scmatsgrp1 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > scmatsgrp1		Structured version Visualization version GIF version

Theorem scmatsgrp1 21266

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 21259	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑥 ∈ 𝑆 → 𝑥 ∈ 𝐷))
8	7	ssrdv 3881	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 ⊆ 𝐷)
9	1, 2, 4, 6	dmatsgrp 21243	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → 𝐷 ∈ (SubGrp‘𝐴))
10	9	ancoms 462	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐷 ∈ (SubGrp‘𝐴))
11		scmatsgrp1.c	. . . . . 6 ⊢ 𝐶 = (𝐴 ↾_s 𝐷)
12	11	subgbas 18394	. . . . 5 ⊢ (𝐷 ∈ (SubGrp‘𝐴) → 𝐷 = (Base‘𝐶))
13	12	eqcomd 2744	. . . 4 ⊢ (𝐷 ∈ (SubGrp‘𝐴) → (Base‘𝐶) = 𝐷)
14	10, 13	syl 17	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (Base‘𝐶) = 𝐷)
15	8, 14	sseqtrrd 3916	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 ⊆ (Base‘𝐶))
16	1, 2, 3, 4, 5	scmatid 21258	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (1_r‘𝐴) ∈ 𝑆)
17	16	ne0d 4222	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 ≠ ∅)
18	10	adantr 484	. . . . 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 484	. . . . . 6 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑥 ∈ 𝐷))
21	20	impcom 411	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝑥 ∈ 𝐷)
22	1, 2, 3, 4, 5, 6	scmatdmat 21259	. . . . . . 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 422	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝑦 ∈ 𝐷)
25		eqid 2738	. . . . . . 7 ⊢ (-_g‘𝐴) = (-_g‘𝐴)
26		eqid 2738	. . . . . . 7 ⊢ (-_g‘𝐶) = (-_g‘𝐶)
27	25, 11, 26	subgsub 18402	. . . . . 6 ⊢ ((𝐷 ∈ (SubGrp‘𝐴) ∧ 𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) → (𝑥(-_g‘𝐴)𝑦) = (𝑥(-_g‘𝐶)𝑦))
28	27	eqcomd 2744	. . . . 5 ⊢ ((𝐷 ∈ (SubGrp‘𝐴) ∧ 𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) → (𝑥(-_g‘𝐶)𝑦) = (𝑥(-_g‘𝐴)𝑦))
29	18, 21, 24, 28	syl3anc 1372	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥(-_g‘𝐶)𝑦) = (𝑥(-_g‘𝐴)𝑦))
30	1, 2, 3, 4, 5	scmatsubcl 21261	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥(-_g‘𝐴)𝑦) ∈ 𝑆)
31	29, 30	eqeltrd 2833	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥(-_g‘𝐶)𝑦) ∈ 𝑆)
32	31	ralrimivva 3103	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(-_g‘𝐶)𝑦) ∈ 𝑆)
33	1, 2, 4, 6	dmatsrng 21245	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → 𝐷 ∈ (SubRing‘𝐴))
34	33	ancoms 462	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐷 ∈ (SubRing‘𝐴))
35	11	subrgring 19650	. . . 4 ⊢ (𝐷 ∈ (SubRing‘𝐴) → 𝐶 ∈ Ring)
36	34, 35	syl 17	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Ring)
37		ringgrp 19414	. . 3 ⊢ (𝐶 ∈ Ring → 𝐶 ∈ Grp)
38		eqid 2738	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
39	38, 26	issubg4 18409	. . 3 ⊢ (𝐶 ∈ Grp → (𝑆 ∈ (SubGrp‘𝐶) ↔ (𝑆 ⊆ (Base‘𝐶) ∧ 𝑆 ≠ ∅ ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(-_g‘𝐶)𝑦) ∈ 𝑆)))
40	36, 37, 39	3syl 18	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑆 ∈ (SubGrp‘𝐶) ↔ (𝑆 ⊆ (Base‘𝐶) ∧ 𝑆 ≠ ∅ ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(-_g‘𝐶)𝑦) ∈ 𝑆)))
41	15, 17, 32, 40	mpbir3and 1343	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 ∈ (SubGrp‘𝐶))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1088 = wceq 1542 ∈ wcel 2113 ≠ wne 2934 ∀wral 3053 ⊆ wss 3841 ∅c0 4209 ‘cfv 6333 (class class class)co 7164 Fincfn 8548 Basecbs 16579 ↾_s cress 16580 0_gc0g 16809 Grpcgrp 18212 -_gcsg 18214 SubGrpcsubg 18384 1_rcur 19363 Ringcrg 19409 SubRingcsubrg 19643 Mat cmat 21151 DMat cdmat 21232 ScMat cscmat 21233

