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Theorem normlem4 21685
Description: Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 29-Jul-1999.) (New usage is discouraged.)
Hypotheses
Ref Expression
normlem1.1  |-  S  e.  CC
normlem1.2  |-  F  e. 
~H
normlem1.3  |-  G  e. 
~H
normlem2.4  |-  B  = 
-u ( ( ( * `  S )  x.  ( F  .ih  G ) )  +  ( S  x.  ( G 
.ih  F ) ) )
normlem3.5  |-  A  =  ( G  .ih  G
)
normlem3.6  |-  C  =  ( F  .ih  F
)
normlem4.7  |-  R  e.  RR
normlem4.8  |-  ( abs `  S )  =  1
Assertion
Ref Expression
normlem4  |-  ( ( F  -h  ( ( S  x.  R )  .h  G ) ) 
.ih  ( F  -h  ( ( S  x.  R )  .h  G
) ) )  =  ( ( ( A  x.  ( R ^
2 ) )  +  ( B  x.  R
) )  +  C
)

Proof of Theorem normlem4
StepHypRef Expression
1 normlem1.1 . . 3  |-  S  e.  CC
2 normlem1.2 . . 3  |-  F  e. 
~H
3 normlem1.3 . . 3  |-  G  e. 
~H
4 normlem4.7 . . 3  |-  R  e.  RR
5 normlem4.8 . . 3  |-  ( abs `  S )  =  1
61, 2, 3, 4, 5normlem1 21682 . 2  |-  ( ( 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
) ) ) )
7 normlem2.4 . . 3  |-  B  = 
-u ( ( ( * `  S )  x.  ( F  .ih  G ) )  +  ( S  x.  ( G 
.ih  F ) ) )
8 normlem3.5 . . 3  |-  A  =  ( G  .ih  G
)
9 normlem3.6 . . 3  |-  C  =  ( F  .ih  F
)
101, 2, 3, 7, 8, 9, 4normlem3 21684 . 2  |-  ( ( ( A  x.  ( R ^ 2 ) )  +  ( B  x.  R ) )  +  C )  =  ( ( ( 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
) ) ) )
116, 10eqtr4i 2308 1  |-  ( ( F  -h  ( ( S  x.  R )  .h  G ) ) 
.ih  ( F  -h  ( ( S  x.  R )  .h  G
) ) )  =  ( ( ( A  x.  ( R ^
2 ) )  +  ( B  x.  R
) )  +  C
)
Colors of variables: wff set class
Syntax hints:    = wceq 1624    e. wcel 1685   ` cfv 5222  (class class class)co 5820   CCcc 8731   RRcr 8732   1c1 8734    + caddc 8736    x. cmul 8738   -ucneg 9034   2c2 9791   ^cexp 11099   *ccj 11576   abscabs 11714   ~Hchil 21492    .h csm 21494    .ih csp 21495    -h cmv 21498
This theorem is referenced by:  normlem5  21686
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 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8789  ax-resscn 8790  ax-1cn 8791  ax-icn 8792  ax-addcl 8793  ax-addrcl 8794  ax-mulcl 8795  ax-mulrcl 8796  ax-mulcom 8797  ax-addass 8798  ax-mulass 8799  ax-distr 8800  ax-i2m1 8801  ax-1ne0 8802  ax-1rid 8803  ax-rnegex 8804  ax-rrecex 8805  ax-cnre 8806  ax-pre-lttri 8807  ax-pre-lttrn 8808  ax-pre-ltadd 8809  ax-pre-mulgt0 8810  ax-pre-sup 8811  ax-hfvadd 21573  ax-hfvmul 21578  ax-hvmulass 21580  ax-hfi 21651  ax-his1 21654  ax-his2 21655  ax-his3 21656
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 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-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  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-fun 5224  df-fn 5225  df-f 5226  df-f1 5227  df-fo 5228  df-f1o 5229  df-fv 5230  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-2nd 6085  df-iota 6253  df-riota 6300  df-recs 6384  df-rdg 6419  df-er 6656  df-en 6860  df-dom 6861  df-sdom 6862  df-sup 7190  df-pnf 8865  df-mnf 8866  df-xr 8867  df-ltxr 8868  df-le 8869  df-sub 9035  df-neg 9036  df-div 9420  df-nn 9743  df-2 9800  df-3 9801  df-n0 9962  df-z 10021  df-uz 10227  df-rp 10351  df-seq 11042  df-exp 11100  df-cj 11579  df-re 11580  df-im 11581  df-sqr 11715  df-abs 11716  df-hvsub 21544
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