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Theorem norm3difi 21672
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 21546 . . . 4  |-  ( A  -h  B )  =  ( A  +h  ( -u 1  .h  B ) )
4 norm3dif.3 . . . . . . 7  |-  C  e. 
~H
51, 4hvsubvali 21546 . . . . . 6  |-  ( A  -h  C )  =  ( A  +h  ( -u 1  .h  C ) )
64, 2hvsubvali 21546 . . . . . 6  |-  ( C  -h  B )  =  ( C  +h  ( -u 1  .h  B ) )
75, 6oveq12i 5790 . . . . 5  |-  ( ( A  -h  C )  +h  ( C  -h  B ) )  =  ( ( A  +h  ( -u 1  .h  C
) )  +h  ( C  +h  ( -u 1  .h  B ) ) )
8 neg1cn 9767 . . . . . . 7  |-  -u 1  e.  CC
98, 4hvmulcli 21540 . . . . . 6  |-  ( -u
1  .h  C )  e.  ~H
108, 2hvmulcli 21540 . . . . . . 7  |-  ( -u
1  .h  B )  e.  ~H
114, 10hvaddcli 21544 . . . . . 6  |-  ( C  +h  ( -u 1  .h  B ) )  e. 
~H
121, 9, 11hvassi 21578 . . . . 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 21578 . . . . . . 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 21545 . . . . . . . . . 10  |-  ( (
-u 1  .h  C
)  +h  C )  =  ( C  +h  ( -u 1  .h  C
) )
154, 4hvsubvali 21546 . . . . . . . . . 10  |-  ( C  -h  C )  =  ( C  +h  ( -u 1  .h  C ) )
16 hvsubid 21551 . . . . . . . . . . 11  |-  ( C  e.  ~H  ->  ( C  -h  C )  =  0h )
174, 16ax-mp 10 . . . . . . . . . 10  |-  ( C  -h  C )  =  0h
1814, 15, 173eqtr2i 2282 . . . . . . . . 9  |-  ( (
-u 1  .h  C
)  +h  C )  =  0h
1918oveq1i 5788 . . . . . . . 8  |-  ( ( ( -u 1  .h  C )  +h  C
)  +h  ( -u
1  .h  B ) )  =  ( 0h 
+h  ( -u 1  .h  B ) )
20 ax-hv0cl 21529 . . . . . . . . 9  |-  0h  e.  ~H
2120, 10hvcomi 21545 . . . . . . . 8  |-  ( 0h 
+h  ( -u 1  .h  B ) )  =  ( ( -u 1  .h  B )  +h  0h )
22 ax-hvaddid 21530 . . . . . . . . 9  |-  ( (
-u 1  .h  B
)  e.  ~H  ->  ( ( -u 1  .h  B )  +h  0h )  =  ( -u 1  .h  B ) )
2310, 22ax-mp 10 . . . . . . . 8  |-  ( (
-u 1  .h  B
)  +h  0h )  =  ( -u 1  .h  B )
2419, 21, 233eqtri 2280 . . . . . . 7  |-  ( ( ( -u 1  .h  C )  +h  C
)  +h  ( -u
1  .h  B ) )  =  ( -u
1  .h  B )
2513, 24eqtr3i 2278 . . . . . 6  |-  ( (
-u 1  .h  C
)  +h  ( C  +h  ( -u 1  .h  B ) ) )  =  ( -u 1  .h  B )
2625oveq2i 5789 . . . . 5  |-  ( A  +h  ( ( -u
1  .h  C )  +h  ( C  +h  ( -u 1  .h  B
) ) ) )  =  ( A  +h  ( -u 1  .h  B
) )
277, 12, 263eqtri 2280 . . . 4  |-  ( ( A  -h  C )  +h  ( C  -h  B ) )  =  ( A  +h  ( -u 1  .h  B ) )
283, 27eqtr4i 2279 . . 3  |-  ( A  -h  B )  =  ( ( A  -h  C )  +h  ( C  -h  B ) )
2928fveq2i 5447 . 2  |-  ( normh `  ( A  -h  B
) )  =  (
normh `  ( ( A  -h  C )  +h  ( C  -h  B
) ) )
301, 4hvsubcli 21547 . . 3  |-  ( A  -h  C )  e. 
~H
314, 2hvsubcli 21547 . . 3  |-  ( C  -h  B )  e. 
~H
3230, 31norm-ii-i 21662 . 2  |-  ( normh `  ( ( A  -h  C )  +h  ( C  -h  B ) ) )  <_  ( ( normh `  ( A  -h  C ) )  +  ( normh `  ( C  -h  B ) ) )
3329, 32eqbrtri 4002 1  |-  ( normh `  ( A  -h  B
) )  <_  (
( normh `  ( A  -h  C ) )  +  ( normh `  ( C  -h  B ) ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1619    e. wcel 1621   class class class wbr 3983   ` cfv 4659  (class class class)co 5778   1c1 8692    + caddc 8694    <_ cle 8822   -ucneg 8992   ~Hchil 21445    +h cva 21446    .h csm 21447   normhcno 21449   0hc0v 21450    -h cmv 21451
This theorem is referenced by:  norm3adifii  21673  norm3lem  21674  norm3dif  21675
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 1927  ax-ext 2237  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470  ax-cnex 8747  ax-resscn 8748  ax-1cn 8749  ax-icn 8750  ax-addcl 8751  ax-addrcl 8752  ax-mulcl 8753  ax-mulrcl 8754  ax-mulcom 8755  ax-addass 8756  ax-mulass 8757  ax-distr 8758  ax-i2m1 8759  ax-1ne0 8760  ax-1rid 8761  ax-rnegex 8762  ax-rrecex 8763  ax-cnre 8764  ax-pre-lttri 8765  ax-pre-lttrn 8766  ax-pre-ltadd 8767  ax-pre-mulgt0 8768  ax-pre-sup 8769  ax-hfvadd 21526  ax-hvcom 21527  ax-hvass 21528  ax-hv0cl 21529  ax-hvaddid 21530  ax-hfvmul 21531  ax-hvmulid 21532  ax-hvmulass 21533  ax-hvdistr2 21535  ax-hvmul0 21536  ax-hfi 21604  ax-his1 21607  ax-his2 21608  ax-his3 21609  ax-his4 21610
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 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2521  df-rex 2522  df-reu 2523  df-rmo 2524  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-pss 3129  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-tp 3608  df-op 3609  df-uni 3788  df-iun 3867  df-br 3984  df-opab 4038  df-mpt 4039  df-tr 4074  df-eprel 4263  df-id 4267  df-po 4272  df-so 4273  df-fr 4310  df-we 4312  df-ord 4353  df-on 4354  df-lim 4355  df-suc 4356  df-om 4615  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-ov 5781  df-oprab 5782  df-mpt2 5783  df-2nd 6043  df-iota 6211  df-riota 6258  df-recs 6342  df-rdg 6377  df-er 6614  df-en 6818  df-dom 6819  df-sdom 6820  df-sup 7148  df-pnf 8823  df-mnf 8824  df-xr 8825  df-ltxr 8826  df-le 8827  df-sub 8993  df-neg 8994  df-div 9378  df-n 9701  df-2 9758  df-3 9759  df-4 9760  df-n0 9919  df-z 9978  df-uz 10184  df-rp 10308  df-seq 10999  df-exp 11057  df-cj 11535  df-re 11536  df-im 11537  df-sqr 11671  df-abs 11672  df-hnorm 21494  df-hvsub 21497
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