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Theorem norm-ii-i 22592
Description: Triangle inequality for norms. Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 11-Aug-1999.) (New usage is discouraged.)
Hypotheses
Ref Expression
norm-ii.1  |-  A  e. 
~H
norm-ii.2  |-  B  e. 
~H
Assertion
Ref Expression
norm-ii-i  |-  ( normh `  ( A  +h  B
) )  <_  (
( normh `  A )  +  ( normh `  B
) )

Proof of Theorem norm-ii-i
StepHypRef Expression
1 1re 9046 . . . . . . . . . . 11  |-  1  e.  RR
2 ax-1cn 9004 . . . . . . . . . . . 12  |-  1  e.  CC
32cjrebi 11934 . . . . . . . . . . 11  |-  ( 1  e.  RR  <->  ( * `  1 )  =  1 )
41, 3mpbi 200 . . . . . . . . . 10  |-  ( * `
 1 )  =  1
54oveq1i 6050 . . . . . . . . 9  |-  ( ( * `  1 )  x.  ( B  .ih  A ) )  =  ( 1  x.  ( B 
.ih  A ) )
6 norm-ii.2 . . . . . . . . . . 11  |-  B  e. 
~H
7 norm-ii.1 . . . . . . . . . . 11  |-  A  e. 
~H
86, 7hicli 22536 . . . . . . . . . 10  |-  ( B 
.ih  A )  e.  CC
98mulid2i 9049 . . . . . . . . 9  |-  ( 1  x.  ( B  .ih  A ) )  =  ( B  .ih  A )
105, 9eqtri 2424 . . . . . . . 8  |-  ( ( * `  1 )  x.  ( B  .ih  A ) )  =  ( B  .ih  A )
117, 6hicli 22536 . . . . . . . . 9  |-  ( A 
.ih  B )  e.  CC
1211mulid2i 9049 . . . . . . . 8  |-  ( 1  x.  ( A  .ih  B ) )  =  ( A  .ih  B )
1310, 12oveq12i 6052 . . . . . . 7  |-  ( ( ( * `  1
)  x.  ( B 
.ih  A ) )  +  ( 1  x.  ( A  .ih  B
) ) )  =  ( ( B  .ih  A )  +  ( A 
.ih  B ) )
14 abs1 12057 . . . . . . . 8  |-  ( abs `  1 )  =  1
152, 6, 7, 14normlem7 22571 . . . . . . 7  |-  ( ( ( * `  1
)  x.  ( B 
.ih  A ) )  +  ( 1  x.  ( A  .ih  B
) ) )  <_ 
( 2  x.  (
( sqr `  ( A  .ih  A ) )  x.  ( sqr `  ( B  .ih  B ) ) ) )
1613, 15eqbrtrri 4193 . . . . . 6  |-  ( ( B  .ih  A )  +  ( A  .ih  B ) )  <_  (
2  x.  ( ( sqr `  ( A 
.ih  A ) )  x.  ( sqr `  ( B  .ih  B ) ) ) )
17 eqid 2404 . . . . . . . . . 10  |-  -u (
( ( * ` 
1 )  x.  ( B  .ih  A ) )  +  ( 1  x.  ( A  .ih  B
) ) )  = 
-u ( ( ( * `  1 )  x.  ( B  .ih  A ) )  +  ( 1  x.  ( A 
.ih  B ) ) )
182, 6, 7, 17normlem2 22566 . . . . . . . . 9  |-  -u (
( ( * ` 
1 )  x.  ( B  .ih  A ) )  +  ( 1  x.  ( A  .ih  B
) ) )  e.  RR
192cjcli 11929 . . . . . . . . . . . 12  |-  ( * `
 1 )  e.  CC
2019, 8mulcli 9051 . . . . . . . . . . 11  |-  ( ( * `  1 )  x.  ( B  .ih  A ) )  e.  CC
212, 11mulcli 9051 . . . . . . . . . . 11  |-  ( 1  x.  ( A  .ih  B ) )  e.  CC
2220, 21addcli 9050 . . . . . . . . . 10  |-  ( ( ( * `  1
)  x.  ( B 
.ih  A ) )  +  ( 1  x.  ( A  .ih  B
) ) )  e.  CC
2322negrebi 9330 . . . . . . . . 9  |-  ( -u ( ( ( * `
 1 )  x.  ( B  .ih  A
) )  +  ( 1  x.  ( A 
.ih  B ) ) )  e.  RR  <->  ( (
( * `  1
)  x.  ( B 
