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Theorem normlem7 22575
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 2408 . . . . . 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 22570 . . . . 5  |-  -u (
( ( * `  S )  x.  ( F  .ih  G ) )  +  ( S  x.  ( G  .ih  F ) ) )  e.  RR
61cjcli 11933 . . . . . . . 8  |-  ( * `
 S )  e.  CC
72, 3hicli 22540 . . . . . . . 8  |-  ( F 
.ih  G )  e.  CC
86, 7mulcli 9055 . . . . . . 7  |-  ( ( * `  S )  x.  ( F  .ih  G ) )  e.  CC
93, 2hicli 22540 . . . . . . . 8  |-  ( G 
.ih  F )  e.  CC
101, 9mulcli 9055 . . . . . . 7  |-  ( S  x.  ( G  .ih  F ) )  e.  CC
118, 10addcli 9054 . . . . . 6  |-  ( ( ( * `  S
)  x.  ( F 
.ih  G ) )  +  ( S  x.  ( G  .ih  F ) ) )  e.  CC
1211negrebi 9334 . . . . 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 12142 . . 3  |-  ( ( ( * `  S
)  x.  ( F 
.ih  G ) )  +  ( S  x.  ( G  .ih  F ) ) )  <_  ( abs `  ( ( ( * `  S )  x.  ( F  .ih  G ) )  +  ( S  x.  ( G 
.ih  F ) ) ) )
1511absnegi 12162 . . 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 4201 . 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 2408 . . 3  |-  ( G 
.ih  G )  =  ( G  .ih  G
)
18 eqid 2408 . . 3  |-  ( F 
.ih  F )  =  ( F  .ih  F
)
19 normlem7.4 . . 3  |-  ( abs `  S )  =  1
201, 2, 3, 4, 17, 18, 19normlem6 22574 . 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 9328 . . . 4  |-  -u (
( ( * `  S )  x.  ( F  .ih  G ) )  +  ( S  x.  ( G  .ih  F ) ) )  e.  CC
2221abscli 12157 . . 3  |-  ( abs `  -u ( ( ( * `  S )  x.  ( F  .ih  G ) )  +  ( S  x.  ( G 
.ih  F ) ) ) )  e.  RR
23 2re 10029 . . . 4  |-  2  e.  RR
24 hiidge0 22557 . . . . . 6  |-  ( G  e.  ~H  ->  0  <_  ( G  .ih  G
) )
25 hiidrcl 22554 . . . . . . . 8  |-  ( G  e.  ~H  ->  ( G  .ih  G )  e.  RR )
263, 25ax-mp 8 . . . . . . 7  |-  ( G 
.ih  G )  e.  RR
2726sqrcli 12134 . . . . . 6  |-  ( 0  <_  ( G  .ih  G )  ->  ( sqr `  ( G  .ih  G
) )  e.  RR )
283, 24, 27mp2b 10 . . . . 5  |-  ( sqr `  ( G  .ih  G
) )  e.  RR
29 hiidge0 22557 . . . . . 6  |-  ( F  e.  ~H  ->  0  <_  ( F  .ih  F
) )
30 hiidrcl 22554 . . . . . . . 8  |-  ( F  e.  ~H  ->  ( F  .ih  F )  e.  RR )
312, 30ax-mp 8 . . . . . . 7  |-  ( F 
.ih  F )  e.  RR
3231sqrcli 12134 . . . . . 6  |-  ( 0  <_  ( F  .ih  F )  ->  ( sqr `  ( F  .ih  F
) )  e.  RR )
332, 29, 32mp2b 10 . . . . 5  |-  ( sqr `  ( F  .ih  F
) )  e.  RR
3428, 33remulcli 9064 . . . 4  |-  ( ( sqr `  ( G 
.ih  G ) )  x.  ( sqr `  ( F  .ih  F ) ) )  e.  RR
3523, 34remulcli 9064 . . 3  |-  ( 2  x.  ( ( sqr `  ( G  .ih  G
) )  x.  ( sqr `  ( F  .ih  F ) ) ) )  e.  RR
3613, 22, 35letri 9162 . 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 4176   ` cfv 5417  (class class class)co 6044   CCcc 8948   RRcr 8949   0cc0 8950   1c1 8951    + caddc 8953    x. cmul 8955    <_ cle 9081   -ucneg 9252   2c2 10009   *ccj 11860   sqrcsqr 11997   abscabs 11998   ~Hchil 22379    .ih csp 22382
This theorem is referenced by:  normlem7tALT  22578  norm-ii-i  22596
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 2389  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-cnex 9006  ax-resscn 9007  ax-1cn 9008  ax-icn 9009  ax-addcl 9010  ax-addrcl 9011  ax-mulcl 9012  ax-mulrcl 9013  ax-mulcom 9014  ax-addass 9015  ax-mulass 9016  ax-distr 9017  ax-i2m1 9018  ax-1ne0 9019  ax-1rid 9020  ax-rnegex 9021  ax-rrecex 9022  ax-cnre 9023  ax-pre-lttri 9024  ax-pre-lttrn 9025  ax-pre-ltadd 9026  ax-pre-mulgt0 9027  ax-pre-sup 9028  ax-hfvadd 22460  ax-hv0cl 22463  ax-hfvmul 22465  ax-hvmulass 22467  ax-hvmul0 22470  ax-hfi 22538  ax-his1 22541  ax-his2 22542  ax-his3 22543  ax-his4 22544
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 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-2nd 6313  df-riota 6512  df-recs 6596  df-rdg 6631  df-er 6868  df-en 7073  df-dom 7074  df-sdom 7075  df-sup 7408  df-pnf 9082  df-mnf 9083  df-xr 9084  df-ltxr 9085  df-le 9086  df-sub 9253  df-neg 9254  df-div 9638  df-nn 9961  df-2 10018  df-3 10019  df-4 10020  df-n0 10182  df-z 10243  df-uz 10449  df-rp 10573  df-seq 11283  df-exp 11342  df-cj 11863  df-re 11864  df-im 11865  df-sqr 11999  df-abs 12000  df-hvsub 22431
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