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Theorem normlem0 21690
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 21596 . . . . 5  |-  ( S  .h  G )  e. 
~H
51, 4hvsubvali 21602 . . . 4  |-  ( F  -h  ( S  .h  G ) )  =  ( F  +h  ( -u 1  .h  ( S  .h  G ) ) )
62mulm1i 9226 . . . . . . 7  |-  ( -u
1  x.  S )  =  -u S
76oveq1i 5870 . . . . . 6  |-  ( (
-u 1  x.  S
)  .h  G )  =  ( -u S  .h  G )
8 neg1cn 9815 . . . . . . 7  |-  -u 1  e.  CC
98, 2, 3hvmulassi 21627 . . . . . 6  |-  ( (
-u 1  x.  S
)  .h  G )  =  ( -u 1  .h  ( S  .h  G
) )
107, 9eqtr3i 2307 . . . . 5  |-  ( -u S  .h  G )  =  ( -u 1  .h  ( S  .h  G
) )
1110oveq2i 5871 . . . 4  |-  ( F  +h  ( -u S  .h  G ) )  =  ( F  +h  ( -u 1  .h  ( S  .h  G ) ) )
125, 11eqtr4i 2308 . . 3  |-  ( F  -h  ( S  .h  G ) )  =  ( F  +h  ( -u S  .h  G ) )
1312, 12oveq12i 5872 . 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 9116 . . . 4  |-  -u S  e.  CC
1514, 3hvmulcli 21596 . . 3  |-  ( -u S  .h  G )  e.  ~H
161, 15hvaddcli 21600 . . 3  |-  ( F  +h  ( -u S  .h  G ) )  e. 
~H
17 ax-his2 21664 . . 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 21671 . . . . 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 21667 . . . . . . 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 11669 . . . . . . 7  |-  ( * `
 -u S )  = 
-u ( * `  S )
2423oveq1i 5870 . . . . . 6  |-  ( ( * `  -u S
)  x.  ( F 
.ih  G ) )  =  ( -u (
* `  S )  x.  ( F  .ih  G
) )
2522, 24eqtri 2305 . . . . 5  |-  ( F 
.ih  ( -u S  .h  G ) )  =  ( -u ( * `
 S )  x.  ( F  .ih  G
) )
2625oveq2i 5871 . . . 4  |-  ( ( F  .ih  F )  +  ( F  .ih  ( -u S  .h  G
) ) )  =  ( ( F  .ih  F )  +  ( -u ( * `  S
)  x.  ( F 
.ih  G ) ) )
2720, 26eqtri 2305 . . 3  |-  ( F 
.ih  ( F  +h  ( -u S  .h  G
) ) )  =  ( ( F  .ih  F )  +  ( -u ( * `  S
)  x.  ( F 
.ih  G ) ) )
28 ax-his3 21665 . . . . 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 21671 . . . . . . 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 21667 . . . . . . . 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 5871 . . . . . 6  |-  ( ( G  .ih  F )  +  ( G  .ih  ( -u S  .h  G
) ) )  =  ( ( G  .ih  F )  +  ( ( * `  -u S
)  x.  ( G 
.ih  G ) ) )
3531, 34eqtri 2305 . . . . 5  |-  ( G 
.ih  ( F  +h  ( -u S  .h  G
) ) )  =  ( ( G  .ih  F )  +  ( ( * `  -u S
)  x.  ( G 
.ih  G ) ) )
3635oveq2i 5871 . . . 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 21662 . . . . . 6  |-  ( G 
.ih  F )  e.  CC
3814cjcli 11656 . . . . . . 7  |-  ( * `
 -u S )  e.  CC
393, 3hicli 21662 . . . . . . 7  |-  ( G 
.ih  G )  e.  CC
4038, 39mulcli 8844 . . . . . 6  |-  ( ( * `  -u S
)  x.  ( G 
.ih  G ) )  e.  CC
4114, 37, 40adddii 8849 . . . . 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 8848 . . . . . . 7  |-  ( (
-u S  x.  (
* `  -u S ) )  x.  ( G 
.ih  G ) )  =  ( -u S  x.  ( ( * `  -u S )  x.  ( G  .ih  G ) ) )
4323oveq2i 5871 . . . . . . . . 9  |-  ( -u S  x.  ( * `  -u S ) )  =  ( -u S  x.  -u ( * `  S ) )
442cjcli 11656 . . . . . . . . . 10  |-  ( * `
 S )  e.  CC
452, 44mul2negi 9229 . . . . . . . . 9  |-  ( -u S  x.  -u ( * `
 S ) )  =  ( S  x.  ( * `  S
) )
4643, 45eqtri 2305 . . . . . . . 8  |-  ( -u S  x.  ( * `  -u S ) )  =  ( S  x.  ( * `  S
) )
4746oveq1i 5870 . . . . . . 7  |-  ( (
-u S  x.  (
* `  -u S ) )  x.  ( G 
.ih  G ) )  =  ( ( S  x.  ( * `  S ) )  x.  ( G  .ih  G
) )
4842, 47eqtr3i 2307 . . . . . 6  |-  ( -u S  x.  ( (
* `  -u S )  x.  ( G  .ih  G ) ) )  =  ( ( S  x.  ( * `  S
) )  x.  ( G  .ih  G ) )
4948oveq2i 5871 . . . . 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 2305 . . . 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 2309 . . 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 5872 . 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 2309 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 1625    e. wcel 1686   ` cfv 5257  (class class class)co 5860   CCcc 8737   1c1 8740    + caddc 8742    x. cmul 8744   -ucneg 9040   *ccj 11583   ~Hchil 21501    +h cva 21502    .h csm 21503    .ih csp 21504    -h cmv 21507
This theorem is referenced by:  normlem1  21691
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-resscn 8796  ax-1cn 8797  ax-icn 8798  ax-addcl 8799  ax-addrcl 8800  ax-mulcl 8801  ax-mulrcl 8802  ax-mulcom 8803  ax-addass 8804  ax-mulass 8805  ax-distr 8806  ax-i2m1 8807  ax-1ne0 8808  ax-1rid 8809  ax-rnegex 8810  ax-rrecex 8811  ax-cnre 8812  ax-pre-lttri 8813  ax-pre-lttrn 8814  ax-pre-ltadd 8815  ax-pre-mulgt0 8816  ax-hfvadd 21582  ax-hfvmul 21587  ax-hvmulass 21589  ax-hfi 21660  ax-his1 21663  ax-his2 21664  ax-his3 21665
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 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4311  df-po 4316  df-so 4317  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-riota 6306  df-er 6662  df-en 6866  df-dom 6867  df-sdom 6868  df-pnf 8871  df-mnf 8872  df-xr 8873  df-ltxr 8874  df-le 8875  df-sub 9041  df-neg 9042  df-div 9426  df-2 9806  df-cj 11586  df-re 11587  df-im 11588  df-hvsub 21553
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