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Theorem cnlnadjlem7 22653
Description: Lemma for cnlnadji 22656. Helper lemma to show that  F is continuous. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.)
Hypotheses
Ref Expression
cnlnadjlem.1  |-  T  e. 
LinOp
cnlnadjlem.2  |-  T  e. 
ConOp
cnlnadjlem.3  |-  G  =  ( g  e.  ~H  |->  ( ( T `  g )  .ih  y
) )
cnlnadjlem.4  |-  B  =  ( iota_ w  e.  ~H A. v  e.  ~H  (
( T `  v
)  .ih  y )  =  ( v  .ih  w ) )
cnlnadjlem.5  |-  F  =  ( y  e.  ~H  |->  B )
Assertion
Ref Expression
cnlnadjlem7  |-  ( A  e.  ~H  ->  ( normh `  ( F `  A ) )  <_ 
( ( normop `  T
)  x.  ( normh `  A ) ) )
Distinct variable groups:    v, g, w, y, A    w, F    T, g, v, w, y   
v, G, w
Allowed substitution hints:    B( y, w, v, g)    F( y, v, g)    G( y, g)

Proof of Theorem cnlnadjlem7
StepHypRef Expression
1 breq1 4026 . 2  |-  ( (
normh `  ( F `  A ) )  =  0  ->  ( ( normh `  ( F `  A ) )  <_ 
( ( normop `  T
)  x.  ( normh `  A ) )  <->  0  <_  ( ( normop `  T )  x.  ( normh `  A )
) ) )
2 cnlnadjlem.1 . . . . . . . . . 10  |-  T  e. 
LinOp
3 cnlnadjlem.2 . . . . . . . . . 10  |-  T  e. 
ConOp
4 cnlnadjlem.3 . . . . . . . . . 10  |-  G  =  ( g  e.  ~H  |->  ( ( T `  g )  .ih  y
) )
5 cnlnadjlem.4 . . . . . . . . . 10  |-  B  =  ( iota_ w  e.  ~H A. v  e.  ~H  (
( T `  v
)  .ih  y )  =  ( v  .ih  w ) )
6 cnlnadjlem.5 . . . . . . . . . 10  |-  F  =  ( y  e.  ~H  |->  B )
72, 3, 4, 5, 6cnlnadjlem4 22650 . . . . . . . . 9  |-  ( A  e.  ~H  ->  ( F `  A )  e.  ~H )
82lnopfi 22549 . . . . . . . . . 10  |-  T : ~H
--> ~H
98ffvelrni 5664 . . . . . . . . 9  |-  ( ( F `  A )  e.  ~H  ->  ( T `  ( F `  A ) )  e. 
~H )
107, 9syl 15 . . . . . . . 8  |-  ( A  e.  ~H  ->  ( T `  ( F `  A ) )  e. 
~H )
11 hicl 21659 . . . . . . . 8  |-  ( ( ( T `  ( F `  A )
)  e.  ~H  /\  A  e.  ~H )  ->  ( ( T `  ( F `  A ) )  .ih  A )  e.  CC )
1210, 11mpancom 650 . . . . . . 7  |-  ( A  e.  ~H  ->  (
( T `  ( F `  A )
)  .ih  A )  e.  CC )
1312abscld 11918 . . . . . 6  |-  ( A  e.  ~H  ->  ( abs `  ( ( T `
 ( F `  A ) )  .ih  A ) )  e.  RR )
14 normcl 21704 . . . . . . . 8  |-  ( ( T `  ( F `
 A ) )  e.  ~H  ->  ( normh `  ( T `  ( F `  A ) ) )  e.  RR )
1510, 14syl 15 . . . . . . 7  |-  ( A  e.  ~H  ->  ( normh `  ( T `  ( F `  A ) ) )  e.  RR )
16 normcl 21704 . . . . . . 7  |-  ( A  e.  ~H  ->  ( normh `  A )  e.  RR )
1715, 16remulcld 8863 . . . . . 6  |-  ( A  e.  ~H  ->  (
( normh `  ( T `  ( F `  A
) ) )  x.  ( normh `  A )
)  e.  RR )
182, 3nmcopexi 22607 . . . . . . . 8  |-  ( normop `  T )  e.  RR
