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Theorem normlem0 21704
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 21610 . . . . 5  |-  ( S  .h  G )  e. 
~H
51, 4hvsubvali 21616 . . . 4  |-  ( F  -h  ( S  .h  G ) )  =  ( F  +h  ( -u 1  .h  ( S  .h  G ) ) )
62mulm1i 9240 . . . . . . 7  |-  ( -u
1  x.  S )  =  -u S
76oveq1i 5884 . . . . . 6  |-  ( (
-u 1  x.  S
)  .h  G )  =  ( -u S  .h  G )
8 neg1cn 9829 . . . . . . 7  |-  -u 1  e.  CC
98, 2, 3hvmulassi 21641 . . . . . 6  |-  ( (
-u 1  x.  S
)  .h  G )  =  ( -u 1  .h  ( S  .h  G
) )
107, 9eqtr3i 2318 . . . . 5  |-  ( -u S  .h  G )  =  ( -u 1  .h  ( S  .h  G
) )
1110oveq2i 5885 . . . 4  |-  ( F  +h  ( -u S  .h  G ) )  =  ( F  +h  ( -u 1  .h  ( S  .h  G ) ) )
125, 11eqtr4i 2319 . . 3  |-  ( F  -h  ( S  .h  G ) )  =  ( F  +h  ( -u S  .h  G ) )
1312, 12oveq12i 5886 . 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 9130 . . . 4  |-  -u S  e.  CC
1514, 3hvmulcli 21610 . . 3  |-  ( -u S  .h  G )  e.  ~H
161, 15hvaddcli 21614 . . 3  |-  ( F  +h  ( -u S  .h  G ) )  e. 
~H
17 ax-his2 21678 . . 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 1277 . 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 21685 . . . . 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 1277 . . . 4  |-  ( F 
.ih  ( F  +h  ( -u S  .h  G
) ) )  =  ( ( F  .ih  F )  +  ( F 
.ih  ( -u S  .h  G ) ) )
21 his5 21681 . . . . . . 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 1277 . . . . . 6  |-  ( F 
.ih  ( -u S  .h  G ) )  =  ( ( * `  -u S )  x.  ( F  .ih  G ) )
232cjnegi 11683 . . . . . . 7  |-  ( * `
 -u S )  = 
-u ( * `  S )
2423oveq1i 5884 . . . . . 6  |-  ( ( * `  -u S
)  x.  ( F 
.ih  G ) )  =  ( -u (
* `  S )  x.  ( F  .ih  G
) )
2522, 24eqtri 2316 . . . . 5  |-  ( F 
.ih  ( -u S  .h  G ) )  =  ( -u ( * `
 S )  x.  ( F  .ih  G
) )
2625oveq2i 5885 . . . 4  |-  ( ( F  .ih  F )  +  ( F  .ih  ( -u S  .h  G
) ) )  =  ( ( F  .ih  F )  +  ( -u ( * `  S
)  x.  ( F 
.ih  G ) ) )
2720, 26eqtri 2316 . . 3  |-  ( F 
.ih  ( F  +h  ( -u S  .h  G
) ) )  =  ( ( F  .ih  F )  +  ( -u ( * `  S
)  x.  ( F 
.ih  G ) ) )
28 ax-his3 21679 . . . . 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 1277 . . . 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 21685 . . . . . . 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 1277 . . . . . 6  |-  ( G 
.ih  ( F  +h  ( -u S  .h  G
) ) )  =  ( ( G  .ih  F )  +  ( G 
.ih  ( -u S  .h  G ) ) )
32 his5 21681 . . . . . . . 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 1277 . . . . . . 7  |-  ( G 
.ih  ( -u S  .h  G ) )  =  ( ( * `  -u S )  x.  ( G  .ih  G ) )
3433oveq2i 5885 . . . . . 6  |-  ( ( G  .ih  F )  +  ( G  .ih  ( -u S  .h  G
) ) )  =  ( ( G  .ih  F )  +  ( ( * `  -u S
)  x.  ( G 
.ih  G ) ) )
3531, 34eqtri 2316 . . . . 5  |-  ( G 
.ih  ( F  +h  ( -u S  .h  G
) ) )  =  ( ( G  .ih  F )  +  ( ( * `  -u S
)  x.  ( G 
.ih  G ) ) )
3635oveq2i 5885 . . . 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 21676 . . . . . 6  |-  ( G 
.ih  F )  e.  CC
3814cjcli 11670 . . . . . . 7  |-  ( * `
 -u S )  e.  CC
393, 3hicli 21676 . . . . . . 7  |-  ( G 
.ih  G )  e.  CC
4038, 39mulcli 8858 . . . . . 6  |-  ( ( * `  -u S
)  x.  ( G 
.ih  G ) )  e.  CC
4114, 37, 40adddii 8863 . . . . 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 8862 . . . . . . 7  |-  ( (
-u S  x.  (
* `  -u S ) )  x.  ( G 
.ih  G ) )  =  ( -u S  x.  ( ( * `  -u S )  x.  ( G  .ih  G ) ) )
4323oveq2i 5885 . . . . . . . . 9  |-  ( -u S  x.  ( * `  -u S ) )  =  ( -u S  x.  -u ( * `  S ) )
442cjcli 11670 . . . . . . . . . 10  |-  ( * `
 S )  e.  CC
452, 44mul2negi 9243 . . . . . . . . 9  |-  ( -u S  x.  -u ( * `
 S ) )  =  ( S  x.  ( * `  S
) )
4643, 45eqtri 2316 . . . . . . . 8  |-  ( -u S  x.  ( * `  -u S ) )  =  ( S  x.  ( * `  S
) )
4746oveq1i 5884 . . . . . . 7  |-  ( (
-u S  x.  (
* `  -u S ) )  x.  ( G 
.ih  G ) )  =  ( ( S  x.  ( * `  S ) )  x.  ( G  .ih  G
) )
4842, 47eqtr3i 2318 . . . . . 6  |-  ( -u S  x.  ( (
* `  -u S )  x.  ( G  .ih  G ) ) )  =  ( ( S  x.  ( * `  S
) )  x.  ( G  .ih  G ) )
4948oveq2i 5885 . . . . 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 2316 . . . 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 2320 . . 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 5886 . 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 2320 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 1632    e. wcel 1696   ` cfv 5271  (class class class)co 5874   CCcc 8751   1c1 8754    + caddc 8756    x. cmul 8758   -ucneg 9054   *ccj 11597   ~Hchil 21515    +h cva 21516    .h csm 21517    .ih csp 21518    -h cmv 21521
This theorem is referenced by:  normlem1  21705
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-hfvadd 21596  ax-hfvmul 21601  ax-hvmulass 21603  ax-hfi 21674  ax-his1 21677  ax-his2 21678  ax-his3 21679
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-2 9820  df-cj 11600  df-re 11601  df-im 11602  df-hvsub 21567
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