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Theorem normlem7 21620
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 2256 . . . . . 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 21615 . . . . 5  |-  -u (
( ( * `  S )  x.  ( F  .ih  G ) )  +  ( S  x.  ( G  .ih  F ) ) )  e.  RR
61cjcli 11584 . . . . . . . 8  |-  ( * `
 S )  e.  CC
72, 3hicli 21585 . . . . . . . 8  |-  ( F 
.ih  G )  e.  CC
86, 7mulcli 8775 . . . . . . 7  |-  ( ( * `  S )  x.  ( F  .ih  G ) )  e.  CC
93, 2hicli 21585 . . . . . . . 8  |-  ( G 
.ih  F )  e.  CC
101, 9mulcli 8775 . . . . . . 7  |-  ( S  x.  ( G  .ih  F ) )  e.  CC
118, 10addcli 8774 . . . . . 6  |-  ( ( ( * `  S
)  x.  ( F 
.ih  G ) )  +  ( S  x.  ( G  .ih  F ) ) )  e.  CC
1211negrebi 9053 . . . . 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 11793 . . 3  |-  ( ( ( * `  S
)  x.  ( F 
.ih  G ) )  +  ( S  x.  ( G  .ih  F ) ) )  <_  ( abs `  ( ( ( * `  S )  x.  ( F  .ih  G ) )  +  ( S  x.  ( G 
.ih  F ) ) ) )
1511absnegi 11813 . . 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 3988 . 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 2256 . . 3  |-  ( G 
.ih  G )  =  ( G  .ih  G
)
18 eqid 2256 . . 3  |-  ( F 
.ih  F )  =  ( F  .ih  F
)
19 normlem7.4 . . 3  |-  ( abs `  S )  =  1
201, 2, 3, 4, 17, 18, 19normlem6 21619 . 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 9047 . . . 4  |-  -u (
( ( * `  S )  x.  ( F  .ih  G ) )  +  ( S  x.  ( G  .ih  F ) ) )  e.  CC
2221abscli 11808 . . 3  |-  ( abs `  -u ( ( ( * `  S )  x.  ( F  .ih  G ) )  +  ( S  x.  ( G 
.ih  F ) ) ) )  e.  RR
23 2re 9748 . . . 4  |-  2  e.  RR
24 hiidge0 21602 . . . . . 6  |-  ( G  e.  ~H  ->  0  <_  ( G  .ih  G
) )
25 hiidrcl 21599 . . . . . . . 8  |-  ( G  e.  ~H  ->  ( G  .ih  G )  e.  RR )
263, 25ax-mp 10 . . . . . . 7  |-  ( G 
.ih  G )  e.  RR
2726sqrcli 11785 . . . . . 6  |-  ( 0  <_  ( G  .ih  G )  ->  ( sqr `  ( G  .ih  G
) )  e.  RR )
283, 24, 27mp2b 11 . . . . 5  |-  ( sqr `  ( G  .ih  G
) )  e.  RR
29 hiidge0 21602 . . . . . 6  |-  ( F  e.  ~H  ->  0  <_  ( F  .ih  F
) )
30 hiidrcl 21599 . . . . . . . 8  |-  ( F  e.  ~H  ->  ( F  .ih  F )  e.  RR )
312, 30ax-mp 10 . . . . . . 7  |-  ( F 
.ih  F )  e.  RR
3231sqrcli 11785 . . . . . 6  |-  ( 0  <_  ( F  .ih  F )  ->  ( sqr `  ( F  .ih  F
) )  e.  RR )
332, 29, 32mp2b 11 . . . . 5  |-  ( sqr `  ( F  .ih  F
) )  e.  RR
3428, 33remulcli 8784 . . . 4  |-  ( ( sqr `  ( G 
.ih  G ) )  x.  ( sqr `  ( F  .ih  F ) ) )  e.  RR
3523, 34remulcli 8784 . . 3  |-  ( 2  x.  ( ( sqr `  ( G  .ih  G
) )  x.  ( sqr `  ( F  .ih  F ) ) ) )  e.  RR
3613, 22, 35letri 8881 . 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 3963   ` cfv 4638  (class class class)co 5757   CCcc 8668   RRcr 8669   0cc0 8670   1c1 8671    + caddc 8673    x. cmul 8675    <_ cle 8801   -ucneg 8971   2c2 9728   *ccj 11511   sqrcsqr 11648   abscabs 11649   ~Hchil 21424    .ih csp 21427
This theorem is referenced by:  normlem7tALT  21623  norm-ii-i  21641
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-hvsub 21476
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