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Theorem normlem0 21688
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 21594 . . . . 5  |-  ( S  .h  G )  e. 
~H
51, 4hvsubvali 21600 . . . 4  |-  ( F  -h  ( S  .h  G ) )  =  ( F  +h  ( -u 1  .h  ( S  .h  G ) ) )
62mulm1i 9224 . . . . . . 7  |-  ( -u
1  x.  S )  =  -u S
76oveq1i 5868 . . . . . 6  |-  ( (
-u 1  x.  S
)  .h  G )  =  ( -u S  .h  G )
8 neg1cn 9813 . . . . . . 7  |-  -u 1  e.  CC
98, 2, 3hvmulassi 21625 . . . . . 6  |-  ( (
-u 1  x.  S
)  .h  G )  =  ( -u 1  .h  ( S  .h  G
) )
107, 9eqtr3i 2305 . . . . 5  |-  ( -u S  .h  G )  =  ( -u 1  .h  ( S  .h  G
) )
1110oveq2i 5869 . . . 4  |-  ( F  +h  ( -u S  .h  G ) )  =  ( F  +h  ( -u 1  .h  ( S  .h  G ) ) )
125, 11eqtr4i 2306 . . 3  |-  ( F  -h  ( S  .h  G ) )  =  ( F  +h  ( -u S  .h  G ) )
1312, 12oveq12i 5870 . 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 9114 . . . 4  |-  -u S  e.  CC
1514, 3hvmulcli 21594 . . 3  |-  ( -u S  .h  G )  e.  ~H
161, 15hvaddcli 21598 . . 3  |-  ( F  +h  ( -u S  .h  G ) )  e. 
~H
17 ax-his2 21662 . . 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 21669 . . . . 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 21665 . . . . . . 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 11667 . . . . . . 7  |-  ( * `
 -u S )  = 
-u ( * `  S )
2423oveq1i 5868 . . . . . 6  |-  ( ( * `  -u S
)  x.  ( F 
.ih  G ) )  =  ( -u (
* `  S )  x.  ( F  .ih  G
) )
2522, 24eqtri 2303 . . . . 5  |-  ( F 
.ih  ( -u S  .h  G ) )  =  ( -u ( * `
 S )  x.  ( F  .ih  G
) )
2625oveq2i 5869 . . . 4  |-  ( ( F  .ih  F )  +  ( F  .ih  ( -u S  .h  G
) ) )  =  ( ( F  .ih  F )  +  ( -u ( * `  S
)  x.  ( F 
.ih  G ) ) )
2720, 26eqtri 2303 . . 3  |-  ( F 
.ih  ( F  +h  ( -u S  .h  G
) ) )  =  ( ( F  .ih  F )  +  ( -u ( * `  S
)  x.  ( F 
.ih  G ) ) )
28 ax-his3 21663 . . . . 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 21669 . . . . . . 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 21665 . . . . . . . 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 5869 . . . . . 6  |-  ( ( G  .ih  F )  +  ( G  .ih  ( -u S  .h  G
) ) )  =  ( ( G  .ih  F )  +  ( ( * `  -u S
)  x.  ( G 
.ih  G ) ) )
3531, 34eqtri 2303 . . . . 5  |-  ( G 
.ih  ( F  +h  ( -u S  .h  G
) ) )  =  ( ( G  .ih  F )  +  ( ( * `  -u S
)  x.  ( G 
.ih  G ) ) )
3635oveq2i 5869 . . . 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 21660 . . . . . 6  |-  ( G 
.ih  F )  e.  CC
3814cjcli 11654 . . . . . . 7  |-  ( * `
 -u S )  e.  CC
393, 3hicli 21660 . . . . . . 7  |-  ( G 
.ih  G )  e.  CC
4038, 39mulcli 8842 . . . . . 6  |-  ( ( * `  -u S
)  x.  ( G 
.ih  G ) )  e.  CC
4114, 37, 40adddii 8847 . . . . 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 8846 . . . . . . 7  |-  ( (
-u S  x.  (
* `  -u S ) )  x.  ( G 
.ih  G ) )  =  ( -u S  x.  ( ( * `  -u S )  x.  ( G  .ih  G ) ) )
4323oveq2i 5869 . . . . . . . . 9  |-  ( -u S  x.  ( * `  -u S ) )  =  ( -u S  x.  -u ( * `  S ) )
442cjcli 11654 . . . . . . . . . 10  |-  ( * `
 S )  e.  CC
452, 44mul2negi 9227 . . . . . . . . 9  |-  ( -u S  x.  -u ( * `
 S ) )  =  ( S  x.  ( * `  S
) )
4643, 45eqtri 2303 . . . . . . . 8  |-  ( -u S  x.  ( * `  -u S ) )  =  ( S  x.  ( * `  S
) )
4746oveq1i 5868 . . . . . . 7  |-  ( (
-u S  x.  (
* `  -u S ) )  x.  ( G 
.ih  G ) )  =  ( ( S  x.  ( * `  S ) )  x.  ( G  .ih  G
) )
4842, 47eqtr3i 2305 . . . . . 6  |-  ( -u S  x.  ( (
* `  -u S )  x.  ( G  .ih  G ) ) )  =  ( ( S  x.  ( * `  S
) )  x.  ( G  .ih  G ) )
4948oveq2i 5869 . . . . 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 2303 . . . 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 2307 . . 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 5870 . 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 2307 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 1623    e. wcel 1684   ` cfv 5255  (class class class)co 5858   CCcc 8735   1c1 8738    + caddc 8740    x. cmul 8742   -ucneg 9038   *ccj 11581   ~Hchil 21499    +h cva 21500    .h csm 21501    .ih csp 21502    -h cmv 21505
This theorem is referenced by:  normlem1  21689
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-hfvadd 21580  ax-hfvmul 21585  ax-hvmulass 21587  ax-hfi 21658  ax-his1 21661  ax-his2 21662  ax-his3 21663
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-2 9804  df-cj 11584  df-re 11585  df-im 11586  df-hvsub 21551
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