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Theorem scmatsgrp1 22496

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 22489	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑥 ∈ 𝑆 → 𝑥 ∈ 𝐷))
8	7	ssrdv 3928	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 ⊆ 𝐷)
9	1, 2, 4, 6	dmatsgrp 22473	. . . . 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 19095	. . . . 5 ⊢ (𝐷 ∈ (SubGrp‘𝐴) → 𝐷 = (Base‘𝐶))
13	12	eqcomd 2743	. . . 4 ⊢ (𝐷 ∈ (SubGrp‘𝐴) → (Base‘𝐶) = 𝐷)
14	10, 13	syl 17	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (Base‘𝐶) = 𝐷)
15	8, 14	sseqtrrd 3960	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 ⊆ (Base‘𝐶))
16	1, 2, 3, 4, 5	scmatid 22488	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (1_r‘𝐴) ∈ 𝑆)
17	16	ne0d 4283	. 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 22489	. . . . . . 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 2737	. . . . . . 7 ⊢ (-_g‘𝐴) = (-_g‘𝐴)
26		eqid 2737	. . . . . . 7 ⊢ (-_g‘𝐶) = (-_g‘𝐶)
27	25, 11, 26	subgsub 19103	. . . . . 6 ⊢ ((𝐷 ∈ (SubGrp‘𝐴) ∧ 𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) → (𝑥(-_g‘𝐴)𝑦) = (𝑥(-_g‘𝐶)𝑦))
28	27	eqcomd 2743	. . . . 5 ⊢ ((𝐷 ∈ (SubGrp‘𝐴) ∧ 𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) → (𝑥(-_g‘𝐶)𝑦) = (𝑥(-_g‘𝐴)𝑦))
29	18, 21, 24, 28	syl3anc 1374	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥(-_g‘𝐶)𝑦) = (𝑥(-_g‘𝐴)𝑦))
30	1, 2, 3, 4, 5	scmatsubcl 22491	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥(-_g‘𝐴)𝑦) ∈ 𝑆)
31	29, 30	eqeltrd 2837	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥(-_g‘𝐶)𝑦) ∈ 𝑆)
32	31	ralrimivva 3181	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(-_g‘𝐶)𝑦) ∈ 𝑆)
33	1, 2, 4, 6	dmatsrng 22475	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → 𝐷 ∈ (SubRing‘𝐴))
34	33	ancoms 458	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐷 ∈ (SubRing‘𝐴))
35	11	subrgring 20540	. . 3 ⊢ (𝐷 ∈ (SubRing‘𝐴) → 𝐶 ∈ Ring)
36		ringgrp 20208	. . 3 ⊢ (𝐶 ∈ Ring → 𝐶 ∈ Grp)
37		eqid 2737	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
38	37, 26	issubg4 19110	. . 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 1344	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 ∈ (SubGrp‘𝐶))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 ∀wral 3052 ⊆ wss 3890 ∅c0 4274 ‘cfv 6490 (class class class)co 7358 Fincfn 8884 Basecbs 17168 ↾_s cress 17189 0_gc0g 17391 Grpcgrp 18898 -_gcsg 18900 SubGrpcsubg 19085 1_rcur 20151 Ringcrg 20203 SubRingcsubrg 20535 Mat cmat 22381 DMat cdmat 22462 ScMat cscmat 22463

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5300 ax-pr 5368 ax-un 7680 ax-cnex 11083 ax-resscn 11084 ax-1cn 11085 ax-icn 11086 ax-addcl 11087 ax-addrcl 11088 ax-mulcl 11089 ax-mulrcl 11090 ax-mulcom 11091 ax-addass 11092 ax-mulass 11093 ax-distr 11094 ax-i2m1 11095 ax-1ne0 11096 ax-1rid 11097 ax-rnegex 11098 ax-rrecex 11099 ax-cnre 11100 ax-pre-lttri 11101 ax-pre-lttrn 11102 ax-pre-ltadd 11103 ax-pre-mulgt0 11104

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-ot 4577 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-se 5576 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-isom 6499 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-om 7809 df-1st 7933 df-2nd 7934 df-supp 8102 df-frecs 8222 df-wrecs 8253 df-recs 8302 df-rdg 8340 df-1o 8396 df-2o 8397 df-er 8634 df-map 8766 df-ixp 8837 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-fsupp 9266 df-sup 9346 df-oi 9416 df-card 9852 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-nn 12164 df-2 12233 df-3 12234 df-4 12235 df-5 12236 df-6 12237 df-7 12238 df-8 12239 df-9 12240 df-n0 12427 df-z 12514 df-dec 12634 df-uz 12778 df-fz 13451 df-fzo 13598 df-seq 13953 df-hash 14282 df-struct 17106 df-sets 17123 df-slot 17141 df-ndx 17153 df-base 17169 df-ress 17190 df-plusg 17222 df-mulr 17223 df-sca 17225 df-vsca 17226 df-ip 17227 df-tset 17228 df-ple 17229 df-ds 17231 df-hom 17233 df-cco 17234 df-0g 17393 df-gsum 17394 df-prds 17399 df-pws 17401 df-mre 17537 df-mrc 17538 df-acs 17540 df-mgm 18597 df-sgrp 18676 df-mnd 18692 df-mhm 18740 df-submnd 18741 df-grp 18901 df-minusg 18902 df-sbg 18903 df-mulg 19033 df-subg 19088 df-ghm 19177 df-cntz 19281 df-cmn 19746 df-abl 19747 df-mgp 20111 df-rng 20123 df-ur 20152 df-ring 20205 df-subrng 20512 df-subrg 20536 df-lmod 20846 df-lss 20916 df-sra 21158 df-rgmod 21159 df-dsmm 21720 df-frlm 21735 df-mamu 22365 df-mat 22382 df-dmat 22464 df-scmat 22465

This theorem is referenced by: scmatsrng1 22497

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