df-tset 17367 df-ple 17368 df-ds 17370 df-unif 17371 df-hom 17372 df-cco 17373 df-0g 17532 df-gsum 17533 df-prds 17538 df-pws 17540 df-mre 17676 df-mrc 17677 df-acs 17679 df-mgm 18736 df-sgrp 18827 df-mnd 18843 df-mhm 18897 df-submnd 18898 df-efmnd 18984 df-grp 19066 df-minusg 19067 df-sbg 19068 df-mulg 19197 df-subg 19252 df-ghm 19347 df-gim 19392 df-cntz 19450 df-oppg 19479 df-symg 19503 df-pmtr 19575 df-psgn 19624 df-cmn 19915 df-abl 19916 df-mgp 20280 df-rng 20294 df-ur 20327 df-ring 20380 df-cring 20381 df-oppr 20484 df-dvdsr 20504 df-unit 20505 df-invr 20535 df-dvr 20548 df-rhm 20619 df-subrng 20714 df-subrg 20738 df-drng 20898 df-lmod 21052 df-lss 21122 df-sra 21363 df-rgmod 21364 df-cnfld 21592 df-zring 21666 df-zrh 21722 df-dsmm 21951 df-frlm 21966 df-ascl 22076 df-psr 22130 df-mvr 22131 df-mpl 22132 df-opsr 22134 df-psr1 22411 df-vr1 22412 df-ply1 22413 df-coe1 22414 df-mamu 22619 df-mat 22636 df-mdet 22813 df-mat2pmat 22938 df-chpmat 23058

This theorem is used by: cpmidpmat 23104

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