cpmatsubgpmat - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > cpmatsubgpmat		Structured version Visualization version GIF version

Theorem cpmatsubgpmat 22635

Description: The set of all constant polynomial matrices over a ring 𝑅 is an additive subgroup of the ring of all polynomial matrices over the ring 𝑅. (Contributed by AV, 15-Nov-2019.)

**Hypotheses**
Ref	Expression
cpmatsrngpmat.s	⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅)
cpmatsrngpmat.p	⊢ 𝑃 = (Poly₁‘𝑅)
cpmatsrngpmat.c	⊢ 𝐶 = (𝑁 Mat 𝑃)

**Assertion**
Ref	Expression
cpmatsubgpmat	⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 ∈ (SubGrp‘𝐶))

Proof of Theorem cpmatsubgpmat Dummy variables 𝑖 𝑗 𝑥 𝑦 𝑘 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cpmatsrngpmat.s	. . . 4 ⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅)
2		cpmatsrngpmat.p	. . . 4 ⊢ 𝑃 = (Poly₁‘𝑅)
3		cpmatsrngpmat.c	. . . 4 ⊢ 𝐶 = (𝑁 Mat 𝑃)
4		eqid 2725	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
5	1, 2, 3, 4	cpmat 22624	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 = {𝑚 ∈ (Base‘𝐶) ∣ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑘 ∈ ℕ ((coe₁‘(𝑖𝑚𝑗))‘𝑘) = (0_g‘𝑅)})
6		ssrab2 4070	. . 3 ⊢ {𝑚 ∈ (Base‘𝐶) ∣ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑘 ∈ ℕ ((coe₁‘(𝑖𝑚𝑗))‘𝑘) = (0_g‘𝑅)} ⊆ (Base‘𝐶)
7	5, 6	eqsstrdi 4028	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 ⊆ (Base‘𝐶))
8	1, 2, 3	1elcpmat 22630	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (1_r‘𝐶) ∈ 𝑆)
9	8	ne0d 4332	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 ≠ ∅)
10	1, 2, 3	cpmatacl 22631	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(+_g‘𝐶)𝑦) ∈ 𝑆)
11	1, 2, 3	cpmatinvcl 22632	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ∀𝑥 ∈ 𝑆 ((inv_g‘𝐶)‘𝑥) ∈ 𝑆)
12		r19.26 3101	. . 3 ⊢ (∀𝑥 ∈ 𝑆 (∀𝑦 ∈ 𝑆 (𝑥(+_g‘𝐶)𝑦) ∈ 𝑆 ∧ ((inv_g‘𝐶)‘𝑥) ∈ 𝑆) ↔ (∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(+_g‘𝐶)𝑦) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 ((inv_g‘𝐶)‘𝑥) ∈ 𝑆))
13	10, 11, 12	sylanbrc 581	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ∀𝑥 ∈ 𝑆 (∀𝑦 ∈ 𝑆 (𝑥(+_g‘𝐶)𝑦) ∈ 𝑆 ∧ ((inv_g‘𝐶)‘𝑥) ∈ 𝑆))
14	2, 3	pmatring 22607	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Ring)
15		ringgrp 20177	. . 3 ⊢ (𝐶 ∈ Ring → 𝐶 ∈ Grp)
16		eqid 2725	. . . 4 ⊢ (+_g‘𝐶) = (+_g‘𝐶)
17		eqid 2725	. . . 4 ⊢ (inv_g‘𝐶) = (inv_g‘𝐶)
18	4, 16, 17	issubg2 19095	. . 3 ⊢ (𝐶 ∈ Grp → (𝑆 ∈ (SubGrp‘𝐶) ↔ (𝑆 ⊆ (Base‘𝐶) ∧ 𝑆 ≠ ∅ ∧ ∀𝑥 ∈ 𝑆 (∀𝑦 ∈ 𝑆 (𝑥(+_g‘𝐶)𝑦) ∈ 𝑆 ∧ ((inv_g‘𝐶)‘𝑥) ∈ 𝑆))))
19	14, 15, 18	3syl 18	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑆 ∈ (SubGrp‘𝐶) ↔ (𝑆 ⊆ (Base‘𝐶) ∧ 𝑆 ≠ ∅ ∧ ∀𝑥 ∈ 𝑆 (∀𝑦 ∈ 𝑆 (𝑥(+_g‘𝐶)𝑦) ∈ 𝑆 ∧ ((inv_g‘𝐶)‘𝑥) ∈ 𝑆))))
20	7, 9, 13, 19	mpbir3and 1339	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 ∈ (SubGrp‘𝐶))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2930 ∀wral 3051 {crab 3419 ⊆ wss 3941 ∅c0 4319 ‘cfv 6543 (class class class)co 7413 Fincfn 8957 ℕcn 12237 Basecbs 17174 +_gcplusg 17227 0_gc0g 17415 Grpcgrp 18889 inv_gcminusg 18890 SubGrpcsubg 19074 1_rcur 20120 Ringcrg 20172 Poly₁cpl1 22099 coe₁cco1 22100 Mat cmat 22320 ConstPolyMat ccpmat 22618

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 2696 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210

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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-ot 4634 df-uni 4905 df-int 4946 df-iun 4994 df-iin 4995 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-se 5629 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-of 7679 df-ofr 7680 df-om 7866 df-1st 7987 df-2nd 7988 df-supp 8159 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-map 8840 df-pm 8841 df-ixp 8910 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-fsupp 9381 df-sup 9460 df-oi 9528 df-card 9957 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-nn 12238 df-2 12300 df-3 12301 df-4 12302 df-5 12303 df-6 12304 df-7 12305 df-8 12306 df-9 12307 df-n0 12498 df-z 12584 df-dec 12703 df-uz 12848 df-fz 13512 df-fzo 13655 df-seq 13994 df-hash 14317 df-struct 17110 df-sets 17127 df-slot 17145 df-ndx 17157 df-base 17175 df-ress 17204 df-plusg 17240 df-mulr 17241 df-sca 17243 df-vsca 17244 df-ip 17245 df-tset 17246 df-ple 17247 df-ds 17249 df-hom 17251 df-cco 17252 df-0g 17417 df-gsum 17418 df-prds 17423 df-pws 17425 df-mre 17560 df-mrc 17561 df-acs 17563 df-mgm 18594 df-sgrp 18673 df-mnd 18689 df-mhm 18734 df-submnd 18735 df-grp 18892 df-minusg 18893 df-sbg 18894 df-mulg 19023 df-subg 19077 df-ghm 19167 df-cntz 19267 df-cmn 19736 df-abl 19737 df-mgp 20074 df-rng 20092 df-ur 20121 df-srg 20126 df-ring 20174 df-subrng 20482 df-subrg 20507 df-lmod 20744 df-lss 20815 df-sra 21057 df-rgmod 21058 df-dsmm 21665 df-frlm 21680 df-ascl 21788 df-psr 21841 df-mvr 21842 df-mpl 21843 df-opsr 21845 df-psr1 22102 df-vr1 22103 df-ply1 22104 df-coe1 22105 df-mamu 22304 df-mat 22321 df-cpmat 22621

This theorem is referenced by: cpmatsrgpmat 22636 0elcpmat 22637 m2cpmghm 22659 chfacfisfcpmat 22770

W3C validator