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Theorem scmatsgrp1 22425

Description: The set of scalar matrices is a subgroup of the group/ring of diagonal matrices. (Contributed by AV, 21-Aug-2019.)

**Hypotheses**
Ref	Expression
scmatid.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
scmatid.b	⊢ 𝐵 = (Base‘𝐴)
scmatid.e	⊢ 𝐸 = (Base‘𝑅)
scmatid.0	⊢ 0 = (0_g‘𝑅)
scmatid.s	⊢ 𝑆 = (𝑁 ScMat 𝑅)
scmatsgrp1.d	⊢ 𝐷 = (𝑁 DMat 𝑅)
scmatsgrp1.c	⊢ 𝐶 = (𝐴 ↾_s 𝐷)

**Assertion**
Ref	Expression
scmatsgrp1	⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 ∈ (SubGrp‘𝐶))

Proof of Theorem scmatsgrp1 Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		scmatid.a	. . . . 5 ⊢ 𝐴 = (𝑁 Mat 𝑅)
2		scmatid.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐴)
3		scmatid.e	. . . . 5 ⊢ 𝐸 = (Base‘𝑅)
4		scmatid.0	. . . . 5 ⊢ 0 = (0_g‘𝑅)
5		scmatid.s	. . . . 5 ⊢ 𝑆 = (𝑁 ScMat 𝑅)
6		scmatsgrp1.d	. . . . 5 ⊢ 𝐷 = (𝑁 DMat 𝑅)
7	1, 2, 3, 4, 5, 6	scmatdmat 22418	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑥 ∈ 𝑆 → 𝑥 ∈ 𝐷))
8	7	ssrdv 3943	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 ⊆ 𝐷)
9	1, 2, 4, 6	dmatsgrp 22402	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → 𝐷 ∈ (SubGrp‘𝐴))
10	9	ancoms 458	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐷 ∈ (SubGrp‘𝐴))
11		scmatsgrp1.c	. . . . . 6 ⊢ 𝐶 = (𝐴 ↾_s 𝐷)
12	11	subgbas 19027	. . . . 5 ⊢ (𝐷 ∈ (SubGrp‘𝐴) → 𝐷 = (Base‘𝐶))
13	12	eqcomd 2735	. . . 4 ⊢ (𝐷 ∈ (SubGrp‘𝐴) → (Base‘𝐶) = 𝐷)
14	10, 13	syl 17	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (Base‘𝐶) = 𝐷)
15	8, 14	sseqtrrd 3975	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 ⊆ (Base‘𝐶))
16	1, 2, 3, 4, 5	scmatid 22417	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (1_r‘𝐴) ∈ 𝑆)
17	16	ne0d 4295	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 ≠ ∅)
18	10	adantr 480	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝐷 ∈ (SubGrp‘𝐴))
19	7	com12 32	. . . . . . 7 ⊢ (𝑥 ∈ 𝑆 → ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑥 ∈ 𝐷))
20	19	adantr 480	. . . . . 6 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑥 ∈ 𝐷))
21	20	impcom 407	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝑥 ∈ 𝐷)
22	1, 2, 3, 4, 5, 6	scmatdmat 22418	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑦 ∈ 𝑆 → 𝑦 ∈ 𝐷))
23	22	a1d 25	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑥 ∈ 𝑆 → (𝑦 ∈ 𝑆 → 𝑦 ∈ 𝐷)))
24	23	imp32 418	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝑦 ∈ 𝐷)
25		eqid 2729	. . . . . . 7 ⊢ (-_g‘𝐴) = (-_g‘𝐴)
26		eqid 2729	. . . . . . 7 ⊢ (-_g‘𝐶) = (-_g‘𝐶)
27	25, 11, 26	subgsub 19035	. . . . . 6 ⊢ ((𝐷 ∈ (SubGrp‘𝐴) ∧ 𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) → (𝑥(-_g‘𝐴)𝑦) = (𝑥(-_g‘𝐶)𝑦))
28	27	eqcomd 2735	. . . . 5 ⊢ ((𝐷 ∈ (SubGrp‘𝐴) ∧ 𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) → (𝑥(-_g‘𝐶)𝑦) = (𝑥(-_g‘𝐴)𝑦))
29	18, 21, 24, 28	syl3anc 1373	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥(-_g‘𝐶)𝑦) = (𝑥(-_g‘𝐴)𝑦))
30	1, 2, 3, 4, 5	scmatsubcl 22420	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥(-_g‘𝐴)𝑦) ∈ 𝑆)
31	29, 30	eqeltrd 2828	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥(-_g‘𝐶)𝑦) ∈ 𝑆)
32	31	ralrimivva 3172	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(-_g‘𝐶)𝑦) ∈ 𝑆)
33	1, 2, 4, 6	dmatsrng 22404	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → 𝐷 ∈ (SubRing‘𝐴))
34	33	ancoms 458	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐷 ∈ (SubRing‘𝐴))
35	11	subrgring 20477	. . 3 ⊢ (𝐷 ∈ (SubRing‘𝐴) → 𝐶 ∈ Ring)
36		ringgrp 20141	. . 3 ⊢ (𝐶 ∈ Ring → 𝐶 ∈ Grp)
37		eqid 2729	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
38	37, 26	issubg4 19042	. . 3 ⊢ (𝐶 ∈ Grp → (𝑆 ∈ (SubGrp‘𝐶) ↔ (𝑆 ⊆ (Base‘𝐶) ∧ 𝑆 ≠ ∅ ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(-_g‘𝐶)𝑦) ∈ 𝑆)))
39	34, 35, 36, 38	4syl 19	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑆 ∈ (SubGrp‘𝐶) ↔ (𝑆 ⊆ (Base‘𝐶) ∧ 𝑆 ≠ ∅ ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(-_g‘𝐶)𝑦) ∈ 𝑆)))
40	15, 17, 32, 39	mpbir3and 1343	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 ∈ (SubGrp‘𝐶))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 ∀wral 3044 ⊆ wss 3905 ∅c0 4286 ‘cfv 6486 (class class class)co 7353 Fincfn 8879 Basecbs 17138 ↾_s cress 17159 0_gc0g 17361 Grpcgrp 18830 -_gcsg 18832 SubGrpcsubg 19017 1_rcur 20084 Ringcrg 20136 SubRingcsubrg 20472 Mat cmat 22310 DMat cdmat 22391 ScMat cscmat 22392

