Theorem hhsssh 26920

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 26917	. . 3 |- ( H e. SH -> W e. ( SubSp ` U ) )
4		shss 26863	. . 3 |- ( H e. SH -> H C_ ~H )
5	3, 4	jca 535	. 2 |- ( H e. SH -> ( W e. ( SubSp ` U ) /\ H C_ ~H ) )
6		eleq1 2517	. . 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 2451	. . . 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 4848	. . . . . . . . . . . . 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 4849	. . . . . . . . . . . . 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 2485	. . . . . . . . . . . 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 4849	. . . . . . . . . . . 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 4174	. . . . . . . . . 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 4174	. . . . . . . . 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 2497	. . . . . . . 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 2513	. . . . . . 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 3453	. . . . . . 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 717	. . . . . 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 4848	. . . . . . . . . . . 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 4849	. . . . . . . . . . . 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 2485	. . . . . . . . . . 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 4849	. . . . . . . . . . 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 4174	. . . . . . . . 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 4174	. . . . . . . 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 2513	. . . . . . 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 3453	. . . . . . 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 717	. . . . . 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 26653	. . . . . . . . . . . 12 |- +h : ( ~H X. ~H ) --> ~H
34		ffn 5728	. . . . . . . . . . . 12 |- ( +h : ( ~H X. ~H ) --> ~H -> +h Fn ( ~H X. ~H ) )
35		fnresdm 5685	. . . . . . . . . . . 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 26658	. . . . . . . . . . . 12 |- .h : ( CC X. ~H ) --> ~H
38		ffn 5728	. . . . . . . . . . . 12 |- ( .h : ( CC X. ~H ) --> ~H -> .h Fn ( CC X. ~H ) )
39		fnresdm 5685	. . . . . . . . . . . 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 4171	. . . . . . . . . 10 |- <. ( +h |` ( ~H X. ~H ) ) , ( .h |` ( CC X. ~H ) ) >. = <. +h , .h >.
42		normf 26776	. . . . . . . . . . 11 |- normh : ~H --> RR
43		ffn 5728	. . . . . . . . . . 11 |- ( normh : ~H --> RR -> normh Fn ~H )
44		fnresdm 5685	. . . . . . . . . . 11 |- ( normh Fn ~H -> ( normh |` ~H ) = normh )
45	42, 43, 44	mp2b 10	. . . . . . . . . 10 |- ( normh |` ~H ) = normh
46	41, 45	opeq12i 4171	. . . . . . . . 9 |- <. <. ( +h |` ( ~H X. ~H ) ) , ( .h |` ( CC X. ~H ) ) >. , ( normh |` ~H ) >. = <. <. +h , .h >. , normh >.
47	46, 1	eqtr4i 2476	. . . . . . . 8 |- <. <. ( +h |` ( ~H X. ~H ) ) , ( .h |` ( CC X. ~H ) ) >. , ( normh |` ~H ) >. = U
48	1	hhnv 26818	. . . . . . . . 9 |- U e. NrmCVec
49		eqid 2451	. . . . . . . . . 10 |- ( SubSp ` U ) = ( SubSp ` U )
50	49	sspid 26364	. . . . . . . . 9 |- ( U e. NrmCVec -> U e. ( SubSp ` U ) )
51	48, 50	ax-mp 5	. . . . . . . 8 |- U e. ( SubSp ` U )
52	47, 51	eqeltri 2525	. . . . . . 7 |- <. <. ( +h |` ( ~H X. ~H ) ) , ( .h |` ( CC X. ~H ) ) >. , ( normh |` ~H ) >. e. ( SubSp ` U )
53		ssid 3451	. . . . . . 7 |- ~H C_ ~H
54	52, 53	pm3.2i 457	. . . . . 6 |- ( <. <. ( +h |` ( ~H X. ~H ) ) , ( .h |` ( CC X. ~H ) ) >. , ( normh |` ~H ) >. e. ( SubSp ` U ) /\ ~H C_ ~H )
55	20, 32, 54	elimhyp 3939	. . . . 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 460	. . . 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 464	. . . 4 |- if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) C_ ~H
58	1, 7, 56, 57	hhshsslem2 26919	. . 3 |- if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) e. SH
59	6, 58	dedth 3932	. 2 |- ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) -> H e. SH )
60	5, 59	impbii 191	1 |- ( H e. SH <-> ( W e. ( SubSp ` U ) /\ H C_ ~H ) )

Syntax hints:   <-> wb 188   /\ wa 371   = wceq 1444   e. wcel 1887   C_ wss 3404   ifcif 3881   <.cop 3974   X. cxp 4832   |` cres 4836   Fn wfn 5577   -->wf 5578   ` cfv 5582   CCcc 9537   RRcr 9538   NrmCVeccnv 26203   SubSpcss 26360   ~Hchil 26572   +h cva 26573   .h csm 26574   normhcno 26576   SHcsh 26581

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617  ax-addf 9618  ax-mulf 9619  ax-hilex 26652  ax-hfvadd 26653  ax-hvcom 26654  ax-hvass 26655  ax-hv0cl 26656  ax-hvaddid 26657  ax-hfvmul 26658  ax-hvmulid 26659  ax-hvmulass 26660  ax-hvdistr1 26661  ax-hvdistr2 26662  ax-hvmul0 26663  ax-hfi 26732  ax-his1 26735  ax-his2 26736  ax-his3 26737  ax-his4 26738

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-er 7363  df-map 7474  df-pm 7475  df-en 7570  df-dom 7571  df-sdom 7572  df-sup 7956  df-inf 7957  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-n0 10870  df-z 10938  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-icc 11642  df-seq 12214  df-exp 12273  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-topgen 15342  df-psmet 18962  df-xmet 18963  df-met 18964  df-bl 18965  df-mopn 18966  df-top 19921  df-bases 19922  df-topon 19923  df-lm 20245  df-haus 20331  df-grpo 25919  df-gid 25920  df-ginv 25921  df-gdiv 25922  df-ablo 26010  df-subgo 26030  df-vc 26165  df-nv 26211  df-va 26214  df-ba 26215  df-sm 26216  df-0v 26217  df-vs 26218  df-nmcv 26219  df-ims 26220  df-ssp 26361  df-hnorm 26621  df-hba 26622  df-hvsub 26624  df-hlim 26625  df-sh 26860  df-ch 26874  df-ch0 26906

This theorem is referenced by:  hhsssh2  26921