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Theorem lawcoslem1 20618
Description: Lemma for Law of Cosines lawcos 20619. 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 12051 . . 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 643 . 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 12220 . . . . . . . 8  |-  ( ph  ->  ( abs `  ( U  /  V ) )  =  ( ( abs `  U )  /  ( abs `  V ) ) )
76oveq2d 6064 . . . . . . 7  |-  ( ph  ->  ( ( Re `  ( U  /  V
) )  /  ( abs `  ( U  /  V ) ) )  =  ( ( Re
`  ( U  /  V ) )  / 
( ( abs `  U
)  /  ( abs `  V ) ) ) )
87oveq2d 6064 . . . . . 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 12201 . . . . . . . . 9  |-  ( ph  ->  ( abs `  U
)  e.  RR )
102abscld 12201 . . . . . . . . 9  |-  ( ph  ->  ( abs `  V
)  e.  RR )
119, 10remulcld 9080 . . . . . . . 8  |-  ( ph  ->  ( ( abs `  U
)  x.  ( abs `  V ) )  e.  RR )
1211recnd 9078 . . . . . . 7  |-  ( ph  ->  ( ( abs `  U
)  x.  ( abs `  V ) )  e.  CC )
131, 2, 5divcld 9754 . . . . . . . . 9  |-  ( ph  ->  ( U  /  V
)  e.  CC )
1413recld 11962 . . . . . . . 8  |-  ( ph  ->  ( Re `  ( U  /  V ) )  e.  RR )
1514recnd 9078 . . . . . . 7  |-  ( ph  ->  ( Re `  ( U  /  V ) )  e.  CC )
169recnd 9078 . . . . . . . 8  |-  ( ph  ->  ( abs `  U
)  e.  CC )
1710recnd 9078 . . . . . . . 8  |-  ( ph  ->  ( abs `  V
)  e.  CC )
182, 5absne0d 12212 . . . . . . . 8  |-  ( ph  ->  ( abs `  V
)  =/=  0 )
1916, 17, 18divcld 9754 . . . . . . 7  |-  ( ph  ->  ( ( abs `  U
)  /  ( abs `  V ) )  e.  CC )
20 lawcoslem1.3 . . . . . . . . 9  |-  ( ph  ->  U  =/=  0 )
211, 20absne0d 12212 . . . . . . . 8  |-  ( ph  ->  ( abs `  U
)  =/=  0 )
2216, 17, 21, 18divne0d 9770 . . . . . . 7  |-  ( ph  ->  ( ( abs `  U
)  /  ( abs `  V ) )  =/=  0 )
2312, 15, 19, 22div12d 9790 . . . . . 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 2444 . . . . 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 9786 . . . . . . 7  |-  ( ph  ->  ( ( ( abs `  U )  x.  ( abs `  V ) )  /  ( ( abs `  U )  /  ( abs `  V ) ) )  =  ( ( ( ( abs `  U
)  x.  ( abs `  V ) )  x.  ( abs `  V
) )  /  ( abs `  U ) ) )
2617sqvald 11483 . . . . . . . . . 10  |-  ( ph  ->  ( ( abs `  V
) ^ 2 )  =  ( ( abs `  V )  x.  ( abs `  V ) ) )
2726oveq1d 6063 . . . . . . . . 9  |-  ( ph  ->  ( ( ( abs `  V ) ^ 2 )  x.  ( abs `  U ) )  =  ( ( ( abs `  V )  x.  ( abs `  V ) )  x.  ( abs `  U
) ) )
2816, 17, 17mul31d 9241 . . . . . . . . 9  |-  ( ph  ->  ( ( ( abs `  U )  x.  ( abs `  V ) )  x.  ( abs `  V
) )  =  ( ( ( abs `  V
)  x.  ( abs `  V ) )  x.  ( abs `  U
) ) )
2927, 28eqtr4d 2447 . . . . . . . 8  |-  ( ph  ->  ( ( ( abs `  V ) ^ 2 )  x.  ( abs `  U ) )  =  ( ( ( abs `  U )  x.  ( abs `  V ) )  x.  ( abs `  V
) ) )
3029oveq1d 6063 . . . . . . 7  |-  ( ph  ->  ( ( ( ( abs `  V ) ^ 2 )  x.  ( abs `  U
) )  /  ( abs `  U ) )  =  ( ( ( ( abs `  U
)  x.  ( abs `  V ) )  x.  ( abs `  V
) )  /  ( abs `  U ) ) )
3117sqcld 11484 . . . . . . . 8  |-  ( ph  ->  ( ( abs `  V
) ^ 2 )  e.  CC )
3231, 16, 21divcan4d 9760 . . . . . . 7  |-  ( ph  ->  ( ( ( ( abs `  V ) ^ 2 )  x.  ( abs `  U
) )  /  ( abs `  U ) )  =  ( ( abs `  V ) ^ 2 ) )
3325, 30, 323eqtr2rd 2451 . . . . . 6  |-  ( ph  ->  ( ( abs `  V
) ^ 2 )  =  ( ( ( abs `  U )  x.  ( abs `  V
) )  /  (
( abs `  U
)  /  ( abs `  V ) ) ) )
3433oveq2d 6064 . . . . 5  |-  ( ph  ->  ( ( Re `  ( U  /  V
) )  x.  (
( abs `  V
) ^ 2 ) )  =  ( ( Re `  ( U  /  V ) )  x.  ( ( ( abs `  U )  x.  ( abs `  V
) )  /  (
( abs `  U
)  /  ( abs `  V ) ) ) ) )
3515, 31mulcomd 9073 . . . . . . 7  |-  ( ph  ->  ( ( Re `  ( U  /  V
) )  x.  (
( abs `  V
) ^ 2 ) )  =  ( ( ( abs `  V