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3345 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-ot 4588 df-uni 4862 df-int 4900 df-iun 4946 df-iin 4947 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-se 5577 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-isom 6495 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-of 7617 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8101 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-er 8632 df-map 8762 df-ixp 8832 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-fsupp 9271 df-sup 9351 df-oi 9421 df-card 9854 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11367 df-neg 11368 df-nn 12147 df-2 12209 df-3 12210 df-4 12211 df-5 12212 df-6 12213 df-7 12214 df-8 12215 df-9 12216 df-n0 12403 df-z 12490 df-dec 12610 df-uz 12754 df-fz 13429 df-fzo 13576 df-seq 13927 df-hash 14256 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17139 df-ress 17160 df-plusg 17192 df-mulr 17193 df-sca 17195 df-vsca 17196 df-ip 17197 df-tset 17198 df-ple 17199 df-ds 17201 df-hom 17203 df-cco 17204 df-0g 17363 df-gsum 17364 df-prds 17369 df-pws 17371 df-mre 17506 df-mrc 17507 df-acs 17509 df-mgm 18532 df-sgrp 18611 df-mnd 18627 df-mhm 18675 df-submnd 18676 df-grp 18833 df-minusg 18834 df-sbg 18835 df-mulg 18965 df-subg 19020 df-ghm 19110 df-cntz 19214 df-cmn 19679 df-abl 19680 df-mgp 20044 df-rng 20056 df-ur 20085 df-ring 20138 df-subrng 20449 df-subrg 20473 df-lmod 20783 df-lss 20853 df-sra 21095 df-rgmod 21096 df-dsmm 21657 df-frlm 21672 df-mamu 22294 df-mat 22311 df-dmat 22393 df-scmat 22394

This theorem is referenced by: scmatsrng1 22426

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