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Theorem normlem2 21520
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 11531 . . . . . . . 8  |-  ( * `
 S )  e.  CC
4 normlem1.2 . . . . . . . . 9  |-  F  e. 
~H
5 normlem1.3 . . . . . . . . 9  |-  G  e. 
~H
64, 5hicli 21490 . . . . . . . 8  |-  ( F 
.ih  G )  e.  CC
73, 6mulcli 8722 . . . . . . 7  |-  ( ( * `  S )  x.  ( F  .ih  G ) )  e.  CC
85, 4hicli 21490 . . . . . . . 8  |-  ( G 
.ih  F )  e.  CC
92, 8mulcli 8722 . . . . . . 7  |-  ( S  x.  ( G  .ih  F ) )  e.  CC
107, 9cjaddi 11550 . . . . . 6  |-  ( * `
 ( ( ( * `  S )  x.  ( F  .ih  G ) )  +  ( S  x.  ( G 
.ih  F ) ) ) )  =  ( ( * `  (
( * `  S
)  x.  ( F 
.ih  G ) ) )  +  ( * `
 ( S  x.  ( G  .ih  F ) ) ) )
112cjcji 11533 . . . . . . . . . 10  |-  ( * `
 ( * `  S ) )  =  S
1211eqcomi 2257 . . . . . . . . 9  |-  S  =  ( * `  (
* `  S )
)
135, 4his1i 21509 . . . . . . . . 9  |-  ( G 
.ih  F )  =  ( * `  ( F  .ih  G ) )
1412, 13oveq12i 5722 . . . . . . . 8  |-  ( S  x.  ( G  .ih  F ) )  =  ( ( * `  (
* `  S )
)  x.  ( * `
 ( F  .ih  G ) ) )
153, 6cjmuli 11551 . . . . . . . 8  |-  ( * `
 ( ( * `
 S )  x.  ( F  .ih  G
) ) )  =  ( ( * `  ( * `  S
) )  x.  (
* `  ( F  .ih  G ) ) )
1614, 15eqtr4i 2276 . . . . . . 7  |-  ( S  x.  ( G  .ih  F ) )  =  ( * `  ( ( * `  S )  x.  ( F  .ih  G ) ) )
174, 5his1i 21509 . . . . . . . . 9  |-  ( F 
.ih  G )  =  ( * `  ( G  .ih  F ) )
1817oveq2i 5721 . . . . . . . 8  |-  ( ( * `  S )  x.  ( F  .ih  G ) )  =  ( ( * `  S
)  x.  ( * `
 ( G  .ih  F ) ) )
192, 8cjmuli 11551 . . . . . . . 8  |-  ( * `
 ( S  x.  ( G  .ih  F ) ) )  =  ( ( * `  S
)  x.  ( * `
 ( G  .ih  F ) ) )
2018, 19eqtr4i 2276 . . . . . . 7  |-  ( ( * `  S )  x.  ( F  .ih  G ) )  =  ( * `  ( S  x.  ( G  .ih  F ) ) )
2116, 20oveq12i 5722 . . . . . 6  |-  ( ( S  x.  ( G 
.ih  F ) )  +  ( ( * `
 S )  x.  ( F  .ih  G
) ) )  =  ( ( * `  ( ( * `  S )  x.  ( F  .ih  G ) ) )  +  ( * `
 ( S  x.  ( G  .ih  F ) ) ) )
2210, 21eqtr4i 2276 . . . . 5  |-  ( * `
 ( ( ( * `  S )  x.  ( F  .ih  G ) )  +  ( S  x.  ( G 
.ih  F ) ) ) )  =  ( ( S  x.  ( G  .ih  F ) )  +  ( ( * `
 S )  x.  ( F  .ih  G
) ) )
237, 9addcomi 8883 . . . . 5  |-  ( ( ( * `  S
)  x.  ( F 
.ih  G ) )  +  ( S  x.  ( G  .ih  F ) ) )  =  ( ( S  x.  ( G  .ih  F ) )  +  ( ( * `
 S )  x.  ( F  .ih  G
) ) )
2422, 23eqtr4i 2276 . . . 4  |-  ( * `
 ( ( ( * `  S )  x.  ( F  .ih  G ) )  +  ( S  x.  ( G 
.ih  F ) ) ) )  =  ( ( ( * `  S )  x.  ( F  .ih  G ) )  +  ( S  x.  ( G  .ih  F ) ) )
257, 9addcli 8721 . . . . 5  |-  ( ( ( * `  S
)  x.  ( F 
.ih  G ) )  +  ( S  x.  ( G  .ih  F ) ) )  e.  CC
2625cjrebi 11536 . . . 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 8988 . 2  |-  -u (
( ( * `  S )  x.  ( F  .ih  G ) )  +  ( S  x.  ( G  .ih  F ) ) )  e.  RR
291, 28eqeltri 2323 1  |-  B  e.  RR
Colors of variables: wff set class
Syntax hints:    = wceq 1619    e. wcel 1621   ` cfv 4592  (class class class)co 5710   CCcc 8615   RRcr 8616    + caddc 8620    x. cmul 8622   -ucneg 8918   *ccj 11458   ~Hchil 21329    .ih csp 21332
This theorem is referenced by:  normlem3  21521  normlem6  21524  normlem7  21525  norm-ii-i  21546
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 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-resscn 8674  ax-1cn 8675  ax-icn 8676  ax-addcl 8677  ax-addrcl 8678  ax-mulcl 8679  ax-mulrcl 8680  ax-mulcom 8681  ax-addass 8682  ax-mulass 8683  ax-distr 8684  ax-i2m1 8685  ax-1ne0 8686  ax-1rid 8687  ax-rnegex 8688  ax-rrecex 8689  ax-cnre 8690  ax-pre-lttri 8691  ax-pre-lttrn 8692  ax-pre-ltadd 8693  ax-pre-mulgt0 8694  ax-hfi 21488  ax-his1 21491
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-po 4207  df-so 4208  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-iota 6143  df-riota 6190  df-er 6546  df-en 6750  df-dom 6751  df-sdom 6752  df-pnf 8749  df-mnf 8750  df-xr 8751  df-ltxr 8752  df-le 8753  df-sub 8919  df-neg 8920  df-div 9304  df-2 9684  df-cj 11461  df-re 11462  df-im 11463
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