Theorem nmcopexi 28886

Description: The norm of a continuous linear Hilbert space operator exists. Theorem 3.5(i) of [Beran] p. 99. (Contributed by NM, 5-Feb-2006.) (Proof shortened by Mario Carneiro, 17-Nov-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
nmcopex.1	|- T e. LinOp
nmcopex.2	|- T e. ContOp

Assertion

Ref	Expression
nmcopexi	|- ( normop ` T ) e. RR

Proof of Theorem nmcopexi

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

Step	Hyp	Ref	Expression
1		nmcopex.2	. . . 4 |- T e. ContOp
2		ax-hv0cl 27860	. . . 4 |- 0h e. ~H
3		1rp 11836	. . . 4 |- 1 e. RR+
4		cnopc 28772	. . . 4 |- ( ( T e. ContOp /\ 0h e. ~H /\ 1 e. RR+ ) -> E. y e. RR+ A. z e. ~H ( ( normh ` ( z -h 0h ) ) < y -> ( normh ` ( ( T ` z ) -h ( T ` 0h ) ) ) < 1 ) )
5	1, 2, 3, 4	mp3an 1424	. . 3 |- E. y e. RR+ A. z e. ~H ( ( normh ` ( z -h 0h ) ) < y -> ( normh ` ( ( T ` z ) -h ( T ` 0h ) ) ) < 1 )
6		hvsub0 27933	. . . . . . . 8 |- ( z e. ~H -> ( z -h 0h ) = z )
7	6	fveq2d 6195	. . . . . . 7 |- ( z e. ~H -> ( normh ` ( z -h 0h ) ) = ( normh ` z ) )
8	7	breq1d 4663	. . . . . 6 |- ( z e. ~H -> ( ( normh ` ( z -h 0h ) ) < y <-> ( normh ` z ) < y ) )
9		nmcopex.1	. . . . . . . . . . 11 |- T e. LinOp
10	9	lnop0i 28829	. . . . . . . . . 10 |- ( T ` 0h ) = 0h
11	10	oveq2i 6661	. . . . . . . . 9 |- ( ( T ` z ) -h ( T ` 0h ) ) = ( ( T ` z ) -h 0h )
12	9	lnopfi 28828	. . . . . . . . . . 11 |- T : ~H --> ~H
13	12	ffvelrni 6358	. . . . . . . . . 10 |- ( z e. ~H -> ( T ` z ) e. ~H )
14		hvsub0 27933	. . . . . . . . . 10 |- ( ( T ` z ) e. ~H -> ( ( T ` z ) -h 0h ) = ( T ` z ) )
15	13, 14	syl 17	. . . . . . . . 9 |- ( z e. ~H -> ( ( T ` z ) -h 0h ) = ( T ` z ) )
16	11, 15	syl5eq 2668	. . . . . . . 8 |- ( z e. ~H -> ( ( T ` z ) -h ( T ` 0h ) ) = ( T ` z ) )
17	16	fveq2d 6195	. . . . . . 7 |- ( z e. ~H -> ( normh ` ( ( T ` z ) -h ( T ` 0h ) ) ) = ( normh ` ( T ` z ) ) )
18	17	breq1d 4663	. . . . . 6 |- ( z e. ~H -> ( ( normh ` ( ( T ` z ) -h ( T ` 0h ) ) ) < 1 <-> ( normh ` ( T ` z ) ) < 1 ) )
19	8, 18	imbi12d 334	. . . . 5 |- ( z e. ~H -> ( ( ( normh ` ( z -h 0h ) ) < y -> ( normh ` ( ( T ` z ) -h ( T ` 0h ) ) ) < 1 ) <-> ( ( normh ` z ) < y -> ( normh ` ( T ` z ) ) < 1 ) ) )
20	19	ralbiia 2979	. . . 4 |- ( A. z e. ~H ( ( normh ` ( z -h 0h ) ) < y -> ( normh ` ( ( T ` z ) -h ( T ` 0h ) ) ) < 1 ) <-> A. z e. ~H ( ( normh ` z ) < y -> ( normh ` ( T ` z ) ) < 1 ) )
21	20	rexbii 3041	. . 3 |- ( E. y e. RR+ A. z e. ~H ( ( normh ` ( z -h 0h ) ) < y -> ( normh ` ( ( T ` z ) -h ( T ` 0h ) ) ) < 1 ) <-> E. y e. RR+ A. z e. ~H ( ( normh ` z ) < y -> ( normh ` ( T ` z ) ) < 1 ) )
22	5, 21	mpbi 220	. 2 |- E. y e. RR+ A. z e. ~H ( ( normh ` z ) < y -> ( normh ` ( T ` z ) ) < 1 )
23		nmopval 28715	. . 3 |- ( T : ~H --> ~H -> ( normop ` T ) = sup ( { m | E. x e. ~H ( ( normh ` x ) <_ 1 /\ m = ( normh ` ( T ` x ) ) ) } , RR* , < ) )
24	12, 23	ax-mp 5	. 2 |- ( normop ` T ) = sup ( { m | E. x e. ~H ( ( normh ` x ) <_ 1 /\ m = ( normh ` ( T ` x ) ) ) } , RR* , < )
25	12	ffvelrni 6358	. . 3 |- ( x e. ~H -> ( T ` x ) e. ~H )
26		normcl 27982	. . 3 |- ( ( T ` x ) e. ~H -> ( normh ` ( T ` x ) ) e. RR )
27	25, 26	syl 17	. 2 |- ( x e. ~H -> ( normh ` ( T ` x ) ) e. RR )
28	10	fveq2i 6194	. . 3 |- ( normh ` ( T ` 0h ) ) = ( normh ` 0h )
29		norm0 27985	. . 3 |- ( normh ` 0h ) = 0
30	28, 29	eqtri 2644	. 2 |- ( normh ` ( T ` 0h ) ) = 0
31		rpcn 11841	. . . . 5 |- ( ( y / 2 ) e. RR+ -> ( y / 2 ) e. CC )
32	9	lnopmuli 28831	. . . . 5 |- ( ( ( y / 2 ) e. CC /\ x e. ~H ) -> ( T ` ( ( y / 2 ) .h x ) ) = ( ( y / 2 ) .h ( T ` x ) ) )
33	31, 32	sylan 488	. . . 4 |- ( ( ( y / 2 ) e. RR+ /\ x e. ~H ) -> ( T ` ( ( y / 2 ) .h x ) ) = ( ( y / 2 ) .h ( T ` x ) ) )
34	33	fveq2d 6195	. . 3 |- ( ( ( y / 2 ) e. RR+ /\ x e. ~H ) -> ( normh ` ( T ` ( ( y / 2 ) .h x ) ) ) = ( normh ` ( ( y / 2 ) .h ( T ` x ) ) ) )
35		norm-iii 27997	. . . 4 |- ( ( ( y / 2 ) e. CC /\ ( T ` x ) e. ~H ) -> ( normh ` ( ( y / 2 ) .h ( T ` x ) ) ) = ( ( abs ` ( y / 2 ) ) x. ( normh ` ( T ` x ) ) ) )
36	31, 25, 35	syl2an 494	. . 3 |- ( ( ( y / 2 ) e. RR+ /\ x e. ~H ) -> ( normh ` ( ( y / 2 ) .h ( T ` x ) ) ) = ( ( abs ` ( y / 2 ) ) x. ( normh ` ( T ` x ) ) ) )
37		rpre 11839	. . . . . 6 |- ( ( y / 2 ) e. RR+ -> ( y / 2 ) e. RR )
38		rpge0 11845	. . . . . 6 |- ( ( y / 2 ) e. RR+ -> 0 <_ ( y / 2 ) )
39	37, 38	absidd 14161	. . . . 5 |- ( ( y / 2 ) e. RR+ -> ( abs ` ( y / 2 ) ) = ( y / 2 ) )
40	39	adantr 481	. . . 4 |- ( ( ( y / 2 ) e. RR+ /\ x e. ~H ) -> ( abs ` ( y / 2 ) ) = ( y / 2 ) )
41	40	oveq1d 6665	. . 3 |- ( ( ( y / 2 ) e. RR+ /\ x e. ~H ) -> ( ( abs ` ( y / 2 ) ) x. ( normh ` ( T ` x ) ) ) = ( ( y / 2 ) x. ( normh ` ( T ` x ) ) ) )
42	34, 36, 41	3eqtrrd 2661	. 2 |- ( ( ( y / 2 ) e. RR+ /\ x e. ~H ) -> ( ( y / 2 ) x. ( normh ` ( T ` x ) ) ) = ( normh ` ( T ` ( ( y / 2 ) .h x ) ) ) )
43	22, 24, 27, 30, 42	nmcexi 28885	1 |- ( normop ` T ) e. RR

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   A.wral 2912   E.wrex 2913   class class class wbr 4653   -->wf 5884   ` cfv 5888  (class class class)co 6650   supcsup 8346   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   x. cmul 9941   RR*cxr 10073   < clt 10074   <_ cle 10075   / cdiv 10684   2c2 11070   RR+crp 11832   abscabs 13974   ~Hchil 27776   .h csm 27778   normhcno 27780   0hc0v 27781   -h cmv 27782   normopcnop 27802   ContOpccop 27803   LinOpclo 27804

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-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-hilex 27856  ax-hfvadd 27857  ax-hvass 27859  ax-hv0cl 27860  ax-hvaddid 27861  ax-hfvmul 27862  ax-hvmulid 27863  ax-hvmulass 27864  ax-hvdistr2 27866  ax-hvmul0 27867  ax-hfi 27936  ax-his1 27939  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-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-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-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-hnorm 27825  df-hvsub 27828  df-nmop 28698  df-cnop 28699  df-lnop 28700

This theorem is referenced by:  nmcoplbi  28887  nmcopex  28888  cnlnadjlem2  28927  cnlnadjlem7  28932  cnlnadjlem8  28933