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Theorem normlem2 22124
Description: Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 27-Jul-1999.) (New usage is discouraged.)
Hypotheses
Ref Expression
normlem1.1  |-  S  e.  CC
normlem1.2  |-  F  e. 
~H
normlem1.3  |-  G  e. 
~H
normlem2.4  |-  B  = 
-u ( ( ( * `  S )  x.  ( F  .ih  G ) )  +  ( S  x.  ( G 
.ih  F ) ) )
Assertion
Ref Expression
normlem2  |-  B  e.  RR

Proof of Theorem normlem2
StepHypRef Expression
1 normlem2.4 . 2  |-  B  = 
-u ( ( ( * `  S )  x.  ( F  .ih  G ) )  +  ( S  x.  ( G 
.ih  F ) ) )
2 normlem1.1 . . . . . . . . 9  |-  S  e.  CC
32cjcli 11861 . . . . . . . 8  |-  ( * `
 S )  e.  CC
4 normlem1.2 . . . . . . . . 9  |-  F  e. 
~H
5 normlem1.3 . . . . . . . . 9  |-  G  e. 
~H
64, 5hicli 22094 . . . . . . . 8  |-  ( F 
.ih  G )  e.  CC
73, 6mulcli 8989 . . . . . . 7  |-  ( ( * `  S )  x.  ( F  .ih  G ) )  e.  CC
85, 4hicli 22094 . . . . . . . 8  |-  ( G 
.ih  F )  e.  CC
92, 8mulcli 8989 . . . . . . 7  |-  ( S  x.  ( G  .ih  F ) )  e.  CC
107, 9cjaddi 11880 . . . . . 6  |-  ( * `
 ( ( ( * `  S )  x.  ( F  .ih  G ) )  +  ( S  x.  ( G 
.ih  F ) ) ) )  =  ( ( * `  (
( * `  S
)  x.  ( F 
.ih  G ) ) )  +  ( * `
 ( S  x.  ( G  .ih  F ) ) ) )
112cjcji 11863 . . . . . . . . . 10  |-  ( * `
 ( * `  S ) )  =  S
1211eqcomi 2370 . . . . . . . . 9  |-  S  =  ( * `  (
* `  S )
)
135, 4his1i 22113 . . . . . . . . 9  |-  ( G 
.ih  F )  =  ( * `  ( F  .ih  G ) )
1412, 13oveq12i 5993 . . . . . . . 8  |-  ( S  x.  ( G  .ih  F ) )  =  ( ( * `  (
* `  S )
)  x.  ( * `
 ( F  .ih  G ) ) )
153, 6cjmuli 11881 . . . . . . . 8  |-  ( * `
 ( ( * `
 S )  x.  ( F  .ih  G
) ) )  =  ( ( * `  ( * `  S
) )  x.  (
* `  ( F  .ih  G ) ) )
1614, 15eqtr4i 2389 . . . . . . 7  |-  ( S  x.  ( G  .ih  F ) )  =  ( * `  ( ( * `  S )  x.  ( F  .ih  G ) ) )
174, 5his1i 22113 . . . . . . . . 9  |-  ( F 
.ih  G )  =  ( * `  ( G  .ih  F ) )
1817oveq2i 5992 . . . . . . . 8  |-  ( ( * `  S )  x.  ( F  .ih  G ) )  =  ( ( * `  S
)  x.  ( * `
 ( G  .ih  F ) ) )
192, 8cjmuli 11881 . . . . . . . 8  |-  ( * `
 ( S  x.  ( G  .ih  F ) ) )  =  ( ( * `  S
)  x.  ( * `
 ( G  .ih  F ) ) )
2018, 19eqtr4i 2389 . . . . . . 7  |-  ( ( * `  S )  x.  ( F  .ih  G ) )  =  ( * `  ( S  x.  ( G  .ih  F ) ) )
2116, 20oveq12i 5993 . . . . . 6  |-  ( ( S  x.  ( G 
.ih  F ) )  +  ( ( * `
 S )  x.  ( F  .ih  G
