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Theorem normlem7 21656
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 2258 . . . . . 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 21651 . . . . 5  |-  -u (
( ( * `  S )  x.  ( F  .ih  G ) )  +  ( S  x.  ( G  .ih  F ) ) )  e.  RR
61cjcli 11620 . . . . . . . 8  |-  ( * `
 S )  e.  CC
72, 3hicli 21621 . . . . . . . 8  |-  ( F 
.ih  G )  e.  CC
86, 7mulcli 8810 . . . . . . 7  |-  ( ( * `  S )  x.  ( F  .ih  G ) )  e.  CC
93, 2hicli 21621 . . . . . . . 8  |-  ( G 
.ih  F )  e.  CC
101, 9mulcli 8810 . . . . . . 7  |-  ( S  x.  ( G  .ih  F ) )  e.  CC
118, 10addcli 8809 . . . . . 6  |-  ( ( ( * `  S
)  x.  ( F 
.ih  G ) )  +  ( S  x.  ( G  .ih  F ) ) )  e.  CC
1211negrebi 9088 . . . . 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 201 . . . 4  |-  ( ( ( * `  S
)  x.  ( F 
.ih  G ) )  +  ( S  x.  ( G  .ih  F ) ) )  e.  RR
1413leabsi 11829 . . 3  |-  ( ( ( * `  S
)  x.  ( F 
.ih  G ) )  +  ( S  x.  ( G  .ih  F ) ) )  <_  ( abs `  ( ( ( * `  S )  x.  ( F  .ih  G ) )  +  ( S  x.  ( G 
.ih  F ) ) ) )
1511absnegi 11849 . . 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 4022 . 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 2258 . . 3  |-  ( G 
.ih  G )  =  ( G  .ih  G
)
18 eqid 2258 . . 3  |-  ( F 
.ih  F )  =  ( F  .ih  F
)
19 normlem7.4 . . 3  |-  ( abs `  S )  =  1
201, 2, 3, 4, 17, 18, 19normlem6 21655 . 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 9082 . . . 4  |-  -u (
( ( * `  S )  x.  ( F  .ih  G ) )  +  ( S  x.  ( G  .ih  F ) ) )  e.  CC
2221abscli 11844 . . 3  |-  ( abs `  -u ( ( ( * `  S )  x.  ( F  .ih  G ) )  +  ( S  x.  ( G 
.ih  F ) ) ) )  e.  RR
23 2re 9783 . . . 4  |-  2  e.  RR
24 hiidge0 21638 . . . . . 6  |-  ( G  e.  ~H  ->  0  <_  ( G  .ih  G
) )
25 hiidrcl 21635 . . . . . . . 8  |-  ( G  e.  ~H  ->  ( G  .ih  G )  e.  RR )
263, 25ax-mp 10 . . . . . . 7  |-  ( G 
.ih  G )  e.  RR
2726sqrcli 11821 . . . . . 6  |-  ( 0  <_  ( G  .ih  G )  ->  ( sqr `  ( G  .ih  G
) )  e.  RR )
283, 24, 27mp2b 11 . . . . 5  |-  ( sqr `  ( G  .ih  G
) )  e.  RR
29 hiidge0 21638 . . . . . 6  |-  ( F  e.  ~H  ->  0  <_  ( F  .ih  F
) )
30 hiidrcl 21635 . . . . . . . 8  |-  ( F  e.  ~H  ->  ( F  .ih  F )  e.  RR )
312, 30ax-mp 10 . . . . . . 7  |-  ( F 
.ih  F )  e.  RR
3231sqrcli 11821 . . . . . 6  |-  ( 0  <_  ( F  .ih  F )  ->  ( sqr `  ( F  .ih  F
) )  e.  RR )
332, 29, 32mp2b 11 . . . . 5  |-  ( sqr `  ( F  .ih  F
) )  e.  RR
3428, 33remulcli 8819 . . . 4  |-  ( ( sqr `  ( G 
.ih  G ) )  x.  ( sqr `  ( F  .ih  F ) ) )  e.  RR
3523, 34remulcli 8819 . . 3  |-  ( 2  x.  ( ( sqr `  ( G  .ih  G
) )  x.  ( sqr `  ( F  .ih  F ) ) ) )  e.  RR
3613, 22, 35letri 8916 . 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 656 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 1619    e. wcel 1621   class class class wbr 3997   ` cfv 4673  (class class class)co 5792   CCcc 8703   RRcr 8704   0cc0 8705   1c1 8706    + caddc 8708    x. cmul 8710    <_ cle 8836   -ucneg 9006   2c2 9763   *ccj 11547   sqrcsqr 11684   abscabs 11685   ~Hchil 21460    .ih csp 21463
This theorem is referenced by:  normlem7tALT  21659  norm-ii-i  21677
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 2239  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-cnex 8761  ax-resscn 8762  ax-1cn 8763  ax-icn 8764  ax-addcl 8765  ax-addrcl 8766  ax-mulcl 8767  ax-mulrcl 8768  ax-mulcom 8769  ax-addass 8770  ax-mulass 8771  ax-distr 8772  ax-i2m1 8773  ax-1ne0 8774  ax-1rid 8775  ax-rnegex 8776  ax-rrecex 8777  ax-cnre 8778  ax-pre-lttri 8779  ax-pre-lttrn 8780  ax-pre-ltadd 8781  ax-pre-mulgt0 8782  ax-pre-sup 8783  ax-hfvadd 21541  ax-hv0cl 21544  ax-hfvmul 21546  ax-hvmulass 21548  ax-hvmul0 21551  ax-hfi 21619  ax-his1 21622  ax-his2 21623  ax-his3 21624  ax-his4 21625
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 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-2nd 6057  df-iota 6225  df-riota 6272  df-recs 6356  df-rdg 6391  df-er 6628  df-en 6832  df-dom 6833  df-sdom 6834  df-sup 7162  df-pnf 8837  df-mnf 8838  df-xr 8839  df-ltxr 8840  df-le 8841  df-sub 9007  df-neg 9008  df-div 9392  df-n 9715  df-2 9772  df-3 9773  df-4 9774  df-n0 9934  df-z 9993  df-uz 10199  df-rp 10323  df-seq 11014  df-exp 11072  df-cj 11550  df-re 11551  df-im 11552  df-sqr 11686  df-abs 11687  df-hvsub 21512
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