Theorem clssubg 21912

Description: The closure of a subgroup in a topological group is a subgroup. (Contributed by Mario Carneiro, 17-Sep-2015.)

Hypothesis

Ref	Expression
subgntr.h	|- J = ( TopOpen ` G )

Assertion

Ref	Expression
clssubg	|- ( ( G e. TopGrp /\ S e. (SubGrp ` G ) ) -> ( ( cls ` J ) ` S ) e. (SubGrp ` G ) )

Proof of Theorem clssubg

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		subgntr.h	. . . . . . 7 |- J = ( TopOpen ` G )
2		eqid 2622	. . . . . . 7 |- ( Base ` G ) = ( Base ` G )
3	1, 2	tgptopon 21886	. . . . . 6 |- ( G e. TopGrp -> J e. (TopOn ` ( Base ` G ) ) )
4	3	adantr 481	. . . . 5 |- ( ( G e. TopGrp /\ S e. (SubGrp ` G ) ) -> J e. (TopOn ` ( Base ` G ) ) )
5		topontop 20718	. . . . 5 |- ( J e. (TopOn ` ( Base ` G ) ) -> J e. Top )
6	4, 5	syl 17	. . . 4 |- ( ( G e. TopGrp /\ S e. (SubGrp ` G ) ) -> J e. Top )
7	2	subgss 17595	. . . . . 6 |- ( S e. (SubGrp ` G ) -> S C_ ( Base ` G ) )
8	7	adantl 482	. . . . 5 |- ( ( G e. TopGrp /\ S e. (SubGrp ` G ) ) -> S C_ ( Base ` G ) )
9		toponuni 20719	. . . . . 6 |- ( J e. (TopOn ` ( Base ` G ) ) -> ( Base ` G ) = U. J )
10	4, 9	syl 17	. . . . 5 |- ( ( G e. TopGrp /\ S e. (SubGrp ` G ) ) -> ( Base ` G ) = U. J )
11	8, 10	sseqtrd 3641	. . . 4 |- ( ( G e. TopGrp /\ S e. (SubGrp ` G ) ) -> S C_ U. J )
12		eqid 2622	. . . . 5 |- U. J = U. J
13	12	clsss3 20863	. . . 4 |- ( ( J e. Top /\ S C_ U. J ) -> ( ( cls ` J ) ` S ) C_ U. J )
14	6, 11, 13	syl2anc 693	. . 3 |- ( ( G e. TopGrp /\ S e. (SubGrp ` G ) ) -> ( ( cls ` J ) ` S ) C_ U. J )
15	14, 10	sseqtr4d 3642	. 2 |- ( ( G e. TopGrp /\ S e. (SubGrp ` G ) ) -> ( ( cls ` J ) ` S ) C_ ( Base ` G ) )
16	12	sscls 20860	. . . 4 |- ( ( J e. Top /\ S C_ U. J ) -> S C_ ( ( cls ` J ) ` S ) )
17	6, 11, 16	syl2anc 693	. . 3 |- ( ( G e. TopGrp /\ S e. (SubGrp ` G ) ) -> S C_ ( ( cls ` J ) ` S ) )
18		eqid 2622	. . . . . 6 |- ( 0g ` G ) = ( 0g ` G )
19	18	subg0cl 17602	. . . . 5 |- ( S e. (SubGrp ` G ) -> ( 0g ` G ) e. S )
20	19	adantl 482	. . . 4 |- ( ( G e. TopGrp /\ S e. (SubGrp ` G ) ) -> ( 0g ` G ) e. S )
21		ne0i 3921	. . . 4 |- ( ( 0g ` G ) e. S -> S =/= (/) )
22	20, 21	syl 17	. . 3 |- ( ( G e. TopGrp /\ S e. (SubGrp ` G ) ) -> S =/= (/) )
23		ssn0 3976	. . 3 |- ( ( S C_ ( ( cls ` J ) ` S ) /\ S =/= (/) ) -> ( ( cls ` J ) ` S ) =/= (/) )
24	17, 22, 23	syl2anc 693	. 2 |- ( ( G e. TopGrp /\ S e. (SubGrp ` G ) ) -> ( ( cls ` J ) ` S ) =/= (/) )
25		df-ov 6653	. . . 4 |- ( x ( -g ` G ) y ) = ( ( -g ` G ) ` <. x , y >. )
