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Theorem norm-ii-i 21732
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 8853 . . . . . . . . . . 11  |-  1  e.  RR
2 ax-1cn 8811 . . . . . . . . . . . 12  |-  1  e.  CC
32cjrebi 11675 . . . . . . . . . . 11  |-  ( 1  e.  RR  <->  ( * `  1 )  =  1 )
41, 3mpbi 199 . . . . . . . . . 10  |-  ( * `
 1 )  =  1
54oveq1i 5884 . . . . . . . . 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 21676 . . . . . . . . . 10  |-  ( B 
.ih  A )  e.  CC
98mulid2i 8856 . . . . . . . . 9  |-  ( 1  x.  ( B  .ih  A ) )  =  ( B  .ih  A )
105, 9eqtri 2316 . . . . . . . 8  |-  ( ( * `  1 )  x.  ( B  .ih  A ) )  =  ( B  .ih  A )
117, 6hicli 21676 . . . . . . . . 9  |-  ( A 
.ih  B )  e.  CC
1211mulid2i 8856 . . . . . . . 8  |-  ( 1  x.  ( A  .ih  B ) )  =  ( A  .ih  B )
1310, 12oveq12i 5886 . . . . . . 7  |-  ( ( ( * `  1
)  x.  ( B 
.ih  A ) )  +  ( 1  x.  ( A  .ih  B
) ) )  =  ( ( B  .ih  A )  +  ( A 
.ih  B ) )
14 abs1 11798 . . . . . . . 8  |-  ( abs `  1 )  =  1
152, 6, 7, 14normlem7 21711 . . . . . . 7  |-  ( ( ( * `  1
)  x.  ( B 
.ih  A ) )  +  ( 1  x.  ( A  .ih  B
) ) )  <_ 
( 2  x.  (
( sqr `  ( A  .ih  A ) )  x.  ( sqr `  ( B  .ih  B ) ) ) )
1613, 15eqbrtrri 4060 . . . . . 6  |-  ( ( B  .ih  A )  +  ( A  .ih  B ) )  <_  (
2  x.  ( ( sqr `  ( A 
.ih  A ) )  x.  ( sqr `  ( B  .ih  B ) ) ) )
17 eqid 2296 . . . . . . . . . 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 21706 . . . . . . . . 9  |-  -u (
( ( * ` 
1 )  x.  ( B  .ih  A ) )  +  ( 1  x.  ( A  .ih  B
) ) )  e.  RR
192cjcli 11670 . . . . . . . . . . . 12  |-  ( * `
 1 )  e.  CC
2019, 8mulcli 8858 . . . . . . . . . . 11  |-  ( ( * `  1 )  x.  ( B  .ih  A ) )  e.  CC
212, 11mulcli 8858 . . . . . . . . . . 11  |-  ( 1  x.  ( A  .ih  B ) )  e.  CC
2220, 21addcli 8857 . . . . . . . . . 10  |-  ( ( ( * `  1
)  x.  ( B 
.ih  A ) )  +  ( 1  x.  ( A  .ih  B
) ) )  e.  CC
2322negrebi 9136 . . . . . . . . 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 2367 . . . . . . 7  |-  ( ( B  .ih  A )  +  ( A  .ih  B ) )  e.  RR
26 2re 9831 . . . . . . . 8  |-  2  e.  RR
27 hiidge0 21693 . . . . . . . . . . 11  |-  ( A  e.  ~H  ->  0  <_  ( A  .ih  A
) )
287, 27ax-mp 8 . . . . . . . . . 10  |-  0  <_  ( A  .ih  A
)
29 hiidrcl 21690 . . . . . . . . . . . 12  |-  ( A  e.  ~H  ->  ( A  .ih  A )  e.  RR )
307, 29ax-mp 8 . . . . . . . . . . 11  |-  ( A 
.ih  A )  e.  RR
3130sqrcli 11871 . . . . . . . . . 10  |-  ( 0  <_  ( A  .ih  A )  ->  ( sqr `  ( A  .ih  A
) )  e.  RR )
3228, 31ax-mp 8 . . . . . . . . 9  |-  ( sqr `  ( A  .ih  A
) )  e.  RR
33 hiidge0 21693 . . . . . . . . . . 11  |-  ( B  e.  ~H  ->  0  <_  ( B  .ih  B