) ) )  =  ( ( * `  ( ( * `  S )  x.  ( F  .ih  G ) ) )  +  ( * `
 ( S  x.  ( G  .ih  F ) ) ) )
2210, 21eqtr4i 2389 . . . . 5  |-  ( * `
 ( ( ( * `  S )  x.  ( F  .ih  G ) )  +  ( S  x.  ( G 
.ih  F ) ) ) )  =  ( ( S  x.  ( G  .ih  F ) )  +  ( ( * `
 S )  x.  ( F  .ih  G
) ) )
237, 9addcomi 9150 . . . . 5  |-  ( ( ( * `  S
)  x.  ( F 
.ih  G ) )  +  ( S  x.  ( G  .ih  F ) ) )  =  ( ( S  x.  ( G  .ih  F ) )  +  ( ( * `
 S )  x.  ( F  .ih  G
) ) )
2422, 23eqtr4i 2389 . . . 4  |-  ( * `
 ( ( ( * `  S )  x.  ( F  .ih  G ) )  +  ( S  x.  ( G 
.ih  F ) ) ) )  =  ( ( ( * `  S )  x.  ( F  .ih  G ) )  +  ( S  x.  ( G  .ih  F ) ) )
257, 9addcli 8988 . . . . 5  |-  ( ( ( * `  S
)  x.  ( F 
.ih  G ) )  +  ( S  x.  ( G  .ih  F ) ) )  e.  CC
2625cjrebi 11866 . . . 4  |-  ( ( ( ( * `  S )  x.  ( F  .ih  G ) )  +  ( S  x.  ( G  .ih  F ) ) )  e.  RR  <->  ( * `  ( ( ( * `  S
)  x.  ( F 
.ih  G ) )  +  ( S  x.  ( G  .ih  F ) ) ) )  =  ( ( ( * `
 S )  x.  ( F  .ih  G
) )  +  ( S  x.  ( G 
.ih  F ) ) ) )
2724, 26mpbir 200 . . 3  |-  ( ( ( * `  S
)  x.  ( F 
.ih  G ) )  +  ( S  x.  ( G  .ih  F ) ) )  e.  RR
2827renegcli 9255 . 2  |-  -u (
( ( * `  S )  x.  ( F  .ih  G ) )  +  ( S  x.  ( G  .ih  F ) ) )  e.  RR
291, 28eqeltri 2436 1  |-  B  e.  RR
Colors of variables: wff set class
Syntax hints:    = wceq 1647    e. wcel 1715   ` cfv 5358  (class class class)co 5981   CCcc 8882   RRcr 8883    + caddc 8887    x. cmul 8889   -ucneg 9185   *ccj 11788   ~Hchil 21933    .ih csp 21936
This theorem is referenced by:  normlem3  22125  normlem6  22128  normlem7  22129  norm-ii-i  22150
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-resscn 8941  ax-1cn 8942  ax-icn 8943  ax-addcl 8944  ax-addrcl 8945  ax-mulcl 8946  ax-mulrcl 8947  ax-mulcom 8948  ax-addass 8949  ax-mulass 8950  ax-distr 8951  ax-i2m1 8952  ax-1ne0 8953  ax-1rid 8954  ax-rnegex 8955  ax-rrecex 8956  ax-cnre 8957  ax-pre-lttri 8958  ax-pre-lttrn 8959  ax-pre-ltadd 8960  ax-pre-mulgt0 8961  ax-hfi 22092  ax-his1 22095
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-po 4417  df-so 4418  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-riota 6446  df-er 6802  df-en 7007  df-dom 7008  df-sdom 7009  df-pnf 9016  df-mnf 9017  df-xr 9018  df-ltxr 9019  df-le 9020  df-sub 9186  df-neg 9187  df-div 9571  df-2 9951  df-cj 11791  df-re 11792  df-im 11793
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