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Theorem norm3difi 26042
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 25915 . . . 4  |-  ( A  -h  B )  =  ( A  +h  ( -u 1  .h  B ) )
4 norm3dif.3 . . . . . . 7  |-  C  e. 
~H
51, 4hvsubvali 25915 . . . . . 6  |-  ( A  -h  C )  =  ( A  +h  ( -u 1  .h  C ) )
64, 2hvsubvali 25915 . . . . . 6  |-  ( C  -h  B )  =  ( C  +h  ( -u 1  .h  B ) )
75, 6oveq12i 6293 . . . . 5  |-  ( ( A  -h  C )  +h  ( C  -h  B ) )  =  ( ( A  +h  ( -u 1  .h  C
) )  +h  ( C  +h  ( -u 1  .h  B ) ) )
8 neg1cn 10646 . . . . . . 7  |-  -u 1  e.  CC
98, 4hvmulcli 25909 . . . . . 6  |-  ( -u
1  .h  C )  e.  ~H
108, 2hvmulcli 25909 . . . . . . 7  |-  ( -u
1  .h  B )  e.  ~H
114, 10hvaddcli 25913 . . . . . 6  |-  ( C  +h  ( -u 1  .h  B ) )  e. 
~H
121, 9, 11hvassi 25948 . . . . 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 25948 . . . . . . 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 25914 . . . . . . . . . 10  |-  ( (
-u 1  .h  C
)  +h  C )  =  ( C  +h  ( -u 1  .h  C
) )
154, 4hvsubvali 25915 . . . . . . . . . 10  |-  ( C  -h  C )  =  ( C  +h  ( -u 1  .h  C ) )
16 hvsubid 25921 . . . . . . . . . . 11  |-  ( C  e.  ~H  ->  ( C  -h  C )  =  0h )
174, 16ax-mp 5 . . . . . . . . . 10  |-  ( C  -h  C )  =  0h
1814, 15, 173eqtr2i 2478 . . . . . . . . 9  |-  ( (
-u 1  .h  C
)  +h  C )  =  0h
1918oveq1i 6291 . . . . . . . 8  |-  ( ( ( -u 1  .h  C )  +h  C
)  +h  ( -u
1  .h  B ) )  =  ( 0h 
+h  ( -u 1  .h  B ) )
20 ax-hv0cl 25898 . . . . . . . . 9  |-  0h  e.  ~H
2120, 10hvcomi 25914 . . . . . . . 8  |-  ( 0h 
+h  ( -u 1  .h  B ) )  =  ( ( -u 1  .h  B )  +h  0h )
22 ax-hvaddid 25899 . . . . . . . . 9  |-  ( (
-u 1  .h  B
)  e.  ~H  ->  ( ( -u 1  .h  B )  +h  0h )  =  ( -u 1  .h  B ) )
2310, 22ax-mp 5 . . . . . . . 8  |-  ( (
-u 1  .h  B
)  +h  0h )  =  ( -u 1  .h  B )
2419, 21, 233eqtri 2476 . . . . . . 7  |-  ( ( ( -u 1  .h  C )  +h  C
)  +h  ( -u
1  .h  B ) )  =  ( -u
1  .h  B )
2513, 24eqtr3i 2474 . . . . . 6  |-  ( (
-u 1  .h  C
)  +h  ( C  +h  ( -u 1  .h  B ) ) )  =  ( -u 1  .h  B )
2625oveq2i 6292 . . . . 5  |-  ( A  +h  ( ( -u
1  .h  C )  +h  ( C  +h  ( -u 1  .h  B
) ) ) )  =  ( A  +h  ( -u 1  .h  B
) )
277, 12, 263eqtri 2476 . . . 4  |-  ( ( A  -h  C )  +h  ( C  -h  B ) )  =  ( A  +h  ( -u 1  .h  B ) )
283, 27eqtr4i 2475 . . 3  |-  ( A  -h  B )  =  ( ( A  -h  C )  +h  ( C  -h  B ) )
2928fveq2i 5859 . 2  |-  ( normh `  ( A  -h  B
) )  =  (
normh `  ( ( A  -h  C )  +h  ( C  -h  B
) ) )
301, 4hvsubcli 25916 . . 3  |-  ( A  -h  C )  e. 
~H
314, 2hvsubcli 25916 . . 3  |-  ( C  -h  B )  e. 
~H
3230, 31norm-ii-i 26032 . 2  |-  ( normh `  ( ( A  -h  C )  +h  ( C  -h  B ) ) )  <_  ( ( normh `  ( A  -h  C ) )  +  ( normh `  ( C  -h  B ) ) )
3329, 32eqbrtri 4456 1  |-  ( normh `  ( A  -h  B
) )  <_  (
( normh `  ( A  -h  C ) )  +  ( normh `  ( C  -h  B ) ) )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1383    e. wcel 1804   class class class wbr 4437   ` cfv 5578  (class class class)co 6281   1c1 9496    + caddc 9498    <_ cle 9632   -ucneg 9811   ~Hchil 25814    +h cva 25815    .h csm 25816   normhcno 25818   0hc0v 25819    -h cmv 25820
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573  ax-hfvadd 25895  ax-hvcom 25896  ax-hvass 25897  ax-hv0cl 25898  ax-hvaddid 25899  ax-hfvmul 25900  ax-hvmulid 25901  ax-hvmulass 25902  ax-hvdistr2 25904  ax-hvmul0 25905  ax-hfi 25974  ax-his1 25977  ax-his2 25978  ax-his3 25979  ax-his4 25980
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-2nd 6786  df-recs 7044  df-rdg 7078  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-sup 7903  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10214  df-nn 10544  df-2 10601  df-3 10602  df-4 10603  df-n0 10803  df-z 10872  df-uz 11093  df-rp 11232  df-seq 12090  df-exp 12149  df-cj 12914  df-re 12915  df-im 12916  df-sqrt 13050  df-abs 13051  df-hnorm 25863  df-hvsub 25866
This theorem is referenced by:  norm3adifii  26043  norm3lem  26044  norm3dif  26045
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