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Theorem cnlnadjlem7 23576
Description: Lemma for cnlnadji 23579. 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 4215 . 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 23573 . . . . . . . . 9  |-  ( A  e.  ~H  ->  ( F `  A )  e.  ~H )
82lnopfi 23472 . . . . . . . . . 10  |-  T : ~H
--> ~H
98ffvelrni 5869 . . . . . . . . 9  |-  ( ( F `  A )  e.  ~H  ->  ( T `  ( F `  A ) )  e. 
~H )
107, 9syl 16 . . . . . . . 8  |-  ( A  e.  ~H  ->  ( T `  ( F `  A ) )  e. 
~H )
11 hicl 22582 . . . . . . . 8  |-  ( ( ( T `  ( F `  A )
)  e.  ~H  /\  A  e.  ~H )  ->  ( ( T `  ( F `  A ) )  .ih  A )  e.  CC )
1210, 11mpancom 651 . . . . . . 7  |-  ( A  e.  ~H  ->  (
( T `  ( F `  A )
)  .ih  A )  e.  CC )
1312abscld 12238 . . . . . 6  |-  ( A  e.  ~H  ->  ( abs `  ( ( T `
 ( F `  A ) )  .ih  A ) )  e.  RR )
14 normcl 22627 . . . . . . . 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 22627 . . . . . . 7  |-  ( A  e.  ~H  ->  ( normh `  A )  e.  RR )
1715, 16remulcld 9116 . . . . . 6  |-  ( A  e.  ~H  ->  (
( normh `  ( T `  ( F `  A
) ) )  x.  ( normh `  A )
)  e.  RR )
182, 3nmcopexi 23530 . . . . . . . 8  |-  ( normop `  T )  e.  RR
19 normcl 22627 . . . . . . . . 9  |-  ( ( F `  A )  e.  ~H  ->  ( normh `  ( F `  A ) )  e.  RR )
207, 19syl 16 . . . . . . . 8  |-  ( A  e.  ~H  ->  ( normh `  ( F `  A ) )  e.  RR )
21 remulcl 9075 . . . . . . . 8  |-  ( ( ( normop `  T )  e.  RR  /\  ( normh `  ( F `  A
) )  e.  RR )  ->  ( ( normop `  T )  x.  ( normh `  ( F `  A ) ) )  e.  RR )
2218, 20, 21sylancr 645 . . . . . . 7  |-  ( A  e.  ~H  ->  (
( normop `  T )  x.  ( normh `  ( F `  A ) ) )  e.  RR )
2322, 16remulcld 9116 . . . . . 6  |-  ( A  e.  ~H  ->  (
( ( normop `  T
)  x.  ( normh `  ( F `  A
) ) )  x.  ( normh `  A )
)  e.  RR )
24 bcs 22683 . . . . . . 7  |-  ( ( ( T `  ( F `  A )
)  e.  ~H  /\  A  e.  ~H )  ->  ( abs `  (
( T `  ( F `  A )
)  .ih  A )
)  <_  ( ( normh `  ( T `  ( F `  A ) ) )  x.  ( normh `  A ) ) )
2510, 24mpancom 651 . . . . . 6  |-  ( A  e.  ~H  ->  ( abs `  ( ( T `
 ( F `  A ) )  .ih  A ) )  <_  (
( normh `  ( T `  ( F `  A
) ) )  x.  ( normh `  A )
) )
26 normge0 22628 . . . . . . 7  |-  ( A  e.  ~H  ->  0  <_  ( normh `  A )
)
272, 3nmcoplbi 23531 . . . . . . . 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 9950 . . . . . 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 9227 . . . . 5  |-  ( A  e.  ~H  ->  ( abs `  ( ( T `
 ( F `  A ) )  .ih  A ) )  <_  (
( ( normop `  T
)  x.  ( normh `  ( F `  A
) ) )  x.  ( normh `  A )
) )
312, 3, 4, 5, 6cnlnadjlem5 23574 . . . . . . . 8  |-  ( ( A  e.  ~H  /\  ( F `  A )  e.  ~H )  -> 
( ( T `  ( F `  A ) )  .ih  A )  =  ( ( F `
 A )  .ih  ( F `  A ) ) )
327, 31mpdan 650 . . . . . . 7  |-  ( A  e.  ~H  ->  (
( T `  ( F `  A )
)  .ih  A )  =  ( ( F `
 A )  .ih  ( F `  A ) ) )
3332fveq2d 5732 . . . . . 6  |-  ( A  e.  ~H  ->  ( abs `  ( ( T `
 ( F `  A ) )  .ih  A ) )  =  ( abs `  ( ( F `  A ) 
.ih  ( F `  A ) ) ) )
34 hiidrcl 22597 . . . . . . . 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 22600 . . . . . . . 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 12225 . . . . . 6  |-  ( A  e.  ~H  ->  ( abs `  ( ( F `
 A )  .ih  ( F `  A ) ) )  =  ( ( F `  A
)  .ih  ( F `  A ) ) )
39 normsq 22636 . . . . . . . 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 9114 . . . . . . . 8  |-  ( A  e.  ~H  ->  ( normh `  ( F `  A ) )  e.  CC )