) )
346, 33ax-mp 8 . . . . . . . . . 10  |-  0  <_  ( B  .ih  B
)
35 hiidrcl 21690 . . . . . . . . . . . 12  |-  ( B  e.  ~H  ->  ( B  .ih  B )  e.  RR )
366, 35ax-mp 8 . . . . . . . . . . 11  |-  ( B 
.ih  B )  e.  RR
3736sqrcli 11871 . . . . . . . . . 10  |-  ( 0  <_  ( B  .ih  B )  ->  ( sqr `  ( B  .ih  B
) )  e.  RR )
3834, 37ax-mp 8 . . . . . . . . 9  |-  ( sqr `  ( B  .ih  B
) )  e.  RR
3932, 38remulcli 8867 . . . . . . . 8  |-  ( ( sqr `  ( A 
.ih  A ) )  x.  ( sqr `  ( B  .ih  B ) ) )  e.  RR
4026, 39remulcli 8867 . . . . . . 7  |-  ( 2  x.  ( ( sqr `  ( A  .ih  A
) )  x.  ( sqr `  ( B  .ih  B ) ) ) )  e.  RR
4130, 36readdcli 8866 . . . . . . 7  |-  ( ( A  .ih  A )  +  ( B  .ih  B ) )  e.  RR
4225, 40, 41leadd2i 9345 . . . . . 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 21712 . . . . . 6  |-  ( ( A  +h  B ) 
.ih  ( A  +h  B ) )  =  ( ( ( A 
.ih  A )  +  ( B  .ih  B
) )  +  ( ( A  .ih  B
)  +  ( B 
.ih  A ) ) )
4511, 8addcomi 9019 . . . . . . 7  |-  ( ( A  .ih  B )  +  ( B  .ih  A ) )  =  ( ( B  .ih  A
)  +  ( A 
.ih  B ) )
4645oveq2i 5885 . . . . . 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 2316 . . . . 5  |-  ( ( A  +h  B ) 
.ih  ( A  +h  B ) )  =  ( ( ( A 
.ih  A )  +  ( B  .ih  B
) )  +  ( ( B  .ih  A
)  +  ( A 
.ih  B ) ) )
4832recni 8865 . . . . . . 7  |-  ( sqr `  ( A  .ih  A
) )  e.  CC
4938recni 8865 . . . . . . 7  |-  ( sqr `  ( B  .ih  B
) )  e.  CC
5048, 49binom2i 11228 . . . . . 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 11200 . . . . . . 7  |-  ( ( sqr `  ( A 
.ih  A ) ) ^ 2 )  e.  CC
52 2cn 9832 . . . . . . . 8  |-  2  e.  CC
5348, 49mulcli 8858 . . . . . . . 8  |-  ( ( sqr `  ( A 
.ih  A ) )  x.  ( sqr `  ( B  .ih  B ) ) )  e.  CC
5452, 53mulcli 8858 . . . . . . 7  |-  ( 2  x.  ( ( sqr `  ( A  .ih  A
) )  x.  ( sqr `  ( B  .ih  B ) ) ) )  e.  CC
5549sqcli 11200 . . . . . . 7  |-  ( ( sqr `  ( B 
.ih  B ) ) ^ 2 )  e.  CC
5651, 54, 55add32i 9046 . . . . . 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 11875 . . . . . . . . 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 11875 . . . . . . . . 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 5886 . . . . . . 7  |-  ( ( ( sqr `  ( A  .ih  A ) ) ^ 2 )  +  ( ( sqr `  ( B  .ih  B ) ) ^ 2 ) )  =  ( ( A 
.ih  A )  +  ( B  .ih  B
) )
6261oveq1i 5884 . . . . . 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 2320 . . . . 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 4067 . . . 4  |-  ( ( A  +h  B ) 
.ih  ( A  +h  B ) )  <_ 
( ( ( sqr `  ( A  .ih  A
) )  +  ( sqr `  ( B 
.ih  B ) ) ) ^ 2 )
657, 6hvaddcli 21614 . . . . . 6  |-  ( A  +h  B )  e. 
