HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  norm3lemt Structured version   Unicode version

Theorem norm3lemt 24566
Description: Lemma involving norm of differences in Hilbert space. (Contributed by NM, 18-Aug-1999.) (New usage is discouraged.)
Assertion
Ref Expression
norm3lemt  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H )  /\  ( C  e.  ~H  /\  D  e.  RR ) )  ->  ( (
( normh `  ( A  -h  C ) )  < 
( D  /  2
)  /\  ( normh `  ( C  -h  B
) )  <  ( D  /  2 ) )  ->  ( normh `  ( A  -h  B ) )  <  D ) )

Proof of Theorem norm3lemt
StepHypRef Expression
1 oveq1 6110 . . . . . 6  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( A  -h  C )  =  ( if ( A  e.  ~H ,  A ,  0h )  -h  C
) )
21fveq2d 5707 . . . . 5  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( normh `  ( A  -h  C ) )  =  ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  C ) ) )
32breq1d 4314 . . . 4  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  (
( normh `  ( A  -h  C ) )  < 
( D  /  2
)  <->  ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  C ) )  < 
( D  /  2
) ) )
43anbi1d 704 . . 3  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  (
( ( normh `  ( A  -h  C ) )  <  ( D  / 
2 )  /\  ( normh `  ( C  -h  B ) )  < 
( D  /  2
) )  <->  ( ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  C ) )  < 
( D  /  2
)  /\  ( normh `  ( C  -h  B
) )  <  ( D  /  2 ) ) ) )
5 oveq1 6110 . . . . 5  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( A  -h  B )  =  ( if ( A  e.  ~H ,  A ,  0h )  -h  B
) )
65fveq2d 5707 . . . 4  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( normh `  ( A  -h  B ) )  =  ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  B ) ) )
76breq1d 4314 . . 3  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  (
( normh `  ( A  -h  B ) )  < 
D  <->  ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  B ) )  < 
D ) )
84, 7imbi12d 320 . 2  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  (
( ( ( normh `  ( A  -h  C
) )  <  ( D  /  2 )  /\  ( normh `  ( C  -h  B ) )  < 
( D  /  2
) )  ->  ( normh `  ( A  -h  B ) )  < 
D )  <->  ( (
( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  C ) )  < 
( D  /  2
)  /\  ( normh `  ( C  -h  B
) )  <  ( D  /  2 ) )  ->  ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  B ) )  < 
D ) ) )
9 oveq2 6111 . . . . . 6  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  ( C  -h  B )  =  ( C  -h  if ( B  e.  ~H ,  B ,  0h )
) )
109fveq2d 5707 . . . . 5  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  ( normh `  ( C  -h  B ) )  =  ( normh `  ( C  -h  if ( B  e. 
~H ,  B ,  0h ) ) ) )
1110breq1d 4314 . . . 4  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  (
( normh `  ( C  -h  B ) )  < 
( D  /  2
)  <->  ( normh `  ( C  -h  if ( B  e.  ~H ,  B ,  0h ) ) )  <  ( D  / 
2 ) ) )
1211anbi2d 703 . . 3  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  (
( ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  C ) )  < 
( D  /  2
)  /\  ( normh `  ( C  -h  B
) )  <  ( D  /  2 ) )  <-> 
( ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  C ) )  < 
( D  /  2
)  /\  ( normh `  ( C  -h  if ( B  e.  ~H ,  B ,  0h )
) )  <  ( D  /  2 ) ) ) )
13 oveq2 6111 . . . . 5  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  ( if ( A  e.  ~H ,  A ,  0h )  -h  B )  =  ( if ( A  e. 
~H ,  A ,  0h )  -h  if ( B  e.  ~H ,  B ,  0h )
) )
1413fveq2d 5707 . . . 4  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  B ) )  =  ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  if ( B  e. 
~H ,  B ,  0h ) ) ) )
1514breq1d 4314 . . 3  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  (
( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  B ) )  < 
D  <->  ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  if ( B  e. 
~H ,  B ,  0h ) ) )  < 
D ) )
1612, 15imbi12d 320 . 2  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  (
( ( ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  C
) )  <  ( D  /  2 )  /\  ( normh `  ( C  -h  B ) )  < 
( D  /  2
) )  ->  ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  B ) )  < 
D )  <->  ( (
( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  C ) )  < 
( D  /  2
)  /\  ( normh `  ( C  -h  if ( B  e.  ~H ,  B ,  0h )
) )  <  ( D  /  2 ) )  ->  ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  if ( B  e. 
~H ,  B ,  0h ) ) )  < 
D ) ) )
17 oveq2 6111 . . . . . 6  |-  ( C  =  if ( C  e.  ~H ,  C ,  0h )  ->  ( if ( A  e.  ~H ,  A ,  0h )  -h  C )  =  ( if ( A  e. 
