Theorem hhcno 28763

Description: The continuous operators of Hilbert space. (Contributed by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)

Hypotheses

Ref	Expression
hhcn.1	|- D = ( normh o. -h )
hhcn.2	|- J = ( MetOpen ` D )

Assertion

Ref	Expression
hhcno	|- ContOp = ( J Cn J )

Proof of Theorem hhcno

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

Step	Hyp	Ref	Expression
1		df-rab 2921	. 2 |- { t e. ( ~H ^m ~H ) | A. x e. ~H A. y e. RR+ E. z e. RR+ A. w e. ~H ( ( normh ` ( w -h x ) ) < z -> ( normh ` ( ( t ` w ) -h ( t ` x ) ) ) < y ) } = { t | ( t e. ( ~H ^m ~H ) /\ A. x e. ~H A. y e. RR+ E. z e. RR+ A. w e. ~H ( ( normh ` ( w -h x ) ) < z -> ( normh ` ( ( t ` w ) -h ( t ` x ) ) ) < y ) ) }
2		df-cnop 28699	. 2 |- ContOp = { t e. ( ~H ^m ~H ) | A. x e. ~H A. y e. RR+ E. z e. RR+ A. w e. ~H ( ( normh ` ( w -h x ) ) < z -> ( normh ` ( ( t ` w ) -h ( t ` x ) ) ) < y ) }
3		hhcn.1	. . . . . . . . . . . . . 14 |- D = ( normh o. -h )
4	3	hilmetdval 28053	. . . . . . . . . . . . 13 |- ( ( x e. ~H /\ w e. ~H ) -> ( x D w ) = ( normh ` ( x -h w ) ) )
5		normsub 28000	. . . . . . . . . . . . 13 |- ( ( x e. ~H /\ w e. ~H ) -> ( normh ` ( x -h w ) ) = ( normh ` ( w -h x ) ) )
6	4, 5	eqtrd 2656	. . . . . . . . . . . 12 |- ( ( x e. ~H /\ w e. ~H ) -> ( x D w ) = ( normh ` ( w -h x ) ) )
7	6	adantll 750	. . . . . . . . . . 11 |- ( ( ( t : ~H --> ~H /\ x e. ~H ) /\ w e. ~H ) -> ( x D w ) = ( normh ` ( w -h x ) ) )
8	7	breq1d 4663	. . . . . . . . . 10 |- ( ( ( t : ~H --> ~H /\ x e. ~H ) /\ w e. ~H ) -> ( ( x D w ) < z <-> ( normh ` ( w -h x ) ) < z ) )
9		ffvelrn 6357	. . . . . . . . . . . . . 14 |- ( ( t : ~H --> ~H /\ x e. ~H ) -> ( t ` x ) e. ~H )
10		ffvelrn 6357	. . . . . . . . . . . . . 14 |- ( ( t : ~H --> ~H /\ w e. ~H ) -> ( t ` w ) e. ~H )
11	9, 10	anim12dan 882	. . . . . . . . . . . . 13 |- ( ( t : ~H --> ~H /\ ( x e. ~H /\ w e. ~H ) ) -> ( ( t ` x ) e. ~H /\ ( t ` w ) e. ~H ) )
12	3	hilmetdval 28053	. . . . . . . . . . . . . 14 |- ( ( ( t ` x ) e. ~H /\ ( t ` w ) e. ~H ) -> ( ( t ` x ) D ( t ` w ) ) = ( normh ` ( ( t ` x ) -h ( t ` w ) ) ) )
13		normsub 28000	. . . . . . . . . . . . . 14 |- ( ( ( t ` x ) e. ~H /\ ( t ` w ) e. ~H ) -> ( normh ` ( ( t ` x ) -h ( t ` w ) ) ) = ( normh ` ( ( t ` w ) -h ( t ` x ) ) ) )
14	12, 13	eqtrd 2656	. . . . . . . . . . . . 13 |- ( ( ( t ` x ) e. ~H /\ ( t ` w ) e. ~H ) -> ( ( t ` x ) D ( t ` w ) ) = ( normh ` ( ( t ` w ) -h ( t ` x ) ) ) )
15	11, 14	syl 17	. . . . . . . . . . . 12 |- ( ( t : ~H --> ~H /\ ( x e. ~H /\ w e. ~H ) ) -> ( ( t ` x ) D ( t ` w ) ) = ( normh ` ( ( t ` w ) -h ( t ` x ) ) ) )
16	15	anassrs 680	. . . . . . . . . . 11 |- ( ( ( t : ~H --> ~H /\ x e. ~H ) /\ w e. ~H ) -> ( ( t ` x ) D ( t ` w ) ) = ( normh ` ( ( t ` w ) -h ( t ` x ) ) ) )
17	16	breq1d 4663	. . . . . . . . . 10 |- ( ( ( t : ~H --> ~H /\ x e. ~H ) /\ w e. ~H ) -> ( ( ( t ` x ) D ( t ` w ) ) < y <-> ( normh ` ( ( t ` w ) -h ( t ` x ) ) ) < y ) )
18	8, 17	imbi12d 334	. . . . . . . . 9 |- ( ( ( t : ~H --> ~H /\ x e. ~H ) /\ w e. ~H ) -> ( ( ( x D w ) < z -> ( ( t ` x ) D ( t ` w ) ) < y ) <-> ( ( normh ` ( w -h x ) ) < z -> ( normh ` ( ( t ` w ) -h ( t ` x ) ) ) < y ) ) )
19	18	ralbidva 2985	. . . . . . . 8 |- ( ( t : ~H --> ~H /\ x e. ~H ) -> ( A. w e. ~H ( ( x D w ) < z -> ( ( t ` x ) D ( t ` w ) ) < y ) <-> A. w e. ~H ( ( normh ` ( w -h x ) ) < z -> ( normh ` ( ( t ` w ) -h ( t ` x ) ) ) < y ) ) )
