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

Theorem normlem3 21521
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 21490 . . . . . . 7  |-  ( G 
.ih  G )  e.  CC
52, 4eqeltri 2323 . . . . . 6  |-  A  e.  CC
6 normlem3.7 . . . . . . . 8  |-  R  e.  RR
76recni 8729 . . . . . . 7  |-  R  e.  CC
87sqcli 11062 . . . . . 6  |-  ( R ^ 2 )  e.  CC
95, 8mulcli 8722 . . . . 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 21520 . . . . . . 7  |-  B  e.  RR
1413recni 8729 . . . . . 6  |-  B  e.  CC
1514, 7mulcli 8722 . . . . 5  |-  ( B  x.  R )  e.  CC
169, 15addcomi 8883 . . . 4  |-  ( ( A  x.  ( R ^ 2 ) )  +  ( B  x.  R ) )  =  ( ( B  x.  R )  +  ( A  x.  ( R ^ 2 ) ) )
1710cjcli 11531 . . . . . . . . . 10  |-  ( * `
 S )  e.  CC
1811, 3hicli 21490 . . . . . . . . . 10  |-  ( F 
.ih  G )  e.  CC
1917, 18mulcli 8722 . . . . . . . . 9  |-  ( ( * `  S )  x.  ( F  .ih  G ) )  e.  CC
203, 11hicli 21490 . . . . . . . . . 10  |-  ( G 
.ih  F )  e.  CC
2110, 20mulcli 8722 . . . . . . . . 9  |-  ( S  x.  ( G  .ih  F ) )  e.  CC
2219, 21addcli 8721 . . . . . . . 8  |-  ( ( ( * `  S
)  x.  ( F 
.ih  G ) )  +  ( S  x.  ( G  .ih  F ) ) )  e.  CC
2322, 7mulneg1i 9105 . . . . . . 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 5720 . . . . . . 7  |-  ( B  x.  R )  =  ( -u ( ( ( * `  S
)  x.  ( F 
.ih  G ) )  +  ( S  x.  ( G  .ih  F ) ) )  x.  R
)
2522, 7mulneg2i 9106 . . . . . . 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 2283 . . . . . 6  |-  ( B  x.  R )  =  ( ( ( ( * `  S )  x.  ( F  .ih  G ) )  +  ( S  x.  ( G 
.ih  F ) ) )  x.  -u R
)
277negcli 8994 . . . . . . 7  |-  -u R  e.  CC
2819, 21, 27adddiri 8728 . . . . . 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 8888 . . . . . . 7  |-  ( ( ( * `  S
)  x.  ( F 
.ih  G ) )  x.  -u R )  =  ( ( ( * `
 S )  x.  -u R )  x.  ( F  .ih  G ) )
3010, 20, 27mul32i 8888 . . . . . . 7  |-  ( ( S  x.  ( G 
.ih  F ) )  x.  -u R )  =  ( ( S  x.  -u R )  x.  ( G  .ih  F ) )
3129, 30oveq12i 5722 . . . . . 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 2277 . . . . 5  |-  ( B  x.  R )  =  ( ( ( ( * `  S )  x.  -u R )  x.  ( F  .ih  G
) )  +  ( ( S  x.  -u R
)  x.  ( G 
.ih  F ) ) )
332oveq2i 5721 . . . . . 6  |-  ( ( R ^ 2 )  x.  A )  =  ( ( R ^
2 )  x.  ( G  .ih  G ) )
348, 5, 33mulcomli 8724 . . . . 5  |-  ( A  x.  ( R ^
2 ) )  =  ( ( R ^
2 )  x.  ( G  .ih  G ) )
3532, 34oveq12i 5722 . . . 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 8722 . . . . . 6  |-  ( ( * `  S )  x.  -u R )  e.  CC
3736, 18mulcli 8722 . . . . 5  |-  ( ( ( * `  S
)  x.  -u R
)  x.  ( F 
.ih  G ) )  e.  CC
3810, 27mulcli 8722 . . . . . 6  |-  ( S  x.  -u R )  e.  CC
3938, 20mulcli 8722 . . . . 5  |-  ( ( S  x.  -u R
)  x.  ( G 
.ih  F ) )  e.  CC
408, 4mulcli 8722 . . . . 5  |-  ( ( R ^ 2 )  x.  ( G  .ih  G ) )  e.  CC
4137, 39, 40addassi 8725 . . . 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 2277 . . 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 5722 . 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 8721 . . 3  |-  ( ( A  x.  ( R ^ 2 ) )  +  ( B  x.  R ) )  e.  CC
4511, 11hicli 21490 . . . 4  |-  ( F 
.ih  F )  e.  CC
461, 45eqeltri 2323 . . 3  |-  C  e.  CC
4744, 46addcomi 8883 . 2  |-  ( ( ( A  x.  ( R ^ 2 ) )  +  ( B  x.  R ) )  +  C )  =  ( C  +  ( ( A  x.  ( R ^ 2 ) )  +  ( B  x.  R ) ) )
4839, 40addcli 8721 . . 3  |-  ( ( ( S  x.  -u R
)  x.  ( G 
.ih  F ) )  +  ( ( R ^ 2 )  x.  ( G  .ih  G
) ) )  e.  CC
4945, 37, 48addassi 8725 . 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 2283 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 1619    e. wcel 1621   ` cfv 4592  (class class class)co 5710   CCcc 8615   RRcr 8616    + caddc 8620    x. cmul 8622   -ucneg 8918   2c2 9675   ^cexp 10982   *ccj 11458   ~Hchil 21329    .ih csp 21332
This theorem is referenced by:  normlem4  21522
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-cnex 8673  ax-resscn 8674  ax-1cn 8675  ax-icn 8676  ax-addcl 8677  ax-addrcl 8678  ax-mulcl 8679  ax-mulrcl 8680  ax-mulcom 8681  ax-addass 8682  ax-mulass 8683  ax-distr 8684  ax-i2m1 8685  ax-1ne0 8686  ax-1rid 8687  ax-rnegex 8688  ax-rrecex 8689  ax-cnre 8690  ax-pre-lttri 8691  ax-pre-lttrn 8692  ax-pre-ltadd 8693  ax-pre-mulgt0 8694  ax-hfi 21488  ax-his1 21491
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-2nd 5975  df-iota 6143  df-riota 6190  df-recs 6274  df-rdg 6309  df-er 6546  df-en 6750  df-dom 6751  df-sdom 6752  df-pnf 8749  df-mnf 8750  df-xr 8751  df-ltxr 8752  df-le 8753  df-sub 8919  df-neg 8920  df-div 9304  df-n 9627  df-2 9684  df-n0 9845  df-z 9904  df-uz 10110  df-seq 10925  df-exp 10983  df-cj 11461  df-re 11462  df-im 11463
  Copyright terms: Public domain W3C validator