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Theorem normlem7 22571
Description: Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 11-Aug-1999.) (New usage is discouraged.)
Hypotheses
Ref Expression
normlem1.1  |-  S  e.  CC
normlem1.2  |-  F  e. 
~H
normlem1.3  |-  G  e. 
~H
normlem7.4  |-  ( abs `  S )  =  1
Assertion
Ref Expression
normlem7  |-  ( ( ( * `  S
)  x.  ( F 
.ih  G ) )  +  ( S  x.  ( G  .ih  F ) ) )  <_  (
2  x.  ( ( sqr `  ( G 
.ih  G ) )  x.  ( sqr `  ( F  .ih  F ) ) ) )

Proof of Theorem normlem7
StepHypRef Expression
1 normlem1.1 . . . . . 6  |-  S  e.  CC
2 normlem1.2 . . . . . 6  |-  F  e. 
~H
3 normlem1.3 . . . . . 6  |-  G  e. 
~H
4 eqid 2404 . . . . . 6  |-  -u (
( ( * `  S )  x.  ( F  .ih  G ) )  +  ( S  x.  ( G  .ih  F ) ) )  =  -u ( ( ( * `
 S )  x.  ( F  .ih  G
) )  +  ( S  x.  ( G 
.ih  F ) ) )
51, 2, 3, 4normlem2 22566 . . . . 5  |-  -u (
( ( * `  S )  x.  ( F  .ih  G ) )  +  ( S  x.  ( G  .ih  F ) ) )  e.  RR
61cjcli 11929 . . . . . . . 8  |-  ( * `
 S )  e.  CC
72, 3hicli 22536 . . . . . . . 8  |-  ( F 
.ih  G )  e.  CC
86, 7mulcli 9051 . . . . . . 7  |-  ( ( * `  S )  x.  ( F  .ih  G ) )  e.  CC
93, 2hicli 22536 . . . . . . . 8  |-  ( G 
.ih  F )  e.  CC
101, 9mulcli 9051 . . . . . . 7  |-  ( S  x.  ( G  .ih  F ) )  e.  CC
118, 10addcli 9050 . . . . . 6  |-  ( ( ( * `  S
)  x.  ( F 
.ih  G ) )  +  ( S  x.  ( G  .ih  F ) ) )  e.  CC
1211negrebi 9330 . . . . 5  |-  ( -u ( ( ( * `
 S )  x.  ( F  .ih  G
) )  +  ( S  x.  ( G 
.ih  F ) ) )  e.  RR  <->  ( (
( * `  S
)  x.  ( F 
.ih  G ) )  +  ( S  x.  ( G  .ih  F ) ) )  e.  RR )
135, 12mpbi 200 . . . 4  |-  ( ( ( * `  S
)  x.  ( F 
.ih  G ) )  +  ( S  x.  ( G  .ih  F ) ) )  e.  RR
1413leabsi 12138 . . 3  |-  ( ( ( * `  S
)  x.  ( F 
.ih  G ) )  +  ( S  x.  ( G  .ih  F ) ) )  <_  ( abs `  ( ( ( * `  S )  x.  ( F  .ih  G ) )  +  ( S  x.  ( G 
.ih  F ) ) ) )
1511absnegi 12158 . . 3  |-  ( abs `  -u ( ( ( * `  S )  x.  ( F  .ih  G ) )  +  ( S  x.  ( G 
.ih  F ) ) ) )  =  ( abs `  ( ( ( * `  S
)  x.  ( F 
.ih  G ) )  +  ( S  x.  ( G  .ih  F ) ) ) )
1614, 15breqtrri 4197 . 2  |-  ( ( ( * `  S
)  x.  ( F 
.ih  G ) )  +  ( S  x.  ( G  .ih  F ) ) )  <_  ( abs `  -u ( ( ( * `  S )  x.  ( F  .ih  G ) )  +  ( S  x.  ( G 
.ih  F ) ) ) )
17 eqid 2404 . . 3  |-  ( G 
.ih  G )  =  ( G  .ih  G
)
18 eqid 2404 . . 3  |-  ( F 
.ih  F )  =  ( F  .ih  F
)
19 normlem7.4 . . 3  |-  ( abs `  S )  =  1
201, 2, 3, 4, 17, 18, 19normlem6 22570 . 2  |-  ( abs `  -u ( ( ( * `  S )  x.  ( F  .ih  G ) )  +  ( S  x.  ( G 
