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Theorem norm3lem 25839
Description: Lemma involving norm of differences in Hilbert space. (Contributed by NM, 18-Aug-1999.) (New usage is discouraged.)
Hypotheses
Ref Expression
norm3dif.1  |-  A  e. 
~H
norm3dif.2  |-  B  e. 
~H
norm3dif.3  |-  C  e. 
~H
norm3lem.4  |-  D  e.  RR
Assertion
Ref Expression
norm3lem  |-  ( ( ( normh `  ( A  -h  C ) )  < 
( D  /  2
)  /\  ( normh `  ( C  -h  B
) )  <  ( D  /  2 ) )  ->  ( normh `  ( A  -h  B ) )  <  D )

Proof of Theorem norm3lem
StepHypRef Expression
1 norm3dif.1 . . . 4  |-  A  e. 
~H
2 norm3dif.2 . . . 4  |-  B  e. 
~H
3 norm3dif.3 . . . 4  |-  C  e. 
~H
41, 2, 3norm3difi 25837 . . 3  |-  ( normh `  ( A  -h  B
) )  <_  (
( normh `  ( A  -h  C ) )  +  ( normh `  ( C  -h  B ) ) )
51, 3hvsubcli 25711 . . . . 5  |-  ( A  -h  C )  e. 
~H
65normcli 25821 . . . 4  |-  ( normh `  ( A  -h  C
) )  e.  RR
73, 2hvsubcli 25711 . . . . 5  |-  ( C  -h  B )  e. 
~H
87normcli 25821 . . . 4  |-  ( normh `  ( C  -h  B
) )  e.  RR
9 norm3lem.4 . . . . 5  |-  D  e.  RR
109rehalfcli 10788 . . . 4  |-  ( D  /  2 )  e.  RR
116, 8, 10, 10lt2addi 10116 . . 3  |-  ( ( ( normh `  ( A  -h  C ) )  < 
( D  /  2
)  /\  ( normh `  ( C  -h  B
) )  <  ( D  /  2 ) )  ->  ( ( normh `  ( A  -h  C
) )  +  (
normh `  ( C  -h  B ) ) )  <  ( ( D  /  2 )  +  ( D  /  2
) ) )
121, 2hvsubcli 25711 . . . . 5  |-  ( A  -h  B )  e. 
~H
1312normcli 25821 . . . 4  |-  ( normh `  ( A  -h  B
) )  e.  RR
146, 8readdcli 9610 . . . 4  |-  ( (
normh `  ( A  -h  C ) )  +  ( normh `  ( C  -h  B ) ) )  e.  RR
1510, 10readdcli 9610 . . . 4  |-  ( ( D  /  2 )  +  ( D  / 
2 ) )  e.  RR
1613, 14, 15lelttri 9712 . . 3  |-  ( ( ( normh `  ( A  -h  B ) )  <_ 
( ( normh `  ( A  -h  C ) )  +  ( normh `  ( C  -h  B ) ) )  /\  ( (
normh `  ( A  -h  C ) )  +  ( normh `  ( C  -h  B ) ) )  <  ( ( D  /  2 )  +  ( D  /  2
) ) )  -> 
( normh `  ( A  -h  B ) )  < 
( ( D  / 
2 )  +  ( D  /  2 ) ) )
174, 11, 16sylancr 663 . 2  |-  ( ( ( normh `  ( A  -h  C ) )  < 
( D  /  2
)  /\  ( normh `  ( C  -h  B
) )  <  ( D  /  2 ) )  ->  ( normh `  ( A  -h  B ) )  <  ( ( D  /  2 )  +  ( D  /  2
) ) )
1810recni 9609 . . . 4  |-  ( D  /  2 )  e.  CC
19182timesi 10657 . . 3  |-  ( 2  x.  ( D  / 
2 ) )  =  ( ( D  / 
2 )  +  ( D  /  2 ) )
209recni 9609 . . . 4  |-  D  e.  CC
21 2cn 10607 . . . 4  |-  2  e.  CC
22 2ne0 10629 . . . 4  |-  2  =/=  0
2320, 21, 22divcan2i 10288 . . 3  |-  ( 2  x.  ( D  / 
2 ) )  =  D
2419, 23eqtr3i 2498 . 2  |-  ( ( D  /  2 )  +  ( D  / 
2 ) )  =  D
2517, 24syl6breq 4486 1  |-  ( ( ( 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    e. wcel 1767   class class class wbr 4447   ` cfv 5588  (class class class)co 6285   RRcr 9492    + caddc 9496    x. cmul 9498    < clt 9629    <_ cle 9630    / cdiv 10207   2c2 10586   ~Hchil 25609   normhcno 25613    -h cmv 25615
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570  ax-pre-sup 9571  ax-hfvadd 25690  ax-hvcom 25691  ax-hvass 25692  ax-hv0cl 25693  ax-hvaddid 25694  ax-hfvmul 25695  ax-hvmulid 25696  ax-hvmulass 25697  ax-hvdistr2 25699  ax-hvmul0 25700  ax-hfi 25769  ax-his1 25772  ax-his2 25773  ax-his3 25774  ax-his4 25775
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-2nd 6786  df-recs 7043  df-rdg 7077  df-er 7312  df-en 7518  df-dom 7519  df-sdom 7520  df-sup 7902  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-div 10208  df-nn 10538  df-2 10595  df-3 10596  df-4 10597  df-n0 10797  df-z 10866  df-uz 11084  df-rp 11222  df-seq 12077  df-exp 12136  df-cj 12898  df-re 12899  df-im 12900  df-sqrt 13034  df-abs 13035  df-hnorm 25658  df-hvsub 25661
This theorem is referenced by:  norm3lemt  25842
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