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Theorem normlem2 21692
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 11656 . . . . . . . 8  |-  ( * `
 S )  e.  CC
4 normlem1.2 . . . . . . . . 9  |-  F  e. 
~H
5 normlem1.3 . . . . . . . . 9  |-  G  e. 
~H
64, 5hicli 21662 . . . . . . . 8  |-  ( F 
.ih  G )  e.  CC
73, 6mulcli 8844 . . . . . . 7  |-  ( ( * `  S )  x.  ( F  .ih  G ) )  e.  CC
85, 4hicli 21662 . . . . . . . 8  |-  ( G 
.ih  F )  e.  CC
92, 8mulcli 8844 . . . . . . 7  |-  ( S  x.  ( G  .ih  F ) )  e.  CC
107, 9cjaddi 11675 . . . . . 6  |-  ( * `
 ( ( ( * `  S )  x.  ( F  .ih  G ) )  +  ( S  x.  ( G 
.ih  F ) ) ) )  =  ( ( * `  (
( * `  S
)  x.  ( F 
.ih  G ) ) )  +  ( * `
 ( S  x.  ( G  .ih  F ) ) ) )
112cjcji 11658 . . . . . . . . . 10  |-  ( * `
 ( * `  S ) )  =  S
1211eqcomi 2289 . . . . . . . . 9  |-  S  =  ( * `  (
* `  S )
)
135, 4his1i 21681 . . . . . . . . 9  |-  ( G 
.ih  F )  =  ( * `  ( F  .ih  G ) )
1412, 13oveq12i 5872 . . . . . . . 8  |-  ( S  x.  ( G  .ih  F ) )  =  ( ( * `  (
* `  S )
)  x.  ( * `
 ( F  .ih  G ) ) )
153, 6cjmuli 11676 . . . . . . . 8  |-  ( * `
 ( ( * `
 S )  x.  ( F  .ih  G
) ) )  =  ( ( * `  ( * `  S
) )  x.  (
* `  ( F  .ih  G ) ) )
1614, 15eqtr4i 2308 . . . . . . 7  |-  ( S  x.  ( G  .ih  F ) )  =  ( * `  ( ( * `  S )  x.  ( F  .ih  G ) ) )
174, 5his1i 21681 . . . . . . . . 9  |-  ( F 
.ih  G )  =  ( * `  ( G  .ih  F ) )
1817oveq2i 5871 . . . . . . . 8  |-  ( ( * `  S )  x.  ( F  .ih  G ) )  =  ( ( * `  S
)  x.  ( * `
 ( G  .ih  F ) ) )
192, 8cjmuli 11676 . . . . . . . 8  |-  ( * `
 ( S  x.  ( G  .ih  F ) ) )  =  ( ( * `  S
)  x.  ( * `
 ( G  .ih  F ) ) )
2018, 19eqtr4i 2308 . . . . . . 7  |-  ( ( * `  S )  x.  ( F  .ih  G ) )  =  ( * `  ( S  x.  ( G  .ih  F ) ) )
2116, 20oveq12i 5872 . . . . . 6  |-  ( ( S  x.  ( G 
.ih  F ) )  +  ( ( * `
 S )  x.  ( F  .ih  G
) ) )  =  ( ( * `  ( ( * `  S )  x.  ( F  .ih  G ) ) )  +  ( * `
 ( S  x.  ( G  .ih  F ) ) ) )
2210, 21eqtr4i 2308 . . . . 5  |-  ( * `
 ( ( ( * `  S )  x.  ( F  .ih  G ) )  +  ( S  x.  ( G 
.ih  F ) ) ) )  =  ( ( S  x.  ( G  .ih  F ) )  +  ( ( * `
 S )  x.  ( F  .ih  G
) ) )
237, 9addcomi 9005 . . . . 5  |-  ( ( ( * `  S
)  x.  ( F 
.ih  G ) )  +  ( S  x.  ( G  .ih  F ) ) )  =  ( ( S  x.  ( G  .ih  F ) )  +  ( ( * `
 S )  x.  ( F  .ih  G
) ) )
2422, 23eqtr4i 2308 . . . 4  |-  ( * `
 ( ( ( * `  S )  x.  ( F  .ih  G ) )  +  ( S  x.  ( G 
.ih  F ) ) ) )  =  ( ( ( * `  S )  x.  ( F  .ih  G ) )  +  ( S  x.  ( G  .ih  F ) ) )
257, 9addcli 8843 . . . . 5  |-  ( ( ( * `  S
)  x.  ( F 
.ih  G ) )  +  ( S  x.  ( G  .ih  F ) ) )  e.  CC
2625cjrebi 11661 . . . 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 9110 . 2  |-  -u (
( ( * `  S )  x.  ( F  .ih  G ) )  +  ( S  x.  ( G  .ih  F ) ) )  e.  RR
291, 28eqeltri 2355 1  |-  B  e.  RR
Colors of variables: wff set class
Syntax hints:    = wceq 1625    e. wcel 1686   ` cfv 5257  (class class class)co 5860   CCcc 8737   RRcr 8738    + caddc 8742    x. cmul 8744   -ucneg 9040   *ccj 11583   ~Hchil 21501    .ih csp 21504
This theorem is referenced by:  normlem3  21693  normlem6  21696  normlem7  21697  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-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-hfi 21660  ax-his1 21663
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-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4311  df-po 4316  df-so 4317  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-riota 6306  df-er 6662  df-en 6866  df-dom 6867  df-sdom 6868  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-2 9806  df-cj 11586  df-re 11587  df-im 11588
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