) ^ 2 )  x.  ( Re `  ( U  /  V
) ) ) )
3610resqcld 11512 . . . . . . . 8  |-  ( ph  ->  ( ( abs `  V
) ^ 2 )  e.  RR )
3736, 13remul2d 11995 . . . . . . 7  |-  ( ph  ->  ( Re `  (
( ( abs `  V
) ^ 2 )  x.  ( U  /  V ) ) )  =  ( ( ( abs `  V ) ^ 2 )  x.  ( Re `  ( U  /  V ) ) ) )
3835, 37eqtr4d 2447 . . . . . 6  |-  ( ph  ->  ( ( Re `  ( U  /  V
) )  x.  (
( abs `  V
) ^ 2 ) )  =  ( Re
`  ( ( ( abs `  V ) ^ 2 )  x.  ( U  /  V
) ) ) )
391, 31, 2, 5div12d 9790 . . . . . . . 8  |-  ( ph  ->  ( U  x.  (
( ( abs `  V
) ^ 2 )  /  V ) )  =  ( ( ( abs `  V ) ^ 2 )  x.  ( U  /  V
) ) )
4031, 2, 5divrecd 9757 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( abs `  V ) ^ 2 )  /  V )  =  ( ( ( abs `  V ) ^ 2 )  x.  ( 1  /  V
) ) )
41 recval 12089 . . . . . . . . . . . . 13  |-  ( ( V  e.  CC  /\  V  =/=  0 )  -> 
( 1  /  V
)  =  ( ( * `  V )  /  ( ( abs `  V ) ^ 2 ) ) )
422, 5, 41syl2anc 643 . . . . . . . . . . . 12  |-  ( ph  ->  ( 1  /  V
)  =  ( ( * `  V )  /  ( ( abs `  V ) ^ 2 ) ) )
4342oveq2d 6064 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( abs `  V ) ^ 2 )  x.  ( 1  /  V ) )  =  ( ( ( abs `  V ) ^ 2 )  x.  ( ( * `  V )  /  (
( abs `  V
) ^ 2 ) ) ) )
442cjcld 11964 . . . . . . . . . . . 12  |-  ( ph  ->  ( * `  V
)  e.  CC )
45 sqne0 11411 . . . . . . . . . . . . . 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 224 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( abs `  V
) ^ 2 )  =/=  0 )
4844, 31, 47divcan2d 9756 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( abs `  V ) ^ 2 )  x.  ( ( * `  V )  /  ( ( abs `  V ) ^ 2 ) ) )  =  ( * `  V
) )
4943, 48eqtrd 2444 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( abs `  V ) ^ 2 )  x.  ( 1  /  V ) )  =  ( * `  V ) )
5040, 49eqtrd 2444 . . . . . . . . 9  |-  ( ph  ->  ( ( ( abs `  V ) ^ 2 )  /  V )  =  ( * `  V ) )
5150oveq2d 6064 . . . . . . . 8  |-  ( ph  ->  ( U  x.  (
( ( abs `  V
) ^ 2 )  /  V ) )  =  ( U  x.  ( * `  V
) ) )
5239, 51eqtr3d 2446 . . . . . . 7  |-  ( ph  ->  ( ( ( abs `  V ) ^ 2 )  x.  ( U  /  V ) )  =  ( U  x.  ( * `  V
) ) )
5352fveq2d 5699 . . . . . 6  |-  ( ph  ->  ( Re `  (
( ( abs `  V
) ^ 2 )  x.  ( U  /  V ) ) )  =  ( Re `  ( U  x.  (
* `  V )
) ) )
5438, 53eqtrd 2444 . . . . 5  |-  ( ph  ->  ( ( Re `  ( U  /  V
) )  x.  (
( abs `  V
) ^ 2 ) )  =  ( Re
`  ( U  x.  ( * `  V
) ) ) )
5524, 34, 543eqtr2rd 2451 . . . 4  |-  ( ph  ->  ( Re `  ( U  x.  ( * `  V ) ) )  =  ( ( ( abs `  U )  x.  ( abs `  V
) )  x.  (
( Re `  ( U  /  V ) )  /  ( abs `  ( U  /  V ) ) ) ) )
5655oveq2d 6064 . . 3  |-  ( ph  ->  ( 2  x.  (
Re `  ( U  x.  ( * `  V
) ) ) )  =  ( 2  x.  ( ( ( abs `  U )  x.  ( abs `  V ) )  x.  ( ( Re
`  ( U  /  V ) )  / 
( abs `  ( U  /  V ) ) ) ) ) )
5756oveq2d 6064 . 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 2444 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 177    = wceq 1649    e. wcel 1721    =/= wne 2575   ` cfv 5421  (class class class)co 6048   CCcc 8952   0cc0 8954   1c1 8955    + caddc 8957    x. cmul 8959    - cmin 9255    / cdiv 9641   2c2 10013   ^cexp 11345   *ccj 11864   Recre 11865   abscabs 12002
This theorem is referenced by:  lawcos  20619
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031  ax-pre-sup 9032
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-er 6872  df-en 7077  df-dom 7078  df-sdom 7079  df-sup 7412  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-div 9642  df-nn 9965  df-2 10022  df-3 10023  df-n0 10186  df-z 10247  df-uz 10453  df-rp 10577  df-seq 11287  df-exp 11346  df-cj 11867  df-re 11868  df-im 11869  df-sqr 12003  df-abs 12004
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