.ih  F ) ) ) )  <_  (
2  x.  ( ( sqr `  ( G 
.ih  G ) )  x.  ( sqr `  ( F  .ih  F ) ) ) )
2111negcli 9324 . . . 4  |-  -u (
( ( * `  S )  x.  ( F  .ih  G ) )  +  ( S  x.  ( G  .ih  F ) ) )  e.  CC
2221abscli 12153 . . 3  |-  ( abs `  -u ( ( ( * `  S )  x.  ( F  .ih  G ) )  +  ( S  x.  ( G 
.ih  F ) ) ) )  e.  RR
23 2re 10025 . . . 4  |-  2  e.  RR
24 hiidge0 22553 . . . . . 6  |-  ( G  e.  ~H  ->  0  <_  ( G  .ih  G
) )
25 hiidrcl 22550 . . . . . . . 8  |-  ( G  e.  ~H  ->  ( G  .ih  G )  e.  RR )
263, 25ax-mp 8 . . . . . . 7  |-  ( G 
.ih  G )  e.  RR
2726sqrcli 12130 . . . . . 6  |-  ( 0  <_  ( G  .ih  G )  ->  ( sqr `  ( G  .ih  G
) )  e.  RR )
283, 24, 27mp2b 10 . . . . 5  |-  ( sqr `  ( G  .ih  G
) )  e.  RR
29 hiidge0 22553 . . . . . 6  |-  ( F  e.  ~H  ->  0  <_  ( F  .ih  F
) )
30 hiidrcl 22550 . . . . . . . 8  |-  ( F  e.  ~H  ->  ( F  .ih  F )  e.  RR )
312, 30ax-mp 8 . . . . . . 7  |-  ( F 
.ih  F )  e.  RR
3231sqrcli 12130 . . . . . 6  |-  ( 0  <_  ( F  .ih  F )  ->  ( sqr `  ( F  .ih  F
) )  e.  RR )
332, 29, 32mp2b 10 . . . . 5  |-  ( sqr `  ( F  .ih  F
) )  e.  RR
3428, 33remulcli 9060 . . . 4  |-  ( ( sqr `  ( G 
.ih  G ) )  x.  ( sqr `  ( F  .ih  F ) ) )  e.  RR
3523, 34remulcli 9060 . . 3  |-  ( 2  x.  ( ( sqr `  ( G  .ih  G
) )  x.  ( sqr `  ( F  .ih  F ) ) ) )  e.  RR
3613, 22, 35letri 9158 . 2  |-  ( ( ( ( ( * `
 S )  x.  ( F  .ih  G
) )  +  ( S  x.  ( G 
.ih  F ) ) )  <_  ( abs `  -u ( ( ( * `
 S )  x.  ( F  .ih  G
) )  +  ( S  x.  ( G 
.ih  F ) ) ) )  /\  ( abs `  -u ( ( ( * `  S )  x.  ( F  .ih  G ) )  +  ( S  x.  ( G 
.ih  F ) ) ) )  <_  (
2  x.  ( ( sqr `  ( G 
.ih  G ) )  x.  ( sqr `  ( F  .ih  F ) ) ) ) )  -> 
( ( ( * `
 S )  x.  ( F  .ih  G
) )  +  ( S  x.  ( G 
.ih  F ) ) )  <_  ( 2  x.  ( ( sqr `  ( G  .ih  G
) )  x.  ( sqr `  ( F  .ih  F ) ) ) ) )
3716, 20, 36mp2an 654 1  |-  ( ( ( * `  S
)  x.  ( F 
.ih  G ) )  +  ( S  x.  ( G  .ih  F ) ) )  <_  (
2  x.  ( ( sqr `  ( G 
.ih  G ) )  x.  ( sqr `  ( F  .ih  F ) ) ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1649    e. wcel 1721   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   CCcc 8944   RRcr 8945   0cc0 8946   1c1 8947    + caddc 8949    x. cmul 8951    <_ cle 9077   -ucneg 9248   2c2 10005   *ccj 11856   sqrcsqr 11993   abscabs 11994   ~Hchil 22375    .ih csp 22378
This theorem is referenced by:  normlem7tALT  22574  norm-ii-i  22592
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-hvsub 22427
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