.ih  A ) )  +  ( 1  x.  ( A  .ih  B
) ) )  e.  RR )
2418, 23mpbi 200 . . . . . . . 8  |-  ( ( ( * `  1
)  x.  ( B 
.ih  A ) )  +  ( 1  x.  ( A  .ih  B
) ) )  e.  RR
2513, 24eqeltrri 2475 . . . . . . 7  |-  ( ( B  .ih  A )  +  ( A  .ih  B ) )  e.  RR
26 2re 10025 . . . . . . . 8  |-  2  e.  RR
27 hiidge0 22553 . . . . . . . . . . 11  |-  ( A  e.  ~H  ->  0  <_  ( A  .ih  A
) )
287, 27ax-mp 8 . . . . . . . . . 10  |-  0  <_  ( A  .ih  A
)
29 hiidrcl 22550 . . . . . . . . . . . 12  |-  ( A  e.  ~H  ->  ( A  .ih  A )  e.  RR )
307, 29ax-mp 8 . . . . . . . . . . 11  |-  ( A 
.ih  A )  e.  RR
3130sqrcli 12130 . . . . . . . . . 10  |-  ( 0  <_  ( A  .ih  A )  ->  ( sqr `  ( A  .ih  A
) )  e.  RR )
3228, 31ax-mp 8 . . . . . . . . 9  |-  ( sqr `  ( A  .ih  A
) )  e.  RR
33 hiidge0 22553 . . . . . . . . . . 11  |-  ( B  e.  ~H  ->  0  <_  ( B  .ih  B
) )
346, 33ax-mp 8 . . . . . . . . . 10  |-  0  <_  ( B  .ih  B
)
35 hiidrcl 22550 . . . . . . . . . . . 12  |-  ( B  e.  ~H  ->  ( B  .ih  B )  e.  RR )
366, 35ax-mp 8 . . . . . . . . . . 11  |-  ( B 
.ih  B )  e.  RR
3736sqrcli 12130 . . . . . . . . . 10  |-  ( 0  <_  ( B  .ih  B )  ->  ( sqr `  ( B  .ih  B
) )  e.  RR )
3834, 37ax-mp 8 . . . . . . . . 9  |-  ( sqr `  ( B  .ih  B
) )  e.  RR
3932, 38remulcli 9060 . . . . . . . 8  |-  ( ( sqr `  ( A 
.ih  A ) )  x.  ( sqr `  ( B  .ih  B ) ) )  e.  RR
4026, 39remulcli 9060 . . . . . . 7  |-  ( 2  x.  ( ( sqr `  ( A  .ih  A
) )  x.  ( sqr `  ( B  .ih  B ) ) ) )  e.  RR
4130, 36readdcli 9059 . . . . . . 7  |-  ( ( A  .ih  A )  +  ( B  .ih  B ) )  e.  RR
4225, 40, 41leadd2i 9539 . . . . . 6  |-  ( ( ( B  .ih  A
)  +  ( A 
.ih  B ) )  <_  ( 2  x.  ( ( sqr `  ( A  .ih  A ) )  x.  ( sqr `  ( B  .ih  B ) ) ) )  <->  ( (
( A  .ih  A
)  +  ( B 
.ih  B ) )  +  ( ( B 
.ih  A )  +  ( A  .ih  B
) ) )  <_ 
( ( ( A 
.ih  A )  +  ( B  .ih  B
) )  +  ( 2  x.  ( ( sqr `  ( A 
.ih  A ) )  x.  ( sqr `  ( B  .ih  B ) ) ) ) ) )
4316, 42mpbi 200 . . . . 5  |-  ( ( ( A  .ih  A
)  +  ( B 
.ih  B ) )  +  ( ( B 
.ih  A )  +  ( A  .ih  B
) ) )  <_ 
( ( ( A 
.ih  A )  +  ( B  .ih  B
) )  +  ( 2  x.  ( ( sqr `  ( A 
.ih  A ) )  x.  ( sqr `  ( B  .ih  B ) ) ) ) )
447, 6, 7, 6normlem8 22572 . . . . . 6  |-  ( ( A  +h  B ) 
.ih  ( A  +h  B ) )  =  ( ( ( A 
.ih  A )  +  ( B  .ih  B
) )  +  ( ( A  .ih  B
)  +  ( B 
.ih  A ) ) )
4511, 8addcomi 9213 . . . . . . 7  |-  ( ( A  .ih  B )  +  ( B  .ih  A ) )  =  ( ( B  .ih  A
)  +  ( A 
.ih  B ) )
4645oveq2i 6051 . . . . . 6  |-  ( ( ( A  .ih  A
)  +  ( B 
.ih  B ) )  +  ( ( A 
.ih  B )  +  ( B  .ih  A
) ) )  =  ( ( ( A 
.ih  A )  +  ( B  .ih  B
) )  +  ( ( B  .ih  A
)  +  ( A 
