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Theorem norm3difi 21726
Description: Norm of differences around common element. Part of Lemma 3.6 of [Beran] p. 101. (Contributed by NM, 16-Aug-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
norm3difi  |-  ( normh `  ( A  -h  B
) )  <_  (
( normh `  ( A  -h  C ) )  +  ( normh `  ( C  -h  B ) ) )

Proof of Theorem norm3difi
StepHypRef Expression
1 norm3dif.1 . . . . 5  |-  A  e. 
~H
2 norm3dif.2 . . . . 5  |-  B  e. 
~H
31, 2hvsubvali 21600 . . . 4  |-  ( A  -h  B )  =  ( A  +h  ( -u 1  .h  B ) )
4 norm3dif.3 . . . . . . 7  |-  C  e. 
~H
51, 4hvsubvali 21600 . . . . . 6  |-  ( A  -h  C )  =  ( A  +h  ( -u 1  .h  C ) )
64, 2hvsubvali 21600 . . . . . 6  |-  ( C  -h  B )  =  ( C  +h  ( -u 1  .h  B ) )
75, 6oveq12i 5870 . . . . 5  |-  ( ( A  -h  C )  +h  ( C  -h  B ) )  =  ( ( A  +h  ( -u 1  .h  C
) )  +h  ( C  +h  ( -u 1  .h  B ) ) )
8 neg1cn 9813 . . . . . . 7  |-  -u 1  e.  CC
98, 4hvmulcli 21594 . . . . . 6  |-  ( -u
1  .h  C )  e.  ~H
108, 2hvmulcli 21594 . . . . . . 7  |-  ( -u
1  .h  B )  e.  ~H
114, 10hvaddcli 21598 . . . . . 6  |-  ( C  +h  ( -u 1  .h  B ) )  e. 
~H
121, 9, 11hvassi 21632 . . . . 5  |-  ( ( A  +h  ( -u
1  .h  C ) )  +h  ( C  +h  ( -u 1  .h  B ) ) )  =  ( A  +h  ( ( -u 1  .h  C )  +h  ( C  +h  ( -u 1  .h  B ) ) ) )
139, 4, 10hvassi 21632 . . . . . . 7  |-  ( ( ( -u 1  .h  C )  +h  C
)  +h  ( -u
1  .h  B ) )  =  ( (
-u 1  .h  C
)  +h  ( C  +h  ( -u 1  .h  B ) ) )
149, 4hvcomi 21599 . . . . . . . . . 10  |-  ( (
-u 1  .h  C
)  +h  C )  =  ( C  +h  ( -u 1  .h  C
) )
154, 4hvsubvali 21600 . . . . . . . . . 10  |-  ( C  -h  C )  =  ( C  +h  ( -u 1  .h  C ) )
16 hvsubid 21605 . . . . . . . . . . 11  |-  ( C  e.  ~H  ->  ( C  -h  C )  =  0h )
174, 16ax-mp 8 . . . . . . . . . 10  |-  ( C  -h  C )  =  0h
1814, 15, 173eqtr2i 2309 . . . . . . . . 9  |-  ( (
-u 1  .h  C
)  +h  C )  =  0h
1918oveq1i 5868 . . . . . . . 8  |-  ( ( ( -u 1  .h  C )  +h  C
)  +h  ( -u
1  .h  B ) )  =  ( 0h 
+h  ( -u 1  .h  B ) )
20 ax-hv0cl 21583 . . . . . . . . 9  |-  0h  e.  ~H
2120, 10hvcomi 21599 . . . . . . . 8  |-  ( 0h 
+h  ( -u 1  .h  B ) )  =  ( ( -u 1  .h  B )  +h  0h )
22 ax-hvaddid 21584 . . . . . . . . 9  |-  ( (
-u 1  .h  B
)  e.  ~H  ->  ( ( -u 1  .h  B )  +h  0h )  =  ( -u 1  .h  B ) )
2310, 22ax-mp 8 . . . . . . . 8  |-  ( (
-u 1  .h  B
)  +h  0h )  =  ( -u 1  .h  B )
2419, 21, 233eqtri 2307 . . . . . . 7  |-  ( ( ( -u 1  .h  C )  +h  C
)  +h  ( -u
1  .h  B ) )  =  ( -u
1  .h  B )
2513, 24eqtr3i 2305 . . . . . 6  |-  ( (
-u 1  .h  C
)  +h  ( C  +h  ( -u 1  .h  B ) ) )  =  ( -u 1  .h  B )
2625oveq2i 5869 . . . . 5  |-  ( A  +h  ( ( -u
1  .h  C )  +h  ( C  +h  ( -u 1  .h  B
) ) ) )  =  ( A  +h  ( -u 1  .h  B
) )
277, 12, 263eqtri 2307 . . . 4  |-  ( ( A  -h  C )  +h  ( C  -h  B ) )  =  ( A  +h  ( -u 1  .h  B ) )
283, 27eqtr4i 2306 . . 3  |-  ( A  -h  B )  =  ( ( A  -h  C )  +h  ( C  -h  B ) )
2928fveq2i 5528 . 2  |-  ( normh `  ( A  -h  B
) )  =  (
normh `  ( ( A  -h  C )  +h  ( C  -h  B
) ) )
301, 4hvsubcli 21601 . . 3  |-  ( A  -h  C )  e. 
~H
314, 2hvsubcli 21601 . . 3  |-  ( C  -h  B )  e. 
~H
3230, 31norm-ii-i 21716 . 2  |-  ( normh `  ( ( A  -h  C )  +h  ( C  -h  B ) ) )  <_  ( ( normh `  ( A  -h  C ) )  +  ( normh `  ( C  -h  B ) ) )
3329, 32eqbrtri 4042 1  |-  ( normh `  ( A  -h  B
) )  <_  (
( normh `  ( A  -h  C ) )  +  ( normh `  ( C  -h  B ) ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1623    e. wcel 1684   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   1c1 8738    + caddc 8740    <_ cle 8868   -ucneg 9038   ~Hchil 21499    +h cva 21500    .h csm 21501   normhcno 21503   0hc0v 21504    -h cmv 21505
This theorem is referenced by:  norm3adifii  21727  norm3lem  21728  norm3dif  21729
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-hfvadd 21580  ax-hvcom 21581  ax-hvass 21582  ax-hv0cl 21583  ax-hvaddid 21584  ax-hfvmul 21585  ax-hvmulid 21586  ax-hvmulass 21587  ax-hvdistr2 21589  ax-hvmul0 21590  ax-hfi 21658  ax-his1 21661  ax-his2 21662  ax-his3 21663  ax-his4 21664
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-hnorm 21548  df-hvsub 21551
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