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Theorem norm3adifii 22603
Description: Norm of differences around common element. Part of Lemma 3.6 of [Beran] p. 101. (Contributed by NM, 30-Sep-1999.) (New usage is discouraged.)
Hypotheses
Ref Expression
norm3dif.1  |-  A  e. 
~H
norm3dif.2  |-  B  e. 
~H
norm3dif.3  |-  C  e. 
~H
Assertion
Ref Expression
norm3adifii  |-  ( abs `  ( ( normh `  ( A  -h  C ) )  -  ( normh `  ( B  -h  C ) ) ) )  <_  ( normh `  ( A  -h  B ) )

Proof of Theorem norm3adifii
StepHypRef Expression
1 norm3dif.1 . . . . . . . 8  |-  A  e. 
~H
2 norm3dif.3 . . . . . . . 8  |-  C  e. 
~H
31, 2hvsubcli 22477 . . . . . . 7  |-  ( A  -h  C )  e. 
~H
43normcli 22586 . . . . . 6  |-  ( normh `  ( A  -h  C
) )  e.  RR
54recni 9058 . . . . 5  |-  ( normh `  ( A  -h  C
) )  e.  CC
6 norm3dif.2 . . . . . . . 8  |-  B  e. 
~H
76, 2hvsubcli 22477 . . . . . . 7  |-  ( B  -h  C )  e. 
~H
87normcli 22586 . . . . . 6  |-  ( normh `  ( B  -h  C
) )  e.  RR
98recni 9058 . . . . 5  |-  ( normh `  ( B  -h  C
) )  e.  CC
105, 9negsubdi2i 9342 . . . 4  |-  -u (
( normh `  ( A  -h  C ) )  -  ( normh `  ( B  -h  C ) ) )  =  ( ( normh `  ( B  -h  C
) )  -  ( normh `  ( A  -h  C ) ) )
116, 2, 1norm3difi 22602 . . . . . 6  |-  ( normh `  ( B  -h  C
) )  <_  (
( normh `  ( B  -h  A ) )  +  ( normh `  ( A  -h  C ) ) )
126, 1normsubi 22596 . . . . . . 7  |-  ( normh `  ( B  -h  A
) )  =  (
normh `  ( A  -h  B ) )
1312oveq1i 6050 . . . . . 6  |-  ( (
normh `  ( B  -h  A ) )  +  ( normh `  ( A  -h  C ) ) )  =  ( ( normh `  ( A  -h  B
) )  +  (
normh `  ( A  -h  C ) ) )
1411, 13breqtri 4195 . . . . 5  |-  ( normh `  ( B  -h  C
) )  <_  (
( normh `  ( A  -h  B ) )  +  ( normh `  ( A  -h  C ) ) )
151, 6hvsubcli 22477 . . . . . . 7  |-  ( A  -h  B )  e. 
~H
1615normcli 22586 . . . . . 6  |-  ( normh `  ( A  -h  B
) )  e.  RR
178, 4, 16lesubaddi 9541 . . . . 5  |-  ( ( ( normh `  ( B  -h  C ) )  -  ( normh `  ( A  -h  C ) ) )  <_  ( normh `  ( A  -h  B ) )  <-> 
( normh `  ( B  -h  C ) )  <_ 
( ( normh `  ( A  -h  B ) )  +  ( normh `  ( A  -h  C ) ) ) )
1814, 17mpbir 201 . . . 4  |-  ( (
normh `  ( B  -h  C ) )  -  ( normh `  ( A  -h  C ) ) )  <_  ( normh `  ( A  -h  B ) )
1910, 18eqbrtri 4191 . . 3  |-  -u (
( normh `  ( A  -h  C ) )  -  ( normh `  ( B  -h  C ) ) )  <_  ( normh `  ( A  -h  B ) )
204, 8resubcli 9319 . . . 4  |-  ( (
normh `  ( A  -h  C ) )  -  ( normh `  ( B  -h  C ) ) )  e.  RR
2120, 16lenegcon1i 9535 . . 3  |-  ( -u ( ( normh `  ( A  -h  C ) )  -  ( normh `  ( B  -h  C ) ) )  <_  ( normh `  ( A  -h  B
) )  <->  -u ( normh `  ( A  -h  B
) )  <_  (
( normh `  ( A  -h  C ) )  -  ( normh `  ( B  -h  C ) ) ) )
2219, 21mpbi 200 . 2  |-  -u ( normh `  ( A  -h  B ) )  <_ 
( ( normh `  ( A  -h  C ) )  -  ( normh `  ( B  -h  C ) ) )
231, 2, 6norm3difi 22602 . . 3  |-  ( normh `  ( A  -h  C
) )  <_  (
( normh `  ( A  -h  B ) )  +  ( normh `  ( B  -h  C ) ) )
244, 8, 16lesubaddi 9541 . . 3  |-  ( ( ( normh `  ( A  -h  C ) )  -  ( normh `  ( B  -h  C ) ) )  <_  ( normh `  ( A  -h  B ) )  <-> 
( normh `  ( A  -h  C ) )  <_ 
( ( normh `  ( A  -h  B ) )  +  ( normh `  ( B  -h  C ) ) ) )
2523, 24mpbir 201 . 2  |-  ( (
normh `  ( A  -h  C ) )  -  ( normh `  ( B  -h  C ) ) )  <_  ( normh `  ( A  -h  B ) )
2620, 16abslei 12150 . 2  |-  ( ( abs `  ( (
normh `  ( A  -h  C ) )  -  ( normh `  ( B  -h  C ) ) ) )  <_  ( normh `  ( A  -h  B
) )  <->  ( -u ( normh `  ( A  -h  B ) )  <_ 
( ( normh `  ( A  -h  C ) )  -  ( normh `  ( B  -h  C ) ) )  /\  ( (
normh `  ( A  -h  C ) )  -  ( normh `  ( B  -h  C ) ) )  <_  ( normh `  ( A  -h  B ) ) ) )
2722, 25, 26mpbir2an 887 1  |-  ( abs `  ( ( normh `  ( A  -h  C ) )  -  ( normh `  ( B  -h  C ) ) ) )  <_  ( normh `  ( A  -h  B ) )
Colors of variables: wff set class
Syntax hints:    e. wcel 1721   class class class wbr 4172   ` cfv 5413  (class class class)co 6040    + caddc 8949    <_ cle 9077    - cmin 9247   -ucneg 9248   abscabs 11994   ~Hchil 22375   normhcno 22379    -h cmv 22381
This theorem is referenced by:  norm3adifi  22608
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-hfvadd 22456  ax-hvcom 22457  ax-hvass 22458  ax-hv0cl 22459  ax-hvaddid 22460  ax-hfvmul 22461  ax-hvmulid 22462  ax-hvmulass 22463  ax-hvdistr1 22464  ax-hvdistr2 22465  ax-hvmul0 22466  ax-hfi 22534  ax-his1 22537  ax-his2 22538  ax-his3 22539  ax-his4 22540
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-seq 11279  df-exp 11338  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-hnorm 22424  df-hvsub 22427
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