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Theorem normlem7 21697
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 2285 . . . . . 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 21692 . . . . 5  |-  -u (
( ( * `  S )  x.  ( F  .ih  G ) )  +  ( S  x.  ( G  .ih  F ) ) )  e.  RR
61cjcli 11656 . . . . . . . 8  |-  ( * `
 S )  e.  CC
72, 3hicli 21662 . . . . . . . 8  |-  ( F 
.ih  G )  e.  CC
86, 7mulcli 8844 . . . . . . 7  |-  ( ( * `  S )  x.  ( F  .ih  G ) )  e.  CC
93, 2hicli 21662 . . . . . . . 8  |-  ( G 
.ih  F )  e.  CC
101, 9mulcli 8844 . . . . . . 7  |-  ( S  x.  ( G  .ih  F ) )  e.  CC
118, 10addcli 8843 . . . . . 6  |-  ( ( ( * `  S
)  x.  ( F 
.ih  G ) )  +  ( S  x.  ( G  .ih  F ) ) )  e.  CC
1211negrebi 9122 . . . . 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 199 . . . 4  |-  ( ( ( * `  S
)  x.  ( F 
.ih  G ) )  +  ( S  x.  ( G  .ih  F ) ) )  e.  RR
1413leabsi 11865 . . 3  |-  ( ( ( * `  S
)  x.  ( F 
.ih  G ) )  +  ( S  x.  ( G  .ih  F ) ) )  <_  ( abs `  ( ( ( * `  S )  x.  ( F  .ih  G ) )  +  ( S  x.  ( G 
.ih  F ) ) ) )
1511absnegi 11885 . . 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 4050 . 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 2285 . . 3  |-  ( G 
.ih  G )  =  ( G  .ih  G
)
18 eqid 2285 . . 3  |-  ( F 
.ih  F )  =  ( F  .ih  F
)
19 normlem7.4 . . 3  |-  ( abs `  S )  =  1
201, 2, 3, 4, 17, 18, 19normlem6 21696 . 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 9116 . . . 4  |-  -u (
( ( * `  S )  x.  ( F  .ih  G ) )  +  ( S  x.  ( G  .ih  F ) ) )  e.  CC
2221abscli 11880 . . 3  |-  ( abs `  -u ( ( ( * `  S )  x.  ( F  .ih  G ) )  +  ( S  x.  ( G 
.ih  F ) ) ) )  e.  RR
23 2re 9817 . . . 4  |-  2  e.  RR
24 hiidge0 21679 . . . . . 6  |-  ( G  e.  ~H  ->  0  <_  ( G  .ih  G
) )
25 hiidrcl 21676 . . . . . . . 8  |-  ( G  e.  ~H  ->  ( G  .ih  G )  e.  RR )
263, 25ax-mp 8 . . . . . . 7  |-  ( G 
.ih  G )  e.  RR
2726sqrcli 11857 . . . . . 6  |-  ( 0  <_  ( G  .ih  G )  ->  ( sqr `  ( G  .ih  G
) )  e.  RR )
283, 24, 27mp2b 9 . . . . 5  |-  ( sqr `  ( G  .ih  G
) )  e.  RR
29 hiidge0 21679 . . . . . 6  |-  ( F  e.  ~H  ->  0  <_  ( F  .ih  F
) )
30 hiidrcl 21676 . . . . . . . 8  |-  ( F  e.  ~H  ->  ( F  .ih  F )  e.  RR )
312, 30ax-mp 8 . . . . . . 7  |-  ( F 
.ih  F )  e.  RR
3231sqrcli 11857 . . . . . 6  |-  ( 0  <_  ( F  .ih  F )  ->  ( sqr `  ( F  .ih  F
) )  e.  RR )
332, 29, 32mp2b 9 . . . . 5  |-  ( sqr `  ( F  .ih  F
) )  e.  RR
3428, 33remulcli 8853 . . . 4  |-  ( ( sqr `  ( G 
.ih  G ) )  x.  ( sqr `  ( F  .ih  F ) ) )  e.  RR
3523, 34remulcli 8853 . . 3  |-  ( 2  x.  ( ( sqr `  ( G  .ih  G
) )  x.  ( sqr `  ( F  .ih  F ) ) ) )  e.  RR
3613, 22, 35letri 8950 . 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 653 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 1625    e. wcel 1686   class class class wbr 4025   ` cfv 5257  (class class class)co 5860   CCcc 8737   RRcr 8738   0cc0 8739   1c1 8740    + caddc 8742    x. cmul 8744    <_ cle 8870   -ucneg 9040   2c2 9797   *ccj 11583   sqrcsqr 11720   abscabs 11721   ~Hchil 21501    .ih csp 21504
This theorem is referenced by:  normlem7tALT  21700  norm-ii-i  21718
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-cnex 8795  ax-resscn 8796  ax-1cn 8797  ax-icn 8798  ax-addcl 8799  ax-addrcl 8800  ax-mulcl 8801  ax-mulrcl 8802  ax-mulcom 8803  ax-addass 8804  ax-mulass 8805  ax-distr 8806  ax-i2m1 8807  ax-1ne0 8808  ax-1rid 8809  ax-rnegex 8810  ax-rrecex 8811  ax-cnre 8812  ax-pre-lttri 8813  ax-pre-lttrn 8814  ax-pre-ltadd 8815  ax-pre-mulgt0 8816  ax-pre-sup 8817  ax-hfvadd 21582  ax-hv0cl 21585  ax-hfvmul 21587  ax-hvmulass 21589  ax-hvmul0 21592  ax-hfi 21660  ax-his1 21663  ax-his2 21664  ax-his3 21665  ax-his4 21666
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-2nd 6125  df-riota 6306  df-recs 6390  df-rdg 6425  df-er 6662  df-en 6866  df-dom 6867  df-sdom 6868  df-sup 7196  df-pnf 8871  df-mnf 8872  df-xr 8873  df-ltxr 8874  df-le 8875  df-sub 9041  df-neg 9042  df-div 9426  df-nn 9749  df-2 9806  df-3 9807  df-4 9808  df-n0 9968  df-z 10027  df-uz 10233  df-rp 10357  df-seq 11049  df-exp 11107  df-cj 11586  df-re 11587  df-im 11588  df-sqr 11722  df-abs 11723  df-hvsub 21553
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