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Theorem norm-ii-i 22622
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 9074 . . . . . . . . . . 11  |-  1  e.  RR
2 ax-1cn 9032 . . . . . . . . . . . 12  |-  1  e.  CC
32cjrebi 11962 . . . . . . . . . . 11  |-  ( 1  e.  RR  <->  ( * `  1 )  =  1 )
41, 3mpbi 200 . . . . . . . . . 10  |-  ( * `
 1 )  =  1
54oveq1i 6077 . . . . . . . . 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 22566 . . . . . . . . . 10  |-  ( B 
.ih  A )  e.  CC
98mulid2i 9077 . . . . . . . . 9  |-  ( 1  x.  ( B  .ih  A ) )  =  ( B  .ih  A )
105, 9eqtri 2450 . . . . . . . 8  |-  ( ( * `  1 )  x.  ( B  .ih  A ) )  =  ( B  .ih  A )
117, 6hicli 22566 . . . . . . . . 9  |-  ( A 
.ih  B )  e.  CC
1211mulid2i 9077 . . . . . . . 8  |-  ( 1  x.  ( A  .ih  B ) )  =  ( A  .ih  B )
1310, 12oveq12i 6079 . . . . . . 7  |-  ( ( ( * `  1
)  x.  ( B 
.ih  A ) )  +  ( 1  x.  ( A  .ih  B
) ) )  =  ( ( B  .ih  A )  +  ( A 
.ih  B ) )
14 abs1 12085 . . . . . . . 8  |-  ( abs `  1 )  =  1
152, 6, 7, 14normlem7 22601 . . . . . . 7  |-  ( ( ( * `  1
)  x.  ( B 
.ih  A ) )  +  ( 1  x.  ( A  .ih  B
) ) )  <_ 
( 2  x.  (
( sqr `  ( A  .ih  A ) )  x.  ( sqr `  ( B  .ih  B ) ) ) )
1613, 15eqbrtrri 4220 . . . . . 6  |-  ( ( B  .ih  A )  +  ( A  .ih  B ) )  <_  (
2  x.  ( ( sqr `  ( A 
.ih  A ) )  x.  ( sqr `  ( B  .ih  B ) ) ) )
17 eqid 2430 . . . . . . . . . 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 22596 . . . . . . . . 9  |-  -u (
( ( * ` 
1 )  x.  ( B  .ih  A ) )  +  ( 1  x.  ( A  .ih  B
) ) )  e.  RR
192cjcli 11957 . . . . . . . . . . . 12  |-  ( * `
 1 )  e.  CC
2019, 8mulcli 9079 . . . . . . . . . . 11  |-  ( ( * `  1 )  x.  ( B  .ih  A ) )  e.  CC
212, 11mulcli 9079 . . . . . . . . . . 11  |-  ( 1  x.  ( A  .ih  B ) )  e.  CC
2220, 21addcli 9078 . . . . . . . . . 10  |-  ( ( ( * `  1
)  x.  ( B 
.ih  A ) )  +  ( 1  x.  ( A  .ih  B
) ) )  e.  CC
2322negrebi 9358 . . . . . . . . 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 2501 . . . . . . 7  |-  ( ( B  .ih  A )  +  ( A  .ih  B ) )  e.  RR
26 2re 10053 . . . . . . . 8  |-  2  e.  RR
27 hiidge0 22583 . . . . . . . . . . 11  |-  ( A  e.  ~H  ->  0  <_  ( A  .ih  A
) )
287, 27ax-mp 8 . . . . . . . . . 10  |-  0  <_  ( A  .ih  A
)
29 hiidrcl 22580 . . . . . . . . . . . 12  |-  ( A  e.  ~H  ->  ( A  .ih  A )  e.  RR )
307, 29ax-mp 8 . . . . . . . . . . 11  |-  ( A 
.ih  A )  e.  RR
3130sqrcli 12158 . . . . . . . . . 10  |-  ( 0  <_  ( A  .ih  A )  ->  ( sqr `  ( A  .ih  A
) )  e.  RR )
3228, 31ax-mp 8 . . . . . . . . 9  |-  ( sqr `  ( A  .ih  A
) )  e.  RR
33 hiidge0 22583 . . . . . . . . . . 11  |-  ( B  e.  ~H  ->  0  <_  ( B  .ih  B
