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Theorem normlem1 21681
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 8844 . . . 4  |-  R  e.  CC
41, 3mulcli 8837 . . 3  |-  ( S  x.  R )  e.  CC
5 normlem1.2 . . 3  |-  F  e. 
~H
6 normlem1.3 . . 3  |-  G  e. 
~H
74, 5, 6normlem0 21680 . 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 11668 . . . . . . . 8  |-  ( * `
 ( S  x.  R ) )  =  ( ( * `  S )  x.  (
* `  R )
)
93cjrebi 11653 . . . . . . . . . 10  |-  ( R  e.  RR  <->  ( * `  R )  =  R )
102, 9mpbi 201 . . . . . . . . 9  |-  ( * `
 R )  =  R
1110oveq2i 5830 . . . . . . . 8  |-  ( ( * `  S )  x.  ( * `  R ) )  =  ( ( * `  S )  x.  R
)
128, 11eqtri 2304 . . . . . . 7  |-  ( * `
 ( S  x.  R ) )  =  ( ( * `  S )  x.  R
)
1312negeqi 9040 . . . . . 6  |-  -u (
* `  ( S  x.  R ) )  = 
-u ( ( * `
 S )  x.  R )
141cjcli 11648 . . . . . . 7  |-  ( * `
 S )  e.  CC
1514, 3mulneg2i 9221 . . . . . 6  |-  ( ( * `  S )  x.  -u R )  = 
-u ( ( * `
 S )  x.  R )
1613, 15eqtr4i 2307 . . . . 5  |-  -u (
* `  ( S  x.  R ) )  =  ( ( * `  S )  x.  -u R
)
1716oveq1i 5829 . . . 4  |-  ( -u ( * `  ( S  x.  R )
)  x.  ( F 
.ih  G ) )  =  ( ( ( * `  S )  x.  -u R )  x.  ( F  .ih  G
) )
1817oveq2i 5830 . . 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 9221 . . . . . 6  |-  ( S  x.  -u R )  = 
-u ( S  x.  R )
2019eqcomi 2288 . . . . 5  |-  -u ( S  x.  R )  =  ( S  x.  -u R )
2120oveq1i 5829 . . . 4  |-  ( -u ( S  x.  R
)  x.  ( G 
.ih  F ) )  =  ( ( S  x.  -u R )  x.  ( G  .ih  F
) )
228oveq2i 5830 . . . . . . 7  |-  ( ( S  x.  R )  x.  ( * `  ( S  x.  R
) ) )  =  ( ( S  x.  R )  x.  (
( * `  S
)  x.  ( * `
 R ) ) )
233cjcli 11648 . . . . . . . . 9  |-  ( * `
 R )  e.  CC
241, 3, 14, 23mul4i 9004 . . . . . . . 8  |-  ( ( S  x.  R )  x.  ( ( * `
 S )  x.  ( * `  R
) ) )  =  ( ( S  x.  ( * `  S
) )  x.  ( R  x.  ( * `  R ) ) )
25 normlem1.5 . . . . . . . . . . . 12  |-  ( abs `  S )  =  1
2625oveq1i 5829 . . . . . . . . . . 11  |-  ( ( abs `  S ) ^ 2 )  =  ( 1 ^ 2 )
271absvalsqi 11870 . . . . . . . . . . 11  |-  ( ( abs `  S ) ^ 2 )  =  ( S  x.  (
* `  S )
)
28 sq1 11192 . . . . . . . . . . 11  |-  ( 1 ^ 2 )  =  1
2926, 27, 283eqtr3i 2312 . . . . . . . . . 10  |-  ( S  x.  ( * `  S ) )  =  1
3010oveq2i 5830 . . . . . . . . . 10  |-  ( R  x.  ( * `  R ) )  =  ( R  x.  R
)
3129, 30oveq12i 5831 . . . . . . . . 9  |-  ( ( S  x.  ( * `
 S ) )  x.  ( R  x.  ( * `  R
) ) )  =  ( 1  x.  ( R  x.  R )
)
323, 3mulcli 8837 . . . . . . . . . 10  |-  ( R  x.  R )  e.  CC
3332mulid2i 8835 . . . . . . . . 9  |-  ( 1  x.  ( R  x.  R ) )  =  ( R  x.  R
)
3431, 33eqtri 2304 . . . . . . . 8  |-  ( ( S  x.  ( * `
 S ) )  x.  ( R  x.  ( * `  R
) ) )  =  ( R  x.  R
)
3524, 34eqtri 2304 . . . . . . 7  |-  ( ( S  x.  R )  x.  ( ( * `
 S )  x.  ( * `  R
) ) )  =  ( R  x.  R
)
3622, 35eqtri 2304 . . . . . 6  |-  ( ( S  x.  R )  x.  ( * `  ( S  x.  R
) ) )  =  ( R  x.  R
)
373sqvali 11177 . . . . . 6  |-  ( R ^ 2 )  =  ( R  x.  R
)
3836, 37eqtr4i 2307 . . . . 5  |-  ( ( S  x.  R )  x.  ( * `  ( S  x.  R
) ) )  =  ( R ^ 2 )
3938oveq1i 5829 . . . 4  |-  ( ( ( S  x.  R
)  x.  ( * `
 ( S  x.  R ) ) )  x.  ( G  .ih  G ) )  =  ( ( R ^ 2 )  x.  ( G 
.ih  G ) )
4021, 39oveq12i 5831 . . 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 5831 . 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 2304 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 1624    e. wcel 1685   ` cfv 5221  (class class class)co 5819   CCcc 8730   RRcr 8731   1c1 8733    + caddc 8735    x. cmul 8737   -ucneg 9033   2c2 9790   ^cexp 11098   *ccj 11575   abscabs 11713   ~Hchil 21491    .h csm 21493    .ih csp 21494    -h cmv 21497
This theorem is referenced by:  normlem4  21684
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-cnex 8788  ax-resscn 8789  ax-1cn 8790  ax-icn 8791  ax-addcl 8792  ax-addrcl 8793  ax-mulcl 8794  ax-mulrcl 8795  ax-mulcom 8796  ax-addass 8797  ax-mulass 8798  ax-distr 8799  ax-i2m1 8800  ax-1ne0 8801  ax-1rid 8802  ax-rnegex 8803  ax-rrecex 8804  ax-cnre 8805  ax-pre-lttri 8806  ax-pre-lttrn 8807  ax-pre-ltadd 8808  ax-pre-mulgt0 8809  ax-pre-sup 8810  ax-hfvadd 21572  ax-hfvmul 21577  ax-hvmulass 21579  ax-hfi 21650  ax-his1 21653  ax-his2 21654  ax-his3 21655
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-2nd 6084  df-iota 6252  df-riota 6299  df-recs 6383  df-rdg 6418  df-er 6655  df-en 6859  df-dom 6860  df-sdom 6861  df-sup 7189  df-pnf 8864  df-mnf 8865  df-xr 8866  df-ltxr 8867  df-le 8868  df-sub 9034  df-neg 9035  df-div 9419  df-nn 9742  df-2 9799  df-3 9800  df-n0 9961  df-z 10020  df-uz 10226  df-rp 10350  df-seq 11041  df-exp 11099  df-cj 11578  df-re 11579  df-im 11580  df-sqr 11714  df-abs 11715  df-hvsub 21543
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