~H ,  A ,  0h )  -h  if ( C  e.  ~H ,  C ,  0h )
) )
1817fveq2d 5707 . . . . 5  |-  ( C  =  if ( C  e.  ~H ,  C ,  0h )  ->  ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  C ) )  =  ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  if ( C  e. 
~H ,  C ,  0h ) ) ) )
1918breq1d 4314 . . . 4  |-  ( C  =  if ( C  e.  ~H ,  C ,  0h )  ->  (
( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  C ) )  < 
( D  /  2
)  <->  ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  if ( C  e. 
~H ,  C ,  0h ) ) )  < 
( D  /  2
) ) )
20 oveq1 6110 . . . . . 6  |-  ( C  =  if ( C  e.  ~H ,  C ,  0h )  ->  ( C  -h  if ( B  e.  ~H ,  B ,  0h ) )  =  ( if ( C  e.  ~H ,  C ,  0h )  -h  if ( B  e.  ~H ,  B ,  0h )
) )
2120fveq2d 5707 . . . . 5  |-  ( C  =  if ( C  e.  ~H ,  C ,  0h )  ->  ( normh `  ( C  -h  if ( B  e.  ~H ,  B ,  0h )
) )  =  (
normh `  ( if ( C  e.  ~H ,  C ,  0h )  -h  if ( B  e. 
~H ,  B ,  0h ) ) ) )
2221breq1d 4314 . . . 4  |-  ( C  =  if ( C  e.  ~H ,  C ,  0h )  ->  (
( normh `  ( C  -h  if ( B  e. 
~H ,  B ,  0h ) ) )  < 
( D  /  2
)  <->  ( normh `  ( if ( C  e.  ~H ,  C ,  0h )  -h  if ( B  e. 
~H ,  B ,  0h ) ) )  < 
( D  /  2
) ) )
2319, 22anbi12d 710 . . 3  |-  ( C  =  if ( C  e.  ~H ,  C ,  0h )  ->  (
( ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  C ) )  < 
( D  /  2
)  /\  ( normh `  ( C  -h  if ( B  e.  ~H ,  B ,  0h )
) )  <  ( D  /  2 ) )  <-> 
( ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  if ( C  e. 
~H ,  C ,  0h ) ) )  < 
( D  /  2
)  /\  ( normh `  ( if ( C  e.  ~H ,  C ,  0h )  -h  if ( B  e.  ~H ,  B ,  0h )
) )  <  ( D  /  2 ) ) ) )
2423imbi1d 317 . 2  |-  ( C  =  if ( C  e.  ~H ,  C ,  0h )  ->  (
( ( ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  C
) )  <  ( D  /  2 )  /\  ( normh `  ( C  -h  if ( B  e. 
~H ,  B ,  0h ) ) )  < 
( D  /  2
) )  ->  ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  if ( B  e. 
~H ,  B ,  0h ) ) )  < 
D )  <->  ( (
( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  if ( C  e. 
~H ,  C ,  0h ) ) )  < 
( D  /  2
)  /\  ( normh `  ( if ( C  e.  ~H ,  C ,  0h )  -h  if ( B  e.  ~H ,  B ,  0h )
) )  <  ( D  /  2 ) )  ->  ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  if ( B  e. 
~H ,  B ,  0h ) ) )  < 
D ) ) )
25 oveq1 6110 . . . . 5  |-  ( D  =  if ( D  e.  RR ,  D ,  2 )  -> 
( D  /  2
)  =  ( if ( D  e.  RR ,  D ,  2 )  /  2 ) )
2625breq2d 4316 . . . 4  |-  ( D  =  if ( D  e.  RR ,  D ,  2 )  -> 
( ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  if ( C  e. 
~H ,  C ,  0h ) ) )  < 
( D  /  2
)  <->  ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  if ( C  e. 
~H ,  C ,  0h ) ) )  < 
( if ( D  e.  RR ,  D ,  2 )  / 
2 ) ) )
2725breq2d 4316 . . . 4  |-  ( D  =  if ( D  e.  RR ,  D ,  2 )  -> 
( ( normh `  ( if ( C  e.  ~H ,  C ,  0h )  -h  if ( B  e. 
~H ,  B ,  0h ) ) )  < 
( D  /  2
)  <->  ( normh `  ( if ( C  e.  ~H ,  C ,  0h )  -h  if ( B  e. 
~H ,  B ,  0h ) ) )  < 
( if ( D  e.  RR ,  D ,  2 )  / 
2 ) ) )
2826, 27anbi12d 710 . . 3  |-  ( D  =  if ( D  e.  RR ,  D ,  2 )  -> 
( ( ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  if ( C  e.  ~H ,  C ,  0h )
) )  <  ( D  /  2 )  /\  ( normh `  ( if ( C  e.  ~H ,  C ,  0h )  -h  if ( B  e. 