.ih  B ) ) )
4744, 46eqtri 2424 . . . . 5  |-  ( ( A  +h  B ) 
.ih  ( A  +h  B ) )  =  ( ( ( A 
.ih  A )  +  ( B  .ih  B
) )  +  ( ( B  .ih  A
)  +  ( A 
.ih  B ) ) )
4832recni 9058 . . . . . . 7  |-  ( sqr `  ( A  .ih  A
) )  e.  CC
4938recni 9058 . . . . . . 7  |-  ( sqr `  ( B  .ih  B
) )  e.  CC
5048, 49binom2i 11445 . . . . . 6  |-  ( ( ( sqr `  ( A  .ih  A ) )  +  ( sqr `  ( B  .ih  B ) ) ) ^ 2 )  =  ( ( ( ( sqr `  ( A  .ih  A ) ) ^ 2 )  +  ( 2  x.  (
( sqr `  ( A  .ih  A ) )  x.  ( sqr `  ( B  .ih  B ) ) ) ) )  +  ( ( sqr `  ( B  .ih  B ) ) ^ 2 ) )
5148sqcli 11417 . . . . . . 7  |-  ( ( sqr `  ( A 
.ih  A ) ) ^ 2 )  e.  CC
52 2cn 10026 . . . . . . . 8  |-  2  e.  CC
5348, 49mulcli 9051 . . . . . . . 8  |-  ( ( sqr `  ( A 
.ih  A ) )  x.  ( sqr `  ( B  .ih  B ) ) )  e.  CC
5452, 53mulcli 9051 . . . . . . 7  |-  ( 2  x.  ( ( sqr `  ( A  .ih  A
) )  x.  ( sqr `  ( B  .ih  B ) ) ) )  e.  CC
5549sqcli 11417 . . . . . . 7  |-  ( ( sqr `  ( B 
.ih  B ) ) ^ 2 )  e.  CC
5651, 54, 55add32i 9240 . . . . . 6  |-  ( ( ( ( sqr `  ( A  .ih  A ) ) ^ 2 )  +  ( 2  x.  (
( sqr `  ( A  .ih  A ) )  x.  ( sqr `  ( B  .ih  B ) ) ) ) )  +  ( ( sqr `  ( B  .ih  B ) ) ^ 2 ) )  =  ( ( ( ( sqr `  ( A  .ih  A ) ) ^ 2 )  +  ( ( sqr `  ( B  .ih  B ) ) ^ 2 ) )  +  ( 2  x.  ( ( sqr `  ( A  .ih  A ) )  x.  ( sqr `  ( B  .ih  B ) ) ) ) )
5730sqsqri 12134 . . . . . . . . 9  |-  ( 0  <_  ( A  .ih  A )  ->  ( ( sqr `  ( A  .ih  A ) ) ^ 2 )  =  ( A 
.ih  A ) )
5828, 57ax-mp 8 . . . . . . . 8  |-  ( ( sqr `  ( A 
.ih  A ) ) ^ 2 )  =  ( A  .ih  A
)
5936sqsqri 12134 . . . . . . . . 9  |-  ( 0  <_  ( B  .ih  B )  ->  ( ( sqr `  ( B  .ih  B ) ) ^ 2 )  =  ( B 
.ih  B ) )
6034, 59ax-mp 8 . . . . . . . 8  |-  ( ( sqr `  ( B 
.ih  B ) ) ^ 2 )  =  ( B  .ih  B
)
6158, 60oveq12i 6052 . . . . . . 7  |-  ( ( ( sqr `  ( A  .ih  A ) ) ^ 2 )  +  ( ( sqr `  ( B  .ih  B ) ) ^ 2 ) )  =  ( ( A 
.ih  A )  +  ( B  .ih  B
) )
6261oveq1i 6050 . . . . . 6  |-  ( ( ( ( sqr `  ( A  .ih  A ) ) ^ 2 )  +  ( ( sqr `  ( B  .ih  B ) ) ^ 2 ) )  +  ( 2  x.  ( ( sqr `  ( A  .ih  A ) )  x.  ( sqr `  ( B  .ih  B ) ) ) ) )  =  ( ( ( A 
.ih  A )  +  ( B  .ih  B
) )  +  ( 2  x.  ( ( sqr `  ( A 
.ih  A ) )  x.  ( sqr `  ( B  .ih  B ) ) ) ) )
6350, 56, 623eqtri 2428 . . . . 5  |-  ( ( ( sqr `  ( A  .ih  A ) )  +  ( sqr `  ( B  .ih  B ) ) ) ^ 2 )  =  ( ( ( A  .ih  A )  +  ( B  .ih  B ) )  +  ( 2  x.  ( ( sqr `  ( A 
.ih  A ) )  x.  ( sqr `  ( B  .ih  B ) ) ) ) )
6443, 47, 633brtr4i 4200 . . . 4  |-  ( ( A  +h  B ) 
.ih  ( A  +h  B ) )  <_ 
( ( ( sqr `  ( A  .ih  A
) )  +  ( sqr `  ( B 
.ih  B ) ) ) ^ 2 )
657, 6hvaddcli 22474 . . . . . 6  |-  ( A  +h  B )  e. 
