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Theorem normlem0 22459
Description: Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 7-Oct-1999.) (New usage is discouraged.)
Hypotheses
Ref Expression
normlem1.1  |-  S  e.  CC
normlem1.2  |-  F  e. 
~H
normlem1.3  |-  G  e. 
~H
Assertion
Ref Expression
normlem0  |-  ( ( F  -h  ( S  .h  G ) ) 
.ih  ( F  -h  ( S  .h  G
) ) )  =  ( ( ( F 
.ih  F )  +  ( -u ( * `
 S )  x.  ( F  .ih  G
) ) )  +  ( ( -u S  x.  ( G  .ih  F
) )  +  ( ( S  x.  (
* `  S )
)  x.  ( G 
.ih  G ) ) ) )

Proof of Theorem normlem0
StepHypRef Expression
1 normlem1.2 . . . . 5  |-  F  e. 
~H
2 normlem1.1 . . . . . 6  |-  S  e.  CC
3 normlem1.3 . . . . . 6  |-  G  e. 
~H
42, 3hvmulcli 22365 . . . . 5  |-  ( S  .h  G )  e. 
~H
51, 4hvsubvali 22371 . . . 4  |-  ( F  -h  ( S  .h  G ) )  =  ( F  +h  ( -u 1  .h  ( S  .h  G ) ) )
62mulm1i 9410 . . . . . . 7  |-  ( -u
1  x.  S )  =  -u S
76oveq1i 6030 . . . . . 6  |-  ( (
-u 1  x.  S
)  .h  G )  =  ( -u S  .h  G )
8 neg1cn 9999 . . . . . . 7  |-  -u 1  e.  CC
98, 2, 3hvmulassi 22396 . . . . . 6  |-  ( (
-u 1  x.  S
)  .h  G )  =  ( -u 1  .h  ( S  .h  G
) )
107, 9eqtr3i 2409 . . . . 5  |-  ( -u S  .h  G )  =  ( -u 1  .h  ( S  .h  G
) )
1110oveq2i 6031 . . . 4  |-  ( F  +h  ( -u S  .h  G ) )  =  ( F  +h  ( -u 1  .h  ( S  .h  G ) ) )
125, 11eqtr4i 2410 . . 3  |-  ( F  -h  ( S  .h  G ) )  =  ( F  +h  ( -u S  .h  G ) )
1312, 12oveq12i 6032 . 2  |-  ( ( F  -h  ( S  .h  G ) ) 
.ih  ( F  -h  ( S  .h  G
) ) )  =  ( ( F  +h  ( -u S  .h  G
) )  .ih  ( F  +h  ( -u S  .h  G ) ) )
142negcli 9300 . . . 4  |-  -u S  e.  CC
1514, 3hvmulcli 22365 . . 3  |-  ( -u S  .h  G )  e.  ~H
161, 15hvaddcli 22369 . . 3  |-  ( F  +h  ( -u S  .h  G ) )  e. 
~H
17 ax-his2 22433 . . 3  |-  ( ( F  e.  ~H  /\  ( -u S  .h  G
)  e.  ~H  /\  ( F  +h  ( -u S  .h  G ) )  e.  ~H )  ->  ( ( F  +h  ( -u S  .h  G
) )  .ih  ( F  +h  ( -u S  .h  G ) ) )  =  ( ( F 
.ih  ( F  +h  ( -u S  .h  G
) ) )  +  ( ( -u S  .h  G )  .ih  ( F  +h  ( -u S  .h  G ) ) ) ) )
181, 15, 16, 17mp3an 1279 . 2  |-  ( ( F  +h  ( -u S  .h  G )
)  .ih  ( F  +h  ( -u S  .h  G ) ) )  =  ( ( F 
.ih  ( F  +h  ( -u S  .h  G
) ) )  +  ( ( -u S  .h  G )  .ih  ( F  +h  ( -u S  .h  G ) ) ) )
19 his7 22440 . . . . 5  |-  ( ( F  e.  ~H  /\  F  e.  ~H  /\  ( -u S  .h  G )  e.  ~H )  -> 
( F  .ih  ( F  +h  ( -u S  .h  G ) ) )  =  ( ( F 
.ih  F )  +  ( F  .ih  ( -u S  .h  G ) ) ) )
201, 1, 15, 19mp3an 1279 . . . 4  |-  ( F 
.ih  ( F  +h  ( -u S  .h  G
) ) )  =  ( ( F  .ih  F )  +  ( F 
.ih  ( -u S  .h  G ) ) )
21 his5 22436 . . . . . . 7  |-  ( (
-u S  e.  CC  /\  F  e.  ~H  /\  G  e.  ~H )  ->  ( F  .ih  ( -u S  .h  G ) )  =  ( ( * `  -u S
