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Theorem nmcoplbi 25579
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. 
ConOp
Assertion
Ref Expression
nmcoplbi  |-  ( A  e.  ~H  ->  ( normh `  ( T `  A ) )  <_ 
( ( normop `  T
)  x.  ( normh `  A ) ) )

Proof of Theorem nmcoplbi
StepHypRef Expression
1 0le0 10517 . . . . 5  |-  0  <_  0
21a1i 11 . . . 4  |-  ( A  =  0h  ->  0  <_  0 )
3 fveq2 5794 . . . . . . 7  |-  ( A  =  0h  ->  ( T `  A )  =  ( T `  0h ) )
4 nmcopex.1 . . . . . . . 8  |-  T  e. 
LinOp
54lnop0i 25521 . . . . . . 7  |-  ( T `
 0h )  =  0h
63, 5syl6eq 2509 . . . . . 6  |-  ( A  =  0h  ->  ( T `  A )  =  0h )
76fveq2d 5798 . . . . 5  |-  ( A  =  0h  ->  ( normh `  ( T `  A ) )  =  ( normh `  0h )
)
8 norm0 24677 . . . . 5  |-  ( normh `  0h )  =  0
97, 8syl6eq 2509 . . . 4  |-  ( A  =  0h  ->  ( normh `  ( T `  A ) )  =  0 )
10 fveq2 5794 . . . . . . 7  |-  ( A  =  0h  ->  ( normh `  A )  =  ( normh `  0h )
)
1110, 8syl6eq 2509 . . . . . 6  |-  ( A  =  0h  ->  ( normh `  A )  =  0 )
1211oveq2d 6211 . . . . 5  |-  ( A  =  0h  ->  (
( normop `  T )  x.  ( normh `  A )
)  =  ( (
normop `  T )  x.  0 ) )
13 nmcopex.2 . . . . . . . 8  |-  T  e. 
ConOp
144, 13nmcopexi 25578 . . . . . . 7  |-  ( normop `  T )  e.  RR
1514recni 9504 . . . . . 6  |-  ( normop `  T )  e.  CC
1615mul01i 9665 . . . . 5  |-  ( (
normop `  T )  x.  0 )  =  0
1712, 16syl6eq 2509 . . . 4  |-  ( A  =  0h  ->  (
( normop `  T )  x.  ( normh `  A )
)  =  0 )
182, 9, 173brtr4d 4425 . . 3  |-  ( A  =  0h  ->  ( normh `  ( T `  A ) )  <_ 
( ( normop `  T
)  x.  ( normh `  A ) ) )
1918adantl 466 . 2  |-  ( ( A  e.  ~H  /\  A  =  0h )  ->  ( normh `  ( T `  A ) )  <_ 
( ( normop `  T
)  x.  ( normh `  A ) ) )
20 normcl 24674 . . . . . . . . 9  |-  ( A  e.  ~H  ->  ( normh `  A )  e.  RR )
2120adantr 465 . . . . . . . 8  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( normh `  A )  e.  RR )
22 normne0 24679 . . . . . . . . 9  |-  ( A  e.  ~H  ->  (
( normh `  A )  =/=  0  <->  A  =/=  0h )
)
2322biimpar 485 . . . . . . . 8  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( normh `  A )  =/=  0 )
2421, 23rereccld 10264 . . . . . . 7  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( 1  /  ( normh `  A ) )  e.  RR )
25 normgt0 24676 . . . . . . . . . 10  |-  ( A  e.  ~H  ->  ( A  =/=  0h  <->  0  <  (
normh `  A ) ) )
2625biimpa 484 . . . . . . . . 9  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
0  <  ( normh `  A ) )
2721, 26recgt0d 10373 . . . . . . . 8  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
0  <  ( 1  /  ( normh `  A
) ) )
28 0re 9492 . . . . . . . . 9  |-  0  e.  RR
29 ltle 9569 . . . . . . . . 9  |-  ( ( 0  e.  RR  /\  ( 1  /  ( normh `  A ) )  e.  RR )  -> 
( 0  <  (
1  /  ( normh `  A ) )  -> 
0  <_  ( 1  /  ( normh `  A
) ) ) )
3028, 29mpan 670 . . . . . . . 8  |-  ( ( 1  /  ( normh `  A ) )  e.  RR  ->  ( 0  <  ( 1  / 
( normh `  A )
)  ->  0  <_  ( 1  /  ( normh `  A ) ) ) )
3124, 27, 30sylc 60 . . . . . . 7  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
0  <_  ( 1  /  ( normh `  A
) ) )
3224, 31absidd 13022 . . . . . 6  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( abs `  (