~H ,  B ,  0h ) ) )  < 
( D  /  2
) )  <->  ( ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  if ( C  e. 
~H ,  C ,  0h ) ) )  < 
( if ( D  e.  RR ,  D ,  2 )  / 
2 )  /\  ( normh `  ( if ( C  e.  ~H ,  C ,  0h )  -h  if ( B  e. 
~H ,  B ,  0h ) ) )  < 
( if ( D  e.  RR ,  D ,  2 )  / 
2 ) ) ) )
29 breq2 4308 . . 3  |-  ( D  =  if ( D  e.  RR ,  D ,  2 )  -> 
( ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  if ( B  e. 
~H ,  B ,  0h ) ) )  < 
D  <->  ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  if ( B  e. 
~H ,  B ,  0h ) ) )  < 
if ( D  e.  RR ,  D , 
2 ) ) )
3028, 29imbi12d 320 . 2  |-  ( D  =  if ( D  e.  RR ,  D ,  2 )  -> 
( ( ( (
normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  if ( C  e. 
~H ,  C ,  0h ) ) )  < 
( D  /  2
)  /\  ( normh `  ( if ( C  e.  ~H ,  C ,  0h )  -h  if ( B  e.  ~H ,  B ,  0h )
) )  <  ( D  /  2 ) )  ->  ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  if ( B  e. 
~H ,  B ,  0h ) ) )  < 
D )  <->  ( (
( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  if ( C  e. 
~H ,  C ,  0h ) ) )  < 
( if ( D  e.  RR ,  D ,  2 )  / 
2 )  /\  ( normh `  ( if ( C  e.  ~H ,  C ,  0h )  -h  if ( B  e. 
~H ,  B ,  0h ) ) )  < 
( if ( D  e.  RR ,  D ,  2 )  / 
2 ) )  -> 
( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  if ( B  e. 
~H ,  B ,  0h ) ) )  < 
if ( D  e.  RR ,  D , 
2 ) ) ) )
31 ifhvhv0 24436 . . 3  |-  if ( A  e.  ~H ,  A ,  0h )  e.  ~H
32 ifhvhv0 24436 . . 3  |-  if ( B  e.  ~H ,  B ,  0h )  e.  ~H
33 ifhvhv0 24436 . . 3  |-  if ( C  e.  ~H ,  C ,  0h )  e.  ~H
34 2re 10403 . . . 4  |-  2  e.  RR
3534elimel 3864 . . 3  |-  if ( D  e.  RR ,  D ,  2 )  e.  RR
3631, 32, 33, 35norm3lem 24563 . 2  |-  ( ( ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  if ( C  e. 
~H ,  C ,  0h ) ) )  < 
( if ( D  e.  RR ,  D ,  2 )  / 
2 )  /\  ( normh `  ( if ( C  e.  ~H ,  C ,  0h )  -h  if ( B  e. 
~H ,  B ,  0h ) ) )  < 
( if ( D  e.  RR ,  D ,  2 )  / 
2 ) )  -> 
( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  if ( B  e. 
~H ,  B ,  0h ) ) )  < 
if ( D  e.  RR ,  D , 
2 ) )
378, 16, 24, 30, 36dedth4h 3856 1  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H )  /\  ( C  e.  ~H  /\  D  e.  RR ) )  ->  ( (
( normh `  ( A  -h  C ) )  < 
( D  /  2
)  /\  ( normh `  ( C  -h  B
) )  <  ( D  /  2 ) )  ->  ( normh `  ( A  -h  B ) )  <  D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   ifcif 3803   class class class wbr 4304   ` cfv 5430  (class class class)co 6103   RRcr 9293    < clt 9430    / cdiv 10005   2c2 10383   ~Hchil 24333   normhcno 24337   0hc0v 24338    -h cmv 24339
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371  ax-pre-sup 9372  ax-hfvadd 24414  ax-hvcom 24415  ax-hvass 24416  ax-hv0cl 24417  ax-hvaddid 24418  ax-hfvmul 24419  ax-hvmulid 24420  ax-hvmulass 24421  ax-hvdistr2 24423  ax-hvmul0 24424  ax-hfi 24493  ax-his1 24496  ax-his2 24497  ax-his3 24498  ax-his4 24499
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-om 6489  df-2nd 6590  df-recs 6844  df-rdg 6878  df-er 7113  df-en 7323  df-dom 7324  df-sdom 7325  df-sup 7703  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610  df-div 10006  df-nn 10335  df-2 10392  df-3 10393  df-4 10394  df-n0 10592  df-z 10659  df-uz 10874  df-rp 11004  df-seq 11819  df-exp 11878  df-cj 12600  df-re 12601  df-im 12602  df-sqr 12736  df-abs 12737  df-hnorm 24382  df-hvsub 24385
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator