Theorem hhsssh 28126

Description: The predicate " H is a subspace of Hilbert space." (Contributed by NM, 25-Mar-2008.) (New usage is discouraged.)

Hypotheses

Ref	Expression
hhsst.1	|- U = <. <. +h , .h >. , normh >.
hhsst.2	|- W = <. <. ( +h |` ( H X. H ) ) , ( .h |` ( CC X. H ) ) >. , ( normh |` H ) >.

Assertion

Ref	Expression
hhsssh	|- ( H e. SH <-> ( W e. ( SubSp ` U ) /\ H C_ ~H ) )

Proof of Theorem hhsssh

Step	Hyp	Ref	Expression
1		hhsst.1	. . . 4 |- U = <. <. +h , .h >. , normh >.
2		hhsst.2	. . . 4 |- W = <. <. ( +h |` ( H X. H ) ) , ( .h |` ( CC X. H ) ) >. , ( normh |` H ) >.
3	1, 2	hhsst 28123	. . 3 |- ( H e. SH -> W e. ( SubSp ` U ) )
4		shss 28067	. . 3 |- ( H e. SH -> H C_ ~H )
5	3, 4	jca 554	. 2 |- ( H e. SH -> ( W e. ( SubSp ` U ) /\ H C_ ~H ) )
6		eleq1 2689	. . 3 |- ( H = if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) -> ( H e. SH <-> if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) e. SH ) )
7		eqid 2622	. . . 4 |- <. <. ( +h |` ( if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) X. if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) ) , ( .h |` ( CC X. if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) ) >. , ( normh |` if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) >. = <. <. ( +h |` ( if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) X. if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) ) , ( .h |` ( CC X. if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) ) >. , ( normh |` if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) >.
8		xpeq1 5128	. . . . . . . . . . . . 13 |- ( H = if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) -> ( H X. H ) = ( if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) X. H ) )
9		xpeq2 5129	. . . . . . . . . . . . 13 |- ( H = if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) -> ( if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) X. H ) = ( if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) X. if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) )
10	8, 9	eqtrd 2656	. . . . . . . . . . . 12 |- ( H = if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) -> ( H X. H ) = ( if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) X. if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) )
11	10	reseq2d 5396	. . . . . . . . . . 11 |- ( H = if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) -> ( +h |` ( H X. H ) ) = ( +h |` ( if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) X. if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) ) )
12		xpeq2 5129	. . . . . . . . . . . 12 |- ( H = if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) -> ( CC X. H ) = ( CC X. if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) )
13	12	reseq2d 5396	. . . . . . . . . . 11 |- ( H = if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) -> ( .h |` ( CC X. H ) ) = ( .h |` ( CC X. if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) ) )
14	11, 13	opeq12d 4410	. . . . . . . . . 10 |- ( H = if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) -> <. ( +h |` ( H X. H ) ) , ( .h |` ( CC X. H ) ) >. = <. ( +h |` ( if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) X. if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) ) , ( .h |` ( CC X. if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) ) >. )
15		reseq2 5391	. . . . . . . . . 10 |- ( H = if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) -> ( normh |` H ) = ( normh |` if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) )
16	14, 15	opeq12d 4410	. . . . . . . . 9 |- ( H = if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) -> <. <. ( +h |` ( H X. H ) ) , ( .h |` ( CC X. H ) ) >. , ( normh |` H ) >. = <. <. ( +h |` ( if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) X. if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) ) , ( .h |` ( CC X. if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) ) >. , ( normh |` if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) >. )
17	2, 16	syl5eq 2668	. . . . . . . 8 |- ( H = if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) -> W = <. <. ( +h |` ( if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) X. if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) ) , ( .h |` ( CC X. if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) ) >. , ( normh |` if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) >. )
