HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  normlem3 Unicode version

Theorem normlem3 21707
Description: Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 21-Aug-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
)
normlem3.7  |-  R  e.  RR
Assertion
Ref Expression
normlem3  |-  ( ( ( 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
) ) ) )

Proof of Theorem normlem3
StepHypRef Expression
1 normlem3.6 . . 3  |-  C  =  ( F  .ih  F
)
2 normlem3.5 . . . . . . 7  |-  A  =  ( G  .ih  G
)
3 normlem1.3 . . . . . . . 8  |-  G  e. 
~H
43, 3hicli 21676 . . . . . . 7  |-  ( G 
.ih  G )  e.  CC
52, 4eqeltri 2366 . . . . . 6  |-  A  e.  CC
6 normlem3.7 . . . . . . . 8  |-  R  e.  RR
76recni 8865 . . . . . . 7  |-  R  e.  CC
87sqcli 11200 . . . . . 6  |-  ( R ^ 2 )  e.  CC
95, 8mulcli 8858 . . . . 5  |-  ( A  x.  ( R ^
2 ) )  e.  CC
10 normlem1.1 . . . . . . . 8  |-  S  e.  CC
11 normlem1.2 . . . . . . . 8  |-  F  e. 
~H
12 normlem2.4 . . . . . . . 8  |-  B  = 
-u ( ( ( * `  S )  x.  ( F  .ih  G ) )  +  ( S  x.  ( G 
.ih  F ) ) )
1310, 11, 3, 12normlem2 21706 . . . . . . 7  |-  B  e.  RR
1413recni 8865 . . . . . 6  |-  B  e.  CC
1514, 7mulcli 8858 . . . . 5  |-  ( B  x.  R )  e.  CC
169, 15addcomi 9019 . . . 4  |-  ( ( A  x.  ( R ^ 2 ) )  +  ( B  x.  R ) )  =  ( ( B  x.  R )  +  ( A  x.  ( R ^ 2 ) ) )
1710cjcli 11670 . . . . . . . . . 10  |-  ( * `
 S )  e.  CC
1811, 3hicli 21676 . . . . . . . . . 10  |-  ( F 
.ih  G )  e.  CC
1917, 18mulcli 8858 . . . . . . . . 9  |-  ( ( * `  S )  x.  ( F  .ih  G ) )  e.  CC
203, 11hicli 21676 . . . . . . . . . 10  |-  ( G 
.ih  F )  e.  CC
2110, 20mulcli 8858 . . . . . . . . 9  |-  ( S  x.  ( G  .ih  F ) )  e.  CC
2219, 21addcli 8857 . . . . . . . 8  |-  ( ( ( * `  S
)  x.  ( F 
.ih  G ) )  +  ( S  x.  ( G  .ih  F ) ) )  e.  CC
2322, 7mulneg1i 9241 . . . . . . 7  |-  ( -u ( ( ( * `
 S )  x.  ( F  .ih  G
) )  +  ( S  x.  ( G 
.ih  F ) ) )  x.  R )  =  -u ( ( ( ( * `  S
)  x.  ( F 
.ih  G ) )  +  ( S  x.  ( G  .ih  F ) ) )  x.  R
)
2412oveq1i 5884 . . . . . . 7  |-  ( B  x.  R )  =  ( -u ( ( ( * `  S
)  x.  ( F 
.ih  G ) )  +  ( S  x.  ( G  .ih  F ) ) )  x.  R
)
2522, 7mulneg2i 9242 . . . . . . 7  |-  ( ( ( ( * `  S )  x.  ( F  .ih  G ) )  +  ( S  x.  ( G  .ih  F ) ) )  x.  -u R
)  =  -u (
( ( ( * `
 S )  x.  ( F  .ih  G
) )  +  ( S  x.  ( G 
.ih  F ) ) )  x.  R )
2623, 24, 253eqtr4i 2326 . . . . . 6  |-  ( B  x.  R )  =  ( ( ( ( * `  S )  x.  ( F  .ih  G ) )  +  ( S  x.  ( G 
.ih  F ) ) )  x.  -u R
)
277negcli 9130 . . . . . . 7  |-  -u R  e.  CC
2819, 21, 27adddiri 8864 . . . . . 6  |-  ( ( ( ( * `  S )  x.  ( F  .ih  G ) )  +  ( S  x.  ( G  .ih  F ) ) )  x.  -u R
)  =  ( ( ( ( * `  S )  x.  ( F  .ih  G ) )  x.  -u R )  +  ( ( S  x.  ( G  .ih  F ) )  x.  -u R
) )
2917, 18, 27mul32i 9024 . . . . . . 7  |-  ( ( ( * `  S
)  x.  ( F 
.ih  G ) )  x.  -u R )  =  ( ( ( * `
 S )  x.  -u R )  x.  ( F  .ih  G ) )
3010, 20, 27mul32i 9024 . . . . . . 7  |-  ( ( S  x.  ( G 
.ih  F ) )  x.  -u R )  =  ( ( S  x.  -u R )  x.  ( G  .ih  F ) )