19 normcl 21704 . . . . . . . . 9  |-  ( ( F `  A )  e.  ~H  ->  ( normh `  ( F `  A ) )  e.  RR )
207, 19syl 15 . . . . . . . 8  |-  ( A  e.  ~H  ->  ( normh `  ( F `  A ) )  e.  RR )
21 remulcl 8822 . . . . . . . 8  |-  ( ( ( normop `  T )  e.  RR  /\  ( normh `  ( F `  A
) )  e.  RR )  ->  ( ( normop `  T )  x.  ( normh `  ( F `  A ) ) )  e.  RR )
2218, 20, 21sylancr 644 . . . . . . 7  |-  ( A  e.  ~H  ->  (
( normop `  T )  x.  ( normh `  ( F `  A ) ) )  e.  RR )
2322, 16remulcld 8863 . . . . . 6  |-  ( A  e.  ~H  ->  (
( ( normop `  T
)  x.  ( normh `  ( F `  A
) ) )  x.  ( normh `  A )
)  e.  RR )
24 bcs 21760 . . . . . . 7  |-  ( ( ( T `  ( F `  A )
)  e.  ~H  /\  A  e.  ~H )  ->  ( abs `  (
( T `  ( F `  A )
)  .ih  A )
)  <_  ( ( normh `  ( T `  ( F `  A ) ) )  x.  ( normh `  A ) ) )
2510, 24mpancom 650 . . . . . 6  |-  ( A  e.  ~H  ->  ( abs `  ( ( T `
 ( F `  A ) )  .ih  A ) )  <_  (
( normh `  ( T `  ( F `  A
) ) )  x.  ( normh `  A )
) )
26 normge0 21705 . . . . . . 7  |-  ( A  e.  ~H  ->  0  <_  ( normh `  A )
)
272, 3nmcoplbi 22608 . . . . . . . 8  |-  ( ( F `  A )  e.  ~H  ->  ( normh `  ( T `  ( F `  A ) ) )  <_  (
( normop `  T )  x.  ( normh `  ( F `  A ) ) ) )
287, 27syl 15 . . . . . . 7  |-  ( A  e.  ~H  ->  ( normh `  ( T `  ( F `  A ) ) )  <_  (
( normop `  T )  x.  ( normh `  ( F `  A ) ) ) )
2915, 22, 16, 26, 28lemul1ad 9696 . . . . . 6  |-  ( A  e.  ~H  ->  (
( normh `  ( T `  ( F `  A
) ) )  x.  ( normh `  A )
)  <_  ( (
( normop `  T )  x.  ( normh `  ( F `  A ) ) )  x.  ( normh `  A
) ) )
3013, 17, 23, 25, 29letrd 8973 . . . . 5  |-  ( A  e.  ~H  ->  ( abs `  ( ( T `
 ( F `  A ) )  .ih  A ) )  <_  (
( ( normop `  T
)  x.  ( normh `  ( F `  A
) ) )  x.  ( normh `  A )
) )
312, 3, 4, 5, 6cnlnadjlem5 22651 . . . . . . . 8  |-  ( ( A  e.  ~H  /\  ( F `  A )  e.  ~H )  -> 
( ( T `  ( F `  A ) )  .ih  A )  =  ( ( F `
 A )  .ih  ( F `  A ) ) )
327, 31mpdan 649 . . . . . . 7  |-  ( A  e.  ~H  ->  (
( T `  ( F `  A )
)  .ih  A )  =  ( ( F `
 A )  .ih  ( F `  A ) ) )
3332fveq2d 5529 . . . . . 6  |-  ( A  e.  ~H  ->  ( abs `  ( ( T `
 ( F `  A ) )  .ih  A ) )  =  ( abs `  ( ( F `  A ) 
.ih  ( F `  A ) ) ) )
34 hiidrcl 21674 . . . . . . . 8  |-  ( ( F `  A )  e.  ~H  ->  (
( F `  A
)  .ih  ( F `  A ) )  e.  RR )
357, 34syl 15 . . . . . . 7  |-  ( A  e.  ~H  ->  (
( F `  A
)  .ih  ( F `  A ) )  e.  RR )
36 hiidge0 21677 . . . . . . . 8  |-  ( ( F `  A )  e.  ~H  ->  0  <_  ( ( F `  A )  .ih  ( F `  A )
) )