18	17	eleq1d 2686	. . . . . . 7 |- ( H = if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) -> ( W e. ( SubSp ` U ) <-> <. <. ( +h |` ( if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) X. if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) ) , ( .h |` ( CC X. if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) ) >. , ( normh |` if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) >. e. ( SubSp ` U ) ) )
19		sseq1 3626	. . . . . . 7 |- ( H = if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) -> ( H C_ ~H <-> if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) C_ ~H ) )
20	18, 19	anbi12d 747	. . . . . 6 |- ( H = if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) -> ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) <-> ( <. <. ( +h |` ( if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) X. if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) ) , ( .h |` ( CC X. if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) ) >. , ( normh |` if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) >. e. ( SubSp ` U ) /\ if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) C_ ~H ) ) )
21		xpeq1 5128	. . . . . . . . . . . 12 |- ( ~H = if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) -> ( ~H X. ~H ) = ( if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) X. ~H ) )
22		xpeq2 5129	. . . . . . . . . . . 12 |- ( ~H = if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) -> ( if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) X. ~H ) = ( if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) X. if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) )
23	21, 22	eqtrd 2656	. . . . . . . . . . 11 |- ( ~H = if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) -> ( ~H X. ~H ) = ( if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) X. if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) )
24	23	reseq2d 5396	. . . . . . . . . 10 |- ( ~H = if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) -> ( +h |` ( ~H X. ~H ) ) = ( +h |` ( if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) X. if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) ) )
25		xpeq2 5129	. . . . . . . . . . 11 |- ( ~H = if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) -> ( CC X. ~H ) = ( CC X. if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) )
26	25	reseq2d 5396	. . . . . . . . . 10 |- ( ~H = if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) -> ( .h |` ( CC X. ~H ) ) = ( .h |` ( CC X. if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) ) )
27	24, 26	opeq12d 4410	. . . . . . . . 9 |- ( ~H = if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) -> <. ( +h |` ( ~H X. ~H ) ) , ( .h |` ( CC X. ~H ) ) >. = <. ( +h |` ( if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) X. if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) ) , ( .h |` ( CC X. if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) ) >. )
28		reseq2 5391	. . . . . . . . 9 |- ( ~H = if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) -> ( normh |` ~H ) = ( normh |` if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) )
29	27, 28	opeq12d 4410	. . . . . . . 8 |- ( ~H = if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) -> <. <. ( +h |` ( ~H X. ~H ) ) , ( .h |` ( CC X. ~H ) ) >. , ( normh |` ~H ) >. = <. <. ( +h |` ( if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) X. if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) ) , ( .h |` ( CC X. if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) ) >. , ( normh |` if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) >. )
30	29	eleq1d 2686	. . . . . . 7 |- ( ~H = if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) -> ( <. <. ( +h |` ( ~H X. ~H ) ) , ( .h |` ( CC X. ~H ) ) >. , ( normh |` ~H ) >. e. ( SubSp ` U ) <-> <. <. ( +h |` ( if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) X. if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) ) , ( .h |` ( CC X. if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) ) >. , ( normh |` if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) >. e. ( SubSp ` U ) ) )
31		sseq1 3626	. . . . . . 7 |- ( ~H = if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) -> ( ~H C_ ~H <-> if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) C_ ~H ) )
32	30, 31	anbi12d 747	. . . . . 6 |- ( ~H = if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) -> ( ( <. <. ( +h |` ( ~H X. ~H ) ) , ( .h |` ( CC X. ~H ) ) >. , ( normh |` ~H ) >. e. ( SubSp ` U ) /\ ~H C_ ~H ) <-> ( <. <. ( +h |` ( if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) X. if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) ) , ( .h |` ( CC X. if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) ) >. , ( normh |` if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) >. e. ( SubSp ` U ) /\ if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) C_ ~H ) ) )
