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Theorem lawcoslem1 20688
Description: Lemma for Law of Cosines lawcos 20689. Here we prove the law for a point at the origin and two distinct points U and V, using an expanded version of the signed angle expression on the complex plane. (Contributed by David A. Wheeler, 11-Jun-2015.)
Hypotheses
Ref Expression
lawcoslem1.1  |-  ( ph  ->  U  e.  CC )
lawcoslem1.2  |-  ( ph  ->  V  e.  CC )
lawcoslem1.3  |-  ( ph  ->  U  =/=  0 )
lawcoslem1.4  |-  ( ph  ->  V  =/=  0 )
Assertion
Ref Expression
lawcoslem1  |-  ( ph  ->  ( ( abs `  ( U  -  V )
) ^ 2 )  =  ( ( ( ( abs `  U
) ^ 2 )  +  ( ( abs `  V ) ^ 2 ) )  -  (
2  x.  ( ( ( abs `  U
)  x.  ( abs `  V ) )  x.  ( ( Re `  ( U  /  V
) )  /  ( abs `  ( U  /  V ) ) ) ) ) ) )

Proof of Theorem lawcoslem1
StepHypRef Expression
1 lawcoslem1.1 . . 3  |-  ( ph  ->  U  e.  CC )
2 lawcoslem1.2 . . 3  |-  ( ph  ->  V  e.  CC )
3 sqabssub 12119 . . 3  |-  ( ( U  e.  CC  /\  V  e.  CC )  ->  ( ( abs `  ( U  -  V )
) ^ 2 )  =  ( ( ( ( abs `  U
) ^ 2 )  +  ( ( abs `  V ) ^ 2 ) )  -  (
2  x.  ( Re
`  ( U  x.  ( * `  V
) ) ) ) ) )
41, 2, 3syl2anc 644 . 2  |-  ( ph  ->  ( ( abs `  ( U  -  V )
) ^ 2 )  =  ( ( ( ( abs `  U
) ^ 2 )  +  ( ( abs `  V ) ^ 2 ) )  -  (
2  x.  ( Re
`  ( U  x.  ( * `  V
) ) ) ) ) )
5 lawcoslem1.4 . . . . . . . . 9  |-  ( ph  ->  V  =/=  0 )
61, 2, 5absdivd 12288 . . . . . . . 8  |-  ( ph  ->  ( abs `  ( U  /  V ) )  =  ( ( abs `  U )  /  ( abs `  V ) ) )
76oveq2d 6126 . . . . . . 7  |-  ( ph  ->  ( ( Re `  ( U  /  V
) )  /  ( abs `  ( U  /  V ) ) )  =  ( ( Re
`  ( U  /  V ) )  / 
( ( abs `  U
)  /  ( abs `  V ) ) ) )
87oveq2d 6126 . . . . . 6  |-  ( ph  ->  ( ( ( abs `  U )  x.  ( abs `  V ) )  x.  ( ( Re
`  ( U  /  V ) )  / 
( abs `  ( U  /  V ) ) ) )  =  ( ( ( abs `  U
)  x.  ( abs `  V ) )  x.  ( ( Re `  ( U  /  V
) )  /  (
( abs `  U
)  /  ( abs `  V ) ) ) ) )
91abscld 12269 . . . . . . . . 9  |-  ( ph  ->  ( abs `  U
)  e.  RR )
102abscld 12269 . . . . . . . . 9  |-  ( ph  ->  ( abs `  V
)  e.  RR )
119, 10remulcld 9147 . . . . . . . 8  |-  ( ph  ->  ( ( abs `  U
)  x.  ( abs `  V ) )  e.  RR )
1211recnd 9145 . . . . . . 7  |-  ( ph  ->  ( ( abs `  U
)  x.  ( abs `  V ) )  e.  CC )
131, 2, 5divcld 9821 . . . . . . . . 9  |-  ( ph  ->  ( U  /  V
)  e.  CC )
1413recld 12030 . . . . . . . 8  |-  ( ph  ->  ( Re `  ( U  /  V ) )  e.  RR )
1514recnd 9145 . . . . . . 7  |-  ( ph  ->  ( Re `  ( U  /  V ) )  e.  CC )
169recnd 9145 . . . . . . . 8  |-  ( ph  ->  ( abs `  U
)  e.  CC )
1710recnd 9145 . . . . . . . 8  |-  ( ph  ->  ( abs `  V
)  e.  CC )
