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Theorem norm-ii-i 21641
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 8770 . . . . . . . . . . 11  |-  1  e.  RR
2 ax-1cn 8728 . . . . . . . . . . . 12  |-  1  e.  CC
32cjrebi 11589 . . . . . . . . . . 11  |-  ( 1  e.  RR  <->  ( * `  1 )  =  1 )
41, 3mpbi 201 . . . . . . . . . 10  |-  ( * `
 1 )  =  1
54oveq1i 5767 . . . . . . . . 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 21585 . . . . . . . . . 10  |-  ( B 
.ih  A )  e.  CC
98mulid2i 8773 . . . . . . . . 9  |-  ( 1  x.  ( B  .ih  A ) )  =  ( B  .ih  A )
105, 9eqtri 2276 . . . . . . . 8  |-  ( ( * `  1 )  x.  ( B  .ih  A ) )  =  ( B  .ih  A )
117, 6hicli 21585 . . . . . . . . 9  |-  ( A 
.ih  B )  e.  CC
1211mulid2i 8773 . . . . . . . 8  |-  ( 1  x.  ( A  .ih  B ) )  =  ( A  .ih  B )
1310, 12oveq12i 5769 . . . . . . 7  |-  ( ( ( * `  1
)  x.  ( B 
.ih  A ) )  +  ( 1  x.  ( A  .ih  B
) ) )  =  ( ( B  .ih  A )  +  ( A 
.ih  B ) )
14 abs1 11712 . . . . . . . 8  |-  ( abs `  1 )  =  1
152, 6, 7, 14normlem7 21620 . . . . . . 7  |-  ( ( ( * `  1
)  x.  ( B 
.ih  A ) )  +  ( 1  x.  ( A  .ih  B
) ) )  <_ 
( 2  x.  (
( sqr `  ( A  .ih  A ) )  x.  ( sqr `  ( B  .ih  B ) ) ) )
1613, 15eqbrtrri 3984 . . . . . 6  |-  ( ( B  .ih  A )  +  ( A  .ih  B ) )  <_  (
2  x.  ( ( sqr `  ( A 
.ih  A ) )  x.  ( sqr `  ( B  .ih  B ) ) ) )
17 eqid 2256 . . . . . . . . . 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 21615 . . . . . . . . 9  |-  -u (
( ( * ` 
1 )  x.  ( B  .ih  A ) )  +  ( 1  x.  ( A  .ih  B
) ) )  e.  RR
192cjcli 11584 . . . . . . . . . . . 12  |-  ( * `
 1 )  e.  CC
2019, 8mulcli 8775 . . . . . . . . . . 11  |-  ( ( * `  1 )  x.  ( B  .ih  A ) )  e.  CC
212, 11mulcli 8775 . . . . . . . . . . 11  |-  ( 1  x.  ( A  .ih  B ) )  e.  CC
2220, 21addcli 8774 . . . . . . . . . 10  |-  ( ( ( * `  1
)  x.  ( B 
.ih  A ) )  +  ( 1  x.  ( A  .ih  B
) ) )  e.  CC
2322negrebi 9053 . . . . . . . . 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 201 . . . . . . . 8  |-  ( ( ( * `  1
)  x.  ( B 
.ih  A ) )  +  ( 1  x.  ( A  .ih  B
) ) )  e.  RR
2513, 24eqeltrri 2327 . . . . . . 7  |-  ( ( B  .ih  A )  +  ( A  .ih  B ) )  e.  RR
26 2re 9748 . . . . . . . 8  |-  2  e.  RR
27 hiidge0 21602 . . . . . . . . . . 11  |-  ( A  e.  ~H  ->  0  <_  ( A  .ih  A
) )
287, 27ax-mp 10 . . . . . . . . . 10  |-  0  <_  ( A  .ih  A
)
29 hiidrcl 21599 . . . . . . . . . . . 12  |-  ( A  e.  ~H  ->  ( A  .ih  A )  e.  RR )
307, 29ax-mp 10 . . . . . . . . . . 11  |-  ( A 
.ih  A )  e.  RR
3130sqrcli 11785 . . . . . . . . . 10  |-  ( 0  <_  ( A  .ih  A )  ->  ( sqr `  ( A  .ih  A
) )  e.  RR )
3228, 31ax-mp 10 . . . . . . . . 9  |-  ( sqr `  ( A  .ih  A
) )  e.  RR
33 hiidge0 21602 . . . . . . . . . . 11  |-  ( B  e.  ~H  ->  0  <_  ( B  .ih  B
) )
346, 33ax-mp 10 . . . . . . . . . 10  |-  0  <_  ( B  .ih  B
)