26		opelxpi 5148	. . . . . . 7 |- ( ( x e. ( ( cls ` J ) ` S ) /\ y e. ( ( cls ` J ) ` S ) ) -> <. x , y >. e. ( ( ( cls ` J ) ` S ) X. ( ( cls ` J ) ` S ) ) )
27		txcls 21407	. . . . . . . . . 10 |- ( ( ( J e. (TopOn ` ( Base ` G ) ) /\ J e. (TopOn ` ( Base ` G ) ) ) /\ ( S C_ ( Base ` G ) /\ S C_ ( Base ` G ) ) ) -> ( ( cls ` ( J tX J ) ) ` ( S X. S ) ) = ( ( ( cls ` J ) ` S ) X. ( ( cls ` J ) ` S ) ) )
28	4, 4, 8, 8, 27	syl22anc 1327	. . . . . . . . 9 |- ( ( G e. TopGrp /\ S e. (SubGrp ` G ) ) -> ( ( cls ` ( J tX J ) ) ` ( S X. S ) ) = ( ( ( cls ` J ) ` S ) X. ( ( cls ` J ) ` S ) ) )
29		txtopon 21394	. . . . . . . . . . . . 13 |- ( ( J e. (TopOn ` ( Base ` G ) ) /\ J e. (TopOn ` ( Base ` G ) ) ) -> ( J tX J ) e. (TopOn ` ( ( Base ` G ) X. ( Base ` G ) ) ) )
30	4, 4, 29	syl2anc 693	. . . . . . . . . . . 12 |- ( ( G e. TopGrp /\ S e. (SubGrp ` G ) ) -> ( J tX J ) e. (TopOn ` ( ( Base ` G ) X. ( Base ` G ) ) ) )
31		topontop 20718	. . . . . . . . . . . 12 |- ( ( J tX J ) e. (TopOn ` ( ( Base ` G ) X. ( Base ` G ) ) ) -> ( J tX J ) e. Top )
32	30, 31	syl 17	. . . . . . . . . . 11 |- ( ( G e. TopGrp /\ S e. (SubGrp ` G ) ) -> ( J tX J ) e. Top )
33		cnvimass 5485	. . . . . . . . . . . . 13 |- ( `' ( -g ` G ) " S ) C_ dom ( -g ` G )
34		tgpgrp 21882	. . . . . . . . . . . . . . . 16 |- ( G e. TopGrp -> G e. Grp )
35	34	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( G e. TopGrp /\ S e. (SubGrp ` G ) ) -> G e. Grp )
36		eqid 2622	. . . . . . . . . . . . . . . 16 |- ( -g ` G ) = ( -g ` G )
37	2, 36	grpsubf 17494	. . . . . . . . . . . . . . 15 |- ( G e. Grp -> ( -g ` G ) : ( ( Base ` G ) X. ( Base ` G ) ) --> ( Base ` G ) )
38	35, 37	syl 17	. . . . . . . . . . . . . 14 |- ( ( G e. TopGrp /\ S e. (SubGrp ` G ) ) -> ( -g ` G ) : ( ( Base ` G ) X. ( Base ` G ) ) --> ( Base ` G ) )
39		fdm 6051	. . . . . . . . . . . . . 14 |- ( ( -g ` G ) : ( ( Base ` G ) X. ( Base ` G ) ) --> ( Base ` G ) -> dom ( -g ` G ) = ( ( Base ` G ) X. ( Base ` G ) ) )
40	38, 39	syl 17	. . . . . . . . . . . . 13 |- ( ( G e. TopGrp /\ S e. (SubGrp ` G ) ) -> dom ( -g ` G ) = ( ( Base ` G ) X. ( Base ` G ) ) )
41	33, 40	syl5sseq 3653	. . . . . . . . . . . 12 |- ( ( G e. TopGrp /\ S e. (SubGrp ` G ) ) -> ( `' ( -g ` G ) " S ) C_ ( ( Base ` G ) X. ( Base ` G ) ) )
42		toponuni 20719	. . . . . . . . . . . . 13 |- ( ( J tX J ) e. (TopOn ` ( ( Base ` G ) X. ( Base ` G ) ) ) -> ( ( Base ` G ) X. ( Base ` G ) ) = U. ( J tX J ) )
43	30, 42	syl 17	. . . . . . . . . . . 12 |- ( ( G e. TopGrp /\ S e. (SubGrp ` G ) ) -> ( ( Base ` G ) X. ( Base ` G ) ) = U. ( J tX J ) )
