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Theorem normlem1 21689
Description: Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 22-Aug-1999.) (New usage is discouraged.)
Hypotheses
Ref Expression
normlem1.1  |-  S  e.  CC
normlem1.2  |-  F  e. 
~H
normlem1.3  |-  G  e. 
~H
normlem1.4  |-  R  e.  RR
normlem1.5  |-  ( abs `  S )  =  1
Assertion
Ref Expression
normlem1  |-  ( ( F  -h  ( ( S  x.  R )  .h  G ) ) 
.ih  ( F  -h  ( ( S  x.  R )  .h  G
) ) )  =  ( ( ( F 
.ih  F )  +  ( ( ( * `
 S )  x.  -u R )  x.  ( F  .ih  G ) ) )  +  ( ( ( S  x.  -u R
)  x.  ( G 
.ih  F ) )  +  ( ( R ^ 2 )  x.  ( G  .ih  G
) ) ) )

Proof of Theorem normlem1
StepHypRef Expression
1 normlem1.1 . . . 4  |-  S  e.  CC
2 normlem1.4 . . . . 5  |-  R  e.  RR
32recni 8849 . . . 4  |-  R  e.  CC
41, 3mulcli 8842 . . 3  |-  ( S  x.  R )  e.  CC
5 normlem1.2 . . 3  |-  F  e. 
~H
6 normlem1.3 . . 3  |-  G  e. 
~H
74, 5, 6normlem0 21688 . 2  |-  ( ( F  -h  ( ( S  x.  R )  .h  G ) ) 
.ih  ( F  -h  ( ( S  x.  R )  .h  G
) ) )  =  ( ( ( F 
.ih  F )  +  ( -u ( * `
 ( S  x.  R ) )  x.  ( F  .ih  G
) ) )  +  ( ( -u ( S  x.  R )  x.  ( G  .ih  F
) )  +  ( ( ( S  x.  R )  x.  (
* `  ( S  x.  R ) ) )  x.  ( G  .ih  G ) ) ) )
81, 3cjmuli 11674 . . . . . . . 8  |-  ( * `
 ( S  x.  R ) )  =  ( ( * `  S )  x.  (
* `  R )
)
93cjrebi 11659 . . . . . . . . . 10  |-  ( R  e.  RR  <->  ( * `  R )  =  R )
102, 9mpbi 199 . . . . . . . . 9  |-  ( * `
 R )  =  R
1110oveq2i 5869 . . . . . . . 8  |-  ( ( * `  S )  x.  ( * `  R ) )  =  ( ( * `  S )  x.  R
)
128, 11eqtri 2303 . . . . . . 7  |-  ( * `
 ( S  x.  R ) )  =  ( ( * `  S )  x.  R
)
1312negeqi 9045 . . . . . 6  |-  -u (
* `  ( S  x.  R ) )  = 
-u ( ( * `
 S )  x.  R )
141cjcli 11654 . . . . . . 7  |-  ( * `
 S )  e.  CC
1514, 3mulneg2i 9226 . . . . . 6  |-  ( ( * `  S )  x.  -u R )  = 
-u ( ( * `
 S )  x.  R )
1613, 15eqtr4i 2306 . . . . 5  |-  -u (
* `  ( S  x.  R ) )  =  ( ( * `  S )  x.  -u R
)
1716oveq1i 5868 . . . 4  |-  ( -u ( * `  ( S  x.  R )
)  x.  ( F 
.ih  G ) )  =  ( ( ( * `  S )  x.  -u R )  x.  ( F  .ih  G
) )
1817oveq2i 5869 . . 3  |-  ( ( F  .ih  F )  +  ( -u (
* `  ( S  x.  R ) )  x.  ( F  .ih  G
) ) )  =  ( ( F  .ih  F )  +  ( ( ( * `  S
)  x.  -u R
)  x.  ( F 
.ih  G ) ) )
191, 3mulneg2i 9226 . . . . . 6  |-  ( S  x.  -u R )  = 
-u ( S  x.  R )
2019eqcomi 2287 . . . . 5  |-  -u ( S  x.  R )  =  ( S  x.  -u R )
2120oveq1i 5868 . . . 4  |-  ( -u ( S  x.  R
)  x.  ( G 
.ih  F ) )  =  ( ( S  x.  -u R )  x.  ( G  .ih  F
) )
228oveq2i 5869 . . . . . . 7  |-  ( ( S  x.  R )  x.  ( * `  ( S  x.  R
) ) )  =  ( ( S  x.  R )  x.  (
( * `  S
)  x.  ( * `
 R ) ) )
233cjcli 11654 . . . . . . . . 9  |-  ( * `
 R )  e.  CC
241, 3, 14, 23mul4i 9009 . . . . . . . 8  |-  ( ( S  x.  R )  x.  ( ( * `
 S )  x.  ( * `  R
) ) )  =  ( ( S  x.  ( * `  S
) )  x.  ( R  x.  ( * `  R ) ) )
25 normlem1.5 . . . . . . . . . . . 12  |-  ( abs `  S )  =  1
