Theorem hhsssh 24621

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 24618	. . 3 |- ( H e. SH -> W e. ( SubSp ` U ) )
4		shss 24563	. . 3 |- ( H e. SH -> H C_ ~H )
5	3, 4	jca 532	. 2 |- ( H e. SH -> ( W e. ( SubSp ` U ) /\ H C_ ~H ) )
6		eleq1 2498	. . 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 2438	. . . 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 4849	. . . . . . . . . . . . 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 4850	. . . . . . . . . . . . 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 2470	. . . . . . . . . . . 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 5105	. . . . . . . . . . 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 4850	. . . . . . . . . . . 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 5105	. . . . . . . . . . 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 4062	. . . . . . . . . 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 5100	. . . . . . . . . 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 4062	. . . . . . . . 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 2482	. . . . . . . 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 2504	. . . . . . 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 3372	. . . . . . 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 710	. . . . . 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 4849	. . . . . . . . . . . 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 4850	. . . . . . . . . . . 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 2470	. . . . . . . . . . 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 5105	. . . . . . . . . 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 4850	. . . . . . . . . . 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 5105	. . . . . . . . . 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 4062	. . . . . . . . 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 5100	. . . . . . . . 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 4062	. . . . . . . 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 2504	. . . . . . 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 3372	. . . . . . 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 710	. . . . . 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 24353	. . . . . . . . . . . 12 |- +h : ( ~H X. ~H ) --> ~H
34		ffn 5554	. . . . . . . . . . . 12 |- ( +h : ( ~H X. ~H ) --> ~H -> +h Fn ( ~H X. ~H ) )
35		fnresdm 5515	. . . . . . . . . . . 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 24358	. . . . . . . . . . . 12 |- .h : ( CC X. ~H ) --> ~H
38		ffn 5554	. . . . . . . . . . . 12 |- ( .h : ( CC X. ~H ) --> ~H -> .h Fn ( CC X. ~H ) )
39		fnresdm 5515	. . . . . . . . . . . 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 4059	. . . . . . . . . 10 |- <. ( +h |` ( ~H X. ~H ) ) , ( .h |` ( CC X. ~H ) ) >. = <. +h , .h >.
42		normf 24476	. . . . . . . . . . 11 |- normh : ~H --> RR
43		ffn 5554	. . . . . . . . . . 11 |- ( normh : ~H --> RR -> normh Fn ~H )
44		fnresdm 5515	. . . . . . . . . . 11 |- ( normh Fn ~H -> ( normh |` ~H ) = normh )
45	42, 43, 44	mp2b 10	. . . . . . . . . 10 |- ( normh |` ~H ) = normh
46	41, 45	opeq12i 4059	. . . . . . . . 9 |- <. <. ( +h |` ( ~H X. ~H ) ) , ( .h |` ( CC X. ~H ) ) >. , ( normh |` ~H ) >. = <. <. +h , .h >. , normh >.
47	46, 1	eqtr4i 2461	. . . . . . . 8 |- <. <. ( +h |` ( ~H X. ~H ) ) , ( .h |` ( CC X. ~H ) ) >. , ( normh |` ~H ) >. = U
48	1	hhnv 24518	. . . . . . . . 9 |- U e. NrmCVec
49		eqid 2438	. . . . . . . . . 10 |- ( SubSp ` U ) = ( SubSp ` U )
50	49	sspid 24074	. . . . . . . . 9 |- ( U e. NrmCVec -> U e. ( SubSp ` U ) )
51	48, 50	ax-mp 5	. . . . . . . 8 |- U e. ( SubSp ` U )
52	47, 51	eqeltri 2508	. . . . . . 7 |- <. <. ( +h |` ( ~H X. ~H ) ) , ( .h |` ( CC X. ~H ) ) >. , ( normh |` ~H ) >. e. ( SubSp ` U )
53		ssid 3370	. . . . . . 7 |- ~H C_ ~H
54	52, 53	pm3.2i 455	. . . . . 6 |- ( <. <. ( +h |` ( ~H X. ~H ) ) , ( .h |` ( CC X. ~H ) ) >. , ( normh |` ~H ) >. e. ( SubSp ` U ) /\ ~H C_ ~H )
55	20, 32, 54	elimhyp 3843	. . . . 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 458	. . . 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 462	. . . 4 |- if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) C_ ~H
58	1, 7, 56, 57	hhshsslem2 24620	. . 3 |- if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) e. SH
59	6, 58	dedth 3836	. 2 |- ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) -> H e. SH )
60	5, 59	impbii 188	1 |- ( H e. SH <-> ( W e. ( SubSp ` U ) /\ H C_ ~H ) )

Syntax hints:   <-> wb 184   /\ wa 369   = wceq 1369   e. wcel 1756   C_ wss 3323   ifcif 3786   <.cop 3878   X. cxp 4833   |` cres 4837   Fn wfn 5408   -->wf 5409   ` cfv 5413   CCcc 9272   RRcr 9273   NrmCVeccnv 23913   SubSpcss 24070   ~Hchil 24272   +h cva 24273   .h csm 24274   normhcno 24276   SHcsh 24281

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352  ax-addf 9353  ax-mulf 9354  ax-hilex 24352  ax-hfvadd 24353  ax-hvcom 24354  ax-hvass 24355  ax-hv0cl 24356  ax-hvaddid 24357  ax-hfvmul 24358  ax-hvmulid 24359  ax-hvmulass 24360  ax-hvdistr1 24361  ax-hvdistr2 24362  ax-hvmul0 24363  ax-hfi 24432  ax-his1 24435  ax-his2 24436  ax-his3 24437  ax-his4 24438

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-er 7093  df-map 7208  df-pm 7209  df-en 7303  df-dom 7304  df-sdom 7305  df-sup 7683  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-n0 10572  df-z 10639  df-uz 10854  df-q 10946  df-rp 10984  df-xneg 11081  df-xadd 11082  df-xmul 11083  df-icc 11299  df-seq 11799  df-exp 11858  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-topgen 14374  df-psmet 17784  df-xmet 17785  df-met 17786  df-bl 17787  df-mopn 17788  df-top 18478  df-bases 18480  df-topon 18481  df-lm 18808  df-haus 18894  df-grpo 23629  df-gid 23630  df-ginv 23631  df-gdiv 23632  df-ablo 23720  df-subgo 23740  df-vc 23875  df-nv 23921  df-va 23924  df-ba 23925  df-sm 23926  df-0v 23927  df-vs 23928  df-nmcv 23929  df-ims 23930  df-ssp 24071  df-hnorm 24321  df-hba 24322  df-hvsub 24324  df-hlim 24325  df-sh 24560  df-ch 24575  df-ch0 24607

This theorem is referenced by:  hhsssh2  24622