Theorem hhsssh 26585

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 26582	. . 3 |- ( H e. SH -> W e. ( SubSp ` U ) )
4		shss 26527	. . 3 |- ( H e. SH -> H C_ ~H )
5	3, 4	jca 530	. 2 |- ( H e. SH -> ( W e. ( SubSp ` U ) /\ H C_ ~H ) )
6		eleq1 2474	. . 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 2402	. . . 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 4836	. . . . . . . . . . . . 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 4837	. . . . . . . . . . . . 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 2443	. . . . . . . . . . . 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 5093	. . . . . . . . . . 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 4837	. . . . . . . . . . . 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 5093	. . . . . . . . . . 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 4166	. . . . . . . . . 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 5088	. . . . . . . . . 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 4166	. . . . . . . . 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 2455	. . . . . . . 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 2471	. . . . . . 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 3462	. . . . . . 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 709	. . . . . 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 4836	. . . . . . . . . . . 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 4837	. . . . . . . . . . . 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 2443	. . . . . . . . . . 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 5093	. . . . . . . . . 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 4837	. . . . . . . . . . 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 5093	. . . . . . . . . 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 4166	. . . . . . . . 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 5088	. . . . . . . . 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 4166	. . . . . . . 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 2471	. . . . . . 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 3462	. . . . . . 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 709	. . . . . 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 26317	. . . . . . . . . . . 12 |- +h : ( ~H X. ~H ) --> ~H
34		ffn 5713	. . . . . . . . . . . 12 |- ( +h : ( ~H X. ~H ) --> ~H -> +h Fn ( ~H X. ~H ) )
35		fnresdm 5670	. . . . . . . . . . . 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 26322	. . . . . . . . . . . 12 |- .h : ( CC X. ~H ) --> ~H
38		ffn 5713	. . . . . . . . . . . 12 |- ( .h : ( CC X. ~H ) --> ~H -> .h Fn ( CC X. ~H ) )
39		fnresdm 5670	. . . . . . . . . . . 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 4163	. . . . . . . . . 10 |- <. ( +h |` ( ~H X. ~H ) ) , ( .h |` ( CC X. ~H ) ) >. = <. +h , .h >.
42		normf 26440	. . . . . . . . . . 11 |- normh : ~H --> RR
43		ffn 5713	. . . . . . . . . . 11 |- ( normh : ~H --> RR -> normh Fn ~H )
44		fnresdm 5670	. . . . . . . . . . 11 |- ( normh Fn ~H -> ( normh |` ~H ) = normh )
45	42, 43, 44	mp2b 10	. . . . . . . . . 10 |- ( normh |` ~H ) = normh
46	41, 45	opeq12i 4163	. . . . . . . . 9 |- <. <. ( +h |` ( ~H X. ~H ) ) , ( .h |` ( CC X. ~H ) ) >. , ( normh |` ~H ) >. = <. <. +h , .h >. , normh >.
47	46, 1	eqtr4i 2434	. . . . . . . 8 |- <. <. ( +h |` ( ~H X. ~H ) ) , ( .h |` ( CC X. ~H ) ) >. , ( normh |` ~H ) >. = U
48	1	hhnv 26482	. . . . . . . . 9 |- U e. NrmCVec
49		eqid 2402	. . . . . . . . . 10 |- ( SubSp ` U ) = ( SubSp ` U )
50	49	sspid 26038	. . . . . . . . 9 |- ( U e. NrmCVec -> U e. ( SubSp ` U ) )
51	48, 50	ax-mp 5	. . . . . . . 8 |- U e. ( SubSp ` U )
52	47, 51	eqeltri 2486	. . . . . . 7 |- <. <. ( +h |` ( ~H X. ~H ) ) , ( .h |` ( CC X. ~H ) ) >. , ( normh |` ~H ) >. e. ( SubSp ` U )
53		ssid 3460	. . . . . . 7 |- ~H C_ ~H
54	52, 53	pm3.2i 453	. . . . . 6 |- ( <. <. ( +h |` ( ~H X. ~H ) ) , ( .h |` ( CC X. ~H ) ) >. , ( normh |` ~H ) >. e. ( SubSp ` U ) /\ ~H C_ ~H )
55	20, 32, 54	elimhyp 3942	. . . . 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 456	. . . 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 460	. . . 4 |- if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) C_ ~H
58	1, 7, 56, 57	hhshsslem2 26584	. . 3 |- if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) e. SH
59	6, 58	dedth 3935	. 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 367   = wceq 1405   e. wcel 1842   C_ wss 3413   ifcif 3884   <.cop 3977   X. cxp 4820   |` cres 4824   Fn wfn 5563   -->wf 5564   ` cfv 5568   CCcc 9519   RRcr 9520   NrmCVeccnv 25877   SubSpcss 26034   ~Hchil 26236   +h cva 26237   .h csm 26238   normhcno 26240   SHcsh 26245

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598  ax-pre-sup 9599  ax-addf 9600  ax-mulf 9601  ax-hilex 26316  ax-hfvadd 26317  ax-hvcom 26318  ax-hvass 26319  ax-hv0cl 26320  ax-hvaddid 26321  ax-hfvmul 26322  ax-hvmulid 26323  ax-hvmulass 26324  ax-hvdistr1 26325  ax-hvdistr2 26326  ax-hvmul0 26327  ax-hfi 26396  ax-his1 26399  ax-his2 26400  ax-his3 26401  ax-his4 26402

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6683  df-1st 6783  df-2nd 6784  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-er 7347  df-map 7458  df-pm 7459  df-en 7554  df-dom 7555  df-sdom 7556  df-sup 7934  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-div 10247  df-nn 10576  df-2 10634  df-3 10635  df-4 10636  df-n0 10836  df-z 10905  df-uz 11127  df-q 11227  df-rp 11265  df-xneg 11370  df-xadd 11371  df-xmul 11372  df-icc 11588  df-seq 12150  df-exp 12209  df-cj 13079  df-re 13080  df-im 13081  df-sqrt 13215  df-abs 13216  df-topgen 15056  df-psmet 18729  df-xmet 18730  df-met 18731  df-bl 18732  df-mopn 18733  df-top 19689  df-bases 19691  df-topon 19692  df-lm 20021  df-haus 20107  df-grpo 25593  df-gid 25594  df-ginv 25595  df-gdiv 25596  df-ablo 25684  df-subgo 25704  df-vc 25839  df-nv 25885  df-va 25888  df-ba 25889  df-sm 25890  df-0v 25891  df-vs 25892  df-nmcv 25893  df-ims 25894  df-ssp 26035  df-hnorm 26285  df-hba 26286  df-hvsub 26288  df-hlim 26289  df-sh 26524  df-ch 26539  df-ch0 26571

This theorem is referenced by:  hhsssh2  26586