377, 36syl 15 . . . . . . 7  |-  ( A  e.  ~H  ->  0  <_  ( ( F `  A )  .ih  ( F `  A )
) )
3835, 37absidd 11905 . . . . . 6  |-  ( A  e.  ~H  ->  ( abs `  ( ( F `
 A )  .ih  ( F `  A ) ) )  =  ( ( F `  A
)  .ih  ( F `  A ) ) )
39 normsq 21713 . . . . . . . 8  |-  ( ( F `  A )  e.  ~H  ->  (
( normh `  ( F `  A ) ) ^
2 )  =  ( ( F `  A
)  .ih  ( F `  A ) ) )
407, 39syl 15 . . . . . . 7  |-  ( A  e.  ~H  ->  (
( normh `  ( F `  A ) ) ^
2 )  =  ( ( F `  A
)  .ih  ( F `  A ) ) )
4120recnd 8861 . . . . . . . 8  |-  ( A  e.  ~H  ->  ( normh `  ( F `  A ) )  e.  CC )
4241sqvald 11242 . . . . . . 7  |-  ( A  e.  ~H  ->  (
( normh `  ( F `  A ) ) ^
2 )  =  ( ( normh `  ( F `  A ) )  x.  ( normh `  ( F `  A ) ) ) )
4340, 42eqtr3d 2317 . . . . . 6  |-  ( A  e.  ~H  ->  (
( F `  A
)  .ih  ( F `  A ) )  =  ( ( normh `  ( F `  A )
)  x.  ( normh `  ( F `  A
) ) ) )
4433, 38, 433eqtrd 2319 . . . . 5  |-  ( A  e.  ~H  ->  ( abs `  ( ( T `
 ( F `  A ) )  .ih  A ) )  =  ( ( normh `  ( F `  A ) )  x.  ( normh `  ( F `  A ) ) ) )
4516recnd 8861 . . . . . 6  |-  ( A  e.  ~H  ->  ( normh `  A )  e.  CC )
4618recni 8849 . . . . . . 7  |-  ( normop `  T )  e.  CC
47 mul32 8979 . . . . . . 7  |-  ( ( ( normop `  T )  e.  CC  /\  ( normh `  ( F `  A
) )  e.  CC  /\  ( normh `  A )  e.  CC )  ->  (
( ( normop `  T
)  x.  ( normh `  ( F `  A
) ) )  x.  ( normh `  A )
)  =  ( ( ( normop `  T )  x.  ( normh `  A )
)  x.  ( normh `  ( F `  A
) ) ) )
4846, 47mp3an1 1264 . . . . . 6  |-  ( ( ( normh `  ( F `  A ) )  e.  CC  /\  ( normh `  A )  e.  CC )  ->  ( ( (
normop `  T )  x.  ( normh `  ( F `  A ) ) )  x.  ( normh `  A
) )  =  ( ( ( normop `  T
)  x.  ( normh `  A ) )  x.  ( normh `  ( F `  A ) ) ) )
4941, 45, 48syl2anc 642 . . . . 5  |-  ( A  e.  ~H  ->  (
( ( normop `  T
)  x.  ( normh `  ( F `  A
) ) )  x.  ( normh `  A )
)  =  ( ( ( normop `  T )  x.  ( normh `  A )
)  x.  ( normh `  ( F `  A
) ) ) )
5030, 44, 493brtr3d 4052 . . . 4  |-  ( A  e.  ~H  ->  (
( normh `  ( F `  A ) )  x.  ( normh `  ( F `  A ) ) )  <_  ( ( (
normop `  T )  x.  ( normh `  A )
)  x.  ( normh `  ( F `  A
) ) ) )
5150adantr 451 . . 3  |-  ( ( A  e.  ~H  /\  ( normh `  ( F `  A ) )  =/=  0 )  ->  (
( normh `  ( F `  A ) )  x.  ( normh `  ( F `  A ) ) )  <_  ( ( (
normop `  T )  x.  ( normh `  A )
)  x.  ( normh `  ( F `  A
) ) ) )
5220adantr 451 . . . 4  |-  ( ( A  e.  ~H  /\  ( normh `  ( F `  A ) )  =/=  0 )  ->  ( normh `  ( F `  A ) )  e.  RR )
53 remulcl 8822 . . . . . 6  |-  ( ( ( normop `  T )  e.  RR  /\  ( normh `  A )  e.  RR )  ->  ( ( normop `  T )  x.  ( normh `  A ) )  e.  RR )