35 hiidrcl 21599 . . . . . . . . . . . 12  |-  ( B  e.  ~H  ->  ( B  .ih  B )  e.  RR )
366, 35ax-mp 10 . . . . . . . . . . 11  |-  ( B 
.ih  B )  e.  RR
3736sqrcli 11785 . . . . . . . . . 10  |-  ( 0  <_  ( B  .ih  B )  ->  ( sqr `  ( B  .ih  B
) )  e.  RR )
3834, 37ax-mp 10 . . . . . . . . 9  |-  ( sqr `  ( B  .ih  B
) )  e.  RR
3932, 38remulcli 8784 . . . . . . . 8  |-  ( ( sqr `  ( A 
.ih  A ) )  x.  ( sqr `  ( B  .ih  B ) ) )  e.  RR
4026, 39remulcli 8784 . . . . . . 7  |-  ( 2  x.  ( ( sqr `  ( A  .ih  A
) )  x.  ( sqr `  ( B  .ih  B ) ) ) )  e.  RR
4130, 36readdcli 8783 . . . . . . 7  |-  ( ( A  .ih  A )  +  ( B  .ih  B ) )  e.  RR
4225, 40, 41leadd2i 9262 . . . . . 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 201 . . . . 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 21621 . . . . . 6  |-  ( ( A  +h  B ) 
.ih  ( A  +h  B ) )  =  ( ( ( A 
.ih  A )  +  ( B  .ih  B
) )  +  ( ( A  .ih  B
)  +  ( B 
.ih  A ) ) )
4511, 8addcomi 8936 . . . . . . 7  |-  ( ( A  .ih  B )  +  ( B  .ih  A ) )  =  ( ( B  .ih  A
)  +  ( A 
.ih  B ) )
4645oveq2i 5768 . . . . . 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 2276 . . . . 5  |-  ( ( A  +h  B ) 
.ih  ( A  +h  B ) )  =  ( ( ( A 
.ih  A )  +  ( B  .ih  B
) )  +  ( ( B  .ih  A
)  +  ( A 
.ih  B ) ) )
4832recni 8782 . . . . . . 7  |-  ( sqr `  ( A  .ih  A
) )  e.  CC
4938recni 8782 . . . . . . 7  |-  ( sqr `  ( B  .ih  B
) )  e.  CC
5048, 49binom2i 11143 . . . . . 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 11115 . . . . . . 7  |-  ( ( sqr `  ( A 
.ih  A ) ) ^ 2 )  e.  CC
52 2cn 9749 . . . . . . . 8  |-  2  e.  CC
5348, 49mulcli 8775 . . . . . . . 8  |-  ( ( sqr `  ( A 
.ih  A ) )  x.  ( sqr `  ( B  .ih  B ) ) )  e.  CC
5452, 53mulcli 8775 . . . . . . 7  |-  ( 2  x.  ( ( sqr `  ( A  .ih  A
) )  x.  ( sqr `  ( B  .ih  B ) ) ) )  e.  CC
5549sqcli 11115 . . . . . . 7  |-  ( ( sqr `  ( B 
.ih  B ) ) ^ 2 )  e.  CC
5651, 54, 55add32i 8963 . . . . . 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 11789 . . . . . . . . 9  |-  ( 0  <_  ( A  .ih  A )  ->  ( ( sqr `  ( A  .ih  A ) ) ^ 2 )  =  ( A 
.ih  A ) )
5828, 57ax-mp 10 . . . . . . . 8  |-  ( ( sqr `  ( A 
.ih  A ) ) ^ 2 )  =  ( A  .ih  A
)
5936sqsqri 11789 . . . . . . . . 9  |-  ( 0  <_  ( B  .ih  B )  ->  ( ( sqr `  ( B  .ih  B ) ) ^ 2 )  =  ( B 
.ih  B ) )
6034, 59ax-mp 10 . . . . . . . 8  |-  ( ( sqr `  ( B 
.ih  B ) ) ^ 2 )  =  ( B  .ih  B
)
6158, 60oveq12i 5769 . . . . . . 7  |-  ( ( ( sqr `  ( A  .ih  A ) ) ^ 2 )  +  ( ( sqr `  ( B  .ih  B ) ) ^ 2 ) )  =  ( ( A 
.ih  A )  +  ( B  .ih  B
) )
6261oveq1i 5767 . . . . . 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 2280 . . . . 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 3991 . . . 4  |-  ( ( A  +h  B ) 
.ih  ( A  +h  B ) )  <_ 
( ( ( sqr `  ( A  .ih  A
) )  +  ( sqr `  ( B 
.ih  B ) ) ) ^ 2 )
657, 6hvaddcli 21523 . . . . . 6  |-  ( A  +h  B )  e. 