44	41, 43	sseqtrd 3641	. . . . . . . . . . 11 |- ( ( G e. TopGrp /\ S e. (SubGrp ` G ) ) -> ( `' ( -g ` G ) " S ) C_ U. ( J tX J ) )
45	36	subgsubcl 17605	. . . . . . . . . . . . . . . 16 |- ( ( S e. (SubGrp ` G ) /\ x e. S /\ y e. S ) -> ( x ( -g ` G ) y ) e. S )
46	45	3expb 1266	. . . . . . . . . . . . . . 15 |- ( ( S e. (SubGrp ` G ) /\ ( x e. S /\ y e. S ) ) -> ( x ( -g ` G ) y ) e. S )
47	46	ralrimivva 2971	. . . . . . . . . . . . . 14 |- ( S e. (SubGrp ` G ) -> A. x e. S A. y e. S ( x ( -g ` G ) y ) e. S )
48		fveq2 6191	. . . . . . . . . . . . . . . . 17 |- ( z = <. x , y >. -> ( ( -g ` G ) ` z ) = ( ( -g ` G ) ` <. x , y >. ) )
49	48, 25	syl6eqr 2674	. . . . . . . . . . . . . . . 16 |- ( z = <. x , y >. -> ( ( -g ` G ) ` z ) = ( x ( -g ` G ) y ) )
50	49	eleq1d 2686	. . . . . . . . . . . . . . 15 |- ( z = <. x , y >. -> ( ( ( -g ` G ) ` z ) e. S <-> ( x ( -g ` G ) y ) e. S ) )
51	50	ralxp 5263	. . . . . . . . . . . . . 14 |- ( A. z e. ( S X. S ) ( ( -g ` G ) ` z ) e. S <-> A. x e. S A. y e. S ( x ( -g ` G ) y ) e. S )
52	47, 51	sylibr 224	. . . . . . . . . . . . 13 |- ( S e. (SubGrp ` G ) -> A. z e. ( S X. S ) ( ( -g ` G ) ` z ) e. S )
53	52	adantl 482	. . . . . . . . . . . 12 |- ( ( G e. TopGrp /\ S e. (SubGrp ` G ) ) -> A. z e. ( S X. S ) ( ( -g ` G ) ` z ) e. S )
54		ffun 6048	. . . . . . . . . . . . . 14 |- ( ( -g ` G ) : ( ( Base ` G ) X. ( Base ` G ) ) --> ( Base ` G ) -> Fun ( -g ` G ) )
55	38, 54	syl 17	. . . . . . . . . . . . 13 |- ( ( G e. TopGrp /\ S e. (SubGrp ` G ) ) -> Fun ( -g ` G ) )
56		xpss12 5225	. . . . . . . . . . . . . . 15 |- ( ( S C_ ( Base ` G ) /\ S C_ ( Base ` G ) ) -> ( S X. S ) C_ ( ( Base ` G ) X. ( Base ` G ) ) )
57	8, 8, 56	syl2anc 693	. . . . . . . . . . . . . 14 |- ( ( G e. TopGrp /\ S e. (SubGrp ` G ) ) -> ( S X. S ) C_ ( ( Base ` G ) X. ( Base ` G ) ) )
58	57, 40	sseqtr4d 3642	. . . . . . . . . . . . 13 |- ( ( G e. TopGrp /\ S e. (SubGrp ` G ) ) -> ( S X. S ) C_ dom ( -g ` G ) )
59		funimass5 6334	. . . . . . . . . . . . 13 |- ( ( Fun ( -g ` G ) /\ ( S X. S ) C_ dom ( -g ` G ) ) -> ( ( S X. S ) C_ ( `' ( -g ` G ) " S ) <-> A. z e. ( S X. S ) ( ( -g ` G ) ` z ) e. S ) )
60	55, 58, 59	syl2anc 693	. . . . . . . . . . . 12 |- ( ( G e. TopGrp /\ S e. (SubGrp ` G ) ) -> ( ( S X. S ) C_ ( `' ( -g ` G ) " S ) <-> A. z e. ( S X. S ) ( ( -g ` G ) ` z ) e. S ) )
61	53, 60	mpbird 247	. . . . . . . . . . 11 |- ( ( G e. TopGrp /\ S e. (SubGrp ` G ) ) -> ( S X. S ) C_ ( `' ( -g ` G ) " S ) )