4241sqvald 11520 . . . . . . 7  |-  ( A  e.  ~H  ->  (
( normh `  ( F `  A ) ) ^
2 )  =  ( ( normh `  ( F `  A ) )  x.  ( normh `  ( F `  A ) ) ) )
4340, 42eqtr3d 2470 . . . . . 6  |-  ( A  e.  ~H  ->  (
( F `  A
)  .ih  ( F `  A ) )  =  ( ( normh `  ( F `  A )
)  x.  ( normh `  ( F `  A
) ) ) )
4433, 38, 433eqtrd 2472 . . . . 5  |-  ( A  e.  ~H  ->  ( abs `  ( ( T `
 ( F `  A ) )  .ih  A ) )  =  ( ( normh `  ( F `  A ) )  x.  ( normh `  ( F `  A ) ) ) )
4516recnd 9114 . . . . . 6  |-  ( A  e.  ~H  ->  ( normh `  A )  e.  CC )
4618recni 9102 . . . . . . 7  |-  ( normop `  T )  e.  CC
47 mul32 9233 . . . . . . 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 1266 . . . . . 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 643 . . . . 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 4241 . . . 4  |-  ( A  e.  ~H  ->  (
( normh `  ( F `  A ) )  x.  ( normh `  ( F `  A ) ) )  <_  ( ( (
normop `  T )  x.  ( normh `  A )
)  x.  ( normh `  ( F `  A
) ) ) )
5150adantr 452 . . 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 452 . . . 4  |-  ( ( A  e.  ~H  /\  ( normh `  ( F `  A ) )  =/=  0 )  ->  ( normh `  ( F `  A ) )  e.  RR )
53 remulcl 9075 . . . . . 6  |-  ( ( ( normop `  T )  e.  RR  /\  ( normh `  A )  e.  RR )  ->  ( ( normop `  T )  x.  ( normh `  A ) )  e.  RR )
5418, 16, 53sylancr 645 . . . . 5  |-  ( A  e.  ~H  ->  (
( normop `  T )  x.  ( normh `  A )
)  e.  RR )
5554adantr 452 . . . 4  |-  ( ( A  e.  ~H  /\  ( normh `  ( F `  A ) )  =/=  0 )  ->  (
( normop `  T )  x.  ( normh `  A )
)  e.  RR )
56 normge0 22628 . . . . . . 7  |-  ( ( F `  A )  e.  ~H  ->  0  <_  ( normh `  ( F `  A ) ) )
57 0re 9091 . . . . . . . 8  |-  0  e.  RR
58 leltne 9164 . . . . . . . 8  |-  ( ( 0  e.  RR  /\  ( normh `  ( F `  A ) )  e.  RR  /\  0  <_ 
( normh `  ( F `  A ) ) )  ->  ( 0  < 
( normh `  ( F `  A ) )  <->  ( normh `  ( F `  A
) )  =/=  0
) )
5957, 58mp3an1 1266 . . . . . . 7  |-  ( ( ( normh `  ( F `  A ) )  e.  RR  /\  0  <_ 
( normh `  ( F `  A ) ) )  ->  ( 0  < 
( normh `  ( F `  A ) )  <->  ( normh `  ( F `  A
) )  =/=  0
) )
6019, 56, 59syl2anc 643 . . . . . 6  |-  ( ( F `  A )  e.  ~H  ->  (
0  <  ( normh `  ( F `  A
) )  <->  ( normh `  ( F `  A
) )  =/=  0
) )
6160biimpar 472 . . . . 5  |-  ( ( ( F `  A
)  e.  ~H  /\  ( normh `  ( F `  A ) )  =/=  0 )  ->  0  <  ( normh `  ( F `  A ) ) )
627, 61sylan 458 . . . 4  |-  ( ( A  e.  ~H  /\  ( normh `  ( F `  A ) )  =/=  0 )  ->  0  <  ( normh `  ( F `  A ) ) )
63 lemul1 9862 . . . 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 1188 . . 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 224 . 2  |-  ( ( A  e.  ~H  /\  ( normh `  ( F `  A ) )  =/=  0 )  ->  ( normh `  ( F `  A ) )  <_ 
( ( normop `  T
)  x.  ( normh `  A ) ) )
66 nmopge0 23414 . . . . 5  |-  ( T : ~H --> ~H  ->  0  <_  ( normop `  T
) )
678, 66ax-mp 8 . . . 4  |-  0  <_  ( normop `  T )
68 mulge0 9545 . . . 4  |-  ( ( ( ( normop `  T
)  e.  RR  /\  0  <_  ( normop `  T
) )  /\  (
( normh `  A )  e.  RR  /\  0  <_ 
( normh `  A )
) )  ->  0  <_  ( ( normop `  T
)  x.  ( normh `  A ) ) )
6918, 67, 68mpanl12 664 . . 3  |-  ( ( ( normh `  A )  e.  RR  /\  0  <_ 
( normh `  A )
)  ->  0  <_  ( ( normop `  T )  x.  ( normh `  A )
) )
7016, 26, 69syl2anc 643 . 2  |-  ( A  e.  ~H  ->  0  <_  ( ( normop `  T
)  x.  ( normh `  A ) ) )
