m2cpmghm - Metamath Proof Explorer

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Theorem m2cpmghm 22666

Description: The transformation of matrices into constant polynomial matrices is an additive group homomorphism. (Contributed by AV, 18-Nov-2019.)

**Hypotheses**
Ref	Expression
m2cpm.s	⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅)
m2cpm.t	⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)
m2cpm.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
m2cpm.b	⊢ 𝐵 = (Base‘𝐴)
m2cpmghm.p	⊢ 𝑃 = (Poly₁‘𝑅)
m2cpmghm.c	⊢ 𝐶 = (𝑁 Mat 𝑃)
m2cpmghm.u	⊢ 𝑈 = (𝐶 ↾_s 𝑆)

**Assertion**
Ref	Expression
m2cpmghm	⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇 ∈ (𝐴 GrpHom 𝑈))

**Proof of Theorem m2cpmghm**
Step	Hyp	Ref	Expression
1		m2cpm.t	. . 3 ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)
2		m2cpm.a	. . 3 ⊢ 𝐴 = (𝑁 Mat 𝑅)
3		m2cpm.b	. . 3 ⊢ 𝐵 = (Base‘𝐴)
4		m2cpmghm.p	. . 3 ⊢ 𝑃 = (Poly₁‘𝑅)
5		m2cpmghm.c	. . 3 ⊢ 𝐶 = (𝑁 Mat 𝑃)
6		eqid 2728	. . 3 ⊢ (Base‘𝐶) = (Base‘𝐶)
7	1, 2, 3, 4, 5, 6	mat2pmatghm 22652	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇 ∈ (𝐴 GrpHom 𝐶))
8		m2cpm.s	. . . 4 ⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅)
9	8, 4, 5	cpmatsubgpmat 22642	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 ∈ (SubGrp‘𝐶))
10	8, 1, 2, 3	m2cpmf 22664	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇:𝐵⟶𝑆)
11	10	frnd 6735	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ran 𝑇 ⊆ 𝑆)
12		m2cpmghm.u	. . . . 5 ⊢ 𝑈 = (𝐶 ↾_s 𝑆)
13	12	resghm2b 19195	. . . 4 ⊢ ((𝑆 ∈ (SubGrp‘𝐶) ∧ ran 𝑇 ⊆ 𝑆) → (𝑇 ∈ (𝐴 GrpHom 𝐶) ↔ 𝑇 ∈ (𝐴 GrpHom 𝑈)))
14	13	bicomd 222	. . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐶) ∧ ran 𝑇 ⊆ 𝑆) → (𝑇 ∈ (𝐴 GrpHom 𝑈) ↔ 𝑇 ∈ (𝐴 GrpHom 𝐶)))
15	9, 11, 14	syl2anc 582	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑇 ∈ (𝐴 GrpHom 𝑈) ↔ 𝑇 ∈ (𝐴 GrpHom 𝐶)))
16	7, 15	mpbird 256	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇 ∈ (𝐴 GrpHom 𝑈))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ⊆ wss 3949 ran crn 5683 ‘cfv 6553 (class class class)co 7426 Fincfn 8970 Basecbs 17187 ↾_s cress 17216 SubGrpcsubg 19082 GrpHom cghm 19174 Ringcrg 20180 Poly₁cpl1 22103 Mat cmat 22327 ConstPolyMat ccpmat 22625 matToPolyMat cmat2pmat 22626

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-ot 4641 df-uni 4913 df-int 4954 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7691 df-ofr 7692 df-om 7877 df-1st 7999 df-2nd 8000 df-supp 8172 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-map 8853 df-pm 8854 df-ixp 8923 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-fsupp 9394 df-sup 9473 df-oi 9541 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-fz 13525 df-fzo 13668 df-seq 14007 df-hash 14330 df-struct 17123 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-mulr 17254 df-sca 17256 df-vsca 17257 df-ip 17258 df-tset 17259 df-ple 17260 df-ds 17262 df-hom 17264 df-cco 17265 df-0g 17430 df-gsum 17431 df-prds 17436 df-pws 17438 df-mre 17573 df-mrc 17574 df-acs 17576 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-mhm 18747 df-submnd 18748 df-grp 18900 df-minusg 18901 df-sbg 18902 df-mulg 19031 df-subg 19085 df-ghm 19175 df-cntz 19275 df-cmn 19744 df-abl 19745 df-mgp 20082 df-rng 20100 df-ur 20129 df-srg 20134 df-ring 20182 df-subrng 20490 df-subrg 20515 df-lmod 20752 df-lss 20823 df-sra 21065 df-rgmod 21066 df-dsmm 21673 df-frlm 21688 df-ascl 21796 df-psr 21849 df-mvr 21850 df-mpl 21851 df-opsr 21853 df-psr1 22106 df-vr1 22107 df-ply1 22108 df-coe1 22109 df-mamu 22306 df-mat 22328 df-cpmat 22628 df-mat2pmat 22629

This theorem is referenced by: m2cpmrhm 22668

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