)  x.  ( F 
.ih  G ) ) )
2214, 1, 3, 21mp3an 1279 . . . . . 6  |-  ( F 
.ih  ( -u S  .h  G ) )  =  ( ( * `  -u S )  x.  ( F  .ih  G ) )
232cjnegi 11914 . . . . . . 7  |-  ( * `
 -u S )  = 
-u ( * `  S )
2423oveq1i 6030 . . . . . 6  |-  ( ( * `  -u S
)  x.  ( F 
.ih  G ) )  =  ( -u (
* `  S )  x.  ( F  .ih  G
) )
2522, 24eqtri 2407 . . . . 5  |-  ( F 
.ih  ( -u S  .h  G ) )  =  ( -u ( * `
 S )  x.  ( F  .ih  G
) )
2625oveq2i 6031 . . . 4  |-  ( ( F  .ih  F )  +  ( F  .ih  ( -u S  .h  G
) ) )  =  ( ( F  .ih  F )  +  ( -u ( * `  S
)  x.  ( F 
.ih  G ) ) )
2720, 26eqtri 2407 . . 3  |-  ( F 
.ih  ( F  +h  ( -u S  .h  G
) ) )  =  ( ( F  .ih  F )  +  ( -u ( * `  S
)  x.  ( F 
.ih  G ) ) )
28 ax-his3 22434 . . . . 5  |-  ( (
-u S  e.  CC  /\  G  e.  ~H  /\  ( F  +h  ( -u S  .h  G ) )  e.  ~H )  ->  ( ( -u S  .h  G )  .ih  ( F  +h  ( -u S  .h  G ) ) )  =  ( -u S  x.  ( G  .ih  ( F  +h  ( -u S  .h  G ) ) ) ) )
2914, 3, 16, 28mp3an 1279 . . . 4  |-  ( (
-u S  .h  G
)  .ih  ( F  +h  ( -u S  .h  G ) ) )  =  ( -u S  x.  ( G  .ih  ( F  +h  ( -u S  .h  G ) ) ) )
30 his7 22440 . . . . . . 7  |-  ( ( G  e.  ~H  /\  F  e.  ~H  /\  ( -u S  .h  G )  e.  ~H )  -> 
( G  .ih  ( F  +h  ( -u S  .h  G ) ) )  =  ( ( G 
.ih  F )  +  ( G  .ih  ( -u S  .h  G ) ) ) )
313, 1, 15, 30mp3an 1279 . . . . . 6  |-  ( G 
.ih  ( F  +h  ( -u S  .h  G
) ) )  =  ( ( G  .ih  F )  +  ( G 
.ih  ( -u S  .h  G ) ) )
32 his5 22436 . . . . . . . 8  |-  ( (
-u S  e.  CC  /\  G  e.  ~H  /\  G  e.  ~H )  ->  ( G  .ih  ( -u S  .h  G ) )  =  ( ( * `  -u S
)  x.  ( G 
.ih  G ) ) )
3314, 3, 3, 32mp3an 1279 . . . . . . 7  |-  ( G 
.ih  ( -u S  .h  G ) )  =  ( ( * `  -u S )  x.  ( G  .ih  G ) )
3433oveq2i 6031 . . . . . 6  |-  ( ( G  .ih  F )  +  ( G  .ih  ( -u S  .h  G
) ) )  =  ( ( G  .ih  F )  +  ( ( * `  -u S
)  x.  ( G 
.ih  G ) ) )
3531, 34eqtri 2407 . . . . 5  |-  ( G 
.ih  ( F  +h  ( -u S  .h  G
) ) )  =  ( ( G  .ih  F )  +  ( ( * `  -u S
)  x.  ( G 
.ih  G ) ) )
3635oveq2i 6031 . . . 4  |-  ( -u S  x.  ( G  .ih  ( F  +h  ( -u S  .h  G ) ) ) )  =  ( -u S  x.  ( ( G  .ih  F )  +  ( ( * `  -u S
)  x.  ( G 
.ih  G ) ) ) )
373, 1hicli 22431 . . . . . 6  |-  ( G 
.ih  F )  e.  CC
3814cjcli 11901 . . . . . . 7  |-  ( * `
 -u S )  e.  CC
393, 3hicli 22431 . . . . . . 7  |-  ( G 
.ih  G )  e.  CC
4038, 39mulcli 9028 . . . . . 6  |-  ( ( * `  -u S
)  x.  ( G 
.ih  G ) )  e.  CC
4114, 37, 40adddii 9033 . . . . 5  |-  ( -u S  x.  ( ( G  .ih  F )  +  ( ( * `  -u S )  x.  ( G  .ih  G ) ) ) )  =  ( ( -u S  x.  ( G  .ih  F ) )  +  ( -u S  x.  ( (
* `  -u S )  x.  ( G  .ih  G ) ) ) )
4214, 38, 39mulassi 9032 . . . . . . 7  |-  ( (
-u S  x.  (
* `  -u S ) )  x.  ( G 
.ih  G ) )  =  ( -u S  x.  ( ( * `  -u S )  x.  ( G  .ih  G ) ) )
4323oveq2i 6031 . . . . . . . . 9  |-  ( -u S  x.  ( * `  -u S ) )  =  ( -u S  x.  -u ( * `  S ) )
442cjcli 11901 . . . . . . . . . 10  |-  ( * `
 S )  e.  CC
452, 44mul2negi 9413 . . . . . . . . 9  |-  ( -u S  x.  -u ( * `
 S ) )  =  ( S  x.  ( * `  S
) )
4643, 45eqtri 2407 . . . . . . . 8  |-  ( -u S  x.  ( * `  -u S ) )  =  ( S  x.  ( * `  S
) )
4746oveq1i 6030 . . . . . . 7  |-  ( (
-u S  x.  (
* `  -u S ) )  x.  ( G 
.ih  G ) )  =  ( ( S  x.  ( * `  S ) )  x.  ( G  .ih  G
) )
4842, 47eqtr3i 2409 . . . . . 6  |-  ( -u S  x.  ( (
* `  -u S )  x.  ( G  .ih  G ) ) )  =  ( ( S  x.  ( * `  S
) )  x.  ( G  .ih  G ) )
4948oveq2i 6031 . . . . 5  |-  ( (
-u S  x.  ( G  .ih  F ) )  +  ( -u S  x.  ( ( * `  -u S )  x.  ( G  .ih  G ) ) ) )  =  ( ( -u S  x.  ( G  .ih  F ) )  +  ( ( S  x.  ( * `
 S ) )  x.  ( G  .ih  G ) ) )
5041, 49eqtri 2407 . . . 4  |-  ( -u S  x.  ( ( G  .ih  F )  +  ( ( * `  -u S )  x.  ( G  .ih  G ) ) ) )  =  ( ( -u S  x.  ( G  .ih  F ) )  +  ( ( S  x.  ( * `
 S ) )  x.  ( G  .ih  G ) ) )
5129, 36, 503eqtri 2411 . . 3  |-  ( (
-u S  .h  G
)  .ih  ( F  +h  ( -u S  .h  G ) ) )  =  ( ( -u S  x.  ( G  .ih  F ) )  +  ( ( S  x.  ( * `  S
) )  x.  ( G  .ih  G ) ) )
5227, 51oveq12i 6032 . 2  |-  ( ( F  .ih  ( F  +h  ( -u S  .h  G ) ) )  +  ( ( -u S  .h  G )  .ih  ( F  +h  ( -u S  .h  G ) ) ) )  =  ( ( ( F 
.ih  F )  +  ( -u ( * `
 S )  x.  ( F  .ih  G
) ) )  +  ( ( -u S  x.  ( G  .ih  F
) )  +  ( ( S  x.  (
* `  S )
)  x.  ( G 
.ih  G ) ) ) )
5313, 18, 523eqtri 2411 1  |-  ( ( F  -h  ( S  .h  G ) ) 
.ih  ( F  -h  ( S  .h  G
) ) )  =  ( ( ( F 
.ih  F )  +  ( -u ( * `
 S )  x.  ( F  .ih  G
) ) )  +  ( ( -u S  x.  ( G  .ih  F
) )  +  ( ( S  x.  (
* `  S )
)  x.  ( G 
.ih  G ) ) ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1649    e. wcel 1717   ` cfv 5394  (class class class)co 6020   CCcc 8921   1c1 8924    + caddc 8926    x. cmul 8928   -ucneg 9224   *ccj 11828   ~Hchil 22270    +h cva 22271    .h csm 22272    .ih csp 22273    -h cmv 22276
This theorem is referenced by:  normlem1  22460
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000  ax-hfvadd 22351  ax-hfvmul 22356  ax-hvmulass 22358  ax-hfi 22429  ax-his1 22432  ax-his2 22433  ax-his3 22434
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-po 4444  df-so 4445  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-riota 6485  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-div 9610  df-2 9990  df-cj 11831  df-re 11832  df-im 11833  df-hvsub 22322
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