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Theorem norm-ii-i 21716
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 8837 . . . . . . . . . . 11  |-  1  e.  RR
2 ax-1cn 8795 . . . . . . . . . . . 12  |-  1  e.  CC
32cjrebi 11659 . . . . . . . . . . 11  |-  ( 1  e.  RR  <->  ( * `  1 )  =  1 )
41, 3mpbi 199 . . . . . . . . . 10  |-  ( * `
 1 )  =  1
54oveq1i 5868 . . . . . . . . 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 21660 . . . . . . . . . 10  |-  ( B 
.ih  A )  e.  CC
98mulid2i 8840 . . . . . . . . 9  |-  ( 1  x.  ( B  .ih  A ) )  =  ( B  .ih  A )
105, 9eqtri 2303 . . . . . . . 8  |-  ( ( * `  1 )  x.  ( B  .ih  A ) )  =  ( B  .ih  A )
117, 6hicli 21660 . . . . . . . . 9  |-  ( A 
.ih  B )  e.  CC
1211mulid2i 8840 . . . . . . . 8  |-  ( 1  x.  ( A  .ih  B ) )  =  ( A  .ih  B )
1310, 12oveq12i 5870 . . . . . . 7  |-  ( ( ( * `  1
)  x.  ( B 
.ih  A ) )  +  ( 1  x.  ( A  .ih  B
) ) )  =  ( ( B  .ih  A )  +  ( A 
.ih  B ) )
14 abs1 11782 . . . . . . . 8  |-  ( abs `  1 )  =  1
152, 6, 7, 14normlem7 21695 . . . . . . 7  |-  ( ( ( * `  1
)  x.  ( B 
.ih  A ) )  +  ( 1  x.  ( A  .ih  B
) ) )  <_ 
( 2  x.  (
( sqr `  ( A  .ih  A ) )  x.  ( sqr `  ( B  .ih  B ) ) ) )
1613, 15eqbrtrri 4044 . . . . . 6  |-  ( ( B  .ih  A )  +  ( A  .ih  B ) )  <_  (
2  x.  ( ( sqr `  ( A 
.ih  A ) )  x.  ( sqr `  ( B  .ih  B ) ) ) )
17 eqid 2283 . . . . . . . . . 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 21690 . . . . . . . . 9  |-  -u (
( ( * ` 
1 )  x.  ( B  .ih  A ) )  +  ( 1  x.  ( A  .ih  B
) ) )  e.  RR
192cjcli 11654 . . . . . . . . . . . 12  |-  ( * `
 1 )  e.  CC
2019, 8mulcli 8842 . . . . . . . . . . 11  |-  ( ( * `  1 )  x.  ( B  .ih  A ) )  e.  CC
212, 11mulcli 8842 . . . . . . . . . . 11  |-  ( 1  x.  ( A  .ih  B ) )  e.  CC
2220, 21addcli 8841 . . . . . . . . . 10  |-  ( ( ( * `  1
)  x.  ( B 
.ih  A ) )  +  ( 1  x.  ( A  .ih  B
) ) )  e.  CC
2322negrebi 9120 . . . . . . . . 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 199 . . . . . . . 8  |-  ( ( ( * `  1
)  x.  ( B 
.ih  A ) )  +  ( 1  x.  ( A  .ih  B
) ) )  e.  RR
2513, 24eqeltrri 2354 . . . . . . 7  |-  ( ( B  .ih  A )  +  ( A  .ih  B ) )  e.  RR
26 2re 9815 . . . . . . . 8  |-  2  e.  RR
27 hiidge0 21677 . . . . . . . . . . 11  |-  ( A  e.  ~H  ->  0  <_  ( A  .ih  A
) )
287, 27ax-mp 8 . . . . . . . . . 10  |-  0  <_  ( A  .ih  A
)
29 hiidrcl 21674 . . . . . . . . . . . 12  |-  ( A  e.  ~H  ->  ( A  .ih  A )  e.  RR )
307, 29ax-mp 8 . . . . . . . . . . 11  |-  ( A 
.ih  A )  e.  RR
3130sqrcli 11855 . . . . . . . . . 10  |-  ( 0  <_  ( A  .ih  A )  ->  ( sqr `  ( A  .ih  A
) )  e.  RR )
3228, 31ax-mp 8 . . . . . . . . 9  |-  ( sqr `  ( A  .ih  A
) )  e.  RR
