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Theorem norm3adifi 26641
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 6312 . . . . . 6  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( A  -h  C )  =  ( if ( A  e.  ~H ,  A ,  0h )  -h  C
) )
21fveq2d 5885 . . . . 5  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( normh `  ( A  -h  C ) )  =  ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  C ) ) )
32oveq1d 6320 . . . 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 5885 . . 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 6312 . . . 4  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( A  -h  B )  =  ( if ( A  e.  ~H ,  A ,  0h )  -h  B
) )
65fveq2d 5885 . . 3  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( normh `  ( A  -h  B ) )  =  ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  B ) ) )
74, 6breq12d 4439 . 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 6312 . . . . . 6  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  ( B  -h  C )  =  ( if ( B  e.  ~H ,  B ,  0h )  -h  C
) )
98fveq2d 5885 . . . . 5  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  ( normh `  ( B  -h  C ) )  =  ( normh `  ( if ( B  e.  ~H ,  B ,  0h )  -h  C ) ) )
109oveq2d 6321 . . . 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 5885 . . 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 6313 . . . 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 5885 . . 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 4439 . 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 ifhvhv0 26510 . . 3  |-  if ( A  e.  ~H ,  A ,  0h )  e.  ~H
16 ifhvhv0 26510 . . 3  |-  if ( B  e.  ~H ,  B ,  0h )  e.  ~H
17 norm3adift.1 . . 3  |-  C  e. 
~H
1815, 16, 17norm3adifii 26636 . 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 )
) )
197, 14, 18dedth2h 3967 1  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( abs `  (
( normh `  ( A  -h  C ) )  -  ( normh `  ( B  -h  C ) ) ) )  <_  ( normh `  ( A  -h  B
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1870   ifcif 3915   class class class wbr 4426   ` cfv 5601  (class class class)co 6305    <_ cle 9675    - cmin 9859   abscabs 13276   ~Hchil 26407   normhcno 26411   0hc0v 26412    -h cmv 26413
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616  ax-hfvadd 26488  ax-hvcom 26489  ax-hvass 26490  ax-hv0cl 26491  ax-hvaddid 26492  ax-hfvmul 26493  ax-hvmulid 26494  ax-hvmulass 26495  ax-hvdistr1 26496  ax-hvdistr2 26497  ax-hvmul0 26498  ax-hfi 26567  ax-his1 26570  ax-his2 26571  ax-his3 26572  ax-his4 26573
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-sup 7962  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-seq 12211  df-exp 12270  df-cj 13141  df-re 13142  df-im 13143  df-sqrt 13277  df-abs 13278  df-hnorm 26456  df-hvsub 26459
This theorem is referenced by: (None)
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