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Theorem normlem2 22618
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 11979 . . . . . . . 8  |-  ( * `
 S )  e.  CC
4 normlem1.2 . . . . . . . . 9  |-  F  e. 
~H
5 normlem1.3 . . . . . . . . 9  |-  G  e. 
~H
64, 5hicli 22588 . . . . . . . 8  |-  ( F 
.ih  G )  e.  CC
73, 6mulcli 9100 . . . . . . 7  |-  ( ( * `  S )  x.  ( F  .ih  G ) )  e.  CC
85, 4hicli 22588 . . . . . . . 8  |-  ( G 
.ih  F )  e.  CC
92, 8mulcli 9100 . . . . . . 7  |-  ( S  x.  ( G  .ih  F ) )  e.  CC
107, 9cjaddi 11998 . . . . . 6  |-  ( * `
 ( ( ( * `  S )  x.  ( F  .ih  G ) )  +  ( S  x.  ( G 
.ih  F ) ) ) )  =  ( ( * `  (
( * `  S
)  x.  ( F 
.ih  G ) ) )  +  ( * `
 ( S  x.  ( G  .ih  F ) ) ) )
112cjcji 11981 . . . . . . . . . 10  |-  ( * `
 ( * `  S ) )  =  S
1211eqcomi 2442 . . . . . . . . 9  |-  S  =  ( * `  (
* `  S )
)
135, 4his1i 22607 . . . . . . . . 9  |-  ( G 
.ih  F )  =  ( * `  ( F  .ih  G ) )
1412, 13oveq12i 6096 . . . . . . . 8  |-  ( S  x.  ( G  .ih  F ) )  =  ( ( * `  (
* `  S )
)  x.  ( * `
 ( F  .ih  G ) ) )
153, 6cjmuli 11999 . . . . . . . 8  |-  ( * `
 ( ( * `
 S )  x.  ( F  .ih  G
) ) )  =  ( ( * `  ( * `  S
) )  x.  (
* `  ( F  .ih  G ) ) )
1614, 15eqtr4i 2461 . . . . . . 7  |-  ( S  x.  ( G  .ih  F ) )  =  ( * `  ( ( * `  S )  x.  ( F  .ih  G ) ) )
174, 5his1i 22607 . . . . . . . . 9  |-  ( F 
.ih  G )  =  ( * `  ( G  .ih  F ) )
1817oveq2i 6095 . . . . . . . 8  |-  ( ( * `  S )  x.  ( F  .ih  G ) )  =  ( ( * `  S
)  x.  ( * `
 ( G  .ih  F ) ) )
192, 8cjmuli 11999 . . . . . . . 8  |-  ( * `
 ( S  x.  ( G  .ih  F ) ) )  =  ( ( * `  S
)  x.  ( * `
 ( G  .ih  F ) ) )
2018, 19eqtr4i 2461 . . . . . . 7  |-  ( ( * `  S )  x.  ( F  .ih  G ) )  =  ( * `  ( S  x.  ( G  .ih  F ) ) )
2116, 20oveq12i 6096 . . . . . 6  |-  ( ( S  x.  ( G 
.ih  F ) )  +  ( ( * `
 S )  x.  ( F  .ih  G
) ) )  =  ( ( * `  ( ( * `  S )  x.  ( F  .ih  G ) ) )  +  ( * `
 ( S  x.  ( G  .ih  F ) ) ) )
2210, 21eqtr4i 2461 . . . . 5  |-  ( * `
 ( ( ( * `  S )  x.  ( F  .ih  G ) )  +  ( S  x.  ( G 
.ih  F ) ) ) )  =  ( ( S  x.  ( G  .ih  F ) )  +  ( ( * `
 S )  x.  ( F  .ih  G
) ) )
237, 9addcomi 9262 . . . . 5  |-  ( ( ( * `  S
)  x.  ( F 
.ih  G ) )  +  ( S  x.  ( G  .ih  F ) ) )  =  ( ( S  x.  ( G  .ih  F ) )  +  ( ( * `
 S )  x.  ( F  .ih  G
) ) )
2422, 23eqtr4i 2461 . . . 4  |-  ( * `
 ( ( ( * `  S )  x.  ( F  .ih  G ) )  +  ( S  x.  ( G 
.ih  F ) ) ) )  =  ( ( ( * `  S )  x.  ( F  .ih  G ) )  +  ( S  x.  ( G  .ih  F ) ) )
257, 9addcli 9099 . . . . 5  |-  ( ( ( * `  S
)  x.  ( F 
.ih  G ) )  +  ( S  x.  ( G  .ih  F ) ) )  e.  CC
2625cjrebi 11984 . . . 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 202 . . 3  |-  ( ( ( * `  S
)  x.  ( F 
.ih  G ) )  +  ( S  x.  ( G  .ih  F ) ) )  e.  RR
2827renegcli 9367 . 2  |-  -u (
( ( * `  S )  x.  ( F  .ih  G ) )  +  ( S  x.  ( G  .ih  F ) ) )  e.  RR
291, 28eqeltri 2508 1  |-  B  e.  RR
Colors of variables: wff set class
Syntax hints:    = wceq 1653    e. wcel 1726   ` cfv 5457  (class class class)co 6084   CCcc 8993   RRcr 8994    + caddc 8998    x. cmul 9000   -ucneg 9297   *ccj 11906   ~Hchil 22427    .ih csp 22430
This theorem is referenced by:  normlem3  22619  normlem6  22622  normlem7  22623  norm-ii-i  22644
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072  ax-hfi 22586  ax-his1 22589
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-po 4506  df-so 4507  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-riota 6552  df-er 6908  df-en 7113  df-dom 7114  df-sdom 7115  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-2 10063  df-cj 11909  df-re 11910  df-im 11911
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