1  /  ( normh `  A ) ) )  =  ( 1  / 
( normh `  A )
) )
3332oveq1d 6210 . . . . 5  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( ( abs `  (
1  /  ( normh `  A ) ) )  x.  ( normh `  ( T `  A )
) )  =  ( ( 1  /  ( normh `  A ) )  x.  ( normh `  ( T `  A )
) ) )
3424recnd 9518 . . . . . . . 8  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( 1  /  ( normh `  A ) )  e.  CC )
35 simpl 457 . . . . . . . 8  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  ->  A  e.  ~H )
364lnopmuli 25523 . . . . . . . 8  |-  ( ( ( 1  /  ( normh `  A ) )  e.  CC  /\  A  e.  ~H )  ->  ( T `  ( (
1  /  ( normh `  A ) )  .h  A ) )  =  ( ( 1  / 
( normh `  A )
)  .h  ( T `
 A ) ) )
3734, 35, 36syl2anc 661 . . . . . . 7  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( T `  (
( 1  /  ( normh `  A ) )  .h  A ) )  =  ( ( 1  /  ( normh `  A
) )  .h  ( T `  A )
) )
3837fveq2d 5798 . . . . . 6  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( normh `  ( T `  ( ( 1  / 
( normh `  A )
)  .h  A ) ) )  =  (
normh `  ( ( 1  /  ( normh `  A
) )  .h  ( T `  A )
) ) )
394lnopfi 25520 . . . . . . . . 9  |-  T : ~H
--> ~H
4039ffvelrni 5946 . . . . . . . 8  |-  ( A  e.  ~H  ->  ( T `  A )  e.  ~H )
4140adantr 465 . . . . . . 7  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( T `  A
)  e.  ~H )
42 norm-iii 24689 . . . . . . 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 )
) ) )
4334, 41, 42syl2anc 661 . . . . . 6  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( normh `  ( (
1  /  ( normh `  A ) )  .h  ( T `  A
) ) )  =  ( ( abs `  (
1  /  ( normh `  A ) ) )  x.  ( normh `  ( T `  A )
) ) )
4438, 43eqtrd 2493 . . . . 5  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( normh `  ( T `  ( ( 1  / 
( normh `  A )
)  .h  A ) ) )  =  ( ( abs `  (
1  /  ( normh `  A ) ) )  x.  ( normh `  ( T `  A )
) ) )
45 normcl 24674 . . . . . . . . 9  |-  ( ( T `  A )  e.  ~H  ->  ( normh `  ( T `  A ) )  e.  RR )
4640, 45syl 16 . . . . . . . 8  |-  ( A  e.  ~H  ->  ( normh `  ( T `  A ) )  e.  RR )
4746adantr 465 . . . . . . 7  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( normh `  ( T `  A ) )  e.  RR )
4847recnd 9518 . . . . . 6  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( normh `  ( T `  A ) )  e.  CC )
4921recnd 9518 . . . . . 6  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( normh `  A )  e.  CC )
5048, 49, 23divrec2d 10217 . . . . 5  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( ( normh `  ( T `  A )
)  /  ( normh `  A ) )  =  ( ( 1  / 
( normh `  A )
)  x.  ( normh `  ( T `  A
) ) ) )
5133, 44, 503eqtr4rd 2504 . . . 4  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( ( normh `  ( T `  A )
)  /  ( normh `  A ) )  =  ( normh `  ( T `  ( ( 1  / 
( normh `  A )
)  .h  A ) ) ) )
52 hvmulcl 24562 . . . . . 6  |-  ( ( ( 1  /  ( normh `  A ) )  e.  CC  /\  A  e.  ~H )  ->  (
( 1  /  ( normh `  A ) )  .h  A )  e. 
~H )
5334, 35, 52syl2anc 661 . . . . 5  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( ( 1  / 
( normh `  A )
)  .h  A )  e.  ~H )
54 normcl 24674 . . . . . . 7  |-  ( ( ( 1  /  ( normh `  A ) )  .h  A )  e. 