~H
66 hiidge0 21602 . . . . . 6  |-  ( ( A  +h  B )  e.  ~H  ->  0  <_  ( ( A  +h  B )  .ih  ( A  +h  B ) ) )
6765, 66ax-mp 10 . . . . 5  |-  0  <_  ( ( A  +h  B )  .ih  ( A  +h  B ) )
6832, 38readdcli 8783 . . . . . 6  |-  ( ( sqr `  ( A 
.ih  A ) )  +  ( sqr `  ( B  .ih  B ) ) )  e.  RR
6968sqge0i 11122 . . . . 5  |-  0  <_  ( ( ( sqr `  ( A  .ih  A
) )  +  ( sqr `  ( B 
.ih  B ) ) ) ^ 2 )
70 hiidrcl 21599 . . . . . . 7  |-  ( ( A  +h  B )  e.  ~H  ->  (
( A  +h  B
)  .ih  ( A  +h  B ) )  e.  RR )
7165, 70ax-mp 10 . . . . . 6  |-  ( ( A  +h  B ) 
.ih  ( A  +h  B ) )  e.  RR
7268resqcli 11120 . . . . . 6  |-  ( ( ( sqr `  ( A  .ih  A ) )  +  ( sqr `  ( B  .ih  B ) ) ) ^ 2 )  e.  RR
7371, 72sqrlei 11802 . . . . 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 656 . . . 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 201 . . 3  |-  ( sqr `  ( ( A  +h  B )  .ih  ( A  +h  B ) ) )  <_  ( sqr `  ( ( ( sqr `  ( A  .ih  A
) )  +  ( sqr `  ( B 
.ih  B ) ) ) ^ 2 ) )
7630sqrge0i 11790 . . . . . 6  |-  ( 0  <_  ( A  .ih  A )  ->  0  <_  ( sqr `  ( A 
.ih  A ) ) )
7728, 76ax-mp 10 . . . . 5  |-  0  <_  ( sqr `  ( A  .ih  A ) )
7836sqrge0i 11790 . . . . . 6  |-  ( 0  <_  ( B  .ih  B )  ->  0  <_  ( sqr `  ( B 
.ih  B ) ) )
7934, 78ax-mp 10 . . . . 5  |-  0  <_  ( sqr `  ( B  .ih  B ) )
8032, 38addge0i 9246 . . . . 5  |-  ( ( 0  <_  ( sqr `  ( A  .ih  A
) )  /\  0  <_  ( sqr `  ( B  .ih  B ) ) )  ->  0  <_  ( ( sqr `  ( A  .ih  A ) )  +  ( sqr `  ( B  .ih  B ) ) ) )
8177, 79, 80mp2an 656 . . . 4  |-  0  <_  ( ( sqr `  ( A  .ih  A ) )  +  ( sqr `  ( B  .ih  B ) ) )
8268sqrsqi 11788 . . . 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 10 . . 3  |-  ( sqr `  ( ( ( sqr `  ( A  .ih  A
) )  +  ( sqr `  ( B 
.ih  B ) ) ) ^ 2 ) )  =  ( ( sqr `  ( A 
.ih  A ) )  +  ( sqr `  ( B  .ih  B ) ) )
8475, 83breqtri 3986 . 2  |-  ( sqr `  ( ( A  +h  B )  .ih  ( A  +h  B ) ) )  <_  ( ( sqr `  ( A  .ih  A ) )  +  ( sqr `  ( B 
.ih  B ) ) )
85 normval 21628 . . 3  |-  ( ( A  +h  B )  e.  ~H  ->  ( normh `  ( A  +h  B ) )  =  ( sqr `  (
( A  +h  B
)  .ih  ( A  +h  B ) ) ) )
8665, 85ax-mp 10 . 2  |-  ( normh `  ( A  +h  B
) )  =  ( sqr `  ( ( A  +h  B ) 
.ih  ( A  +h  B ) ) )
87 normval 21628 . . . 4  |-  ( A  e.  ~H  ->  ( normh `  A )  =  ( sqr `  ( A  .ih  A ) ) )
887, 87ax-mp 10 . . 3  |-  ( normh `  A )  =  ( sqr `  ( A 
