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Theorem cnlnadjlem7 27108
Description: Lemma for cnlnadji 27111. 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 4370 . 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 27105 . . . . . . . . 9  |-  ( A  e.  ~H  ->  ( F `  A )  e.  ~H )
82lnopfi 27004 . . . . . . . . . 10  |-  T : ~H
--> ~H
98ffvelrni 5932 . . . . . . . . 9  |-  ( ( F `  A )  e.  ~H  ->  ( T `  ( F `  A ) )  e. 
~H )
107, 9syl 16 . . . . . . . 8  |-  ( A  e.  ~H  ->  ( T `  ( F `  A ) )  e. 
~H )
11 hicl 26114 . . . . . . . 8  |-  ( ( ( T `  ( F `  A )
)  e.  ~H  /\  A  e.  ~H )  ->  ( ( T `  ( F `  A ) )  .ih  A )  e.  CC )
1210, 11mpancom 667 . . . . . . 7  |-  ( A  e.  ~H  ->  (
( T `  ( F `  A )
)  .ih  A )  e.  CC )
1312abscld 13269 . . . . . 6  |-  ( A  e.  ~H  ->  ( abs `  ( ( T `
 ( F `  A ) )  .ih  A ) )  e.  RR )
14 normcl 26159 . . . . . . . 8  |-  ( ( T `  ( F `
 A ) )  e.  ~H  ->  ( normh `  ( T `  ( F `  A ) ) )  e.  RR )
1510, 14syl 16 . . . . . . 7  |-  ( A  e.  ~H  ->  ( normh `  ( T `  ( F `  A ) ) )  e.  RR )
16 normcl 26159 . . . . . . 7  |-  ( A  e.  ~H  ->  ( normh `  A )  e.  RR )
1715, 16remulcld 9535 . . . . . 6  |-  ( A  e.  ~H  ->  (
( normh `  ( T `  ( F `  A
) ) )  x.  ( normh `  A )
)  e.  RR )
182, 3nmcopexi 27062 . . . . . . . 8  |-  ( normop `  T )  e.  RR
19 normcl 26159 . . . . . . . . 9  |-  ( ( F `  A )  e.  ~H  ->  ( normh `  ( F `  A ) )  e.  RR )
207, 19syl 16 . . . . . . . 8  |-  ( A  e.  ~H  ->  ( normh `  ( F `  A ) )  e.  RR )
21 remulcl 9488 . . . . . . . 8  |-  ( ( ( normop `  T )  e.  RR  /\  ( normh `  ( F `  A
) )  e.  RR )  ->  ( ( normop `  T )  x.  ( normh `  ( F `  A ) ) )  e.  RR )
2218, 20, 21sylancr 661 . . . . . . 7  |-  ( A  e.  ~H  ->  (
( normop `  T )  x.  ( normh `  ( F `  A ) ) )  e.  RR )
2322, 16remulcld 9535 . . . . . 6  |-  ( A  e.  ~H  ->  (
( ( normop `  T
)  x.  ( normh `  ( F `  A
) ) )  x.  ( normh `  A )
)  e.  RR )
24 bcs 26215 . . . . . . 7  |-  ( ( ( T `  ( F `  A )
)  e.  ~H  /\  A  e.  ~H )  ->  ( abs `  (
( T `  ( F `  A )
)  .ih  A )
)  <_  ( ( normh `  ( T `  ( F `  A ) ) )  x.  ( normh `  A ) ) )
2510, 24mpancom 667 . . . . . 6  |-  ( A  e.  ~H  ->  ( abs `  ( ( T `
 ( F `  A ) )  .ih  A ) )  <_  (
( normh `  ( T `  ( F `  A
) ) )  x.  ( normh `  A )
) )
26 normge0 26160 . . . . . . 7  |-  ( A  e.  ~H  ->  0  <_  ( normh `  A )
)
272, 3nmcoplbi 27063 . . . . . . . 8  |-  ( ( F `  A )  e.  ~H  ->  ( normh `  ( T `  ( F `  A ) ) )  <_  (
( normop `  T )  x.  ( normh `  ( F `  A ) ) ) )
287, 27syl 16 . . . . . . 7  |-  ( A  e.  ~H  ->  ( normh `  ( T `  ( F `  A ) ) )  <_  (
( normop `  T )  x.  ( normh `  ( F `  A ) ) ) )
2915, 22, 16, 26, 28lemul1ad 10401 . . . . . 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 9650 . . . . 5  |-  ( A  e.  ~H  ->  ( abs `  ( ( T `
 ( F `  A ) )  .ih  A ) )  <_  (
( ( normop `  T