~H  ->  ( normh `  (
( 1  /  ( normh `  A ) )  .h  A ) )  e.  RR )
5553, 54syl 16 . . . . . 6  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( normh `  ( (
1  /  ( normh `  A ) )  .h  A ) )  e.  RR )
56 norm1 24799 . . . . . 6  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( normh `  ( (
1  /  ( normh `  A ) )  .h  A ) )  =  1 )
57 eqle 9583 . . . . . 6  |-  ( ( ( normh `  ( (
1  /  ( normh `  A ) )  .h  A ) )  e.  RR  /\  ( normh `  ( ( 1  / 
( normh `  A )
)  .h  A ) )  =  1 )  ->  ( normh `  (
( 1  /  ( normh `  A ) )  .h  A ) )  <_  1 )
5855, 56, 57syl2anc 661 . . . . 5  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( normh `  ( (
1  /  ( normh `  A ) )  .h  A ) )  <_ 
1 )
59 nmoplb 25458 . . . . . 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
) )
6039, 59mp3an1 1302 . . . . 5  |-  ( ( ( ( 1  / 
( normh `  A )
)  .h  A )  e.  ~H  /\  ( normh `  ( ( 1  /  ( normh `  A
) )  .h  A
) )  <_  1
)  ->  ( normh `  ( T `  (
( 1  /  ( normh `  A ) )  .h  A ) ) )  <_  ( normop `  T
) )
6153, 58, 60syl2anc 661 . . . 4  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( normh `  ( T `  ( ( 1  / 
( normh `  A )
)  .h  A ) ) )  <_  ( normop `  T ) )
6251, 61eqbrtrd 4415 . . 3  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( ( normh `  ( T `  A )
)  /  ( normh `  A ) )  <_ 
( normop `  T )
)
6314a1i 11 . . . 4  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( normop `  T )  e.  RR )
64 ledivmul2 10315 . . . 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 ) ) ) )
6547, 63, 21, 26, 64syl112anc 1223 . . 3  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( ( ( normh `  ( T `  A
) )  /  ( normh `  A ) )  <_  ( normop `  T
)  <->  ( normh `  ( T `  A )
)  <_  ( ( normop `  T )  x.  ( normh `  A ) ) ) )
6662, 65mpbid 210 . 2  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( normh `  ( T `  A ) )  <_ 
( ( normop `  T
)  x.  ( normh `  A ) ) )
6719, 66pm2.61dane 2767 1  |-  ( A  e.  ~H  ->  ( normh `  ( T `  A ) )  <_ 
( ( normop `  T
)  x.  ( normh `  A ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2645   class class class wbr 4395   -->wf 5517   ` cfv 5521  (class class class)co 6195   CCcc 9386   RRcr 9387   0cc0 9388   1c1 9389    x. cmul 9393    < clt 9524    <_ cle 9525    / cdiv 10099   abscabs 12836   ~Hchil 24468    .h csm 24470   normhcno 24472   0hc0v 24473   normopcnop 24494   ConOpccop 24495   LinOpclo 24496
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-cnex 9444  ax-resscn 9445  ax-1cn 9446  ax-icn 9447  ax-addcl 9448  ax-addrcl 9449  ax-mulcl 9450  ax-mulrcl 9451  ax-mulcom 9452  ax-addass 9453  ax-mulass 9454  ax-distr 9455  ax-i2m1 9456  ax-1ne0 9457  ax-1rid 9458  ax-rnegex 9459  ax-rrecex 9460  ax-cnre 9461  ax-pre-lttri 9462  ax-pre-lttrn 9463  ax-pre-ltadd 9464  ax-pre-mulgt0 9465  ax-pre-sup 9466  ax-hilex 24548  ax-hfvadd 24549  ax-hvcom 24550  ax-hvass 24551  ax-hv0cl 24552  ax-hvaddid 24553  ax-hfvmul 24554  ax-hvmulid 24555  ax-hvmulass 24556  ax-hvdistr1 24557  ax-hvdistr2 24558  ax-hvmul0 24559  ax-hfi 24628  ax-his1 24631  ax-his2 24632  ax-his3 24633  ax-his4 24634
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-reu 2803  df-rmo 2804  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-om 6582  df-1st 6682  df-2nd 6683  df-recs 6937  df-rdg 6971  df-er 7206  df-map 7321  df-en 7416  df-dom 7417  df-sdom 7418  df-sup 7797  df-pnf 9526  df-mnf 9527  df-xr 9528  df-ltxr 9529  df-le 9530  df-sub 9703  df-neg 9704  df-div 10100  df-nn 10429  df-2 10486  df-3 10487  df-4 10488  df-n0 10686  df-z 10753  df-uz 10968  df-rp 11098  df-seq 11919  df-exp 11978  df-cj 12701  df-re 12702  df-im 12703  df-sqr 12837  df-abs 12838  df-grpo 23825  df-gid 23826  df-ablo 23916  df-vc 24071  df-nv 24117  df-va 24120  df-ba 24121  df-sm 24122  df-0v 24123  df-nmcv 24125  df-hnorm 24517  df-hba 24518  df-hvsub 24520  df-nmop 25390  df-cnop 25391  df-lnop 25392
This theorem is referenced by:  nmcoplb  25581  cnlnadjlem2  25619  cnlnadjlem7  25624
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