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Theorem norm3dif 21559
Description: Norm of differences around common element. Part of Lemma 3.6 of [Beran] p. 101. (Contributed by NM, 20-Apr-2006.) (New usage is discouraged.)
Assertion
Ref Expression
norm3dif  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( normh `  ( A  -h  B ) )  <_ 
( ( normh `  ( A  -h  C ) )  +  ( normh `  ( C  -h  B ) ) ) )

Proof of Theorem norm3dif
StepHypRef Expression
1 oveq1 5717 . . . 4  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( A  -h  B )  =  ( if ( A  e.  ~H ,  A ,  0h )  -h  B
) )
21fveq2d 5381 . . 3  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( normh `  ( A  -h  B ) )  =  ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  B ) ) )
3 oveq1 5717 . . . . 5  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( A  -h  C )  =  ( if ( A  e.  ~H ,  A ,  0h )  -h  C
) )
43fveq2d 5381 . . . 4  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( normh `  ( A  -h  C ) )  =  ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  C ) ) )
54oveq1d 5725 . . 3  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  (
( normh `  ( A  -h  C ) )  +  ( normh `  ( C  -h  B ) ) )  =  ( ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  C
) )  +  (
normh `  ( C  -h  B ) ) ) )
62, 5breq12d 3933 . 2  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  (
( normh `  ( A  -h  B ) )  <_ 
( ( normh `  ( A  -h  C ) )  +  ( normh `  ( C  -h  B ) ) )  <->  ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  B ) )  <_ 
( ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  C ) )  +  ( normh `  ( C  -h  B ) ) ) ) )
7 oveq2 5718 . . . 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 )
) )
87fveq2d 5381 . . 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 ) ) ) )
9 oveq2 5718 . . . . 5  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  ( C  -h  B )  =  ( C  -h  if ( B  e.  ~H ,  B ,  0h )
) )
109fveq2d 5381 . . . 4  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  ( normh `  ( C  -h  B ) )  =  ( normh `  ( C  -h  if ( B  e. 
~H ,  B ,  0h ) ) ) )
1110oveq2d 5726 . . 3  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  (
( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  C ) )  +  ( normh `  ( C  -h  B ) ) )  =  ( ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  C
) )  +  (
normh `  ( C  -h  if ( B  e.  ~H ,  B ,  0h )
) ) ) )
128, 11breq12d 3933 . 2  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  (
( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  B ) )  <_ 
( ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  C ) )  +  ( normh `  ( C  -h  B ) ) )  <-> 
( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  if ( B  e. 
~H ,  B ,  0h ) ) )  <_ 
( ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  C ) )  +  ( normh `  ( C  -h  if ( B  e. 
~H ,  B ,  0h ) ) ) ) ) )
13 oveq2 5718 . . . . 5  |-  ( C  =  if ( C  e.  ~H ,  C ,  0h )  ->  ( if ( A  e.  ~H ,  A ,  0h )  -h  C )  =  ( if ( A  e. 
~H ,  A ,  0h )  -h  if ( C  e.  ~H ,  C ,  0h )
) )
1413fveq2d 5381 . . . 4  |-  ( C  =  if ( C  e.  ~H ,  C ,  0h )  ->  ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  C ) )  =  ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  if ( C  e. 
~H ,  C ,  0h ) ) ) )
15 oveq1 5717 . . . . 5  |-  ( C  =  if ( C  e.  ~H ,  C ,  0h )  ->  ( C  -h  if ( B  e.  ~H ,  B ,  0h ) )  =  ( if ( C  e.  ~H ,  C ,  0h )  -h  if ( B  e.  ~H ,  B ,  0h )
) )
1615fveq2d 5381 . . . 4  |-  ( C  =  if ( C  e.  ~H ,  C ,  0h )  ->  ( normh `  ( C  -h  if ( B  e.  ~H ,  B ,  0h )
) )  =  (
normh `  ( if ( C  e.  ~H ,  C ,  0h )  -h  if ( B  e. 
~H ,  B ,  0h ) ) ) )
1714, 16oveq12d 5728 . . 3  |-  ( C  =  if ( C  e.  ~H ,  C ,  0h )  ->  (
( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  C ) )  +  ( normh `  ( C  -h  if ( B  e. 