)  x.  ( normh `  ( F `  A
) ) )  x.  ( normh `  A )
) )
312, 3, 4, 5, 6cnlnadjlem5 27106 . . . . . . . 8  |-  ( ( A  e.  ~H  /\  ( F `  A )  e.  ~H )  -> 
( ( T `  ( F `  A ) )  .ih  A )  =  ( ( F `
 A )  .ih  ( F `  A ) ) )
327, 31mpdan 666 . . . . . . 7  |-  ( A  e.  ~H  ->  (
( T `  ( F `  A )
)  .ih  A )  =  ( ( F `
 A )  .ih  ( F `  A ) ) )
3332fveq2d 5778 . . . . . 6  |-  ( A  e.  ~H  ->  ( abs `  ( ( T `
 ( F `  A ) )  .ih  A ) )  =  ( abs `  ( ( F `  A ) 
.ih  ( F `  A ) ) ) )
34 hiidrcl 26129 . . . . . . . 8  |-  ( ( F `  A )  e.  ~H  ->  (
( F `  A
)  .ih  ( F `  A ) )  e.  RR )
357, 34syl 16 . . . . . . 7  |-  ( A  e.  ~H  ->  (
( F `  A
)  .ih  ( F `  A ) )  e.  RR )
36 hiidge0 26132 . . . . . . . 8  |-  ( ( F `  A )  e.  ~H  ->  0  <_  ( ( F `  A )  .ih  ( F `  A )
) )
377, 36syl 16 . . . . . . 7  |-  ( A  e.  ~H  ->  0  <_  ( ( F `  A )  .ih  ( F `  A )
) )
3835, 37absidd 13256 . . . . . 6  |-  ( A  e.  ~H  ->  ( abs `  ( ( F `
 A )  .ih  ( F `  A ) ) )  =  ( ( F `  A
)  .ih  ( F `  A ) ) )
39 normsq 26168 . . . . . . . 8  |-  ( ( F `  A )  e.  ~H  ->  (
( normh `  ( F `  A ) ) ^
2 )  =  ( ( F `  A
)  .ih  ( F `  A ) ) )
407, 39syl 16 . . . . . . 7  |-  ( A  e.  ~H  ->  (
( normh `  ( F `  A ) ) ^
2 )  =  ( ( F `  A
)  .ih  ( F `  A ) ) )
4120recnd 9533 . . . . . . . 8  |-  ( A  e.  ~H  ->  ( normh `  ( F `  A ) )  e.  CC )
4241sqvald 12209 . . . . . . 7  |-  ( A  e.  ~H  ->  (
( normh `  ( F `  A ) ) ^
2 )  =  ( ( normh `  ( F `  A ) )  x.  ( normh `  ( F `  A ) ) ) )
4340, 42eqtr3d 2425 . . . . . 6  |-  ( A  e.  ~H  ->  (
( F `  A
)  .ih  ( F `  A ) )  =  ( ( normh `  ( F `  A )
)  x.  ( normh `  ( F `  A
) ) ) )
4433, 38, 433eqtrd 2427 . . . . 5  |-  ( A  e.  ~H  ->  ( abs `  ( ( T `
 ( F `  A ) )  .ih  A ) )  =  ( ( normh `  ( F `  A ) )  x.  ( normh `  ( F `  A ) ) ) )
4516recnd 9533 . . . . . 6  |-  ( A  e.  ~H  ->  ( normh `  A )  e.  CC )
4618recni 9519 . . . . . . 7  |-  ( normop `  T )  e.  CC
47 mul32 9658 . . . . . . 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 1309 . . . . . 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 659 . . . . 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 4396 . . . 4  |-  ( A  e.  ~H  ->  (
( normh `  ( F `  A ) )  x.  ( normh `  ( F `  A ) ) )  <_  ( ( (
normop `  T )  x.  ( normh `  A )
)  x.  ( normh `  ( F `  A
) ) ) )
5150adantr 463 . . 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 463 . . . 4  |-  ( ( A  e.  ~H  /\  ( normh `  ( F `  A ) )  =/=  0 )  ->  ( normh `  ( F `  A ) )  e.  RR )
53 remulcl 9488 . . . . . 6  |-  ( ( ( normop `  T )  e.  RR  /\  ( normh `  A )  e.  RR )  ->  ( ( normop `  T )  x.  ( normh `  A ) )  e.  RR )
5418, 16, 53sylancr 661 . . . . 5  |-  ( A  e.  ~H  ->  (
( normop `  T )  x.  ( normh `  A )
)  e.  RR )
5554adantr 463 . . . 4  |-  ( ( A  e.  ~H  /\  ( normh `  ( F `  A ) )  =/=  0 )  ->  (
