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Theorem norm3lem 25938
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 25936 . . 3  |-  ( normh `  ( A  -h  B
) )  <_  (
( normh `  ( A  -h  C ) )  +  ( normh `  ( C  -h  B ) ) )
51, 3hvsubcli 25810 . . . . 5  |-  ( A  -h  C )  e. 
~H
65normcli 25920 . . . 4  |-  ( normh `  ( A  -h  C
) )  e.  RR
73, 2hvsubcli 25810 . . . . 5  |-  ( C  -h  B )  e. 
~H
87normcli 25920 . . . 4  |-  ( normh `  ( C  -h  B
) )  e.  RR
9 norm3lem.4 . . . . 5  |-  D  e.  RR
109rehalfcli 10793 . . . 4  |-  ( D  /  2 )  e.  RR
116, 8, 10, 10lt2addi 10121 . . 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 25810 . . . . 5  |-  ( A  -h  B )  e. 
~H
1312normcli 25920 . . . 4  |-  ( normh `  ( A  -h  B
) )  e.  RR
146, 8readdcli 9612 . . . 4  |-  ( (
normh `  ( A  -h  C ) )  +  ( normh `  ( C  -h  B ) ) )  e.  RR
1510, 10readdcli 9612 . . . 4  |-  ( ( D  /  2 )  +  ( D  / 
2 ) )  e.  RR
1613, 14, 15lelttri 9714 . . 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 9611 . . . 4  |-  ( D  /  2 )  e.  CC
19182timesi 10662 . . 3  |-  ( 2  x.  ( D  / 
2 ) )  =  ( ( D  / 
2 )  +  ( D  /  2 ) )
209recni 9611 . . . 4  |-  D  e.  CC
21 2cn 10612 . . . 4  |-  2  e.  CC
22 2ne0 10634 . . . 4  |-  2  =/=  0
2320, 21, 22divcan2i 10293 . . 3  |-  ( 2  x.  ( D  / 
2 ) )  =  D
2419, 23eqtr3i 2474 . 2  |-  ( ( D  /  2 )  +  ( D  / 
2 ) )  =  D
2517, 24syl6breq 4476 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 1804   class class class wbr 4437   ` cfv 5578  (class class class)co 6281   RRcr 9494    + caddc 9498    x. cmul 9500    < clt 9631    <_ cle 9632    / cdiv 10212   2c2 10591   ~Hchil 25708   normhcno 25712    -h cmv 25714
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573  ax-hfvadd 25789  ax-hvcom 25790  ax-hvass 25791  ax-hv0cl 25792  ax-hvaddid 25793  ax-hfvmul 25794  ax-hvmulid 25795  ax-hvmulass 25796  ax-hvdistr2 25798  ax-hvmul0 25799  ax-hfi 25868  ax-his1 25871  ax-his2 25872  ax-his3 25873  ax-his4 25874
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-2nd 6786  df-recs 7044  df-rdg 7078  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-sup 7903  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10213  df-nn 10543  df-2 10600  df-3 10601  df-4 10602  df-n0 10802  df-z 10871  df-uz 11091  df-rp 11230  df-seq 12087  df-exp 12146  df-cj 12911  df-re 12912  df-im 12913  df-sqrt 13047  df-abs 13048  df-hnorm 25757  df-hvsub 25760
This theorem is referenced by:  norm3lemt  25941
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