~H
66 hiidge0 22553 . . . . . 6  |-  ( ( A  +h  B )  e.  ~H  ->  0  <_  ( ( A  +h  B )  .ih  ( A  +h  B ) ) )
6765, 66ax-mp 8 . . . . 5  |-  0  <_  ( ( A  +h  B )  .ih  ( A  +h  B ) )
6832, 38readdcli 9059 . . . . . 6  |-  ( ( sqr `  ( A 
.ih  A ) )  +  ( sqr `  ( B  .ih  B ) ) )  e.  RR
6968sqge0i 11424 . . . . 5  |-  0  <_  ( ( ( sqr `  ( A  .ih  A
) )  +  ( sqr `  ( B 
.ih  B ) ) ) ^ 2 )
70 hiidrcl 22550 . . . . . . 7  |-  ( ( A  +h  B )  e.  ~H  ->  (
( A  +h  B
)  .ih  ( A  +h  B ) )  e.  RR )
7165, 70ax-mp 8 . . . . . 6  |-  ( ( A  +h  B ) 
.ih  ( A  +h  B ) )  e.  RR
7268resqcli 11422 . . . . . 6  |-  ( ( ( sqr `  ( A  .ih  A ) )  +  ( sqr `  ( B  .ih  B ) ) ) ^ 2 )  e.  RR
7371, 72sqrlei 12147 . . . . 5  |-  ( ( 0  <_  ( ( A  +h  B )  .ih  ( A  +h  B
) )  /\  0  <_  ( ( ( sqr `  ( A  .ih  A
) )  +  ( sqr `  ( B 
.ih  B ) ) ) ^ 2 ) )  ->  ( (
( A  +h  B
)  .ih  ( A  +h  B ) )  <_ 
( ( ( sqr `  ( A  .ih  A
) )  +  ( sqr `  ( B 
.ih  B ) ) ) ^ 2 )  <-> 
( sqr `  (
( A  +h  B
)  .ih  ( A  +h  B ) ) )  <_  ( sqr `  (
( ( sqr `  ( A  .ih  A ) )  +  ( sqr `  ( B  .ih  B ) ) ) ^ 2 ) ) ) )
7467, 69, 73mp2an 654 . . . 4  |-  ( ( ( A  +h  B
)  .ih  ( A  +h  B ) )  <_ 
( ( ( sqr `  ( A  .ih  A
) )  +  ( sqr `  ( B 
.ih  B ) ) ) ^ 2 )  <-> 
( sqr `  (
( A  +h  B
)  .ih  ( A  +h  B ) ) )  <_  ( sqr `  (
( ( sqr `  ( A  .ih  A ) )  +  ( sqr `  ( B  .ih  B ) ) ) ^ 2 ) ) )
7564, 74mpbi 200 . . 3  |-  ( sqr `  ( ( A  +h  B )  .ih  ( A  +h  B ) ) )  <_  ( sqr `  ( ( ( sqr `  ( A  .ih  A
) )  +  ( sqr `  ( B 
.ih  B ) ) ) ^ 2 ) )
7630sqrge0i 12135 . . . . . 6  |-  ( 0  <_  ( A  .ih  A )  ->  0  <_  ( sqr `  ( A 
.ih  A ) ) )
7728, 76ax-mp 8 . . . . 5  |-  0  <_  ( sqr `  ( A  .ih  A ) )
7836sqrge0i 12135 . . . . . 6  |-  ( 0  <_  ( B  .ih  B )  ->  0  <_  ( sqr `  ( B 
.ih  B ) ) )
7934, 78ax-mp 8 . . . . 5  |-  0  <_  ( sqr `  ( B  .ih  B ) )
8032, 38addge0i 9523 . . . . 5  |-  ( ( 0  <_  ( sqr `  ( A  .ih  A
) )  /\  0  <_  ( sqr `  ( B  .ih  B ) ) )  ->  0  <_  ( ( sqr `  ( A  .ih  A ) )  +  ( sqr `  ( B  .ih  B ) ) ) )
8177, 79, 80mp2an 654 . . . 4  |-  0  <_  ( ( sqr `  ( A  .ih  A ) )  +  ( sqr `  ( B  .ih  B ) ) )