( normop `  T )  x.  ( normh `  A )
)  e.  RR )
56 normge0 26160 . . . . . . 7  |-  ( ( F `  A )  e.  ~H  ->  0  <_  ( normh `  ( F `  A ) ) )
57 0re 9507 . . . . . . . 8  |-  0  e.  RR
58 leltne 9585 . . . . . . . 8  |-  ( ( 0  e.  RR  /\  ( normh `  ( F `  A ) )  e.  RR  /\  0  <_ 
( normh `  ( F `  A ) ) )  ->  ( 0  < 
( normh `  ( F `  A ) )  <->  ( normh `  ( F `  A
) )  =/=  0
) )
5957, 58mp3an1 1309 . . . . . . 7  |-  ( ( ( normh `  ( F `  A ) )  e.  RR  /\  0  <_ 
( normh `  ( F `  A ) ) )  ->  ( 0  < 
( normh `  ( F `  A ) )  <->  ( normh `  ( F `  A
) )  =/=  0
) )
6019, 56, 59syl2anc 659 . . . . . 6  |-  ( ( F `  A )  e.  ~H  ->  (
0  <  ( normh `  ( F `  A
) )  <->  ( normh `  ( F `  A
) )  =/=  0
) )
6160biimpar 483 . . . . 5  |-  ( ( ( F `  A
)  e.  ~H  /\  ( normh `  ( F `  A ) )  =/=  0 )  ->  0  <  ( normh `  ( F `  A ) ) )
627, 61sylan 469 . . . 4  |-  ( ( A  e.  ~H  /\  ( normh `  ( F `  A ) )  =/=  0 )  ->  0  <  ( normh `  ( F `  A ) ) )
63 lemul1 10311 . . . 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 1230 . . 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 232 . 2  |-  ( ( A  e.  ~H  /\  ( normh `  ( F `  A ) )  =/=  0 )  ->  ( normh `  ( F `  A ) )  <_ 
( ( normop `  T
)  x.  ( normh `  A ) ) )
66 nmopge0 26946 . . . . 5  |-  ( T : ~H --> ~H  ->  0  <_  ( normop `  T
) )
678, 66ax-mp 5 . . . 4  |-  0  <_  ( normop `  T )
68 mulge0 9988 . . . 4  |-  ( ( ( ( normop `  T
)  e.  RR  /\  0  <_  ( normop `  T
) )  /\  (
( normh `  A )  e.  RR  /\  0  <_ 
( normh `  A )
) )  ->  0  <_  ( ( normop `  T
)  x.  ( normh `  A ) ) )
6918, 67, 68mpanl12 680 . . 3  |-  ( ( ( normh `  A )  e.  RR  /\  0  <_ 
( normh `  A )
)  ->  0  <_  ( ( normop `  T )  x.  ( normh `  A )
) )
7016, 26, 69syl2anc 659 . 2  |-  ( A  e.  ~H  ->  0  <_  ( ( normop `  T
)  x.  ( normh `  A ) ) )
711, 65, 70pm2.61ne 2697 1  |-  ( A  e.  ~H  ->  ( normh `  ( F `  A ) )  <_ 
( ( normop `  T
)  x.  ( normh `  A ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1399    e. wcel 1826    =/= wne 2577   A.wral 2732   class class class wbr 4367    |-> cmpt 4425   -->wf 5492   ` cfv 5496   iota_crio 6157  (class class class)co 6196   CCcc 9401   RRcr 9402   0cc0 9403    x. cmul 9408    < clt 9539    <_ cle 9540   2c2 10502   ^cexp 12069   abscabs 13069   ~Hchil 25953    .ih csp 25956   normhcno 25957   normopcnop 25979   ConOpccop 25980   LinOpclo 25981
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-inf2 7972  ax-cc 8728  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480  ax-pre-sup 9481  ax-addf 9482  ax-mulf 9483  ax-hilex 26033  ax-hfvadd 26034  ax-hvcom 26035  ax-hvass 26036  ax-hv0cl 26037  ax-hvaddid 26038  ax-hfvmul 26039  ax-hvmulid 26040  ax-hvmulass 26041  ax-hvdistr1 26042  ax-hvdistr2 26043  ax-hvmul0 26044  ax-hfi 26113  ax-his1 26116  ax-his2 26117  ax-his3 26118  ax-his4 26119  ax-hcompl 26236
