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Theorem norm3adifii 21557
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 21431 . . . . . . 7  |-  ( A  -h  C )  e. 
~H
43normcli 21540 . . . . . 6  |-  ( normh `  ( A  -h  C
) )  e.  RR
54recni 8729 . . . . 5  |-  ( normh `  ( A  -h  C
) )  e.  CC
6 norm3dif.2 . . . . . . . 8  |-  B  e. 
~H
76, 2hvsubcli 21431 . . . . . . 7  |-  ( B  -h  C )  e. 
~H
87normcli 21540 . . . . . 6  |-  ( normh `  ( B  -h  C
) )  e.  RR
98recni 8729 . . . . 5  |-  ( normh `  ( B  -h  C
) )  e.  CC
105, 9negsubdi2i 9012 . . . 4  |-  -u (
( normh `  ( A  -h  C ) )  -  ( normh `  ( B  -h  C ) ) )  =  ( ( normh `  ( B  -h  C
) )  -  ( normh `  ( A  -h  C ) ) )
116, 2, 1norm3difi 21556 . . . . . 6  |-  ( normh `  ( B  -h  C
) )  <_  (
( normh `  ( B  -h  A ) )  +  ( normh `  ( A  -h  C ) ) )
126, 1normsubi 21550 . . . . . . 7  |-  ( normh `  ( B  -h  A
) )  =  (
normh `  ( A  -h  B ) )
1312oveq1i 5720 . . . . . 6  |-  ( (
normh `  ( B  -h  A ) )  +  ( normh `  ( A  -h  C ) ) )  =  ( ( normh `  ( A  -h  B
) )  +  (
normh `  ( A  -h  C ) ) )
1411, 13breqtri 3943 . . . . 5  |-  ( normh `  ( B  -h  C
) )  <_  (
( normh `  ( A  -h  B ) )  +  ( normh `  ( A  -h  C ) ) )
151, 6hvsubcli 21431 . . . . . . 7  |-  ( A  -h  B )  e. 
~H
1615normcli 21540 . . . . . 6  |-  ( normh `  ( A  -h  B
) )  e.  RR
178, 4, 16lesubaddi 9211 . . . . 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 3939 . . 3  |-  -u (
( normh `  ( A  -h  C ) )  -  ( normh `  ( B  -h  C ) ) )  <_  ( normh `  ( A  -h  B ) )
204, 8resubcli 8989 . . . 4  |-  ( (
normh `  ( A  -h  C ) )  -  ( normh `  ( B  -h  C ) ) )  e.  RR
2120, 16lenegcon1i 9205 . . 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 21556 . . 3  |-  ( normh `  ( A  -h  C
) )  <_  (
( normh `  ( A  -h  B ) )  +  ( normh `  ( B  -h  C ) ) )
244, 8, 16lesubaddi 9211 . . 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 11752 . 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 891 1  |-  ( abs `  ( ( normh `  ( A  -h  C ) )  -  ( normh `  ( B  -h  C ) ) ) )  <_  ( normh `  ( A  -h  B ) )
Colors of variables: wff set class
Syntax hints:    e. wcel 1621   class class class wbr 3920   ` cfv 4592  (class class class)co 5710    + caddc 8620    <_ cle 8748    - cmin 8917   -ucneg 8918   abscabs 11596   ~Hchil 21329   normhcno 21333    -h cmv 21335
This theorem is referenced by:  norm3adifi  21562
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-hvdistr1 21418  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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