182, 5absne0d 12280 . . . . . . . 8  |-  ( ph  ->  ( abs `  V
)  =/=  0 )
1916, 17, 18divcld 9821 . . . . . . 7  |-  ( ph  ->  ( ( abs `  U
)  /  ( abs `  V ) )  e.  CC )
20 lawcoslem1.3 . . . . . . . . 9  |-  ( ph  ->  U  =/=  0 )
211, 20absne0d 12280 . . . . . . . 8  |-  ( ph  ->  ( abs `  U
)  =/=  0 )
2216, 17, 21, 18divne0d 9837 . . . . . . 7  |-  ( ph  ->  ( ( abs `  U
)  /  ( abs `  V ) )  =/=  0 )
2312, 15, 19, 22div12d 9857 . . . . . 6  |-  ( ph  ->  ( ( ( abs `  U )  x.  ( abs `  V ) )  x.  ( ( Re
`  ( U  /  V ) )  / 
( ( abs `  U
)  /  ( abs `  V ) ) ) )  =  ( ( Re `  ( U  /  V ) )  x.  ( ( ( abs `  U )  x.  ( abs `  V
) )  /  (
( abs `  U
)  /  ( abs `  V ) ) ) ) )
248, 23eqtrd 2474 . . . . 5  |-  ( ph  ->  ( ( ( abs `  U )  x.  ( abs `  V ) )  x.  ( ( Re
`  ( U  /  V ) )  / 
( abs `  ( U  /  V ) ) ) )  =  ( ( Re `  ( U  /  V ) )  x.  ( ( ( abs `  U )  x.  ( abs `  V
) )  /  (
( abs `  U
)  /  ( abs `  V ) ) ) ) )
2512, 16, 17, 21, 18divdiv2d 9853 . . . . . . 7  |-  ( ph  ->  ( ( ( abs `  U )  x.  ( abs `  V ) )  /  ( ( abs `  U )  /  ( abs `  V ) ) )  =  ( ( ( ( abs `  U
)  x.  ( abs `  V ) )  x.  ( abs `  V
) )  /  ( abs `  U ) ) )
2617sqvald 11551 . . . . . . . . . 10  |-  ( ph  ->  ( ( abs `  V
) ^ 2 )  =  ( ( abs `  V )  x.  ( abs `  V ) ) )
2726oveq1d 6125 . . . . . . . . 9  |-  ( ph  ->  ( ( ( abs `  V ) ^ 2 )  x.  ( abs `  U ) )  =  ( ( ( abs `  V )  x.  ( abs `  V ) )  x.  ( abs `  U
) ) )
2816, 17, 17mul31d 9308 . . . . . . . . 9  |-  ( ph  ->  ( ( ( abs `  U )  x.  ( abs `  V ) )  x.  ( abs `  V
) )  =  ( ( ( abs `  V
)  x.  ( abs `  V ) )  x.  ( abs `  U
) ) )
2927, 28eqtr4d 2477 . . . . . . . 8  |-  ( ph  ->  ( ( ( abs `  V ) ^ 2 )  x.  ( abs `  U ) )  =  ( ( ( abs `  U )  x.  ( abs `  V ) )  x.  ( abs `  V
) ) )
3029oveq1d 6125 . . . . . . 7  |-  ( ph  ->  ( ( ( ( abs `  V ) ^ 2 )  x.  ( abs `  U
) )  /  ( abs `  U ) )  =  ( ( ( ( abs `  U
)  x.  ( abs `  V ) )  x.  ( abs `  V
) )  /  ( abs `  U ) ) )
3117sqcld 11552 . . . . . . . 8  |-  ( ph  ->  ( ( abs `  V
) ^ 2 )  e.  CC )
3231, 16, 21divcan4d 9827 . . . . . . 7  |-  ( ph  ->  ( ( ( ( abs `  V ) ^ 2 )  x.  ( abs `  U
) )  /  ( abs `  U ) )  =  ( ( abs `  V ) ^ 2 ) )
3325, 30, 323eqtr2rd 2481 . . . . . 6  |-  ( ph  ->  ( ( abs `  V
) ^ 2 )  =  ( ( ( abs `  U )  x.  ( abs `  V
) )  /  (
( abs `  U
)  /  ( abs `  V ) ) ) )
3433oveq2d 6126 . . . . 5  |-  ( ph  ->  ( ( Re `  ( U  /  V
) )  x.  (
( abs `  V
) ^ 2 ) )  =  ( ( Re `  ( U  /  V ) )  x.  ( ( ( abs `  U )  x.  ( abs `  V
) )  /  (
( abs `  U
)  /  ( abs `  V ) ) ) ) )
3515, 31mulcomd 9140 . . . . . . 7  |-  ( ph  ->  ( ( Re `  ( U  /  V
