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Theorem normlem1 22600
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 9091 . . . 4  |-  R  e.  CC
41, 3mulcli 9084 . . 3  |-  ( S  x.  R )  e.  CC
5 normlem1.2 . . 3  |-  F  e. 
~H
6 normlem1.3 . . 3  |-  G  e. 
~H
74, 5, 6normlem0 22599 . 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 11982 . . . . . . . 8  |-  ( * `
 ( S  x.  R ) )  =  ( ( * `  S )  x.  (
* `  R )
)
93cjrebi 11967 . . . . . . . . . 10  |-  ( R  e.  RR  <->  ( * `  R )  =  R )
102, 9mpbi 200 . . . . . . . . 9  |-  ( * `
 R )  =  R
1110oveq2i 6083 . . . . . . . 8  |-  ( ( * `  S )  x.  ( * `  R ) )  =  ( ( * `  S )  x.  R
)
128, 11eqtri 2455 . . . . . . 7  |-  ( * `
 ( S  x.  R ) )  =  ( ( * `  S )  x.  R
)
1312negeqi 9288 . . . . . 6  |-  -u (
* `  ( S  x.  R ) )  = 
-u ( ( * `
 S )  x.  R )
141cjcli 11962 . . . . . . 7  |-  ( * `
 S )  e.  CC
1514, 3mulneg2i 9469 . . . . . 6  |-  ( ( * `  S )  x.  -u R )  = 
-u ( ( * `
 S )  x.  R )
1613, 15eqtr4i 2458 . . . . 5  |-  -u (
* `  ( S  x.  R ) )  =  ( ( * `  S )  x.  -u R
)
1716oveq1i 6082 . . . 4  |-  ( -u ( * `  ( S  x.  R )
)  x.  ( F 
.ih  G ) )  =  ( ( ( * `  S )  x.  -u R )  x.  ( F  .ih  G
) )
1817oveq2i 6083 . . 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 9469 . . . . . 6  |-  ( S  x.  -u R )  = 
-u ( S  x.  R )
2019eqcomi 2439 . . . . 5  |-  -u ( S  x.  R )  =  ( S  x.  -u R )
2120oveq1i 6082 . . . 4  |-  ( -u ( S  x.  R
)  x.  ( G 
.ih  F ) )  =  ( ( S  x.  -u R )  x.  ( G  .ih  F
) )
228oveq2i 6083 . . . . . . 7  |-  ( ( S  x.  R )  x.  ( * `  ( S  x.  R
) ) )  =  ( ( S  x.  R )  x.  (
( * `  S
)  x.  ( * `
 R ) ) )
233cjcli 11962 . . . . . . . . 9  |-  ( * `
 R )  e.  CC
241, 3, 14, 23mul4i 9252 . . . . . . . 8  |-  ( ( S  x.  R )  x.  ( ( * `
 S )  x.  ( * `  R
) ) )  =  ( ( S  x.  ( * `  S
) )  x.  ( R  x.  ( * `  R ) ) )
25 normlem1.5 . . . . . . . . . . . 12  |-  ( abs `  S )  =  1
2625oveq1i 6082 . . . . . . . . . . 11  |-  ( ( abs `  S ) ^ 2 )  =  ( 1 ^ 2 )
271absvalsqi 12184 . . . . . . . . . . 11  |-  ( ( abs `  S ) ^ 2 )  =  ( S  x.  (
* `  S )
)
28 sq1 11464 . . . . . . . . . . 11  |-  ( 1 ^ 2 )  =  1
2926, 27, 283eqtr3i 2463 . . . . . . . . . 10  |-  ( S  x.  ( * `  S ) )  =  1
3010oveq2i 6083 . . . . . . . . . 10  |-  ( R  x.  ( * `  R ) )  =  ( R  x.  R
)
3129, 30oveq12i 6084 . . . . . . . . 9  |-  ( ( S  x.  ( * `
 S ) )  x.  ( R  x.  ( * `  R
) ) )  =  ( 1  x.  ( R  x.  R )
)
323, 3mulcli 9084 . . . . . . . . . 10  |-  ( R  x.  R )  e.  CC
3332mulid2i 9082 . . . . . . . . 9  |-  ( 1  x.  ( R  x.  R ) )  =  ( R  x.  R
)
3431, 33eqtri 2455 . . . . . . . 8  |-  ( ( S  x.  ( * `
 S ) )  x.  ( R  x.  ( * `  R
) ) )  =  ( R  x.  R
)
3524, 34eqtri 2455 . . . . . . 7  |-  ( ( S  x.  R )  x.  ( ( * `
 S )  x.  ( * `  R
) ) )  =  ( R  x.  R
)
3622, 35eqtri 2455 . . . . . 6  |-  ( ( S  x.  R )  x.  ( * `  ( S  x.  R
) ) )  =  ( R  x.  R
)
373sqvali 11449 . . . . . 6  |-  ( R ^ 2 )  =  ( R  x.  R
)
3836, 37eqtr4i 2458 . . . . 5  |-  ( ( S  x.  R )  x.  ( * `  ( S  x.  R
) ) )  =  ( R ^ 2 )
3938oveq1i 6082 . . . 4  |-  ( ( ( S  x.  R
)  x.  ( * `
 ( S  x.  R ) ) )  x.  ( G  .ih  G ) )  =  ( ( R ^ 2 )  x.  ( G 
.ih  G ) )
4021, 39oveq12i 6084 . . 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 6084 . 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 2455 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 1652    e. wcel 1725   ` cfv 5445  (class class class)co 6072   CCcc 8977   RRcr 8978   1c1 8980    + caddc 8982    x. cmul 8984   -ucneg 9281   2c2 10038   ^cexp 11370   *ccj 11889   abscabs 12027   ~Hchil 22410    .h csm 22412    .ih csp 22413    -h cmv 22416
This theorem is referenced by:  normlem4  22603
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692  ax-cnex 9035  ax-resscn 9036  ax-1cn 9037  ax-icn 9038  ax-addcl 9039  ax-addrcl 9040  ax-mulcl 9041  ax-mulrcl 9042  ax-mulcom 9043  ax-addass 9044  ax-mulass 9045  ax-distr 9046  ax-i2m1 9047  ax-1ne0 9048  ax-1rid 9049  ax-rnegex 9050  ax-rrecex 9051  ax-cnre 9052  ax-pre-lttri 9053  ax-pre-lttrn 9054  ax-pre-ltadd 9055  ax-pre-mulgt0 9056  ax-pre-sup 9057  ax-hfvadd 22491  ax-hfvmul 22496  ax-hvmulass 22498  ax-hfi 22569  ax-his1 22572  ax-his2 22573  ax-his3 22574
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4837  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-2nd 6341  df-riota 6540  df-recs 6624  df-rdg 6659  df-er 6896  df-en 7101  df-dom 7102  df-sdom 7103  df-sup 7437  df-pnf 9111  df-mnf 9112  df-xr 9113  df-ltxr 9114  df-le 9115  df-sub 9282  df-neg 9283  df-div 9667  df-nn 9990  df-2 10047  df-3 10048  df-n0 10211  df-z 10272  df-uz 10478  df-rp 10602  df-seq 11312  df-exp 11371  df-cj 11892  df-re 11893  df-im 11894  df-sqr 12028  df-abs 12029  df-hvsub 22462
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