33		ax-hfvadd 27857	. . . . . . . . . . . 12 |- +h : ( ~H X. ~H ) --> ~H
34		ffn 6045	. . . . . . . . . . . 12 |- ( +h : ( ~H X. ~H ) --> ~H -> +h Fn ( ~H X. ~H ) )
35		fnresdm 6000	. . . . . . . . . . . 12 |- ( +h Fn ( ~H X. ~H ) -> ( +h |` ( ~H X. ~H ) ) = +h )
36	33, 34, 35	mp2b 10	. . . . . . . . . . 11 |- ( +h |` ( ~H X. ~H ) ) = +h
37		ax-hfvmul 27862	. . . . . . . . . . . 12 |- .h : ( CC X. ~H ) --> ~H
38		ffn 6045	. . . . . . . . . . . 12 |- ( .h : ( CC X. ~H ) --> ~H -> .h Fn ( CC X. ~H ) )
39		fnresdm 6000	. . . . . . . . . . . 12 |- ( .h Fn ( CC X. ~H ) -> ( .h |` ( CC X. ~H ) ) = .h )
40	37, 38, 39	mp2b 10	. . . . . . . . . . 11 |- ( .h |` ( CC X. ~H ) ) = .h
41	36, 40	opeq12i 4407	. . . . . . . . . 10 |- <. ( +h |` ( ~H X. ~H ) ) , ( .h |` ( CC X. ~H ) ) >. = <. +h , .h >.
42		normf 27980	. . . . . . . . . . 11 |- normh : ~H --> RR
43		ffn 6045	. . . . . . . . . . 11 |- ( normh : ~H --> RR -> normh Fn ~H )
44		fnresdm 6000	. . . . . . . . . . 11 |- ( normh Fn ~H -> ( normh |` ~H ) = normh )
45	42, 43, 44	mp2b 10	. . . . . . . . . 10 |- ( normh |` ~H ) = normh
46	41, 45	opeq12i 4407	. . . . . . . . 9 |- <. <. ( +h |` ( ~H X. ~H ) ) , ( .h |` ( CC X. ~H ) ) >. , ( normh |` ~H ) >. = <. <. +h , .h >. , normh >.
47	46, 1	eqtr4i 2647	. . . . . . . 8 |- <. <. ( +h |` ( ~H X. ~H ) ) , ( .h |` ( CC X. ~H ) ) >. , ( normh |` ~H ) >. = U
48	1	hhnv 28022	. . . . . . . . 9 |- U e. NrmCVec
49		eqid 2622	. . . . . . . . . 10 |- ( SubSp ` U ) = ( SubSp ` U )
50	49	sspid 27580	. . . . . . . . 9 |- ( U e. NrmCVec -> U e. ( SubSp ` U ) )
51	48, 50	ax-mp 5	. . . . . . . 8 |- U e. ( SubSp ` U )
52	47, 51	eqeltri 2697	. . . . . . 7 |- <. <. ( +h |` ( ~H X. ~H ) ) , ( .h |` ( CC X. ~H ) ) >. , ( normh |` ~H ) >. e. ( SubSp ` U )
53		ssid 3624	. . . . . . 7 |- ~H C_ ~H
54	52, 53	pm3.2i 471	. . . . . 6 |- ( <. <. ( +h |` ( ~H X. ~H ) ) , ( .h |` ( CC X. ~H ) ) >. , ( normh |` ~H ) >. e. ( SubSp ` U ) /\ ~H C_ ~H )
55	20, 32, 54	elimhyp 4146	. . . . 5 |- ( <. <. ( +h |` ( if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) X. if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) ) , ( .h |` ( CC X. if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) ) >. , ( normh |` if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) >. e. ( SubSp ` U ) /\ if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) C_ ~H )
56	55	simpli 474	. . . 4 |- <. <. ( +h |` ( if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) X. if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) ) , ( .h |` ( CC X. if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) ) >. , ( normh |` if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) >. e. ( SubSp ` U )
57	55	simpri 478	. . . 4 |- if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) C_ ~H
58	1, 7, 56, 57	hhshsslem2 28125	. . 3 |- if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) e. SH
59	6, 58	dedth 4139	. 2 |- ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) -> H e. SH )
60	5, 59	impbii 199	1 |- ( H e. SH <-> ( W e. ( SubSp ` U ) /\ H C_ ~H ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   C_ wss 3574   ifcif 4086   <.cop 4183   X. cxp 5112   |` cres 5116   Fn wfn 5883   -->wf 5884   ` cfv 5888   CCcc 9934   RRcr 9935   NrmCVeccnv 27439   SubSpcss 27576   ~Hchil 27776   +h cva 27777   .h csm 27778   normhcno 27780   SHcsh 27785

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  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016  ax-hilex 27856  ax-hfvadd 27857  ax-hvcom 27858  ax-hvass 27859  ax-hv0cl 27860  ax-hvaddid 27861  ax-hfvmul 27862  ax-hvmulid 27863  ax-hvmulass 27864  ax-hvdistr1 27865  ax-hvdistr2 27866  ax-hvmul0 27867  ax-hfi 27936  ax-his1 27939  ax-his2 27940  ax-his3 27941  ax-his4 27942

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-iun 4522  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-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-icc 12182  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-topgen 16104  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-top 20699  df-topon 20716  df-bases 20750  df-lm 21033  df-haus 21119  df-grpo 27347  df-gid 27348  df-ginv 27349  df-gdiv 27350  df-ablo 27399  df-vc 27414  df-nv 27447  df-va 27450  df-ba 27451  df-sm 27452  df-0v 27453  df-vs 27454  df-nmcv 27455  df-ims 27456  df-ssp 27577  df-hnorm 27825  df-hba 27826  df-hvsub 27828  df-hlim 27829  df-sh 28064  df-ch 28078  df-ch0 28110

This theorem is referenced by:  hhsssh2  28127