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Theorem norm3adifi 25732
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 6282 . . . . . 6  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( A  -h  C )  =  ( if ( A  e.  ~H ,  A ,  0h )  -h  C
) )
21fveq2d 5861 . . . . 5  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( normh `  ( A  -h  C ) )  =  ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  C ) ) )
32oveq1d 6290 . . . 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 5861 . . 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 6282 . . . 4  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( A  -h  B )  =  ( if ( A  e.  ~H ,  A ,  0h )  -h  B
) )
65fveq2d 5861 . . 3  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( normh `  ( A  -h  B ) )  =  ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  B ) ) )
74, 6breq12d 4453 . 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 6282 . . . . . 6  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  ( B  -h  C )  =  ( if ( B  e.  ~H ,  B ,  0h )  -h  C
) )
98fveq2d 5861 . . . . 5  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  ( normh `  ( B  -h  C ) )  =  ( normh `  ( if ( B  e.  ~H ,  B ,  0h )  -h  C ) ) )
109oveq2d 6291 . . . 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 5861 . . 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 6283 . . . 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 5861 . . 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 4453 . 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 25601 . . 3  |-  if ( A  e.  ~H ,  A ,  0h )  e.  ~H
16 ifhvhv0 25601 . . 3  |-  if ( B  e.  ~H ,  B ,  0h )  e.  ~H
17 norm3adift.1 . . 3  |-  C  e. 
~H
1815, 16, 17norm3adifii 25727 . 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 3985 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 369    = wceq 1374    e. wcel 1762   ifcif 3932   class class class wbr 4440   ` cfv 5579  (class class class)co 6275    <_ cle 9618    - cmin 9794   abscabs 13017   ~Hchil 25498   normhcno 25502   0hc0v 25503    -h cmv 25504
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559  ax-hfvadd 25579  ax-hvcom 25580  ax-hvass 25581  ax-hv0cl 25582  ax-hvaddid 25583  ax-hfvmul 25584  ax-hvmulid 25585  ax-hvmulass 25586  ax-hvdistr1 25587  ax-hvdistr2 25588  ax-hvmul0 25589  ax-hfi 25658  ax-his1 25661  ax-his2 25662  ax-his3 25663  ax-his4 25664
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-2nd 6775  df-recs 7032  df-rdg 7066  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-sup 7890  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-2 10583  df-3 10584  df-4 10585  df-n0 10785  df-z 10854  df-uz 11072  df-rp 11210  df-seq 12064  df-exp 12123  df-cj 12882  df-re 12883  df-im 12884  df-sqr 13018  df-abs 13019  df-hnorm 25547  df-hvsub 25550
This theorem is referenced by: (None)
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