8268sqrsqi 12133 . . . 4  |-  ( 0  <_  ( ( sqr `  ( A  .ih  A
) )  +  ( sqr `  ( B 
.ih  B ) ) )  ->  ( sqr `  ( ( ( sqr `  ( A  .ih  A
) )  +  ( sqr `  ( B 
.ih  B ) ) ) ^ 2 ) )  =  ( ( sqr `  ( A 
.ih  A ) )  +  ( sqr `  ( B  .ih  B ) ) ) )
8381, 82ax-mp 8 . . 3  |-  ( sqr `  ( ( ( sqr `  ( A  .ih  A
) )  +  ( sqr `  ( B 
.ih  B ) ) ) ^ 2 ) )  =  ( ( sqr `  ( A 
.ih  A ) )  +  ( sqr `  ( B  .ih  B ) ) )
8475, 83breqtri 4195 . 2  |-  ( sqr `  ( ( A  +h  B )  .ih  ( A  +h  B ) ) )  <_  ( ( sqr `  ( A  .ih  A ) )  +  ( sqr `  ( B 
.ih  B ) ) )
85 normval 22579 . . 3  |-  ( ( A  +h  B )  e.  ~H  ->  ( normh `  ( A  +h  B ) )  =  ( sqr `  (
( A  +h  B
)  .ih  ( A  +h  B ) ) ) )
8665, 85ax-mp 8 . 2  |-  ( normh `  ( A  +h  B
) )  =  ( sqr `  ( ( A  +h  B ) 
.ih  ( A  +h  B ) ) )
87 normval 22579 . . . 4  |-  ( A  e.  ~H  ->  ( normh `  A )  =  ( sqr `  ( A  .ih  A ) ) )
887, 87ax-mp 8 . . 3  |-  ( normh `  A )  =  ( sqr `  ( A 
.ih  A ) )
89 normval 22579 . . . 4  |-  ( B  e.  ~H  ->  ( normh `  B )  =  ( sqr `  ( B  .ih  B ) ) )
906, 89ax-mp 8 . . 3  |-  ( normh `  B )  =  ( sqr `  ( B 
.ih  B ) )
9188, 90oveq12i 6052 . 2  |-  ( (
normh `  A )  +  ( normh `  B )
)  =  ( ( sqr `  ( A 
.ih  A ) )  +  ( sqr `  ( B  .ih  B ) ) )
9284, 86, 913brtr4i 4200 1  |-  ( normh `  ( A  +h  B
) )  <_  (
( normh `  A )  +  ( normh `  B
) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    = wceq 1649    e. wcel 1721   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   RRcr 8945   0cc0 8946   1c1 8947    + caddc 8949    x. cmul 8951    <_ cle 9077   -ucneg 9248   2c2 10005   ^cexp 11337   *ccj 11856   sqrcsqr 11993   ~Hchil 22375    +h cva 22376    .ih csp 22378   normhcno 22379
This theorem is referenced by:  norm-ii  22593  norm3difi  22602
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-hfvadd 22456  ax-hv0cl 22459  ax-hfvmul 22461  ax-hvmulass 22463  ax-hvmul0 22466  ax-hfi 22534  ax-his1 22537  ax-his2 22538  ax-his3 22539  ax-his4 22540
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-seq 11279  df-exp 11338  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-hnorm 22424  df-hvsub 22427
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