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Theorem normlem2 21686
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 11650 . . . . . . . 8  |-  ( * `
 S )  e.  CC
4 normlem1.2 . . . . . . . . 9  |-  F  e. 
~H
5 normlem1.3 . . . . . . . . 9  |-  G  e. 
~H
64, 5hicli 21656 . . . . . . . 8  |-  ( F 
.ih  G )  e.  CC
73, 6mulcli 8838 . . . . . . 7  |-  ( ( * `  S )  x.  ( F  .ih  G ) )  e.  CC
85, 4hicli 21656 . . . . . . . 8  |-  ( G 
.ih  F )  e.  CC
92, 8mulcli 8838 . . . . . . 7  |-  ( S  x.  ( G  .ih  F ) )  e.  CC
107, 9cjaddi 11669 . . . . . 6  |-  ( * `
 ( ( ( * `  S )  x.  ( F  .ih  G ) )  +  ( S  x.  ( G 
.ih  F ) ) ) )  =  ( ( * `  (
( * `  S
)  x.  ( F 
.ih  G ) ) )  +  ( * `
 ( S  x.  ( G  .ih  F ) ) ) )
112cjcji 11652 . . . . . . . . . 10  |-  ( * `
 ( * `  S ) )  =  S
1211eqcomi 2288 . . . . . . . . 9  |-  S  =  ( * `  (
* `  S )
)
135, 4his1i 21675 . . . . . . . . 9  |-  ( G 
.ih  F )  =  ( * `  ( F  .ih  G ) )
1412, 13oveq12i 5832 . . . . . . . 8  |-  ( S  x.  ( G  .ih  F ) )  =  ( ( * `  (
* `  S )
)  x.  ( * `
 ( F  .ih  G ) ) )
153, 6cjmuli 11670 . . . . . . . 8  |-  ( * `
 ( ( * `
 S )  x.  ( F  .ih  G
) ) )  =  ( ( * `  ( * `  S
) )  x.  (
* `  ( F  .ih  G ) ) )
1614, 15eqtr4i 2307 . . . . . . 7  |-  ( S  x.  ( G  .ih  F ) )  =  ( * `  ( ( * `  S )  x.  ( F  .ih  G ) ) )
174, 5his1i 21675 . . . . . . . . 9  |-  ( F 
.ih  G )  =  ( * `  ( G  .ih  F ) )
1817oveq2i 5831 . . . . . . . 8  |-  ( ( * `  S )  x.  ( F  .ih  G ) )  =  ( ( * `  S
)  x.  ( * `
 ( G  .ih  F ) ) )
192, 8cjmuli 11670 . . . . . . . 8  |-  ( * `
 ( S  x.  ( G  .ih  F ) ) )  =  ( ( * `  S
)  x.  ( * `
 ( G  .ih  F ) ) )
2018, 19eqtr4i 2307 . . . . . . 7  |-  ( ( * `  S )  x.  ( F  .ih  G ) )  =  ( * `  ( S  x.  ( G  .ih  F ) ) )
2116, 20oveq12i 5832 . . . . . 6  |-  ( ( S  x.  ( G 
.ih  F ) )  +  ( ( * `
 S )  x.  ( F  .ih  G
) ) )  =  ( ( * `  ( ( * `  S )  x.  ( F  .ih  G ) ) )  +  ( * `
 ( S  x.  ( G  .ih  F ) ) ) )
2210, 21eqtr4i 2307 . . . . 5  |-  ( * `
 ( ( ( * `  S )  x.  ( F  .ih  G ) )  +  ( S  x.  ( G 
.ih  F ) ) ) )  =  ( ( S  x.  ( G  .ih  F ) )  +  ( ( * `
 S )  x.  ( F  .ih  G
) ) )
237, 9addcomi 8999 . . . . 5  |-  ( ( ( * `  S
)  x.  ( F 
.ih  G ) )  +  ( S  x.  ( G  .ih  F ) ) )  =  ( ( S  x.  ( G  .ih  F ) )  +  ( ( * `
 S )  x.  ( F  .ih  G
) ) )
2422, 23eqtr4i 2307 . . . 4  |-  ( * `
 ( ( ( * `  S )  x.  ( F  .ih  G ) )  +  ( S  x.  ( G 
.ih  F ) ) ) )  =  ( ( ( * `  S )  x.  ( F  .ih  G ) )  +  ( S  x.  ( G  .ih  F ) ) )
257, 9addcli 8837 . . . . 5  |-  ( ( ( * `  S
)  x.  ( F 
.ih  G ) )  +  ( S  x.  ( G  .ih  F ) ) )  e.  CC
2625cjrebi 11655 . . . 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 9104 . 2  |-  -u (
( ( * `  S )  x.  ( F  .ih  G ) )  +  ( S  x.  ( G  .ih  F ) ) )  e.  RR
291, 28eqeltri 2354 1  |-  B  e.  RR
Colors of variables: wff set class
Syntax hints:    = wceq 1623    e. wcel 1685   ` cfv 5221  (class class class)co 5820   CCcc 8731   RRcr 8732    + caddc 8736    x. cmul 8738   -ucneg 9034   *ccj 11577   ~Hchil 21495    .ih csp 21498
This theorem is referenced by:  normlem3  21687  normlem6  21690  normlem7  21691  norm-ii-i  21712
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-resscn 8790  ax-1cn 8791  ax-icn 8792  ax-addcl 8793  ax-addrcl 8794  ax-mulcl 8795  ax-mulrcl 8796  ax-mulcom 8797  ax-addass 8798  ax-mulass 8799  ax-distr 8800  ax-i2m1 8801  ax-1ne0 8802  ax-1rid 8803  ax-rnegex 8804  ax-rrecex 8805  ax-cnre 8806  ax-pre-lttri 8807  ax-pre-lttrn 8808  ax-pre-ltadd 8809  ax-pre-mulgt0 8810  ax-hfi 21654  ax-his1 21657
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 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-po 4313  df-so 4314  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-iota 6253  df-riota 6300  df-er 6656  df-en 6860  df-dom 6861  df-sdom 6862  df-pnf 8865  df-mnf 8866  df-xr 8867  df-ltxr 8868  df-le 8869  df-sub 9035  df-neg 9036  df-div 9420  df-2 9800  df-cj 11580  df-re 11581  df-im 11582
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