3129, 30oveq12i 5886 . . . . . 6  |-  ( ( ( ( * `  S )  x.  ( F  .ih  G ) )  x.  -u R )  +  ( ( S  x.  ( G  .ih  F ) )  x.  -u R
) )  =  ( ( ( ( * `
 S )  x.  -u R )  x.  ( F  .ih  G ) )  +  ( ( S  x.  -u R )  x.  ( G  .ih  F
) ) )
3226, 28, 313eqtri 2320 . . . . 5  |-  ( B  x.  R )  =  ( ( ( ( * `  S )  x.  -u R )  x.  ( F  .ih  G
) )  +  ( ( S  x.  -u R
)  x.  ( G 
.ih  F ) ) )
332oveq2i 5885 . . . . . 6  |-  ( ( R ^ 2 )  x.  A )  =  ( ( R ^
2 )  x.  ( G  .ih  G ) )
348, 5, 33mulcomli 8860 . . . . 5  |-  ( A  x.  ( R ^
2 ) )  =  ( ( R ^
2 )  x.  ( G  .ih  G ) )
3532, 34oveq12i 5886 . . . 4  |-  ( ( B  x.  R )  +  ( A  x.  ( R ^ 2 ) ) )  =  ( ( ( ( ( * `  S )  x.  -u R )  x.  ( F  .ih  G
) )  +  ( ( S  x.  -u R
)  x.  ( G 
.ih  F ) ) )  +  ( ( R ^ 2 )  x.  ( G  .ih  G ) ) )
3617, 27mulcli 8858 . . . . . 6  |-  ( ( * `  S )  x.  -u R )  e.  CC
3736, 18mulcli 8858 . . . . 5  |-  ( ( ( * `  S
)  x.  -u R
)  x.  ( F 
.ih  G ) )  e.  CC
3810, 27mulcli 8858 . . . . . 6  |-  ( S  x.  -u R )  e.  CC
3938, 20mulcli 8858 . . . . 5  |-  ( ( S  x.  -u R
)  x.  ( G 
.ih  F ) )  e.  CC
408, 4mulcli 8858 . . . . 5  |-  ( ( R ^ 2 )  x.  ( G  .ih  G ) )  e.  CC
4137, 39, 40addassi 8861 . . . 4  |-  ( ( ( ( ( * `
 S )  x.  -u R )  x.  ( F  .ih  G ) )  +  ( ( S  x.  -u R )  x.  ( G  .ih  F
) ) )  +  ( ( R ^
2 )  x.  ( G  .ih  G ) ) )  =  ( ( ( ( * `  S )  x.  -u R
)  x.  ( F 
.ih  G ) )  +  ( ( ( S  x.  -u R
)  x.  ( G 
.ih  F ) )  +  ( ( R ^ 2 )  x.  ( G  .ih  G
) ) ) )
4216, 35, 413eqtri 2320 . . 3  |-  ( ( A  x.  ( R ^ 2 ) )  +  ( B  x.  R ) )  =  ( ( ( ( * `  S )  x.  -u R )  x.  ( F  .ih  G
) )  +  ( ( ( S  x.  -u R )  x.  ( G  .ih  F ) )  +  ( ( R ^ 2 )  x.  ( G  .ih  G
) ) ) )
431, 42oveq12i 5886 . 2  |-  ( C  +  ( ( A  x.  ( R ^
2 ) )  +  ( B  x.  R
) ) )  =  ( ( 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
) ) ) ) )
449, 15addcli 8857 . . 3  |-  ( ( A  x.  ( R ^ 2 ) )  +  ( B  x.  R ) )  e.  CC
4511, 11hicli 21676 . . . 4  |-  ( F 
.ih  F )  e.  CC
461, 45eqeltri 2366 . . 3  |-  C  e.  CC
4744, 46addcomi 9019 . 2  |-  ( ( ( A  x.  ( R ^ 2 ) )  +  ( B  x.  R ) )  +  C )  =  ( C  +  ( ( A  x.  ( R ^ 2 ) )  +  ( B  x.  R ) ) )
4839, 40addcli 8857 . . 3  |-  ( ( ( S  x.  -u R
)  x.  ( G 
.ih  F ) )  +  ( ( R ^ 2 )  x.  ( G  .ih  G
) ) )  e.  CC
4945, 37, 48addassi 8861 . 2  |-  ( ( ( 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
) ) ) )  =  ( ( 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
) ) ) ) )
5043, 47, 493eqtr4i 2326 1  |-  ( ( ( 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
) ) ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1632    e. wcel 1696   ` cfv 5271  (class class class)co 5874   CCcc 8751   RRcr 8752    + caddc 8756    x. cmul 8758   -ucneg 9054   2c2 9811   ^cexp 11120   *ccj 11597   ~Hchil 21515    .ih csp 21518
This theorem is referenced by:  normlem4  21708
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-hfi 21674  ax-his1 21677
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-n0 9982  df-z 10041  df-uz 10247  df-seq 11063  df-exp 11121  df-cj 11600  df-re 11601  df-im 11602
  Copyright terms: Public domain W3C validator