62		eqid 2622	. . . . . . . . . . . 12 |- U. ( J tX J ) = U. ( J tX J )
63	62	clsss 20858	. . . . . . . . . . 11 |- ( ( ( J tX J ) e. Top /\ ( `' ( -g ` G ) " S ) C_ U. ( J tX J ) /\ ( S X. S ) C_ ( `' ( -g ` G ) " S ) ) -> ( ( cls ` ( J tX J ) ) ` ( S X. S ) ) C_ ( ( cls ` ( J tX J ) ) ` ( `' ( -g ` G ) " S ) ) )
64	32, 44, 61, 63	syl3anc 1326	. . . . . . . . . 10 |- ( ( G e. TopGrp /\ S e. (SubGrp ` G ) ) -> ( ( cls ` ( J tX J ) ) ` ( S X. S ) ) C_ ( ( cls ` ( J tX J ) ) ` ( `' ( -g ` G ) " S ) ) )
65	1, 36	tgpsubcn 21894	. . . . . . . . . . . 12 |- ( G e. TopGrp -> ( -g ` G ) e. ( ( J tX J ) Cn J ) )
66	65	adantr 481	. . . . . . . . . . 11 |- ( ( G e. TopGrp /\ S e. (SubGrp ` G ) ) -> ( -g ` G ) e. ( ( J tX J ) Cn J ) )
67	12	cncls2i 21074	. . . . . . . . . . 11 |- ( ( ( -g ` G ) e. ( ( J tX J ) Cn J ) /\ S C_ U. J ) -> ( ( cls ` ( J tX J ) ) ` ( `' ( -g ` G ) " S ) ) C_ ( `' ( -g ` G ) " ( ( cls ` J ) ` S ) ) )
68	66, 11, 67	syl2anc 693	. . . . . . . . . 10 |- ( ( G e. TopGrp /\ S e. (SubGrp ` G ) ) -> ( ( cls ` ( J tX J ) ) ` ( `' ( -g ` G ) " S ) ) C_ ( `' ( -g ` G ) " ( ( cls ` J ) ` S ) ) )
69	64, 68	sstrd 3613	. . . . . . . . 9 |- ( ( G e. TopGrp /\ S e. (SubGrp ` G ) ) -> ( ( cls ` ( J tX J ) ) ` ( S X. S ) ) C_ ( `' ( -g ` G ) " ( ( cls ` J ) ` S ) ) )
70	28, 69	eqsstr3d 3640	. . . . . . . 8 |- ( ( G e. TopGrp /\ S e. (SubGrp ` G ) ) -> ( ( ( cls ` J ) ` S ) X. ( ( cls ` J ) ` S ) ) C_ ( `' ( -g ` G ) " ( ( cls ` J ) ` S ) ) )
71	70	sselda 3603	. . . . . . 7 |- ( ( ( G e. TopGrp /\ S e. (SubGrp ` G ) ) /\ <. x , y >. e. ( ( ( cls ` J ) ` S ) X. ( ( cls ` J ) ` S ) ) ) -> <. x , y >. e. ( `' ( -g ` G ) " ( ( cls ` J ) ` S ) ) )
72	26, 71	sylan2 491	. . . . . 6 |- ( ( ( G e. TopGrp /\ S e. (SubGrp ` G ) ) /\ ( x e. ( ( cls ` J ) ` S ) /\ y e. ( ( cls ` J ) ` S ) ) ) -> <. x , y >. e. ( `' ( -g ` G ) " ( ( cls ` J ) ` S ) ) )
73	34	ad2antrr 762	. . . . . . 7 |- ( ( ( G e. TopGrp /\ S e. (SubGrp ` G ) ) /\ ( x e. ( ( cls ` J ) ` S ) /\ y e. ( ( cls ` J ) ` S ) ) ) -> G e. Grp )
74		ffn 6045	. . . . . . 7 |- ( ( -g ` G ) : ( ( Base ` G ) X. ( Base ` G ) ) --> ( Base ` G ) -> ( -g ` G ) Fn ( ( Base ` G ) X. ( Base ` G ) ) )
75		elpreima 6337	. . . . . . 7 |- ( ( -g ` G ) Fn ( ( Base ` G ) X. ( Base ` G ) ) -> ( <. x , y >. e. ( `' ( -g ` G ) " ( ( cls ` J ) ` S ) ) <-> ( <. x , y >. e. ( ( Base ` G ) X. ( Base ` G ) ) /\ ( ( -g ` G ) ` <. x , y >. ) e. ( ( cls ` J ) ` S ) ) ) )
