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

Theorem norm3lemt 22654
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 6088 . . . . . 6  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( A  -h  C )  =  ( if ( A  e.  ~H ,  A ,  0h )  -h  C
) )
21fveq2d 5732 . . . . 5  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( normh `  ( A  -h  C ) )  =  ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  C ) ) )
32breq1d 4222 . . . 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 686 . . 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 6088 . . . . 5  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( A  -h  B )  =  ( if ( A  e.  ~H ,  A ,  0h )  -h  B
) )
65fveq2d 5732 . . . 4  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( normh `  ( A  -h  B ) )  =  ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  B ) ) )
76breq1d 4222 . . 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 312 . 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 6089 . . . . . 6  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  ( C  -h  B )  =  ( C  -h  if ( B  e.  ~H ,  B ,  0h )
) )
109fveq2d 5732 . . . . 5  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  ( normh `  ( C  -h  B ) )  =  ( normh `  ( C  -h  if ( B  e. 
~H ,  B ,  0h ) ) ) )
1110breq1d 4222 . . . 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 685 . . 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 6089 . . . . 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 5732 . . . 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 4222 . . 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 312 . 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 6089 . . . . . 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 5732 . . . . 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 4222 . . . 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 6088 . . . . . 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 5732 . . . . 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 4222 . . . 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 692 . . 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 309 . 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 6088 . . . . 5  |-  ( D  =  if ( D  e.  RR ,  D ,  2 )  -> 
( D  /  2
)  =  ( if ( D  e.  RR ,  D ,  2 )  /  2 ) )
2625breq2d 4224 . . . 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 4224 . . . 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 692 . . 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 4216 . . 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 312 . 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 ax-hv0cl 22506 . . . 4  |-  0h  e.  ~H
3231elimel 3791 . . 3  |-  if ( A  e.  ~H ,  A ,  0h )  e.  ~H
3331elimel 3791 . . 3  |-  if ( B  e.  ~H ,  B ,  0h )  e.  ~H
3431elimel 3791 . . 3  |-  if ( C  e.  ~H ,  C ,  0h )  e.  ~H
35 2re 10069 . . . 4  |-  2  e.  RR
3635elimel 3791 . . 3  |-  if ( D  e.  RR ,  D ,  2 )  e.  RR
3732, 33, 34, 36norm3lem 22651 . 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 ) )
388, 16, 24, 30, 37dedth4h 3783 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 set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   ifcif 3739   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   RRcr 8989    < clt 9120    / cdiv 9677   2c2 10049   ~Hchil 22422   normhcno 22426   0hc0v 22427    -h cmv 22428
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-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  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-hfvadd 22503  ax-hvcom 22504  ax-hvass 22505  ax-hv0cl 22506  ax-hvaddid 22507  ax-hfvmul 22508  ax-hvmulid 22509  ax-hvmulass 22510  ax-hvdistr2 22512  ax-hvmul0 22513  ax-hfi 22581  ax-his1 22584  ax-his2 22585  ax-his3 22586  ax-his4 22587
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-iun 4095  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-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-ov 6084  df-oprab 6085  df-mpt2 6086  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-sup 7446  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-n0 10222  df-z 10283  df-uz 10489  df-rp 10613  df-seq 11324  df-exp 11383  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-hnorm 22471  df-hvsub 22474
  Copyright terms: Public domain W3C validator