Theorem nmcoplbi 28887

Description: A lower bound for the norm of a continuous linear operator. Theorem 3.5(ii) of [Beran] p. 99. (Contributed by NM, 7-Feb-2006.) (Revised 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
nmcoplbi	|- ( A e. ~H -> ( normh ` ( T ` A ) ) <_ ( ( normop ` T ) x. ( normh ` A ) ) )

Proof of Theorem nmcoplbi

Step	Hyp	Ref	Expression
1		0le0 11110	. . . . 5 |- 0 <_ 0
2	1	a1i 11	. . . 4 |- ( A = 0h -> 0 <_ 0 )
3		fveq2 6191	. . . . . . 7 |- ( A = 0h -> ( T ` A ) = ( T ` 0h ) )
4		nmcopex.1	. . . . . . . 8 |- T e. LinOp
5	4	lnop0i 28829	. . . . . . 7 |- ( T ` 0h ) = 0h
6	3, 5	syl6eq 2672	. . . . . 6 |- ( A = 0h -> ( T ` A ) = 0h )
7	6	fveq2d 6195	. . . . 5 |- ( A = 0h -> ( normh ` ( T ` A ) ) = ( normh ` 0h ) )
8		norm0 27985	. . . . 5 |- ( normh ` 0h ) = 0
9	7, 8	syl6eq 2672	. . . 4 |- ( A = 0h -> ( normh ` ( T ` A ) ) = 0 )
10		fveq2 6191	. . . . . . 7 |- ( A = 0h -> ( normh ` A ) = ( normh ` 0h ) )
11	10, 8	syl6eq 2672	. . . . . 6 |- ( A = 0h -> ( normh ` A ) = 0 )
12	11	oveq2d 6666	. . . . 5 |- ( A = 0h -> ( ( normop ` T ) x. ( normh ` A ) ) = ( ( normop ` T ) x. 0 ) )
13		nmcopex.2	. . . . . . . 8 |- T e. ContOp
14	4, 13	nmcopexi 28886	. . . . . . 7 |- ( normop ` T ) e. RR
15	14	recni 10052	. . . . . 6 |- ( normop ` T ) e. CC
16	15	mul01i 10226	. . . . 5 |- ( ( normop ` T ) x. 0 ) = 0
17	12, 16	syl6eq 2672	. . . 4 |- ( A = 0h -> ( ( normop ` T ) x. ( normh ` A ) ) = 0 )
18	2, 9, 17	3brtr4d 4685	. . 3 |- ( A = 0h -> ( normh ` ( T ` A ) ) <_ ( ( normop ` T ) x. ( normh ` A ) ) )
19	18	adantl 482	. 2 |- ( ( A e. ~H /\ A = 0h ) -> ( normh ` ( T ` A ) ) <_ ( ( normop ` T ) x. ( normh ` A ) ) )
20		normcl 27982	. . . . . . . . 9 |- ( A e. ~H -> ( normh ` A ) e. RR )
21	20	adantr 481	. . . . . . . 8 |- ( ( A e. ~H /\ A =/= 0h ) -> ( normh ` A ) e. RR )
22		normne0 27987	. . . . . . . . 9 |- ( A e. ~H -> ( ( normh ` A ) =/= 0 <-> A =/= 0h ) )
23	22	biimpar 502	. . . . . . . 8 |- ( ( A e. ~H /\ A =/= 0h ) -> ( normh ` A ) =/= 0 )
24	21, 23	rereccld 10852	. . . . . . 7 |- ( ( A e. ~H /\ A =/= 0h ) -> ( 1 / ( normh ` A ) ) e. RR )
25		normgt0 27984	. . . . . . . . . 10 |- ( A e. ~H -> ( A =/= 0h <-> 0 < ( normh ` A ) ) )
26	25	biimpa 501	. . . . . . . . 9 |- ( ( A e. ~H /\ A =/= 0h ) -> 0 < ( normh ` A ) )
27	21, 26	recgt0d 10958	. . . . . . . 8 |- ( ( A e. ~H /\ A =/= 0h ) -> 0 < ( 1 / ( normh ` A ) ) )
28		0re 10040	. . . . . . . . 9 |- 0 e. RR
29		ltle 10126	. . . . . . . . 9 |- ( ( 0 e. RR /\ ( 1 / ( normh ` A ) ) e. RR ) -> ( 0 < ( 1 / ( normh ` A ) ) -> 0 <_ ( 1 / ( normh ` A ) ) ) )
30	28, 29	mpan 706	. . . . . . . 8 |- ( ( 1 / ( normh ` A ) ) e. RR -> ( 0 < ( 1 / ( normh ` A ) ) -> 0 <_ ( 1 / ( normh ` A ) ) ) )
31	24, 27, 30	sylc 65	. . . . . . 7 |- ( ( A e. ~H /\ A =/= 0h ) -> 0 <_ ( 1 / ( normh ` A ) ) )
32	24, 31	absidd 14161	. . . . . 6 |- ( ( A e. ~H /\ A =/= 0h ) -> ( abs ` ( 1 / ( normh ` A ) ) ) = ( 1 / ( normh ` A ) ) )
33	32	oveq1d 6665	. . . . 5 |- ( ( A e. ~H /\ A =/= 0h ) -> ( ( abs ` ( 1 / ( normh ` A ) ) ) x. ( normh ` ( T ` A ) ) ) = ( ( 1 / ( normh ` A ) ) x. ( normh ` ( T ` A ) ) ) )
34	24	recnd 10068	. . . . . . . 8 |- ( ( A e. ~H /\ A =/= 0h ) -> ( 1 / ( normh ` A ) ) e. CC )
35		simpl 473	. . . . . . . 8 |- ( ( A e. ~H /\ A =/= 0h ) -> A e. ~H )
36	4	lnopmuli 28831	. . . . . . . 8 |- ( ( ( 1 / ( normh ` A ) ) e. CC /\ A e. ~H ) -> ( T ` ( ( 1 / ( normh ` A ) ) .h A ) ) = ( ( 1 / ( normh ` A ) ) .h ( T ` A ) ) )
37	34, 35, 36	syl2anc 693	. . . . . . 7 |- ( ( A e. ~H /\ A =/= 0h ) -> ( T ` ( ( 1 / ( normh ` A ) ) .h A ) ) = ( ( 1 / ( normh ` A ) ) .h ( T ` A ) ) )
38	37	fveq2d 6195	. . . . . 6 |- ( ( A e. ~H /\ A =/= 0h ) -> ( normh ` ( T ` ( ( 1 / ( normh ` A ) ) .h A ) ) ) = ( normh ` ( ( 1 / ( normh ` A ) ) .h ( T ` A ) ) ) )
39	4	lnopfi 28828	. . . . . . . . 9 |- T : ~H --> ~H
40	39	ffvelrni 6358	. . . . . . . 8 |- ( A e. ~H -> ( T ` A ) e. ~H )
41	40	adantr 481	. . . . . . 7 |- ( ( A e. ~H /\ A =/= 0h ) -> ( T ` A ) e. ~H )
42		norm-iii 27997	. . . . . . 7 |- ( ( ( 1 / ( normh ` A ) ) e. CC /\ ( T ` A ) e. ~H ) -> ( normh ` ( ( 1 / ( normh ` A ) ) .h ( T ` A ) ) ) = ( ( abs ` ( 1 / ( normh ` A ) ) ) x. ( normh ` ( T ` A ) ) ) )
43	34, 41, 42	syl2anc 693	. . . . . 6 |- ( ( A e. ~H /\ A =/= 0h ) -> ( normh ` ( ( 1 / ( normh ` A ) ) .h ( T ` A ) ) ) = ( ( abs ` ( 1 / ( normh ` A ) ) ) x. ( normh ` ( T ` A ) ) ) )
44	38, 43	eqtrd 2656	. . . . 5 |- ( ( A e. ~H /\ A =/= 0h ) -> ( normh ` ( T ` ( ( 1 / ( normh ` A ) ) .h A ) ) ) = ( ( abs ` ( 1 / ( normh ` A ) ) ) x. ( normh ` ( T ` A ) ) ) )
45		normcl 27982	. . . . . . . . 9 |- ( ( T ` A ) e. ~H -> ( normh ` ( T ` A ) ) e. RR )
46	40, 45	syl 17	. . . . . . . 8 |- ( A e. ~H -> ( normh ` ( T ` A ) ) e. RR )
47	46	adantr 481	. . . . . . 7 |- ( ( A e. ~H /\ A =/= 0h ) -> ( normh ` ( T ` A ) ) e. RR )
48	47	recnd 10068	. . . . . 6 |- ( ( A e. ~H /\ A =/= 0h ) -> ( normh ` ( T ` A ) ) e. CC )
49	21	recnd 10068	. . . . . 6 |- ( ( A e. ~H /\ A =/= 0h ) -> ( normh ` A ) e. CC )
50	48, 49, 23	divrec2d 10805	. . . . 5 |- ( ( A e. ~H /\ A =/= 0h ) -> ( ( normh ` ( T ` A ) ) / ( normh ` A ) ) = ( ( 1 / ( normh ` A ) ) x. ( normh ` ( T ` A ) ) ) )
51	33, 44, 50	3eqtr4rd 2667	. . . 4 |- ( ( A e. ~H /\ A =/= 0h ) -> ( ( normh ` ( T ` A ) ) / ( normh ` A ) ) = ( normh ` ( T ` ( ( 1 / ( normh ` A ) ) .h A ) ) ) )
52		hvmulcl 27870	. . . . . 6 |- ( ( ( 1 / ( normh ` A ) ) e. CC /\ A e. ~H ) -> ( ( 1 / ( normh ` A ) ) .h A ) e. ~H )
53	34, 35, 52	syl2anc 693	. . . . 5 |- ( ( A e. ~H /\ A =/= 0h ) -> ( ( 1 / ( normh ` A ) ) .h A ) e. ~H )
54		normcl 27982	. . . . . . 7 |- ( ( ( 1 / ( normh ` A ) ) .h A ) e. ~H -> ( normh ` ( ( 1 / ( normh ` A ) ) .h A ) ) e. RR )
