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Theorem norm3adifii 21720
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 21594 . . . . . . 7  |-  ( A  -h  C )  e. 
~H
43normcli 21703 . . . . . 6  |-  ( normh `  ( A  -h  C
) )  e.  RR
54recni 8845 . . . . 5  |-  ( normh `  ( A  -h  C
) )  e.  CC
6 norm3dif.2 . . . . . . . 8  |-  B  e. 
~H
76, 2hvsubcli 21594 . . . . . . 7  |-  ( B  -h  C )  e. 
~H
87normcli 21703 . . . . . 6  |-  ( normh `  ( B  -h  C
) )  e.  RR
98recni 8845 . . . . 5  |-  ( normh `  ( B  -h  C
) )  e.  CC
105, 9negsubdi2i 9128 . . . 4  |-  -u (
( normh `  ( A  -h  C ) )  -  ( normh `  ( B  -h  C ) ) )  =  ( ( normh `  ( B  -h  C
) )  -  ( normh `  ( A  -h  C ) ) )
116, 2, 1norm3difi 21719 . . . . . 6  |-  ( normh `  ( B  -h  C
) )  <_  (
( normh `  ( B  -h  A ) )  +  ( normh `  ( A  -h  C ) ) )
126, 1normsubi 21713 . . . . . . 7  |-  ( normh `  ( B  -h  A
) )  =  (
normh `  ( A  -h  B ) )
1312oveq1i 5830 . . . . . 6  |-  ( (
normh `  ( B  -h  A ) )  +  ( normh `  ( A  -h  C ) ) )  =  ( ( normh `  ( A  -h  B
) )  +  (
normh `  ( A  -h  C ) ) )
1411, 13breqtri 4048 . . . . 5  |-  ( normh `  ( B  -h  C
) )  <_  (
( normh `  ( A  -h  B ) )  +  ( normh `  ( A  -h  C ) ) )
151, 6hvsubcli 21594 . . . . . . 7  |-  ( A  -h  B )  e. 
~H
1615normcli 21703 . . . . . 6  |-  ( normh `  ( A  -h  B
) )  e.  RR
178, 4, 16lesubaddi 9327 . . . . 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 202 . . . 4  |-  ( (
normh `  ( B  -h  C ) )  -  ( normh `  ( A  -h  C ) ) )  <_  ( normh `  ( A  -h  B ) )
1910, 18eqbrtri 4044 . . 3  |-  -u (
( normh `  ( A  -h  C ) )  -  ( normh `  ( B  -h  C ) ) )  <_  ( normh `  ( A  -h  B ) )
204, 8resubcli 9105 . . . 4  |-  ( (
normh `  ( A  -h  C ) )  -  ( normh `  ( B  -h  C ) ) )  e.  RR
2120, 16lenegcon1i 9321 . . 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 201 . 2  |-  -u ( normh `  ( A  -h  B ) )  <_ 
( ( normh `  ( A  -h  C ) )  -  ( normh `  ( B  -h  C ) ) )
231, 2, 6norm3difi 21719 . . 3  |-  ( normh `  ( A  -h  C
) )  <_  (
( normh `  ( A  -h  B ) )  +  ( normh `  ( B  -h  C ) ) )
244, 8, 16lesubaddi 9327 . . 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 202 . 2  |-  ( (
normh `  ( A  -h  C ) )  -  ( normh `  ( B  -h  C ) ) )  <_  ( normh `  ( A  -h  B ) )
2620, 16abslei 11870 . 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 888 1  |-  ( abs `  ( ( normh `  ( A  -h  C ) )  -  ( normh `  ( B  -h  C ) ) ) )  <_  ( normh `  ( A  -h  B ) )
Colors of variables: wff set class
Syntax hints:    e. wcel 1685   class class class wbr 4025   ` cfv 5222  (class class class)co 5820    + caddc 8736    <_ cle 8864    - cmin 9033   -ucneg 9034   abscabs 11714   ~Hchil 21492   normhcno 21496    -h cmv 21498
This theorem is referenced by:  norm3adifi  21725
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8789  ax-resscn 8790  ax-1cn 8791  ax-icn 8792  ax-addcl 8793  ax-addrcl 8794  ax-mulcl 8795  ax-mulrcl 8796  ax-mulcom 8797  ax-addass 8798  ax-mulass 8799  ax-distr 8800  ax-i2m1 8801  ax-1ne0 8802  ax-1rid 8803  ax-rnegex 8804  ax-rrecex 8805  ax-cnre 8806  ax-pre-lttri 8807  ax-pre-lttrn 8808  ax-pre-ltadd 8809  ax-pre-mulgt0 8810  ax-pre-sup 8811  ax-hfvadd 21573  ax-hvcom 21574  ax-hvass 21575  ax-hv0cl 21576  ax-hvaddid 21577  ax-hfvmul 21578  ax-hvmulid 21579  ax-hvmulass 21580  ax-hvdistr1 21581  ax-hvdistr2 21582  ax-hvmul0 21583  ax-hfi 21651  ax-his1 21654  ax-his2 21655  ax-his3 21656  ax-his4 21657
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fun 5224  df-fn 5225  df-f 5226  df-f1 5227  df-fo 5228  df-f1o 5229  df-fv 5230  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-2nd 6085  df-iota 6253  df-riota 6300  df-recs 6384  df-rdg 6419  df-er 6656  df-en 6860  df-dom 6861  df-sdom 6862  df-sup 7190  df-pnf 8865  df-mnf 8866  df-xr 8867  df-ltxr 8868  df-le 8869  df-sub 9035  df-neg 9036  df-div 9420  df-nn 9743  df-2 9800  df-3 9801  df-4 9802  df-n0 9962  df-z 10021  df-uz 10227  df-rp 10351  df-seq 11042  df-exp 11100  df-cj 11579  df-re 11580  df-im 11581  df-sqr 11715  df-abs 11716  df-hnorm 21541  df-hvsub 21544
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