) )  x.  (
( abs `  V
) ^ 2 ) )  =  ( ( ( abs `  V
) ^ 2 )  x.  ( Re `  ( U  /  V
) ) ) )
3610resqcld 11580 . . . . . . . 8  |-  ( ph  ->  ( ( abs `  V
) ^ 2 )  e.  RR )
3736, 13remul2d 12063 . . . . . . 7  |-  ( ph  ->  ( Re `  (
( ( abs `  V
) ^ 2 )  x.  ( U  /  V ) ) )  =  ( ( ( abs `  V ) ^ 2 )  x.  ( Re `  ( U  /  V ) ) ) )
3835, 37eqtr4d 2477 . . . . . 6  |-  ( ph  ->  ( ( Re `  ( U  /  V
) )  x.  (
( abs `  V
) ^ 2 ) )  =  ( Re
`  ( ( ( abs `  V ) ^ 2 )  x.  ( U  /  V
) ) ) )
391, 31, 2, 5div12d 9857 . . . . . . . 8  |-  ( ph  ->  ( U  x.  (
( ( abs `  V
) ^ 2 )  /  V ) )  =  ( ( ( abs `  V ) ^ 2 )  x.  ( U  /  V
) ) )
4031, 2, 5divrecd 9824 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( abs `  V ) ^ 2 )  /  V )  =  ( ( ( abs `  V ) ^ 2 )  x.  ( 1  /  V
) ) )
41 recval 12157 . . . . . . . . . . . . 13  |-  ( ( V  e.  CC  /\  V  =/=  0 )  -> 
( 1  /  V
)  =  ( ( * `  V )  /  ( ( abs `  V ) ^ 2 ) ) )
422, 5, 41syl2anc 644 . . . . . . . . . . . 12  |-  ( ph  ->  ( 1  /  V
)  =  ( ( * `  V )  /  ( ( abs `  V ) ^ 2 ) ) )
4342oveq2d 6126 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( abs `  V ) ^ 2 )  x.  ( 1  /  V ) )  =  ( ( ( abs `  V ) ^ 2 )  x.  ( ( * `  V )  /  (
( abs `  V
) ^ 2 ) ) ) )
442cjcld 12032 . . . . . . . . . . . 12  |-  ( ph  ->  ( * `  V
)  e.  CC )
45 sqne0 11479 . . . . . . . . . . . . . 14  |-  ( ( abs `  V )  e.  CC  ->  (
( ( abs `  V
) ^ 2 )  =/=  0  <->  ( abs `  V )  =/=  0
) )
4617, 45syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( ( abs `  V ) ^ 2 )  =/=  0  <->  ( abs `  V )  =/=  0 ) )
4718, 46mpbird 225 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( abs `  V
) ^ 2 )  =/=  0 )
4844, 31, 47divcan2d 9823 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( abs `  V ) ^ 2 )  x.  ( ( * `  V )  /  ( ( abs `  V ) ^ 2 ) ) )  =  ( * `  V
) )
4943, 48eqtrd 2474 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( abs `  V ) ^ 2 )  x.  ( 1  /  V ) )  =  ( * `  V ) )
5040, 49eqtrd 2474 . . . . . . . . 9  |-  ( ph  ->  ( ( ( abs `  V ) ^ 2 )  /  V )  =  ( * `  V ) )
5150oveq2d 6126 . . . . . . . 8  |-  ( ph  ->  ( U  x.  (
( ( abs `  V
) ^ 2 )  /  V ) )  =  ( U  x.  ( * `  V
) ) )
5239, 51eqtr3d 2476 . . . . . . 7  |-  ( ph  ->  ( ( ( abs `  V ) ^ 2 )  x.  ( U  /  V ) )  =  ( U  x.  ( * `  V
) ) )
5352fveq2d 5761 . . . . . 6  |-  ( ph  ->  ( Re `  (
( ( abs `  V
) ^ 2 )  x.  ( U  /  V ) ) )  =  ( Re `  ( U  x.  (
* `  V )
) ) )
5438, 53eqtrd 2474 . . . . 5  |-  ( ph  ->  ( ( Re `  ( U  /  V
) )  x.  (
( abs `  V
) ^ 2 ) )  =  ( Re
`  ( U  x.  ( * `  V
) ) ) )