) )
346, 33ax-mp 8 . . . . . . . . . 10  |-  0  <_  ( B  .ih  B
)
35 hiidrcl 22580 . . . . . . . . . . . 12  |-  ( B  e.  ~H  ->  ( B  .ih  B )  e.  RR )
366, 35ax-mp 8 . . . . . . . . . . 11  |-  ( B 
.ih  B )  e.  RR
3736sqrcli 12158 . . . . . . . . . 10  |-  ( 0  <_  ( B  .ih  B )  ->  ( sqr `  ( B  .ih  B
) )  e.  RR )
3834, 37ax-mp 8 . . . . . . . . 9  |-  ( sqr `  ( B  .ih  B
) )  e.  RR
3932, 38remulcli 9088 . . . . . . . 8  |-  ( ( sqr `  ( A 
.ih  A ) )  x.  ( sqr `  ( B  .ih  B ) ) )  e.  RR
4026, 39remulcli 9088 . . . . . . 7  |-  ( 2  x.  ( ( sqr `  ( A  .ih  A
) )  x.  ( sqr `  ( B  .ih  B ) ) ) )  e.  RR
4130, 36readdcli 9087 . . . . . . 7  |-  ( ( A  .ih  A )  +  ( B  .ih  B ) )  e.  RR
4225, 40, 41leadd2i 9567 . . . . . 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 22602 . . . . . 6  |-  ( ( A  +h  B ) 
.ih  ( A  +h  B ) )  =  ( ( ( A 
.ih  A )  +  ( B  .ih  B
) )  +  ( ( A  .ih  B
)  +  ( B 
.ih  A ) ) )
4511, 8addcomi 9241 . . . . . . 7  |-  ( ( A  .ih  B )  +  ( B  .ih  A ) )  =  ( ( B  .ih  A
)  +  ( A 
.ih  B ) )
4645oveq2i 6078 . . . . . 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 2450 . . . . 5  |-  ( ( A  +h  B ) 
.ih  ( A  +h  B ) )  =  ( ( ( A 
.ih  A )  +  ( B  .ih  B
) )  +  ( ( B  .ih  A
)  +  ( A 
.ih  B ) ) )
4832recni 9086 . . . . . . 7  |-  ( sqr `  ( A  .ih  A
) )  e.  CC
4938recni 9086 . . . . . . 7  |-  ( sqr `  ( B  .ih  B
) )  e.  CC
5048, 49binom2i 11473 . . . . . 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 11445 . . . . . . 7  |-  ( ( sqr `  ( A 
.ih  A ) ) ^ 2 )  e.  CC
52 2cn 10054 . . . . . . . 8  |-  2  e.  CC
5348, 49mulcli 9079 . . . . . . . 8  |-  ( ( sqr `  ( A 
.ih  A ) )  x.  ( sqr `  ( B  .ih  B ) ) )  e.  CC
5452, 53mulcli 9079 . . . . . . 7  |-  ( 2  x.  ( ( sqr `  ( A  .ih  A
) )  x.  ( sqr `  ( B  .ih  B ) ) ) )  e.  CC
5549sqcli 11445 . . . . . . 7  |-  ( ( sqr `  ( B 
.ih  B ) ) ^ 2 )  e.  CC
5651, 54, 55add32i 9268 . . . . . 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 12162 . . . . . . . . 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 12162 . . . . . . . . 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 6079 . . . . . . 7  |-  ( ( ( sqr `  ( A  .ih  A ) ) ^ 2 )  +  ( ( sqr `  ( B  .ih  B ) ) ^ 2 ) )  =  ( ( A 
.ih  A )  +  ( B  .ih  B
) )
6261oveq1i 6077 . . . . . 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 2454 . . . . 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 4227 . . . 4  |-  ( ( A  +h  B ) 
.ih  ( A  +h  B ) )  <_ 
( ( ( sqr `  ( A  .ih  A
) )  +  ( sqr `  ( B 
.ih  B ) ) ) ^ 2 )
657, 6hvaddcli 22504 . . . . . 6  |-  ( A  +h  B )  e. 