20	19	rexbidv 3052	. . . . . . 7 |- ( ( t : ~H --> ~H /\ x e. ~H ) -> ( E. z e. RR+ A. w e. ~H ( ( x D w ) < z -> ( ( t ` x ) D ( t ` w ) ) < y ) <-> E. z e. RR+ A. w e. ~H ( ( normh ` ( w -h x ) ) < z -> ( normh ` ( ( t ` w ) -h ( t ` x ) ) ) < y ) ) )
21	20	ralbidv 2986	. . . . . 6 |- ( ( t : ~H --> ~H /\ x e. ~H ) -> ( A. y e. RR+ E. z e. RR+ A. w e. ~H ( ( x D w ) < z -> ( ( t ` x ) D ( t ` w ) ) < y ) <-> A. y e. RR+ E. z e. RR+ A. w e. ~H ( ( normh ` ( w -h x ) ) < z -> ( normh ` ( ( t ` w ) -h ( t ` x ) ) ) < y ) ) )
22	21	ralbidva 2985	. . . . 5 |- ( t : ~H --> ~H -> ( A. x e. ~H A. y e. RR+ E. z e. RR+ A. w e. ~H ( ( x D w ) < z -> ( ( t ` x ) D ( t ` w ) ) < y ) <-> A. x e. ~H A. y e. RR+ E. z e. RR+ A. w e. ~H ( ( normh ` ( w -h x ) ) < z -> ( normh ` ( ( t ` w ) -h ( t ` x ) ) ) < y ) ) )
23	22	pm5.32i 669	. . . 4 |- ( ( t : ~H --> ~H /\ A. x e. ~H A. y e. RR+ E. z e. RR+ A. w e. ~H ( ( x D w ) < z -> ( ( t ` x ) D ( t ` w ) ) < y ) ) <-> ( t : ~H --> ~H /\ A. x e. ~H A. y e. RR+ E. z e. RR+ A. w e. ~H ( ( normh ` ( w -h x ) ) < z -> ( normh ` ( ( t ` w ) -h ( t ` x ) ) ) < y ) ) )
24	3	hilxmet 28052	. . . . 5 |- D e. ( *Met ` ~H )
25		hhcn.2	. . . . . 6 |- J = ( MetOpen ` D )
26	25, 25	metcn 22348	. . . . 5 |- ( ( D e. ( *Met ` ~H ) /\ D e. ( *Met ` ~H ) ) -> ( t e. ( J Cn J ) <-> ( t : ~H --> ~H /\ A. x e. ~H A. y e. RR+ E. z e. RR+ A. w e. ~H ( ( x D w ) < z -> ( ( t ` x ) D ( t ` w ) ) < y ) ) ) )
27	24, 24, 26	mp2an 708	. . . 4 |- ( t e. ( J Cn J ) <-> ( t : ~H --> ~H /\ A. x e. ~H A. y e. RR+ E. z e. RR+ A. w e. ~H ( ( x D w ) < z -> ( ( t ` x ) D ( t ` w ) ) < y ) ) )
28		ax-hilex 27856	. . . . . 6 |- ~H e. _V
29	28, 28	elmap 7886	. . . . 5 |- ( t e. ( ~H ^m ~H ) <-> t : ~H --> ~H )
30	29	anbi1i 731	. . . 4 |- ( ( t e. ( ~H ^m ~H ) /\ A. x e. ~H A. y e. RR+ E. z e. RR+ A. w e. ~H ( ( normh ` ( w -h x ) ) < z -> ( normh ` ( ( t ` w ) -h ( t ` x ) ) ) < y ) ) <-> ( t : ~H --> ~H /\ A. x e. ~H A. y e. RR+ E. z e. RR+ A. w e. ~H ( ( normh ` ( w -h x ) ) < z -> ( normh ` ( ( t ` w ) -h ( t ` x ) ) ) < y ) ) )
31	23, 27, 30	3bitr4i 292	. . 3 |- ( t e. ( J Cn J ) <-> ( t e. ( ~H ^m ~H ) /\ A. x e. ~H A. y e. RR+ E. z e. RR+ A. w e. ~H ( ( normh ` ( w -h x ) ) < z -> ( normh ` ( ( t ` w ) -h ( t ` x ) ) ) < y ) ) )
32	31	abbi2i 2738	. 2 |- ( J Cn J ) = { t | ( t e. ( ~H ^m ~H ) /\ A. x e. ~H A. y e. RR+ E. z e. RR+ A. w e. ~H ( ( normh ` ( w -h x ) ) < z -> ( normh ` ( ( t ` w ) -h ( t ` x ) ) ) < y ) ) }
33	1, 2, 32	3eqtr4i 2654	1 |- ContOp = ( J Cn J )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   A.wral 2912   E.wrex 2913   {crab 2916   class class class wbr 4653   o. ccom 5118   -->wf 5884   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   < clt 10074   RR+crp 11832   *Metcxmt 19731   MetOpencmopn 19736   Cn ccn 21028   ~Hchil 27776   normhcno 27780   -h cmv 27782   ContOpccop 27803

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-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-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-cn 21031  df-cnp 21032  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-hnorm 27825  df-hvsub 27828  df-cnop 28699

This theorem is referenced by:  hmopidmchi  29010