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-rep 5151 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 ax-cnex 10664 ax-resscn 10665 ax-1cn 10666 ax-icn 10667 ax-addcl 10668 ax-addrcl 10669 ax-mulcl 10670 ax-mulrcl 10671 ax-mulcom 10672 ax-addass 10673 ax-mulass 10674 ax-distr 10675 ax-i2m1 10676 ax-1ne0 10677 ax-1rid 10678 ax-rnegex 10679 ax-rrecex 10680 ax-cnre 10681 ax-pre-lttri 10682 ax-pre-lttrn 10683 ax-pre-ltadd 10684 ax-pre-mulgt0 10685

This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-pss 3860 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-tp 4518 df-op 4520 df-ot 4522 df-uni 4794 df-int 4834 df-iun 4880 df-iin 4881 df-br 5028 df-opab 5090 df-mpt 5108 df-tr 5134 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6123 df-ord 6169 df-on 6170 df-lim 6171 df-suc 6172 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-isom 6342 df-riota 7121 df-ov 7167 df-oprab 7168 df-mpo 7169 df-of 7419 df-om 7594 df-1st 7707 df-2nd 7708 df-supp 7850 df-wrecs 7969 df-recs 8030 df-rdg 8068 df-1o 8124 df-er 8313 df-map 8432 df-ixp 8501 df-en 8549 df-dom 8550 df-sdom 8551 df-fin 8552 df-fsupp 8900 df-sup 8972 df-oi 9040 df-card 9434 df-pnf 10748 df-mnf 10749 df-xr 10750 df-ltxr 10751 df-le 10752 df-sub 10943 df-neg 10944 df-nn 11710 df-2 11772 df-3 11773 df-4 11774 df-5 11775 df-6 11776 df-7 11777 df-8 11778 df-9 11779 df-n0 11970 df-z 12056 df-dec 12173 df-uz 12318 df-fz 12975 df-fzo 13118 df-seq 13454 df-hash 13776 df-struct 16581 df-ndx 16582 df-slot 16583 df-base 16585 df-sets 16586 df-ress 16587 df-plusg 16674 df-mulr 16675 df-sca 16677 df-vsca 16678 df-ip 16679 df-tset 16680 df-ple 16681 df-ds 16683 df-hom 16685 df-cco 16686 df-0g 16811 df-gsum 16812 df-prds 16817 df-pws 16819 df-mre 16953 df-mrc 16954 df-acs 16956 df-mgm 17961 df-sgrp 18010 df-mnd 18021 df-mhm 18065 df-submnd 18066 df-grp 18215 df-minusg 18216 df-sbg 18217 df-mulg 18336 df-subg 18387 df-ghm 18467 df-cntz 18558 df-cmn 19019 df-abl 19020 df-mgp 19352 df-ur 19364 df-ring 19411 df-subrg 19645 df-lmod 19748 df-lss 19816 df-sra 20056 df-rgmod 20057 df-dsmm 20541 df-frlm 20556 df-mamu 21130 df-mat 21152 df-dmat 21234 df-scmat 21235

This theorem is referenced by: scmatsrng1 21267

W3C validator