33 hiidge0 21677 . . . . . . . . . . 11  |-  ( B  e.  ~H  ->  0  <_  ( B  .ih  B
) )
346, 33ax-mp 8 . . . . . . . . . 10  |-  0  <_  ( B  .ih  B
)
35 hiidrcl 21674 . . . . . . . . . . . 12  |-  ( B  e.  ~H  ->  ( B  .ih  B )  e.  RR )
366, 35ax-mp 8 . . . . . . . . . . 11  |-  ( B 
.ih  B )  e.  RR
3736sqrcli 11855 . . . . . . . . . 10  |-  ( 0  <_  ( B  .ih  B )  ->  ( sqr `  ( B  .ih  B
) )  e.  RR )
3834, 37ax-mp 8 . . . . . . . . 9  |-  ( sqr `  ( B  .ih  B
) )  e.  RR
3932, 38remulcli 8851 . . . . . . . 8  |-  ( ( sqr `  ( A 
.ih  A ) )  x.  ( sqr `  ( B  .ih  B ) ) )  e.  RR
4026, 39remulcli 8851 . . . . . . 7  |-  ( 2  x.  ( ( sqr `  ( A  .ih  A
) )  x.  ( sqr `  ( B  .ih  B ) ) ) )  e.  RR
4130, 36readdcli 8850 . . . . . . 7  |-  ( ( A  .ih  A )  +  ( B  .ih  B ) )  e.  RR
4225, 40, 41leadd2i 9329 . . . . . 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 199 . . . . 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 21696 . . . . . 6  |-  ( ( A  +h  B ) 
.ih  ( A  +h  B ) )  =  ( ( ( A 
.ih  A )  +  ( B  .ih  B
) )  +  ( ( A  .ih  B
)  +  ( B 
.ih  A ) ) )
4511, 8addcomi 9003 . . . . . . 7  |-  ( ( A  .ih  B )  +  ( B  .ih  A ) )  =  ( ( B  .ih  A
)  +  ( A 
.ih  B ) )
4645oveq2i 5869 . . . . . 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 2303 . . . . 5  |-  ( ( A  +h  B ) 
.ih  ( A  +h  B ) )  =  ( ( ( A 
.ih  A )  +  ( B  .ih  B
) )  +  ( ( B  .ih  A
)  +  ( A 
.ih  B ) ) )
4832recni 8849 . . . . . . 7  |-  ( sqr `  ( A  .ih  A
) )  e.  CC
4938recni 8849 . . . . . . 7  |-  ( sqr `  ( B  .ih  B
) )  e.  CC
5048, 49binom2i 11212 . . . . . 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 11184 . . . . . . 7  |-  ( ( sqr `  ( A 
.ih  A ) ) ^ 2 )  e.  CC
52 2cn 9816 . . . . . . . 8  |-  2  e.  CC
5348, 49mulcli 8842 . . . . . . . 8  |-  ( ( sqr `  ( A 
.ih  A ) )  x.  ( sqr `  ( B  .ih  B ) ) )  e.  CC
5452, 53mulcli 8842 . . . . . . 7  |-  ( 2  x.  ( ( sqr `  ( A  .ih  A
) )  x.  ( sqr `  ( B  .ih  B ) ) ) )  e.  CC
5549sqcli 11184 . . . . . . 7  |-  ( ( sqr `  ( B 
.ih  B ) ) ^ 2 )  e.  CC
5651, 54, 55add32i 9030 . . . . . 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 11859 . . . . . . . . 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 11859 . . . . . . . . 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 5870 . . . . . . 7  |-  ( ( ( sqr `  ( A  .ih  A ) ) ^ 2 )  +  ( ( sqr `  ( B  .ih  B ) ) ^ 2 ) )  =  ( ( A 
.ih  A )  +  ( B  .ih  B
) )
6261oveq1i 5868 . . . . . 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 2307 . . . . 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 4051 . . . 4  |-  ( ( A  +h  B ) 
.ih  ( A  +h  B ) )  <_ 
( ( ( sqr `  ( A  .ih  A
) )  +  ( sqr `  ( B 
.ih  B ) ) ) ^ 2 )
657, 6hvaddcli 21598 . . . . . 6  |-  ( A  +h  B )  e. 
