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Theorem norm3adifi 22643
Description: Norm of differences around common element. Part of Lemma 3.6 of [Beran] p. 101. (Contributed by NM, 3-Oct-1999.) (New usage is discouraged.)
Hypothesis
Ref Expression
norm3adift.1  |-  C  e. 
~H
Assertion
Ref Expression
norm3adifi  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( abs `  (
( normh `  ( A  -h  C ) )  -  ( normh `  ( B  -h  C ) ) ) )  <_  ( normh `  ( A  -h  B
) ) )

Proof of Theorem norm3adifi
StepHypRef Expression
1 oveq1 6079 . . . . . 6  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( A  -h  C )  =  ( if ( A  e.  ~H ,  A ,  0h )  -h  C
) )
21fveq2d 5723 . . . . 5  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( normh `  ( A  -h  C ) )  =  ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  C ) ) )
32oveq1d 6087 . . . 4  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  (
( normh `  ( A  -h  C ) )  -  ( normh `  ( B  -h  C ) ) )  =  ( ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  C
) )  -  ( normh `  ( B  -h  C ) ) ) )
43fveq2d 5723 . . 3  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( abs `  ( ( normh `  ( A  -h  C
) )  -  ( normh `  ( B  -h  C ) ) ) )  =  ( abs `  ( ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  C ) )  -  ( normh `  ( B  -h  C ) ) ) ) )
5 oveq1 6079 . . . 4  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( A  -h  B )  =  ( if ( A  e.  ~H ,  A ,  0h )  -h  B
) )
65fveq2d 5723 . . 3  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( normh `  ( A  -h  B ) )  =  ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  B ) ) )
74, 6breq12d 4217 . 2  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  (
( abs `  (
( normh `  ( A  -h  C ) )  -  ( normh `  ( B  -h  C ) ) ) )  <_  ( normh `  ( A  -h  B
) )  <->  ( abs `  ( ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  C ) )  -  ( normh `  ( B  -h  C ) ) ) )  <_  ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  B
) ) ) )
8 oveq1 6079 . . . . . 6  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  ( B  -h  C )  =  ( if ( B  e.  ~H ,  B ,  0h )  -h  C
) )
98fveq2d 5723 . . . . 5  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  ( normh `  ( B  -h  C ) )  =  ( normh `  ( if ( B  e.  ~H ,  B ,  0h )  -h  C ) ) )
109oveq2d 6088 . . . 4  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  (
( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  C ) )  -  ( normh `  ( B  -h  C ) ) )  =  ( ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  C
) )  -  ( normh `  ( if ( B  e.  ~H ,  B ,  0h )  -h  C ) ) ) )
1110fveq2d 5723 . . 3  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  ( abs `  ( ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  C
) )  -  ( normh `  ( B  -h  C ) ) ) )  =  ( abs `  ( ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  C ) )  -  ( normh `  ( if ( B  e.  ~H ,  B ,  0h )  -h  C ) ) ) ) )
12 oveq2 6080 . . . 4  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  ( if ( A  e.  ~H ,  A ,  0h )  -h  B )  =  ( if ( A  e. 
~H ,  A ,  0h )  -h  if ( B  e.  ~H ,  B ,  0h )
) )
1312fveq2d 5723 . . 3  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  B ) )  =  ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  if ( B  e. 
~H ,  B ,  0h ) ) ) )
1411, 13breq12d 4217 . 2  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  (
( abs `  (
( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  C ) )  -  ( normh `  ( B  -h  C ) ) ) )  <_  ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  B
) )  <->  ( abs `  ( ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  C ) )  -  ( normh `  ( if ( B  e.  ~H ,  B ,  0h )  -h  C ) ) ) )  <_  ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  if ( B  e.  ~H ,  B ,  0h )
) ) ) )
15 ax-hv0cl 22494 . . . 4  |-  0h  e.  ~H
1615elimel 3783 . . 3  |-  if ( A  e.  ~H ,  A ,  0h )  e.  ~H
1715elimel 3783 . . 3  |-  if ( B  e.  ~H ,  B ,  0h )  e.  ~H
18 norm3adift.1 . . 3  |-  C  e. 
~H
1916, 17, 18norm3adifii 22638 . 2  |-  ( abs `  ( ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  C ) )  -  ( normh `  ( if ( B  e.  ~H ,  B ,  0h )  -h  C ) ) ) )  <_  ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  if ( B  e.  ~H ,  B ,  0h )
) )
207, 14, 19dedth2h 3773 1  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( abs `  (
( normh `  ( A  -h  C ) )  -  ( normh `  ( B  -h  C ) ) ) )  <_  ( normh `  ( A  -h  B
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   ifcif 3731   class class class wbr 4204   ` cfv 5445  (class class class)co 6072    <_ cle 9110    - cmin 9280   abscabs 12027   ~Hchil 22410   normhcno 22414   0hc0v 22415    -h cmv 22416
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692  ax-cnex 9035  ax-resscn 9036  ax-1cn 9037  ax-icn 9038  ax-addcl 9039  ax-addrcl 9040  ax-mulcl 9041  ax-mulrcl 9042  ax-mulcom 9043  ax-addass 9044  ax-mulass 9045  ax-distr 9046  ax-i2m1 9047  ax-1ne0 9048  ax-1rid 9049  ax-rnegex 9050  ax-rrecex 9051  ax-cnre 9052  ax-pre-lttri 9053  ax-pre-lttrn 9054  ax-pre-ltadd 9055  ax-pre-mulgt0 9056  ax-pre-sup 9057  ax-hfvadd 22491  ax-hvcom 22492  ax-hvass 22493  ax-hv0cl 22494  ax-hvaddid 22495  ax-hfvmul 22496  ax-hvmulid 22497  ax-hvmulass 22498  ax-hvdistr1 22499  ax-hvdistr2 22500  ax-hvmul0 22501  ax-hfi 22569  ax-his1 22572  ax-his2 22573  ax-his3 22574  ax-his4 22575
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4837  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-2nd 6341  df-riota 6540  df-recs 6624  df-rdg 6659  df-er 6896  df-en 7101  df-dom 7102  df-sdom 7103  df-sup 7437  df-pnf 9111  df-mnf 9112  df-xr 9113  df-ltxr 9114  df-le 9115  df-sub 9282  df-neg 9283  df-div 9667  df-nn 9990  df-2 10047  df-3 10048  df-4 10049  df-n0 10211  df-z 10272  df-uz 10478  df-rp 10602  df-seq 11312  df-exp 11371  df-cj 11892  df-re 11893  df-im 11894  df-sqr 12028  df-abs 12029  df-hnorm 22459  df-hvsub 22462
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