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Theorem norm3adifii 25856
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 25729 . . . . . . 7  |-  ( A  -h  C )  e. 
~H
43normcli 25839 . . . . . 6  |-  ( normh `  ( A  -h  C
) )  e.  RR
54recni 9618 . . . . 5  |-  ( normh `  ( A  -h  C
) )  e.  CC
6 norm3dif.2 . . . . . . . 8  |-  B  e. 
~H
76, 2hvsubcli 25729 . . . . . . 7  |-  ( B  -h  C )  e. 
~H
87normcli 25839 . . . . . 6  |-  ( normh `  ( B  -h  C
) )  e.  RR
98recni 9618 . . . . 5  |-  ( normh `  ( B  -h  C
) )  e.  CC
105, 9negsubdi2i 9915 . . . 4  |-  -u (
( normh `  ( A  -h  C ) )  -  ( normh `  ( B  -h  C ) ) )  =  ( ( normh `  ( B  -h  C
) )  -  ( normh `  ( A  -h  C ) ) )
116, 2, 1norm3difi 25855 . . . . . 6  |-  ( normh `  ( B  -h  C
) )  <_  (
( normh `  ( B  -h  A ) )  +  ( normh `  ( A  -h  C ) ) )
126, 1normsubi 25849 . . . . . . 7  |-  ( normh `  ( B  -h  A
) )  =  (
normh `  ( A  -h  B ) )
1312oveq1i 6304 . . . . . 6  |-  ( (
normh `  ( B  -h  A ) )  +  ( normh `  ( A  -h  C ) ) )  =  ( ( normh `  ( A  -h  B
) )  +  (
normh `  ( A  -h  C ) ) )
1411, 13breqtri 4475 . . . . 5  |-  ( normh `  ( B  -h  C
) )  <_  (
( normh `  ( A  -h  B ) )  +  ( normh `  ( A  -h  C ) ) )
151, 6hvsubcli 25729 . . . . . . 7  |-  ( A  -h  B )  e. 
~H
1615normcli 25839 . . . . . 6  |-  ( normh `  ( A  -h  B
) )  e.  RR
178, 4, 16lesubaddi 10121 . . . . 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 209 . . . 4  |-  ( (
normh `  ( B  -h  C ) )  -  ( normh `  ( A  -h  C ) ) )  <_  ( normh `  ( A  -h  B ) )
1910, 18eqbrtri 4471 . . 3  |-  -u (
( normh `  ( A  -h  C ) )  -  ( normh `  ( B  -h  C ) ) )  <_  ( normh `  ( A  -h  B ) )
204, 8resubcli 9891 . . . 4  |-  ( (
normh `  ( A  -h  C ) )  -  ( normh `  ( B  -h  C ) ) )  e.  RR
2120, 16lenegcon1i 10115 . . 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 208 . 2  |-  -u ( normh `  ( A  -h  B ) )  <_ 
( ( normh `  ( A  -h  C ) )  -  ( normh `  ( B  -h  C ) ) )
231, 2, 6norm3difi 25855 . . 3  |-  ( normh `  ( A  -h  C
) )  <_  (
( normh `  ( A  -h  B ) )  +  ( normh `  ( B  -h  C ) ) )
244, 8, 16lesubaddi 10121 . . 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 209 . 2  |-  ( (
normh `  ( A  -h  C ) )  -  ( normh `  ( B  -h  C ) ) )  <_  ( normh `  ( A  -h  B ) )
2620, 16abslei 13199 . 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 918 1  |-  ( abs `  ( ( normh `  ( A  -h  C ) )  -  ( normh `  ( B  -h  C ) ) ) )  <_  ( normh `  ( A  -h  B ) )
Colors of variables: wff setvar class
Syntax hints:    e. wcel 1767   class class class wbr 4452   ` cfv 5593  (class class class)co 6294    + caddc 9505    <_ cle 9639    - cmin 9815   -ucneg 9816   abscabs 13042   ~Hchil 25627   normhcno 25631    -h cmv 25633
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 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586  ax-cnex 9558  ax-resscn 9559  ax-1cn 9560  ax-icn 9561  ax-addcl 9562  ax-addrcl 9563  ax-mulcl 9564  ax-mulrcl 9565  ax-mulcom 9566  ax-addass 9567  ax-mulass 9568  ax-distr 9569  ax-i2m1 9570  ax-1ne0 9571  ax-1rid 9572  ax-rnegex 9573  ax-rrecex 9574  ax-cnre 9575  ax-pre-lttri 9576  ax-pre-lttrn 9577  ax-pre-ltadd 9578  ax-pre-mulgt0 9579  ax-pre-sup 9580  ax-hfvadd 25708  ax-hvcom 25709  ax-hvass 25710  ax-hv0cl 25711  ax-hvaddid 25712  ax-hfvmul 25713  ax-hvmulid 25714  ax-hvmulass 25715  ax-hvdistr1 25716  ax-hvdistr2 25717  ax-hvmul0 25718  ax-hfi 25787  ax-his1 25790  ax-his2 25791  ax-his3 25792  ax-his4 25793
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 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4251  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-tr 4546  df-eprel 4796  df-id 4800  df-po 4805  df-so 4806  df-fr 4843  df-we 4845  df-ord 4886  df-on 4887  df-lim 4888  df-suc 4889  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-riota 6255  df-ov 6297  df-oprab 6298  df-mpt2 6299  df-om 6695  df-2nd 6795  df-recs 7052  df-rdg 7086  df-er 7321  df-en 7527  df-dom 7528  df-sdom 7529  df-sup 7911  df-pnf 9640  df-mnf 9641  df-xr 9642  df-ltxr 9643  df-le 9644  df-sub 9817  df-neg 9818  df-div 10217  df-nn 10547  df-2 10604  df-3 10605  df-4 10606  df-n0 10806  df-z 10875  df-uz 11093  df-rp 11231  df-seq 12086  df-exp 12145  df-cj 12907  df-re 12908  df-im 12909  df-sqrt 13043  df-abs 13044  df-hnorm 25676  df-hvsub 25679
This theorem is referenced by:  norm3adifi  25861
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