5524, 34, 543eqtr2rd 2481 . . . 4  |-  ( ph  ->  ( Re `  ( U  x.  ( * `  V ) ) )  =  ( ( ( abs `  U )  x.  ( abs `  V
) )  x.  (
( Re `  ( U  /  V ) )  /  ( abs `  ( U  /  V ) ) ) ) )
5655oveq2d 6126 . . 3  |-  ( ph  ->  ( 2  x.  (
Re `  ( U  x.  ( * `  V
) ) ) )  =  ( 2  x.  ( ( ( abs `  U )  x.  ( abs `  V ) )  x.  ( ( Re
`  ( U  /  V ) )  / 
( abs `  ( U  /  V ) ) ) ) ) )
5756oveq2d 6126 . 2  |-  ( ph  ->  ( ( ( ( abs `  U ) ^ 2 )  +  ( ( abs `  V
) ^ 2 ) )  -  ( 2  x.  ( Re `  ( U  x.  (
* `  V )
) ) ) )  =  ( ( ( ( abs `  U
) ^ 2 )  +  ( ( abs `  V ) ^ 2 ) )  -  (
2  x.  ( ( ( abs `  U
)  x.  ( abs `  V ) )  x.  ( ( Re `  ( U  /  V
) )  /  ( abs `  ( U  /  V ) ) ) ) ) ) )
584, 57eqtrd 2474 1  |-  ( ph  ->  ( ( abs `  ( U  -  V )
) ^ 2 )  =  ( ( ( ( abs `  U
) ^ 2 )  +  ( ( abs `  V ) ^ 2 ) )  -  (
2  x.  ( ( ( abs `  U
)  x.  ( abs `  V ) )  x.  ( ( Re `  ( U  /  V
) )  /  ( abs `  ( U  /  V ) ) ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    = wceq 1653    e. wcel 1727    =/= wne 2605   ` cfv 5483  (class class class)co 6110   CCcc 9019   0cc0 9021   1c1 9022    + caddc 9024    x. cmul 9026    - cmin 9322    / cdiv 9708   2c2 10080   ^cexp 11413   *ccj 11932   Recre 11933   abscabs 12070
This theorem is referenced by:  lawcos  20689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730  ax-cnex 9077  ax-resscn 9078  ax-1cn 9079  ax-icn 9080  ax-addcl 9081  ax-addrcl 9082  ax-mulcl 9083  ax-mulrcl 9084  ax-mulcom 9085  ax-addass 9086  ax-mulass 9087  ax-distr 9088  ax-i2m1 9089  ax-1ne0 9090  ax-1rid 9091  ax-rnegex 9092  ax-rrecex 9093  ax-cnre 9094  ax-pre-lttri 9095  ax-pre-lttrn 9096  ax-pre-ltadd 9097  ax-pre-mulgt0 9098  ax-pre-sup 9099
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2716  df-rex 2717  df-reu 2718  df-rmo 2719  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-pss 3322  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-tp 3846  df-op 3847  df-uni 4040  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-tr 4328  df-eprel 4523  df-id 4527  df-po 4532  df-so 4533  df-fr 4570  df-we 4572  df-ord 4613  df-on 4614  df-lim 4615  df-suc 4616  df-om 4875  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-2nd 6379  df-riota 6578  df-recs 6662  df-rdg 6697  df-er 6934  df-en 7139  df-dom 7140  df-sdom 7141  df-sup 7475  df-pnf 9153  df-mnf 9154  df-xr 9155  df-ltxr 9156  df-le 9157  df-sub 9324  df-neg 9325  df-div 9709  df-nn 10032  df-2 10089  df-3 10090  df-n0 10253  df-z 10314  df-uz 10520  df-rp 10644  df-seq 11355  df-exp 11414  df-cj 11935  df-re 11936  df-im 11937  df-sqr 12071  df-abs 12072
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