Theorem idcnop 28840

Description: The identity function (restricted to Hilbert space) is a continuous operator. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.)

Assertion

Ref	Expression
idcnop	|- ( _I |` ~H ) e. ContOp

Proof of Theorem idcnop

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

Step	Hyp	Ref	Expression
1		f1oi 6174	. . 3 |- ( _I |` ~H ) : ~H -1-1-onto-> ~H
2		f1of 6137	. . 3 |- ( ( _I |` ~H ) : ~H -1-1-onto-> ~H -> ( _I |` ~H ) : ~H --> ~H )
3	1, 2	ax-mp 5	. 2 |- ( _I |` ~H ) : ~H --> ~H
4		id 22	. . . 4 |- ( y e. RR+ -> y e. RR+ )
5		fvresi 6439	. . . . . . . . 9 |- ( w e. ~H -> ( ( _I |` ~H ) ` w ) = w )
6		fvresi 6439	. . . . . . . . 9 |- ( x e. ~H -> ( ( _I |` ~H ) ` x ) = x )
7	5, 6	oveqan12rd 6670	. . . . . . . 8 |- ( ( x e. ~H /\ w e. ~H ) -> ( ( ( _I |` ~H ) ` w ) -h ( ( _I |` ~H ) ` x ) ) = ( w -h x ) )
8	7	fveq2d 6195	. . . . . . 7 |- ( ( x e. ~H /\ w e. ~H ) -> ( normh ` ( ( ( _I |` ~H ) ` w ) -h ( ( _I |` ~H ) ` x ) ) ) = ( normh ` ( w -h x ) ) )
9	8	breq1d 4663	. . . . . 6 |- ( ( x e. ~H /\ w e. ~H ) -> ( ( normh ` ( ( ( _I |` ~H ) ` w ) -h ( ( _I |` ~H ) ` x ) ) ) < y <-> ( normh ` ( w -h x ) ) < y ) )
10	9	biimprd 238	. . . . 5 |- ( ( x e. ~H /\ w e. ~H ) -> ( ( normh ` ( w -h x ) ) < y -> ( normh ` ( ( ( _I |` ~H ) ` w ) -h ( ( _I |` ~H ) ` x ) ) ) < y ) )
11	10	ralrimiva 2966	. . . 4 |- ( x e. ~H -> A. w e. ~H ( ( normh ` ( w -h x ) ) < y -> ( normh ` ( ( ( _I |` ~H ) ` w ) -h ( ( _I |` ~H ) ` x ) ) ) < y ) )
12		breq2 4657	. . . . . . 7 |- ( z = y -> ( ( normh ` ( w -h x ) ) < z <-> ( normh ` ( w -h x ) ) < y ) )
13	12	imbi1d 331	. . . . . 6 |- ( z = y -> ( ( ( normh ` ( w -h x ) ) < z -> ( normh ` ( ( ( _I |` ~H ) ` w ) -h ( ( _I |` ~H ) ` x ) ) ) < y ) <-> ( ( normh ` ( w -h x ) ) < y -> ( normh ` ( ( ( _I |` ~H ) ` w ) -h ( ( _I |` ~H ) ` x ) ) ) < y ) ) )
14	13	ralbidv 2986	. . . . 5 |- ( z = y -> ( A. w e. ~H ( ( normh ` ( w -h x ) ) < z -> ( normh ` ( ( ( _I |` ~H ) ` w ) -h ( ( _I |` ~H ) ` x ) ) ) < y ) <-> A. w e. ~H ( ( normh ` ( w -h x ) ) < y -> ( normh ` ( ( ( _I |` ~H ) ` w ) -h ( ( _I |` ~H ) ` x ) ) ) < y ) ) )
15	14	rspcev 3309	. . . 4 |- ( ( y e. RR+ /\ A. w e. ~H ( ( normh ` ( w -h x ) ) < y -> ( normh ` ( ( ( _I |` ~H ) ` w ) -h ( ( _I |` ~H ) ` x ) ) ) < y ) ) -> E. z e. RR+ A. w e. ~H ( ( normh ` ( w -h x ) ) < z -> ( normh ` ( ( ( _I |` ~H ) ` w ) -h ( ( _I |` ~H ) ` x ) ) ) < y ) )
16	4, 11, 15	syl2anr 495	. . 3 |- ( ( x e. ~H /\ y e. RR+ ) -> E. z e. RR+ A. w e. ~H ( ( normh ` ( w -h x ) ) < z -> ( normh ` ( ( ( _I |` ~H ) ` w ) -h ( ( _I |` ~H ) ` x ) ) ) < y ) )
17	16	rgen2 2975	. 2 |- A. x e. ~H A. y e. RR+ E. z e. RR+ A. w e. ~H ( ( normh ` ( w -h x ) ) < z -> ( normh ` ( ( ( _I |` ~H ) ` w ) -h ( ( _I |` ~H ) ` x ) ) ) < y )
18		elcnop 28716	. 2 |- ( ( _I |` ~H ) e. ContOp <-> ( ( _I |` ~H ) : ~H --> ~H /\ A. x e. ~H A. y e. RR+ E. z e. RR+ A. w e. ~H ( ( normh ` ( w -h x ) ) < z -> ( normh ` ( ( ( _I |` ~H ) ` w ) -h ( ( _I |` ~H ) ` x ) ) ) < y ) ) )
19	3, 17, 18	mpbir2an 955	1 |- ( _I |` ~H ) e. ContOp

Syntax hints:   -> wi 4   /\ wa 384   e. wcel 1990   A.wral 2912   E.wrex 2913   class class class wbr 4653   _I cid 5023   |` cres 5116   -->wf 5884   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   < clt 10074   RR+crp 11832   ~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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-hilex 27856

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  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-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-cnop 28699

This theorem is referenced by:  nmcopex  28888  nmcoplb  28889