711, 65, 70pm2.61ne 2679 1  |-  ( A  e.  ~H  ->  ( normh `  ( F `  A ) )  <_ 
( ( normop `  T
)  x.  ( normh `  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599   A.wral 2705   class class class wbr 4212    e. cmpt 4266   -->wf 5450   ` cfv 5454  (class class class)co 6081   iota_crio 6542   CCcc 8988   RRcr 8989   0cc0 8990    x. cmul 8995    < clt 9120    <_ cle 9121   2c2 10049   ^cexp 11382   abscabs 12039   ~Hchil 22422    .ih csp 22425   normhcno 22426   normopcnop 22448   ConOpccop 22449   LinOpclo 22450
This theorem is referenced by:  cnlnadjlem8  23577  nmopadjlei  23591
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596  ax-cc 8315  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068  ax-addf 9069  ax-mulf 9070  ax-hilex 22502  ax-hfvadd 22503  ax-hvcom 22504  ax-hvass 22505  ax-hv0cl 22506  ax-hvaddid 22507  ax-hfvmul 22508  ax-hvmulid 22509  ax-hvmulass 22510  ax-hvdistr1 22511  ax-hvdistr2 22512  ax-hvmul0 22513  ax-hfi 22581  ax-his1 22584  ax-his2 22585  ax-his3 22586  ax-his4 22587  ax-hcompl 22704
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-iin 4096  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-of 6305  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-2o 6725  df-oadd 6728  df-omul 6729  df-er 6905  df-map 7020  df-pm 7021  df-ixp 7064  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-fi 7416  df-sup 7446  df-oi 7479  df-card 7826  df-acn 7829  df-cda 8048  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-5 10061  df-6 10062  df-7 10063  df-8 10064  df-9 10065  df-10 10066  df-n0 10222  df-z 10283  df-dec 10383  df-uz 10489  df-q 10575  df-rp 10613  df-xneg 10710  df-xadd 10711  df-xmul 10712  df-ioo 10920  df-ico 10922  df-icc 10923  df-fz 11044  df-fzo 11136  df-fl 11202  df-seq 11324  df-exp 11383  df-hash 11619  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-clim 12282  df-rlim 12283  df-sum 12480  df-struct 13471  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-mulr 13543  df-starv 13544  df-sca 13545  df-vsca 13546  df-tset 13548  df-ple 13549  df-ds 13551  df-unif 13552  df-hom 13553  df-cco 13554  df-rest 13650  df-topn 13651  df-topgen 13667  df-pt 13668  df-prds 13671  df-xrs 13726  df-0g 13727  df-gsum 13728  df-qtop 13733  df-imas 13734  df-xps 13736  df-mre 13811  df-mrc 13812  df-acs 13814  df-mnd 14690  df-submnd 14739  df-mulg 14815  df-cntz 15116  df-cmn 15414  df-psmet 16694  df-xmet 16695  df-met 16696  df-bl 16697  df-mopn 16698  df-fbas 16699  df-fg 16700  df-cnfld 16704  df-top 16963  df-bases 16965  df-topon 16966  df-topsp 16967  df-cld 17083  df-ntr 17084  df-cls 17085  df-nei 17162  df-cn 17291  df-cnp 17292  df-lm 17293  df-t1 17378  df-haus 17379  df-tx 17594  df-hmeo 17787  df-fil 17878  df-fm 17970  df-flim 17971  df-flf 17972  df-xms 18350  df-ms 18351  df-tms 18352  df-cfil 19208  df-cau 19209  df-cmet 19210  df-grpo 21779  df-gid 21780  df-ginv 21781  df-gdiv 21782  df-ablo 21870  df-subgo 21890  df-vc 22025  df-nv 22071  df-va 22074  df-ba 22075  df-sm 22076  df-0v 22077  df-vs 22078  df-nmcv 22079  df-ims 22080  df-dip 22197  df-ssp 22221  df-ph 22314  df-cbn 22365  df-hnorm 22471  df-hba 22472  df-hvsub 22474  df-hlim 22475  df-hcau 22476  df-sh 22709  df-ch 22724  df-oc 22754  df-ch0 22755  df-nmop 23342  df-cnop 23343  df-lnop 23344  df-nmfn 23348  df-nlfn 23349  df-cnfn 23350  df-lnfn 23351
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