5418, 16, 53sylancr 644 . . . . 5  |-  ( A  e.  ~H  ->  (
( normop `  T )  x.  ( normh `  A )
)  e.  RR )
5554adantr 451 . . . 4  |-  ( ( A  e.  ~H  /\  ( normh `  ( F `  A ) )  =/=  0 )  ->  (
( normop `  T )  x.  ( normh `  A )
)  e.  RR )
56 normge0 21705 . . . . . . 7  |-  ( ( F `  A )  e.  ~H  ->  0  <_  ( normh `  ( F `  A ) ) )
57 0re 8838 . . . . . . . 8  |-  0  e.  RR
58 leltne 8911 . . . . . . . 8  |-  ( ( 0  e.  RR  /\  ( normh `  ( F `  A ) )  e.  RR  /\  0  <_ 
( normh `  ( F `  A ) ) )  ->  ( 0  < 
( normh `  ( F `  A ) )  <->  ( normh `  ( F `  A
) )  =/=  0
) )
5957, 58mp3an1 1264 . . . . . . 7  |-  ( ( ( normh `  ( F `  A ) )  e.  RR  /\  0  <_ 
( normh `  ( F `  A ) ) )  ->  ( 0  < 
( normh `  ( F `  A ) )  <->  ( normh `  ( F `  A
) )  =/=  0
) )
6019, 56, 59syl2anc 642 . . . . . 6  |-  ( ( F `  A )  e.  ~H  ->  (
0  <  ( normh `  ( F `  A
) )  <->  ( normh `  ( F `  A
) )  =/=  0
) )
6160biimpar 471 . . . . 5  |-  ( ( ( F `  A
)  e.  ~H  /\  ( normh `  ( F `  A ) )  =/=  0 )  ->  0  <  ( normh `  ( F `  A ) ) )
627, 61sylan 457 . . . 4  |-  ( ( A  e.  ~H  /\  ( normh `  ( F `  A ) )  =/=  0 )  ->  0  <  ( normh `  ( F `  A ) ) )
63 lemul1 9608 . . . 4  |-  ( ( ( normh `  ( F `  A ) )  e.  RR  /\  ( (
normop `  T )  x.  ( normh `  A )
)  e.  RR  /\  ( ( normh `  ( F `  A )
)  e.  RR  /\  0  <  ( normh `  ( F `  A )
) ) )  -> 
( ( normh `  ( F `  A )
)  <_  ( ( normop `  T )  x.  ( normh `  A ) )  <-> 
( ( normh `  ( F `  A )
)  x.  ( normh `  ( F `  A
) ) )  <_ 
( ( ( normop `  T )  x.  ( normh `  A ) )  x.  ( normh `  ( F `  A )
) ) ) )
6452, 55, 52, 62, 63syl112anc 1186 . . 3  |-  ( ( A  e.  ~H  /\  ( normh `  ( F `  A ) )  =/=  0 )  ->  (
( normh `  ( F `  A ) )  <_ 
( ( normop `  T
)  x.  ( normh `  A ) )  <->  ( ( normh `  ( F `  A ) )  x.  ( normh `  ( F `  A ) ) )  <_  ( ( (
normop `  T )  x.  ( normh `  A )
)  x.  ( normh `  ( F `  A
) ) ) ) )
6551, 64mpbird 223 . 2  |-  ( ( A  e.  ~H  /\  ( normh `  ( F `  A ) )  =/=  0 )  ->  ( normh `  ( F `  A ) )  <_ 
( ( normop `  T
)  x.  ( normh `  A ) ) )
66 nmopge0 22491 . . . . 5  |-  ( T : ~H --> ~H  ->  0  <_  ( normop `  T
) )
678, 66ax-mp 8 . . . 4  |-  0  <_  ( normop `  T )
68 mulge0 9291 . . . 4  |-  ( ( ( ( normop `  T
)  e.  RR  /\  0  <_  ( normop `  T
) )  /\  (
( normh `  A )  e.  RR  /\  0  <_ 
( normh `  A )
) )  ->  0  <_  ( ( normop `  T
)  x.  ( normh `  A ) ) )
6918, 67, 68mpanl12 663 . . 3  |-  ( ( ( normh `  A )  e.  RR  /\  0  <_ 
( normh `  A )
