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Theorem normlem0 22564
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 22470 . . . . 5  |-  ( S  .h  G )  e. 
~H
51, 4hvsubvali 22476 . . . 4  |-  ( F  -h  ( S  .h  G ) )  =  ( F  +h  ( -u 1  .h  ( S  .h  G ) ) )
62mulm1i 9434 . . . . . . 7  |-  ( -u
1  x.  S )  =  -u S
76oveq1i 6050 . . . . . 6  |-  ( (
-u 1  x.  S
)  .h  G )  =  ( -u S  .h  G )
8 neg1cn 10023 . . . . . . 7  |-  -u 1  e.  CC
98, 2, 3hvmulassi 22501 . . . . . 6  |-  ( (
-u 1  x.  S
)  .h  G )  =  ( -u 1  .h  ( S  .h  G
) )
107, 9eqtr3i 2426 . . . . 5  |-  ( -u S  .h  G )  =  ( -u 1  .h  ( S  .h  G
) )
1110oveq2i 6051 . . . 4  |-  ( F  +h  ( -u S  .h  G ) )  =  ( F  +h  ( -u 1  .h  ( S  .h  G ) ) )
125, 11eqtr4i 2427 . . 3  |-  ( F  -h  ( S  .h  G ) )  =  ( F  +h  ( -u S  .h  G ) )
1312, 12oveq12i 6052 . 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 9324 . . . 4  |-  -u S  e.  CC
1514, 3hvmulcli 22470 . . 3  |-  ( -u S  .h  G )  e.  ~H
161, 15hvaddcli 22474 . . 3  |-  ( F  +h  ( -u S  .h  G ) )  e. 
~H
17 ax-his2 22538 . . 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 22545 . . . . 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 22541 . . . . . . 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 11942 . . . . . . 7  |-  ( * `
 -u S )  = 
-u ( * `  S )
2423oveq1i 6050 . . . . . 6  |-  ( ( * `  -u S
)  x.  ( F 
.ih  G ) )  =  ( -u (
* `  S )  x.  ( F  .ih  G
) )
2522, 24eqtri 2424 . . . . 5  |-  ( F 
.ih  ( -u S  .h  G ) )  =  ( -u ( * `
 S )  x.  ( F  .ih  G
) )
2625oveq2i 6051 . . . 4  |-  ( ( F  .ih  F )  +  ( F  .ih  ( -u S  .h  G
) ) )  =  ( ( F  .ih  F )  +  ( -u ( * `  S
)  x.  ( F 
.ih  G ) ) )
2720, 26eqtri 2424 . . 3  |-  ( F 
.ih  ( F  +h  ( -u S  .h  G
) ) )  =  ( ( F  .ih  F )  +  ( -u ( * `  S
)  x.  ( F 
.ih  G ) ) )
28 ax-his3 22539 . . . . 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 22545 . . . . . . 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 22541 . . . . . . . 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 6051 . . . . . 6  |-  ( ( G  .ih  F )  +  ( G  .ih  ( -u S  .h  G
) ) )  =  ( ( G  .ih  F )  +  ( ( * `  -u S
)  x.  ( G 
.ih  G ) ) )
3531, 34eqtri 2424 . . . . 5  |-  ( G 
.ih  ( F  +h  ( -u S  .h  G
) ) )  =  ( ( G  .ih  F )  +  ( ( * `  -u S
)  x.  ( G 
.ih  G ) ) )
3635oveq2i 6051 . . . 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 22536 . . . . . 6  |-  ( G 
.ih  F )  e.  CC
3814cjcli 11929 . . . . . . 7  |-  ( * `
 -u S )  e.  CC
393, 3hicli 22536 . . . . . . 7  |-  ( G 
.ih  G )  e.  CC
4038, 39mulcli 9051 . . . . . 6  |-  ( ( * `  -u S
)  x.  ( G 
.ih  G ) )  e.  CC
4114, 37, 40adddii 9056 . . . . 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 9055 . . . . . . 7  |-  ( (
-u S  x.  (
* `  -u S ) )  x.  ( G 
.ih  G ) )  =  ( -u S  x.  ( ( * `  -u S )  x.  ( G  .ih  G ) ) )
4323oveq2i 6051 . . . . . . . . 9  |-  ( -u S  x.  ( * `  -u S ) )  =  ( -u S  x.  -u ( * `  S ) )
442cjcli 11929 . . . . . . . . . 10  |-  ( * `
 S )  e.  CC
452, 44mul2negi 9437 . . . . . . . . 9  |-  ( -u S  x.  -u ( * `
 S ) )  =  ( S  x.  ( * `  S
) )
4643, 45eqtri 2424 . . . . . . . 8  |-  ( -u S  x.  ( * `  -u S ) )  =  ( S  x.  ( * `  S
) )
4746oveq1i 6050 . . . . . . 7  |-  ( (
-u S  x.  (
* `  -u S ) )  x.  ( G 
.ih  G ) )  =  ( ( S  x.  ( * `  S ) )  x.  ( G  .ih  G
) )
4842, 47eqtr3i 2426 . . . . . 6  |-  ( -u S  x.  ( (
* `  -u S )  x.  ( G  .ih  G ) ) )  =  ( ( S  x.  ( * `  S
) )  x.  ( G  .ih  G ) )
4948oveq2i 6051 . . . . 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 2424 . . . 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 2428 . . 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 6052 . 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 2428 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 1721   ` cfv 5413  (class class class)co 6040   CCcc 8944   1c1 8947    + caddc 8949    x. cmul 8951   -ucneg 9248   *ccj 11856   ~Hchil 22375    +h cva 22376    .h csm 22377    .ih csp 22378    -h cmv 22381
This theorem is referenced by:  normlem1  22565
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-hfvadd 22456  ax-hfvmul 22461  ax-hvmulass 22463  ax-hfi 22534  ax-his1 22537  ax-his2 22538  ax-his3 22539
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-po 4463  df-so 4464  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-riota 6508  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-2 10014  df-cj 11859  df-re 11860  df-im 11861  df-hvsub 22427
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