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-fal 1405  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-int 4200  df-iun 4245  df-iin 4246  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-se 4753  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-isom 5505  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-of 6439  df-om 6600  df-1st 6699  df-2nd 6700  df-supp 6818  df-recs 6960  df-rdg 6994  df-1o 7048  df-2o 7049  df-oadd 7052  df-omul 7053  df-er 7229  df-map 7340  df-pm 7341  df-ixp 7389  df-en 7436  df-dom 7437  df-sdom 7438  df-fin 7439  df-fsupp 7745  df-fi 7786  df-sup 7816  df-oi 7850  df-card 8233  df-acn 8236  df-cda 8461  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-div 10124  df-nn 10453  df-2 10511  df-3 10512  df-4 10513  df-5 10514  df-6 10515  df-7 10516  df-8 10517  df-9 10518  df-10 10519  df-n0 10713  df-z 10782  df-dec 10896  df-uz 11002  df-q 11102  df-rp 11140  df-xneg 11239  df-xadd 11240  df-xmul 11241  df-ioo 11454  df-ico 11456  df-icc 11457  df-fz 11594  df-fzo 11718  df-fl 11828  df-seq 12011  df-exp 12070  df-hash 12308  df-cj 12934  df-re 12935  df-im 12936  df-sqrt 13070  df-abs 13071  df-clim 13313  df-rlim 13314  df-sum 13511  df-struct 14636  df-ndx 14637  df-slot 14638  df-base 14639  df-sets 14640  df-ress 14641  df-plusg 14715  df-mulr 14716  df-starv 14717  df-sca 14718  df-vsca 14719  df-ip 14720  df-tset 14721  df-ple 14722  df-ds 14724  df-unif 14725  df-hom 14726  df-cco 14727  df-rest 14830  df-topn 14831  df-0g 14849  df-gsum 14850  df-topgen 14851  df-pt 14852  df-prds 14855  df-xrs 14909  df-qtop 14914  df-imas 14915  df-xps 14917  df-mre 14993  df-mrc 14994  df-acs 14996  df-mgm 15989  df-sgrp 16028  df-mnd 16038  df-submnd 16084  df-mulg 16177  df-cntz 16472  df-cmn 16917  df-psmet 18524  df-xmet 18525  df-met 18526  df-bl 18527  df-mopn 18528  df-fbas 18529  df-fg 18530  df-cnfld 18534  df-top 19484  df-bases 19486  df-topon 19487  df-topsp 19488  df-cld 19605  df-ntr 19606  df-cls 19607  df-nei 19685  df-cn 19814  df-cnp 19815  df-lm 19816  df-t1 19901  df-haus 19902  df-tx 20148  df-hmeo 20341  df-fil 20432  df-fm 20524  df-flim 20525  df-flf 20526  df-xms 20908  df-ms 20909  df-tms 20910  df-cfil 21779  df-cau 21780  df-cmet 21781  df-grpo 25310  df-gid 25311  df-ginv 25312  df-gdiv 25313  df-ablo 25401  df-subgo 25421  df-vc 25556  df-nv 25602  df-va 25605  df-ba 25606  df-sm 25607  df-0v 25608  df-vs 25609  df-nmcv 25610  df-ims 25611  df-dip 25728  df-ssp 25752  df-ph 25845  df-cbn 25896  df-hnorm 26002  df-hba 26003  df-hvsub 26005  df-hlim 26006  df-hcau 26007  df-sh 26241  df-ch 26256  df-oc 26287  df-ch0 26288  df-nmop 26874  df-cnop 26875  df-lnop 26876  df-nmfn 26880  df-nlfn 26881  df-cnfn 26882  df-lnfn 26883
This theorem is referenced by:  cnlnadjlem8  27109  nmopadjlei  27123
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