)  ->  0  <_  ( ( normop `  T )  x.  ( normh `  A )
) )
7016, 26, 69syl2anc 642 . 2  |-  ( A  e.  ~H  ->  0  <_  ( ( normop `  T
)  x.  ( normh `  A ) ) )
711, 65, 70pm2.61ne 2521 1  |-  ( A  e.  ~H  ->  ( normh `  ( F `  A ) )  <_ 
( ( normop `  T
)  x.  ( normh `  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   class class class wbr 4023    e. cmpt 4077   -->wf 5251   ` cfv 5255  (class class class)co 5858   iota_crio 6297   CCcc 8735   RRcr 8736   0cc0 8737    x. cmul 8742    < clt 8867    <_ cle 8868   2c2 9795   ^cexp 11104   abscabs 11719   ~Hchil 21499    .ih csp 21502   normhcno 21503   normopcnop 21525   ConOpccop 21526   LinOpclo 21527
This theorem is referenced by:  cnlnadjlem8  22654  nmopadjlei  22668
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cc 8061  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817  ax-hilex 21579  ax-hfvadd 21580  ax-hvcom 21581  ax-hvass 21582  ax-hv0cl 21583  ax-hvaddid 21584  ax-hfvmul 21585  ax-hvmulid 21586  ax-hvmulass 21587  ax-hvdistr1 21588  ax-hvdistr2 21589  ax-hvmul0 21590  ax-hfi 21658  ax-his1 21661  ax-his2 21662  ax-his3 21663  ax-his4 21664  ax-hcompl 21781
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-omul 6484  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-acn 7575  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-fl 10925  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-rlim 11963  df-sum 12159  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-pt 13345  df-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  df-qtop 13410  df-imas 13411  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-mulg 14492  df-cntz 14793  df-cmn 15091  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-cn 16957  df-cnp 16958  df-lm 16959  df-t1 17042  df-haus 17043  df-tx 17257  df-hmeo 17446  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633  df-flim 17634  df-flf 17635  df-xms 17885  df-ms 17886  df-tms 17887  df-cfil 18681  df-cau 18682  df-cmet 18683  df-grpo 20858  df-gid 20859  df-ginv 20860  df-gdiv 20861  df-ablo 20949  df-subgo 20969  df-vc 21102  df-nv 21148  df-va 21151  df-ba 21152  df-sm 21153  df-0v 21154  df-vs 21155  df-nmcv 21156  df-ims 21157  df-dip 21274  df-ssp 21298  df-ph 21391  df-cbn 21442  df-hnorm 21548  df-hba 21549  df-hvsub 21551  df-hlim 21552  df-hcau 21553  df-sh 21786  df-ch 21801  df-oc 21831  df-ch0 21832  df-nmop 22419  df-cnop 22420  df-lnop 22421  df-nmfn 22425  df-nlfn 22426  df-cnfn 22427  df-lnfn 22428
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