~H
66 hiidge0 21693 . . . . . 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 8866 . . . . . 6  |-  ( ( sqr `  ( A 
.ih  A ) )  +  ( sqr `  ( B  .ih  B ) ) )  e.  RR
6968sqge0i 11207 . . . . 5  |-  0  <_  ( ( ( sqr `  ( A  .ih  A
) )  +  ( sqr `  ( B 
.ih  B ) ) ) ^ 2 )
70 hiidrcl 21690 . . . . . . 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 11205 . . . . . 6  |-  ( ( ( sqr `  ( A  .ih  A ) )  +  ( sqr `  ( B  .ih  B ) ) ) ^ 2 )  e.  RR
7371, 72sqrlei 11888 . . . . 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 11876 . . . . . 6  |-  ( 0  <_  ( A  .ih  A )  ->  0  <_  ( sqr `  ( A 
.ih  A ) ) )
7728, 76ax-mp 8 . . . . 5  |-  0  <_  ( sqr `  ( A  .ih  A ) )
7836sqrge0i 11876 . . . . . 6  |-  ( 0  <_  ( B  .ih  B )  ->  0  <_  ( sqr `  ( B 
.ih  B ) ) )
7934, 78ax-mp 8 . . . . 5  |-  0  <_  ( sqr `  ( B  .ih  B ) )
8032, 38addge0i 9329 . . . . 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 11874 . . . 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 4062 . 2  |-  ( sqr `  ( ( A  +h  B )  .ih  ( A  +h  B ) ) )  <_  ( ( sqr `  ( A  .ih  A ) )  +  ( sqr `  ( B 
.ih  B ) ) )
85 normval 21719 . . 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 21719 . . . 4  |-  ( A  e.  ~H  ->  ( normh `  A )  =  ( sqr `  ( A  .ih  A ) ) )
887, 87ax-mp 8 . . 3  |-  ( normh `  A )  =  ( sqr `  ( A 
.ih  A ) )
89 normval 21719 . . . 4  |-  ( B  e.  ~H  ->  ( normh `  B )  =  ( sqr `  ( B  .ih  B ) ) )
906, 89ax-mp 8 . . 3  |-  ( normh `  B )  =  ( sqr `  ( B 
.ih  B ) )
9188, 90oveq12i 5886 . 2  |-  ( (
normh `  A )  +  ( normh `  B )
)  =  ( ( sqr `  ( A 
.ih  A ) )  +  ( sqr `  ( B  .ih  B ) ) )
9284, 86, 913brtr4i 4067 1  |-  ( normh `  ( A  +h  B
) )  <_  (
( normh `  A )  +  ( normh `  B
) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1632    e. wcel 1696   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758    <_ cle 8884   -ucneg 9054   2c2 9811   ^cexp 11120   *ccj 11597   sqrcsqr 11734   ~Hchil 21515    +h cva 21516    .ih csp 21518   normhcno 21519
This theorem is referenced by:  norm-ii  21733  norm3difi  21742
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-hfvadd 21596  ax-hv0cl 21599  ax-hfvmul 21601  ax-hvmulass 21603  ax-hvmul0 21606  ax-hfi 21674  ax-his1 21677  ax-his2 21678  ax-his3 21679  ax-his4 21680
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-seq 11063  df-exp 11121  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-hnorm 21564  df-hvsub 21567
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