2625oveq1i 5868 . . . . . . . . . . 11  |-  ( ( abs `  S ) ^ 2 )  =  ( 1 ^ 2 )
271absvalsqi 11876 . . . . . . . . . . 11  |-  ( ( abs `  S ) ^ 2 )  =  ( S  x.  (
* `  S )
)
28 sq1 11198 . . . . . . . . . . 11  |-  ( 1 ^ 2 )  =  1
2926, 27, 283eqtr3i 2311 . . . . . . . . . 10  |-  ( S  x.  ( * `  S ) )  =  1
3010oveq2i 5869 . . . . . . . . . 10  |-  ( R  x.  ( * `  R ) )  =  ( R  x.  R
)
3129, 30oveq12i 5870 . . . . . . . . 9  |-  ( ( S  x.  ( * `
 S ) )  x.  ( R  x.  ( * `  R
) ) )  =  ( 1  x.  ( R  x.  R )
)
323, 3mulcli 8842 . . . . . . . . . 10  |-  ( R  x.  R )  e.  CC
3332mulid2i 8840 . . . . . . . . 9  |-  ( 1  x.  ( R  x.  R ) )  =  ( R  x.  R
)
3431, 33eqtri 2303 . . . . . . . 8  |-  ( ( S  x.  ( * `
 S ) )  x.  ( R  x.  ( * `  R
) ) )  =  ( R  x.  R
)
3524, 34eqtri 2303 . . . . . . 7  |-  ( ( S  x.  R )  x.  ( ( * `
 S )  x.  ( * `  R
) ) )  =  ( R  x.  R
)
3622, 35eqtri 2303 . . . . . 6  |-  ( ( S  x.  R )  x.  ( * `  ( S  x.  R
) ) )  =  ( R  x.  R
)
373sqvali 11183 . . . . . 6  |-  ( R ^ 2 )  =  ( R  x.  R
)
3836, 37eqtr4i 2306 . . . . 5  |-  ( ( S  x.  R )  x.  ( * `  ( S  x.  R
) ) )  =  ( R ^ 2 )
3938oveq1i 5868 . . . 4  |-  ( ( ( S  x.  R
)  x.  ( * `
 ( S  x.  R ) ) )  x.  ( G  .ih  G ) )  =  ( ( R ^ 2 )  x.  ( G 
.ih  G ) )
4021, 39oveq12i 5870 . . 3  |-  ( (
-u ( S  x.  R )  x.  ( G  .ih  F ) )  +  ( ( ( S  x.  R )  x.  ( * `  ( S  x.  R
) ) )  x.  ( G  .ih  G
) ) )  =  ( ( ( S  x.  -u R )  x.  ( G  .ih  F
) )  +  ( ( R ^ 2 )  x.  ( G 
.ih  G ) ) )
4118, 40oveq12i 5870 . 2  |-  ( ( ( F  .ih  F
)  +  ( -u ( * `  ( S  x.  R )
)  x.  ( F 
.ih  G ) ) )  +  ( (
-u ( S  x.  R )  x.  ( G  .ih  F ) )  +  ( ( ( S  x.  R )  x.  ( * `  ( S  x.  R
) ) )  x.  ( G  .ih  G
) ) ) )  =  ( ( ( F  .ih  F )  +  ( ( ( * `  S )  x.  -u R )  x.  ( F  .ih  G
) ) )  +  ( ( ( S  x.  -u R )  x.  ( G  .ih  F
) )  +  ( ( R ^ 2 )  x.  ( G 
.ih  G ) ) ) )
427, 41eqtri 2303 1  |-  ( ( F  -h  ( ( S  x.  R )  .h  G ) ) 
.ih  ( F  -h  ( ( S  x.  R )  .h  G
) ) )  =  ( ( ( F 
.ih  F )  +  ( ( ( * `
 S )  x.  -u R )  x.  ( F  .ih  G ) ) )  +  ( ( ( S  x.  -u R
)  x.  ( G 
.ih  F ) )  +  ( ( R ^ 2 )  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   RRcr 8736   1c1 8738    + caddc 8740    x. cmul 8742   -ucneg 9038   2c2 9795   ^cexp 11104   *ccj 11581   abscabs 11719   ~Hchil 21499    .h csm 21501    .ih csp 21502    -h cmv 21505
This theorem is referenced by:  normlem4  21692
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-cnex 8793  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-pre-sup 8815  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-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  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-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  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-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-hvsub 21551
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