~H
66 hiidge0 22583 . . . . . 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 9087 . . . . . 6  |-  ( ( sqr `  ( A 
.ih  A ) )  +  ( sqr `  ( B  .ih  B ) ) )  e.  RR
6968sqge0i 11452 . . . . 5  |-  0  <_  ( ( ( sqr `  ( A  .ih  A
) )  +  ( sqr `  ( B 
.ih  B ) ) ) ^ 2 )
70 hiidrcl 22580 . . . . . . 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 11450 . . . . . 6  |-  ( ( ( sqr `  ( A  .ih  A ) )  +  ( sqr `  ( B  .ih  B ) ) ) ^ 2 )  e.  RR
7371, 72sqrlei 12175 . . . . 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 12163 . . . . . 6  |-  ( 0  <_  ( A  .ih  A )  ->  0  <_  ( sqr `  ( A 
.ih  A ) ) )
7728, 76ax-mp 8 . . . . 5  |-  0  <_  ( sqr `  ( A  .ih  A ) )
7836sqrge0i 12163 . . . . . 6  |-  ( 0  <_  ( B  .ih  B )  ->  0  <_  ( sqr `  ( B 
.ih  B ) ) )
7934, 78ax-mp 8 . . . . 5  |-  0  <_  ( sqr `  ( B  .ih  B ) )
8032, 38addge0i 9551 . . . . 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 12161 . . . 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 4222 . 2  |-  ( sqr `  ( ( A  +h  B )  .ih  ( A  +h  B ) ) )  <_  ( ( sqr `  ( A  .ih  A ) )  +  ( sqr `  ( B 
.ih  B ) ) )
85 normval 22609 . . 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 22609 . . . 4  |-  ( A  e.  ~H  ->  ( normh `  A )  =  ( sqr `  ( A  .ih  A ) ) )
887, 87ax-mp 8 . . 3  |-  ( normh `  A )  =  ( sqr `  ( A 
.ih  A ) )
89 normval 22609 . . . 4  |-  ( B  e.  ~H  ->  ( normh `  B )  =  ( sqr `  ( B  .ih  B ) ) )
906, 89ax-mp 8 . . 3  |-  ( normh `  B )  =  ( sqr `  ( B 
.ih  B ) )
9188, 90oveq12i 6079 . 2  |-  ( (
normh `  A )  +  ( normh `  B )
)  =  ( ( sqr `  ( A 
.ih  A ) )  +  ( sqr `  ( B  .ih  B ) ) )
9284, 86, 913brtr4i 4227 1  |-  ( normh `  ( A  +h  B
) )  <_  (
( normh `  A )  +  ( normh `  B
) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    = wceq 1652    e. wcel 1725   class class class wbr 4199   ` cfv 5440  (class class class)co 6067   RRcr 8973   0cc0 8974   1c1 8975    + caddc 8977    x. cmul 8979    <_ cle 9105   -ucneg 9276   2c2 10033   ^cexp 11365   *ccj 11884   sqrcsqr 12021   ~Hchil 22405    +h cva 22406    .ih csp 22408   normhcno 22409
This theorem is referenced by:  norm-ii  22623  norm3difi  22632
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2411  ax-sep 4317  ax-nul 4325  ax-pow 4364  ax-pr 4390  ax-un 4687  ax-cnex 9030  ax-resscn 9031  ax-1cn 9032  ax-icn 9033  ax-addcl 9034  ax-addrcl 9035  ax-mulcl 9036  ax-mulrcl 9037  ax-mulcom 9038  ax-addass 9039  ax-mulass 9040  ax-distr 9041  ax-i2m1 9042  ax-1ne0 9043  ax-1rid 9044  ax-rnegex 9045  ax-rrecex 9046  ax-cnre 9047  ax-pre-lttri 9048  ax-pre-lttrn 9049  ax-pre-ltadd 9050  ax-pre-mulgt0 9051  ax-pre-sup 9052  ax-hfvadd 22486  ax-hv0cl 22489  ax-hfvmul 22491  ax-hvmulass 22493  ax-hvmul0 22496  ax-hfi 22564  ax-his1 22567  ax-his2 22568  ax-his3 22569  ax-his4 22570
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-ne 2595  df-nel 2596  df-ral 2697  df-rex 2698  df-reu 2699  df-rmo 2700  df-rab 2701  df-v 2945  df-sbc 3149  df-csb 3239  df-dif 3310  df-un 3312  df-in 3314  df-ss 3321  df-pss 3323  df-nul 3616  df-if 3727  df-pw 3788  df-sn 3807  df-pr 3808  df-tp 3809  df-op 3810  df-uni 4003  df-iun 4082  df-br 4200  df-opab 4254  df-mpt 4255  df-tr 4290  df-eprel 4481  df-id 4485  df-po 4490  df-so 4491  df-fr 4528  df-we 4530  df-ord 4571  df-on 4572  df-lim 4573  df-suc 4574  df-om 4832  df-xp 4870  df-rel 4871  df-cnv 4872  df-co 4873  df-dm 4874  df-rn 4875  df-res 4876  df-ima 4877  df-iota 5404  df-fun 5442  df-fn 5443  df-f 5444  df-f1 5445  df-fo 5446  df-f1o 5447  df-fv 5448  df-ov 6070  df-oprab 6071  df-mpt2 6072  df-2nd 6336  df-riota 6535  df-recs 6619  df-rdg 6654  df-er 6891  df-en 7096  df-dom 7097  df-sdom 7098  df-sup 7432  df-pnf 9106  df-mnf 9107  df-xr 9108  df-ltxr 9109  df-le 9110  df-sub 9277  df-neg 9278  df-div 9662  df-nn 9985  df-2 10042  df-3 10043  df-4 10044  df-n0 10206  df-z 10267  df-uz 10473  df-rp 10597  df-seq 11307  df-exp 11366  df-cj 11887  df-re 11888  df-im 11889  df-sqr 12023  df-abs 12024  df-hnorm 22454  df-hvsub 22457
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