.ih  A ) )
89 normval 21628 . . . 4  |-  ( B  e.  ~H  ->  ( normh `  B )  =  ( sqr `  ( B  .ih  B ) ) )
906, 89ax-mp 10 . . 3  |-  ( normh `  B )  =  ( sqr `  ( B 
.ih  B ) )
9188, 90oveq12i 5769 . 2  |-  ( (
normh `  A )  +  ( normh `  B )
)  =  ( ( sqr `  ( A 
.ih  A ) )  +  ( sqr `  ( B  .ih  B ) ) )
9284, 86, 913brtr4i 3991 1  |-  ( normh `  ( A  +h  B
) )  <_  (
( normh `  A )  +  ( normh `  B
) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    = wceq 1619    e. wcel 1621   class class class wbr 3963   ` cfv 4638  (class class class)co 5757   RRcr 8669   0cc0 8670   1c1 8671    + caddc 8673    x. cmul 8675    <_ cle 8801   -ucneg 8971   2c2 9728   ^cexp 11035   *ccj 11511   sqrcsqr 11648   ~Hchil 21424    +h cva 21425    .ih csp 21427   normhcno 21428
This theorem is referenced by:  norm-ii  21642  norm3difi  21651
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 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449  ax-cnex 8726  ax-resscn 8727  ax-1cn 8728  ax-icn 8729  ax-addcl 8730  ax-addrcl 8731  ax-mulcl 8732  ax-mulrcl 8733  ax-mulcom 8734  ax-addass 8735  ax-mulass 8736  ax-distr 8737  ax-i2m1 8738  ax-1ne0 8739  ax-1rid 8740  ax-rnegex 8741  ax-rrecex 8742  ax-cnre 8743  ax-pre-lttri 8744  ax-pre-lttrn 8745  ax-pre-ltadd 8746  ax-pre-mulgt0 8747  ax-pre-sup 8748  ax-hfvadd 21505  ax-hv0cl 21508  ax-hfvmul 21510  ax-hvmulass 21512  ax-hvmul0 21515  ax-hfi 21583  ax-his1 21586  ax-his2 21587  ax-his3 21588  ax-his4 21589
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 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-iun 3848  df-br 3964  df-opab 4018  df-mpt 4019  df-tr 4054  df-eprel 4242  df-id 4246  df-po 4251  df-so 4252  df-fr 4289  df-we 4291  df-ord 4332  df-on 4333  df-lim 4334  df-suc 4335  df-om 4594  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-2nd 6022  df-iota 6190  df-riota 6237  df-recs 6321  df-rdg 6356  df-er 6593  df-en 6797  df-dom 6798  df-sdom 6799  df-sup 7127  df-pnf 8802  df-mnf 8803  df-xr 8804  df-ltxr 8805  df-le 8806  df-sub 8972  df-neg 8973  df-div 9357  df-n 9680  df-2 9737  df-3 9738  df-4 9739  df-n0 9898  df-z 9957  df-uz 10163  df-rp 10287  df-seq 10978  df-exp 11036  df-cj 11514  df-re 11515  df-im 11516  df-sqr 11650  df-abs 11651  df-hnorm 21473  df-hvsub 21476
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