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Theorem scmatsgrp1 22478

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 22471	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑥 ∈ 𝑆 → 𝑥 ∈ 𝐷))
8	7	ssrdv 3941	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 ⊆ 𝐷)
9	1, 2, 4, 6	dmatsgrp 22455	. . . . 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 19072	. . . . 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 3973	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 ⊆ (Base‘𝐶))
16	1, 2, 3, 4, 5	scmatid 22470	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (1_r‘𝐴) ∈ 𝑆)
17	16	ne0d 4296	. 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 22471	. . . . . . 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 19080	. . . . . 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 22473	. . . 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 22457	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → 𝐷 ∈ (SubRing‘𝐴))
34	33	ancoms 458	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐷 ∈ (SubRing‘𝐴))
35	11	subrgring 20519	. . 3 ⊢ (𝐷 ∈ (SubRing‘𝐴) → 𝐶 ∈ Ring)
36		ringgrp 20185	. . 3 ⊢ (𝐶 ∈ Ring → 𝐶 ∈ Grp)
37		eqid 2737	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
38	37, 26	issubg4 19087	. . 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 3903 ∅c0 4287 ‘cfv 6500 (class class class)co 7368 Fincfn 8895 Basecbs 17148 ↾_s cress 17169 0_gc0g 17371 Grpcgrp 18875 -_gcsg 18877 SubGrpcsubg 19062 1_rcur 20128 Ringcrg 20180 SubRingcsubrg 20514 Mat cmat 22363 DMat cdmat 22444 ScMat cscmat 22445

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 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115

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 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-ot 4591 df-uni 4866 df-int 4905 df-iun 4950 df-iin 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-se 5586 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-of 7632 df-om 7819 df-1st 7943 df-2nd 7944 df-supp 8113 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-2o 8408 df-er 8645 df-map 8777 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9277 df-sup 9357 df-oi 9427 df-card 9863 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-nn 12158 df-2 12220 df-3 12221 df-4 12222 df-5 12223 df-6 12224 df-7 12225 df-8 12226 df-9 12227 df-n0 12414 df-z 12501 df-dec 12620 df-uz 12764 df-fz 13436 df-fzo 13583 df-seq 13937 df-hash 14266 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17149 df-ress 17170 df-plusg 17202 df-mulr 17203 df-sca 17205 df-vsca 17206 df-ip 17207 df-tset 17208 df-ple 17209 df-ds 17211 df-hom 17213 df-cco 17214 df-0g 17373 df-gsum 17374 df-prds 17379 df-pws 17381 df-mre 17517 df-mrc 17518 df-acs 17520 df-mgm 18577 df-sgrp 18656 df-mnd 18672 df-mhm 18720 df-submnd 18721 df-grp 18878 df-minusg 18879 df-sbg 18880 df-mulg 19010 df-subg 19065 df-ghm 19154 df-cntz 19258 df-cmn 19723 df-abl 19724 df-mgp 20088 df-rng 20100 df-ur 20129 df-ring 20182 df-subrng 20491 df-subrg 20515 df-lmod 20825 df-lss 20895 df-sra 21137 df-rgmod 21138 df-dsmm 21699 df-frlm 21714 df-mamu 22347 df-mat 22364 df-dmat 22446 df-scmat 22447

This theorem is referenced by: scmatsrng1 22479

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