~H
66 hiidge0 21677 . . . . . 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 8850 . . . . . 6  |-  ( ( sqr `  ( A 
.ih  A ) )  +  ( sqr `  ( B  .ih  B ) ) )  e.  RR
6968sqge0i 11191 . . . . 5  |-  0  <_  ( ( ( sqr `  ( A  .ih  A
) )  +  ( sqr `  ( B 
.ih  B ) ) ) ^ 2 )
70 hiidrcl 21674 . . . . . . 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 11189 . . . . . 6  |-  ( ( ( sqr `  ( A  .ih  A ) )  +  ( sqr `  ( B  .ih  B ) ) ) ^ 2 )  e.  RR
7371, 72sqrlei 11872 . . . . 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 653 . . . 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 199 . . 3  |-  ( sqr `  ( ( A  +h  B )  .ih  ( A  +h  B ) ) )  <_  ( sqr `  ( ( ( sqr `  ( A  .ih  A
) )  +  ( sqr `  ( B 
.ih  B ) ) ) ^ 2 ) )
7630sqrge0i 11860 . . . . . 6  |-  ( 0  <_  ( A  .ih  A )  ->  0  <_  ( sqr `  ( A 
.ih  A ) ) )
7728, 76ax-mp 8 . . . . 5  |-  0  <_  ( sqr `  ( A  .ih  A ) )
7836sqrge0i 11860 . . . . . 6  |-  ( 0  <_  ( B  .ih  B )  ->  0  <_  ( sqr `  ( B 
.ih  B ) ) )
7934, 78ax-mp 8 . . . . 5  |-  0  <_  ( sqr `  ( B  .ih  B ) )
8032, 38addge0i 9313 . . . . 5  |-  ( ( 0  <_  ( sqr `  ( A  .ih  A
) )  /\  0  <_  ( sqr `  ( B  .ih  B ) ) )  ->  0  <_  ( ( sqr `  ( A  .ih  A ) )  +  ( sqr `  ( B  .ih  B ) ) ) )
8177, 79, 80mp2an 653 . . . 4  |-  0  <_  ( ( sqr `  ( A  .ih  A ) )  +  ( sqr `  ( B  .ih  B ) ) )
8268sqrsqi 11858 . . . 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 4046 . 2  |-  ( sqr `  ( ( A  +h  B )  .ih  ( A  +h  B ) ) )  <_  ( ( sqr `  ( A  .ih  A ) )  +  ( sqr `  ( B 
.ih  B ) ) )
85 normval 21703 . . 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 21703 . . . 4  |-  ( A  e.  ~H  ->  ( normh `  A )  =  ( sqr `  ( A  .ih  A ) ) )
887, 87ax-mp 8 . . 3  |-  ( normh `  A )  =  ( sqr `  ( A 
.ih  A ) )
89 normval 21703 . . . 4  |-  ( B  e.  ~H  ->  ( normh `  B )  =  ( sqr `  ( B  .ih  B ) ) )
906, 89ax-mp 8 . . 3  |-  ( normh `  B )  =  ( sqr `  ( B 
.ih  B ) )
9188, 90oveq12i 5870 . 2  |-  ( (
normh `  A )  +  ( normh `  B )
)  =  ( ( sqr `  ( A 
.ih  A ) )  +  ( sqr `  ( B  .ih  B ) ) )
9284, 86, 913brtr4i 4051 1  |-  ( normh `  ( A  +h  B
) )  <_  (
( normh `  A )  +  ( normh `  B
) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1623    e. wcel 1684   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742    <_ cle 8868   -ucneg 9038   2c2 9795   ^cexp 11104   *ccj 11581   sqrcsqr 11718   ~Hchil 21499    +h cva 21500    .ih csp 21502   normhcno 21503
This theorem is referenced by:  norm-ii  21717  norm3difi  21726
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-hv0cl 21583  ax-hfvmul 21585  ax-hvmulass 21587  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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