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Theorem normlem3 21637
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 21606 . . . . . . 7  |-  ( G 
.ih  G )  e.  CC
52, 4eqeltri 2326 . . . . . 6  |-  A  e.  CC
6 normlem3.7 . . . . . . . 8  |-  R  e.  RR
76recni 8803 . . . . . . 7  |-  R  e.  CC
87sqcli 11136 . . . . . 6  |-  ( R ^ 2 )  e.  CC
95, 8mulcli 8796 . . . . 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 21636 . . . . . . 7  |-  B  e.  RR
1413recni 8803 . . . . . 6  |-  B  e.  CC
1514, 7mulcli 8796 . . . . 5  |-  ( B  x.  R )  e.  CC
169, 15addcomi 8957 . . . 4  |-  ( ( A  x.  ( R ^ 2 ) )  +  ( B  x.  R ) )  =  ( ( B  x.  R )  +  ( A  x.  ( R ^ 2 ) ) )
1710cjcli 11605 . . . . . . . . . 10  |-  ( * `
 S )  e.  CC
1811, 3hicli 21606 . . . . . . . . . 10  |-  ( F 
.ih  G )  e.  CC
1917, 18mulcli 8796 . . . . . . . . 9  |-  ( ( * `  S )  x.  ( F  .ih  G ) )  e.  CC
203, 11hicli 21606 . . . . . . . . . 10  |-  ( G 
.ih  F )  e.  CC
2110, 20mulcli 8796 . . . . . . . . 9  |-  ( S  x.  ( G  .ih  F ) )  e.  CC
2219, 21addcli 8795 . . . . . . . 8  |-  ( ( ( * `  S
)  x.  ( F 
.ih  G ) )  +  ( S  x.  ( G  .ih  F ) ) )  e.  CC
2322, 7mulneg1i 9179 . . . . . . 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 5788 . . . . . . 7  |-  ( B  x.  R )  =  ( -u ( ( ( * `  S
)  x.  ( F 
.ih  G ) )  +  ( S  x.  ( G  .ih  F ) ) )  x.  R
)
2522, 7mulneg2i 9180 . . . . . . 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 2286 . . . . . 6  |-  ( B  x.  R )  =  ( ( ( ( * `  S )  x.  ( F  .ih  G ) )  +  ( S  x.  ( G 
.ih  F ) ) )  x.  -u R
)
277negcli 9068 . . . . . . 7  |-  -u R  e.  CC
2819, 21, 27adddiri 8802 . . . . . 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 8962 . . . . . . 7  |-  ( ( ( * `  S
)  x.  ( F 
.ih  G ) )  x.  -u R )  =  ( ( ( * `
 S )  x.  -u R )  x.  ( F  .ih  G ) )
3010, 20, 27mul32i 8962 . . . . . . 7  |-  ( ( S  x.  ( G 
.ih  F ) )  x.  -u R )  =  ( ( S  x.  -u R )  x.  ( G  .ih  F ) )
3129, 30oveq12i 5790 . . . . . 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 2280 . . . . 5  |-  ( B  x.  R )  =  ( ( ( ( * `  S )  x.  -u R )  x.  ( F  .ih  G
) )  +  ( ( S  x.  -u R
)  x.  ( G 
.ih  F ) ) )
332oveq2i 5789 . . . . . 6  |-  ( ( R ^ 2 )  x.  A )  =  ( ( R ^
2 )  x.  ( G  .ih  G ) )
348, 5, 33mulcomli 8798 . . . . 5  |-  ( A  x.  ( R ^
2 ) )  =  ( ( R ^
2 )  x.  ( G  .ih  G ) )
3532, 34oveq12i 5790 . . . 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 8796 . . . . . 6  |-  ( ( * `  S )  x.  -u R )  e.  CC
3736, 18mulcli 8796 . . . . 5  |-  ( ( ( * `  S
)  x.  -u R
)  x.  ( F 
.ih  G ) )  e.  CC
3810, 27mulcli 8796 . . . . . 6  |-  ( S  x.  -u R )  e.  CC
3938, 20mulcli 8796 . . . . 5  |-  ( ( S  x.  -u R
)  x.  ( G 
.ih  F ) )  e.  CC
408, 4mulcli 8796 . . . . 5  |-  ( ( R ^ 2 )  x.  ( G  .ih  G ) )  e.  CC
4137, 39, 40addassi 8799 . . . 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 2280 . . 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 5790 . 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 8795 . . 3  |-  ( ( A  x.  ( R ^ 2 ) )  +  ( B  x.  R ) )  e.  CC
4511, 11hicli 21606 . . . 4  |-  ( F 
.ih  F )  e.  CC
461, 45eqeltri 2326 . . 3  |-  C  e.  CC
4744, 46addcomi 8957 . 2  |-  ( ( ( A  x.  ( R ^ 2 ) )  +  ( B  x.  R ) )  +  C )  =  ( C  +  ( ( A  x.  ( R ^ 2 ) )  +  ( B  x.  R ) ) )
4839, 40addcli 8795 . . 3  |-  ( ( ( S  x.  -u R
)  x.  ( G 
.ih  F ) )  +  ( ( R ^ 2 )  x.  ( G  .ih  G
) ) )  e.  CC
4945, 37, 48addassi 8799 . 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 2286 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 4659  (class class class)co 5778   CCcc 8689   RRcr 8690    + caddc 8694    x. cmul 8696   -ucneg 8992   2c2 9749   ^cexp 11056   *ccj 11532   ~Hchil 21445    .ih csp 21448
This theorem is referenced by:  normlem4  21638
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 1927  ax-ext 2237  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470  ax-cnex 8747  ax-resscn 8748  ax-1cn 8749  ax-icn 8750  ax-addcl 8751  ax-addrcl 8752  ax-mulcl 8753  ax-mulrcl 8754  ax-mulcom 8755  ax-addass 8756  ax-mulass 8757  ax-distr 8758  ax-i2m1 8759  ax-1ne0 8760  ax-1rid 8761  ax-rnegex 8762  ax-rrecex 8763  ax-cnre 8764  ax-pre-lttri 8765  ax-pre-lttrn 8766  ax-pre-ltadd 8767  ax-pre-mulgt0 8768  ax-hfi 21604  ax-his1 21607
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 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2521  df-rex 2522  df-reu 2523  df-rmo 2524  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-pss 3129  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-tp 3608  df-op 3609  df-uni 3788  df-iun 3867  df-br 3984  df-opab 4038  df-mpt 4039  df-tr 4074  df-eprel 4263  df-id 4267  df-po 4272  df-so 4273  df-fr 4310  df-we 4312  df-ord 4353  df-on 4354  df-lim 4355  df-suc 4356  df-om 4615  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-ov 5781  df-oprab 5782  df-mpt2 5783  df-2nd 6043  df-iota 6211  df-riota 6258  df-recs 6342  df-rdg 6377  df-er 6614  df-en 6818  df-dom 6819  df-sdom 6820  df-pnf 8823  df-mnf 8824  df-xr 8825  df-ltxr 8826  df-le 8827  df-sub 8993  df-neg 8994  df-div 9378  df-n 9701  df-2 9758  df-n0 9919  df-z 9978  df-uz 10184  df-seq 10999  df-exp 11057  df-cj 11535  df-re 11536  df-im 11537
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