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Theorem normlem1 21635
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 8803 . . . 4  |-  R  e.  CC
41, 3mulcli 8796 . . 3  |-  ( S  x.  R )  e.  CC
5 normlem1.2 . . 3  |-  F  e. 
~H
6 normlem1.3 . . 3  |-  G  e. 
~H
74, 5, 6normlem0 21634 . 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 11625 . . . . . . . 8  |-  ( * `
 ( S  x.  R ) )  =  ( ( * `  S )  x.  (
* `  R )
)
93cjrebi 11610 . . . . . . . . . 10  |-  ( R  e.  RR  <->  ( * `  R )  =  R )
102, 9mpbi 201 . . . . . . . . 9  |-  ( * `
 R )  =  R
1110oveq2i 5789 . . . . . . . 8  |-  ( ( * `  S )  x.  ( * `  R ) )  =  ( ( * `  S )  x.  R
)
128, 11eqtri 2276 . . . . . . 7  |-  ( * `
 ( S  x.  R ) )  =  ( ( * `  S )  x.  R
)
1312negeqi 8999 . . . . . 6  |-  -u (
* `  ( S  x.  R ) )  = 
-u ( ( * `
 S )  x.  R )
141cjcli 11605 . . . . . . 7  |-  ( * `
 S )  e.  CC
1514, 3mulneg2i 9180 . . . . . 6  |-  ( ( * `  S )  x.  -u R )  = 
-u ( ( * `
 S )  x.  R )
1613, 15eqtr4i 2279 . . . . 5  |-  -u (
* `  ( S  x.  R ) )  =  ( ( * `  S )  x.  -u R
)
1716oveq1i 5788 . . . 4  |-  ( -u ( * `  ( S  x.  R )
)  x.  ( F 
.ih  G ) )  =  ( ( ( * `  S )  x.  -u R )  x.  ( F  .ih  G
) )
1817oveq2i 5789 . . 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 9180 . . . . . 6  |-  ( S  x.  -u R )  = 
-u ( S  x.  R )
2019eqcomi 2260 . . . . 5  |-  -u ( S  x.  R )  =  ( S  x.  -u R )
2120oveq1i 5788 . . . 4  |-  ( -u ( S  x.  R
)  x.  ( G 
.ih  F ) )  =  ( ( S  x.  -u R )  x.  ( G  .ih  F
) )
228oveq2i 5789 . . . . . . 7  |-  ( ( S  x.  R )  x.  ( * `  ( S  x.  R
) ) )  =  ( ( S  x.  R )  x.  (
( * `  S
)  x.  ( * `
 R ) ) )
233cjcli 11605 . . . . . . . . 9  |-  ( * `
 R )  e.  CC
241, 3, 14, 23mul4i 8963 . . . . . . . 8  |-  ( ( S  x.  R )  x.  ( ( * `
 S )  x.  ( * `  R
) ) )  =  ( ( S  x.  ( * `  S
) )  x.  ( R  x.  ( * `  R ) ) )
25 normlem1.5 . . . . . . . . . . . 12  |-  ( abs `  S )  =  1
2625oveq1i 5788 . . . . . . . . . . 11  |-  ( ( abs `  S ) ^ 2 )  =  ( 1 ^ 2 )
271absvalsqi 11827 . . . . . . . . . . 11  |-  ( ( abs `  S ) ^ 2 )  =  ( S  x.  (
* `  S )
)
28 sq1 11150 . . . . . . . . . . 11  |-  ( 1 ^ 2 )  =  1
2926, 27, 283eqtr3i 2284 . . . . . . . . . 10  |-  ( S  x.  ( * `  S ) )  =  1
3010oveq2i 5789 . . . . . . . . . 10  |-  ( R  x.  ( * `  R ) )  =  ( R  x.  R
)
3129, 30oveq12i 5790 . . . . . . . . 9  |-  ( ( S  x.  ( * `
 S ) )  x.  ( R  x.  ( * `  R
) ) )  =  ( 1  x.  ( R  x.  R )
)
323, 3mulcli 8796 . . . . . . . . . 10  |-  ( R  x.  R )  e.  CC
3332mulid2i 8794 . . . . . . . . 9  |-  ( 1  x.  ( R  x.  R ) )  =  ( R  x.  R
)
3431, 33eqtri 2276 . . . . . . . 8  |-  ( ( S  x.  ( * `
 S ) )  x.  ( R  x.  ( * `  R
) ) )  =  ( R  x.  R
)
3524, 34eqtri 2276 . . . . . . 7  |-  ( ( S  x.  R )  x.  ( ( * `
 S )  x.  ( * `  R
) ) )  =  ( R  x.  R
)
3622, 35eqtri 2276 . . . . . 6  |-  ( ( S  x.  R )  x.  ( * `  ( S  x.  R
) ) )  =  ( R  x.  R
)
373sqvali 11135 . . . . . 6  |-  ( R ^ 2 )  =  ( R  x.  R
)
3836, 37eqtr4i 2279 . . . . 5  |-  ( ( S  x.  R )  x.  ( * `  ( S  x.  R
) ) )  =  ( R ^ 2 )
3938oveq1i 5788 . . . 4  |-  ( ( ( S  x.  R
)  x.  ( * `
 ( S  x.  R ) ) )  x.  ( G  .ih  G ) )  =  ( ( R ^ 2 )  x.  ( G 
.ih  G ) )
4021, 39oveq12i 5790 . . 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 5790 . 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 2276 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 1619    e. wcel 1621   ` cfv 4659  (class class class)co 5778   CCcc 8689   RRcr 8690   1c1 8692    + caddc 8694    x. cmul 8696   -ucneg 8992   2c2 9749   ^cexp 11056   *ccj 11532   abscabs 11670   ~Hchil 21445    .h csm 21447    .ih csp 21448    -h cmv 21451
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-pre-sup 8769  ax-hfvadd 21526  ax-hfvmul 21531  ax-hvmulass 21533  ax-hfi 21604  ax-his1 21607  ax-his2 21608  ax-his3 21609
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-sup 7148  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-3 9759  df-n0 9919  df-z 9978  df-uz 10184  df-rp 10308  df-seq 10999  df-exp 11057  df-cj 11535  df-re 11536  df-im 11537  df-sqr 11671  df-abs 11672  df-hvsub 21497
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