76	73, 37, 74, 75	4syl 19	. . . . . 6 |- ( ( ( G e. TopGrp /\ S e. (SubGrp ` G ) ) /\ ( x e. ( ( cls ` J ) ` S ) /\ y e. ( ( cls ` J ) ` S ) ) ) -> ( <. x , y >. e. ( `' ( -g ` G ) " ( ( cls ` J ) ` S ) ) <-> ( <. x , y >. e. ( ( Base ` G ) X. ( Base ` G ) ) /\ ( ( -g ` G ) ` <. x , y >. ) e. ( ( cls ` J ) ` S ) ) ) )
77	72, 76	mpbid 222	. . . . 5 |- ( ( ( G e. TopGrp /\ S e. (SubGrp ` G ) ) /\ ( x e. ( ( cls ` J ) ` S ) /\ y e. ( ( cls ` J ) ` S ) ) ) -> ( <. x , y >. e. ( ( Base ` G ) X. ( Base ` G ) ) /\ ( ( -g ` G ) ` <. x , y >. ) e. ( ( cls ` J ) ` S ) ) )
78	77	simprd 479	. . . 4 |- ( ( ( G e. TopGrp /\ S e. (SubGrp ` G ) ) /\ ( x e. ( ( cls ` J ) ` S ) /\ y e. ( ( cls ` J ) ` S ) ) ) -> ( ( -g ` G ) ` <. x , y >. ) e. ( ( cls ` J ) ` S ) )
79	25, 78	syl5eqel 2705	. . 3 |- ( ( ( G e. TopGrp /\ S e. (SubGrp ` G ) ) /\ ( x e. ( ( cls ` J ) ` S ) /\ y e. ( ( cls ` J ) ` S ) ) ) -> ( x ( -g ` G ) y ) e. ( ( cls ` J ) ` S ) )
80	79	ralrimivva 2971	. 2 |- ( ( G e. TopGrp /\ S e. (SubGrp ` G ) ) -> A. x e. ( ( cls ` J ) ` S ) A. y e. ( ( cls ` J ) ` S ) ( x ( -g ` G ) y ) e. ( ( cls ` J ) ` S ) )
81	2, 36	issubg4 17613	. . 3 |- ( G e. Grp -> ( ( ( cls ` J ) ` S ) e. (SubGrp ` G ) <-> ( ( ( cls ` J ) ` S ) C_ ( Base ` G ) /\ ( ( cls ` J ) ` S ) =/= (/) /\ A. x e. ( ( cls ` J ) ` S ) A. y e. ( ( cls ` J ) ` S ) ( x ( -g ` G ) y ) e. ( ( cls ` J ) ` S ) ) ) )
82	35, 81	syl 17	. 2 |- ( ( G e. TopGrp /\ S e. (SubGrp ` G ) ) -> ( ( ( cls ` J ) ` S ) e. (SubGrp ` G ) <-> ( ( ( cls ` J ) ` S ) C_ ( Base ` G ) /\ ( ( cls ` J ) ` S ) =/= (/) /\ A. x e. ( ( cls ` J ) ` S ) A. y e. ( ( cls ` J ) ` S ) ( x ( -g ` G ) y ) e. ( ( cls ` J ) ` S ) ) ) )
83	15, 24, 80, 82	mpbir3and 1245	1 |- ( ( G e. TopGrp /\ S e. (SubGrp ` G ) ) -> ( ( cls ` J ) ` S ) e. (SubGrp ` G ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   C_ wss 3574   (/)c0 3915   <.cop 4183   U.cuni 4436   X. cxp 5112   `'ccnv 5113   dom cdm 5114   "cima 5117   Fun wfun 5882   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   Basecbs 15857   TopOpenctopn 16082   0gc0g 16100   Grpcgrp 17422   -gcsg 17424  SubGrpcsubg 17588   Topctop 20698  TopOnctopon 20715   clsccl 20822   Cn ccn 21028   tX ctx 21363   TopGrpctgp 21875

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-0g 16102  df-topgen 16104  df-plusf 17241  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-sbg 17427  df-subg 17591  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-cn 21031  df-tx 21365  df-tmd 21876  df-tgp 21877

This theorem is referenced by:  clsnsg  21913  tgptsmscls  21953