~H ,  B ,  0h ) ) ) )  =  ( ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  if ( C  e.  ~H ,  C ,  0h )
) )  +  (
normh `  ( if ( C  e.  ~H ,  C ,  0h )  -h  if ( B  e. 
~H ,  B ,  0h ) ) ) ) )
1817breq2d 3932 . 2  |-  ( C  =  if ( C  e.  ~H ,  C ,  0h )  ->  (
( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  if ( B  e. 
~H ,  B ,  0h ) ) )  <_ 
( ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  C ) )  +  ( normh `  ( C  -h  if ( B  e. 
~H ,  B ,  0h ) ) ) )  <-> 
( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  if ( B  e. 
~H ,  B ,  0h ) ) )  <_ 
( ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  if ( C  e. 
~H ,  C ,  0h ) ) )  +  ( normh `  ( if ( C  e.  ~H ,  C ,  0h )  -h  if ( B  e. 
~H ,  B ,  0h ) ) ) ) ) )
19 ax-hv0cl 21413 . . . 4  |-  0h  e.  ~H
2019elimel 3522 . . 3  |-  if ( A  e.  ~H ,  A ,  0h )  e.  ~H
2119elimel 3522 . . 3  |-  if ( B  e.  ~H ,  B ,  0h )  e.  ~H
2219elimel 3522 . . 3  |-  if ( C  e.  ~H ,  C ,  0h )  e.  ~H
2320, 21, 22norm3difi 21556 . 2  |-  ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  if ( B  e.  ~H ,  B ,  0h )
) )  <_  (
( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  if ( C  e. 
~H ,  C ,  0h ) ) )  +  ( normh `  ( if ( C  e.  ~H ,  C ,  0h )  -h  if ( B  e. 
~H ,  B ,  0h ) ) ) )
246, 12, 18, 23dedth3h 3513 1  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( normh `  ( A  -h  B ) )  <_ 
( ( normh `  ( A  -h  C ) )  +  ( normh `  ( C  -h  B ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ w3a 939    = wceq 1619    e. wcel 1621   ifcif 3470   class class class wbr 3920   ` cfv 4592  (class class class)co 5710    + caddc 8620    <_ cle 8748   ~Hchil 21329   normhcno 21333   0hc0v 21334    -h cmv 21335
This theorem is referenced by:  norm3dif2  21560
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-cnex 8673  ax-resscn 8674  ax-1cn 8675  ax-icn 8676  ax-addcl 8677  ax-addrcl 8678  ax-mulcl 8679  ax-mulrcl 8680  ax-mulcom 8681  ax-addass 8682  ax-mulass 8683  ax-distr 8684  ax-i2m1 8685  ax-1ne0 8686  ax-1rid 8687  ax-rnegex 8688  ax-rrecex 8689  ax-cnre 8690  ax-pre-lttri 8691  ax-pre-lttrn 8692  ax-pre-ltadd 8693  ax-pre-mulgt0 8694  ax-pre-sup 8695  ax-hfvadd 21410  ax-hvcom 21411  ax-hvass 21412  ax-hv0cl 21413  ax-hvaddid 21414  ax-hfvmul 21415  ax-hvmulid 21416  ax-hvmulass 21417  ax-hvdistr2 21419  ax-hvmul0 21420  ax-hfi 21488  ax-his1 21491  ax-his2 21492  ax-his3 21493  ax-his4 21494
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-2nd 5975  df-iota 6143  df-riota 6190  df-recs 6274  df-rdg 6309  df-er 6546  df-en 6750  df-dom 6751  df-sdom 6752  df-sup 7078  df-pnf 8749  df-mnf 8750  df-xr 8751  df-ltxr 8752  df-le 8753  df-sub 8919  df-neg 8920  df-div 9304  df-n 9627  df-2 9684  df-3 9685  df-4 9686  df-n0 9845  df-z 9904  df-uz 10110  df-rp 10234  df-seq 10925  df-exp 10983  df-cj 11461  df-re 11462  df-im 11463  df-sqr 11597  df-abs 11598  df-hnorm 21378  df-hvsub 21381
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