55	53, 54	syl 17	. . . . . 6 |- ( ( A e. ~H /\ A =/= 0h ) -> ( normh ` ( ( 1 / ( normh ` A ) ) .h A ) ) e. RR )
56		norm1 28106	. . . . . 6 |- ( ( A e. ~H /\ A =/= 0h ) -> ( normh ` ( ( 1 / ( normh ` A ) ) .h A ) ) = 1 )
57		eqle 10139	. . . . . 6 |- ( ( ( normh ` ( ( 1 / ( normh ` A ) ) .h A ) ) e. RR /\ ( normh ` ( ( 1 / ( normh ` A ) ) .h A ) ) = 1 ) -> ( normh ` ( ( 1 / ( normh ` A ) ) .h A ) ) <_ 1 )
58	55, 56, 57	syl2anc 693	. . . . 5 |- ( ( A e. ~H /\ A =/= 0h ) -> ( normh ` ( ( 1 / ( normh ` A ) ) .h A ) ) <_ 1 )
59		nmoplb 28766	. . . . . 6 |- ( ( T : ~H --> ~H /\ ( ( 1 / ( normh ` A ) ) .h A ) e. ~H /\ ( normh ` ( ( 1 / ( normh ` A ) ) .h A ) ) <_ 1 ) -> ( normh ` ( T ` ( ( 1 / ( normh ` A ) ) .h A ) ) ) <_ ( normop ` T ) )
60	39, 59	mp3an1 1411	. . . . 5 |- ( ( ( ( 1 / ( normh ` A ) ) .h A ) e. ~H /\ ( normh ` ( ( 1 / ( normh ` A ) ) .h A ) ) <_ 1 ) -> ( normh ` ( T ` ( ( 1 / ( normh ` A ) ) .h A ) ) ) <_ ( normop ` T ) )
61	53, 58, 60	syl2anc 693	. . . 4 |- ( ( A e. ~H /\ A =/= 0h ) -> ( normh ` ( T ` ( ( 1 / ( normh ` A ) ) .h A ) ) ) <_ ( normop ` T ) )
62	51, 61	eqbrtrd 4675	. . 3 |- ( ( A e. ~H /\ A =/= 0h ) -> ( ( normh ` ( T ` A ) ) / ( normh ` A ) ) <_ ( normop ` T ) )
63	14	a1i 11	. . . 4 |- ( ( A e. ~H /\ A =/= 0h ) -> ( normop ` T ) e. RR )
64		ledivmul2 10902	. . . 4 |- ( ( ( normh ` ( T ` A ) ) e. RR /\ ( normop ` T ) e. RR /\ ( ( normh ` A ) e. RR /\ 0 < ( normh ` A ) ) ) -> ( ( ( normh ` ( T ` A ) ) / ( normh ` A ) ) <_ ( normop ` T ) <-> ( normh ` ( T ` A ) ) <_ ( ( normop ` T ) x. ( normh ` A ) ) ) )
65	47, 63, 21, 26, 64	syl112anc 1330	. . 3 |- ( ( A e. ~H /\ A =/= 0h ) -> ( ( ( normh ` ( T ` A ) ) / ( normh ` A ) ) <_ ( normop ` T ) <-> ( normh ` ( T ` A ) ) <_ ( ( normop ` T ) x. ( normh ` A ) ) ) )
66	62, 65	mpbid 222	. 2 |- ( ( A e. ~H /\ A =/= 0h ) -> ( normh ` ( T ` A ) ) <_ ( ( normop ` T ) x. ( normh ` A ) ) )
67	19, 66	pm2.61dane 2881	1 |- ( A e. ~H -> ( normh ` ( T ` A ) ) <_ ( ( normop ` T ) x. ( normh ` A ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   class class class wbr 4653   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   x. cmul 9941   < clt 10074   <_ cle 10075   / cdiv 10684   abscabs 13974   ~Hchil 27776   .h csm 27778   normhcno 27780   0hc0v 27781   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-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-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-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-rp 11833  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-grpo 27347  df-gid 27348  df-ablo 27399  df-vc 27414  df-nv 27447  df-va 27450  df-ba 27451  df-sm 27452  df-0v 27453  df-nmcv 27455  df-hnorm 27825  df-hba 27826  df-hvsub 27828  df-nmop 28698  df-cnop 28699  df-lnop 28700

This theorem is referenced by:  nmcoplb  28889  cnlnadjlem2  28927  cnlnadjlem7  28932