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Theorem lawcoslem1 20129
Description: Lemma for Law of Cosines lawcos 20130. 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 11784 . . 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 642 . 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 11953 . . . . . . . 8  |-  ( ph  ->  ( abs `  ( U  /  V ) )  =  ( ( abs `  U )  /  ( abs `  V ) ) )
76oveq2d 5890 . . . . . . 7  |-  ( ph  ->  ( ( Re `  ( U  /  V
) )  /  ( abs `  ( U  /  V ) ) )  =  ( ( Re
`  ( U  /  V ) )  / 
( ( abs `  U
)  /  ( abs `  V ) ) ) )
87oveq2d 5890 . . . . . 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 11934 . . . . . . . . 9  |-  ( ph  ->  ( abs `  U
)  e.  RR )
102abscld 11934 . . . . . . . . 9  |-  ( ph  ->  ( abs `  V
)  e.  RR )
119, 10remulcld 8879 . . . . . . . 8  |-  ( ph  ->  ( ( abs `  U
)  x.  ( abs `  V ) )  e.  RR )
1211recnd 8877 . . . . . . 7  |-  ( ph  ->  ( ( abs `  U
)  x.  ( abs `  V ) )  e.  CC )
131, 2, 5divcld 9552 . . . . . . . . 9  |-  ( ph  ->  ( U  /  V
)  e.  CC )
1413recld 11695 . . . . . . . 8  |-  ( ph  ->  ( Re `  ( U  /  V ) )  e.  RR )
1514recnd 8877 . . . . . . 7  |-  ( ph  ->  ( Re `  ( U  /  V ) )  e.  CC )
169recnd 8877 . . . . . . . 8  |-  ( ph  ->  ( abs `  U
)  e.  CC )
1710recnd 8877 . . . . . . . 8  |-  ( ph  ->  ( abs `  V
)  e.  CC )
182, 5absne0d 11945 . . . . . . . 8  |-  ( ph  ->  ( abs `  V
)  =/=  0 )
1916, 17, 18divcld 9552 . . . . . . 7  |-  ( ph  ->  ( ( abs `  U
)  /  ( abs `  V ) )  e.  CC )
20 lawcoslem1.3 . . . . . . . . 9  |-  ( ph  ->  U  =/=  0 )
211, 20absne0d 11945 . . . . . . . 8  |-  ( ph  ->  ( abs `  U
)  =/=  0 )
2216, 17, 21, 18divne0d 9568 . . . . . . 7  |-  ( ph  ->  ( ( abs `  U
)  /  ( abs `  V ) )  =/=  0 )
2312, 15, 19, 22div12d 9588 . . . . . 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 2328 . . . . 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 9584 . . . . . . 7  |-  ( ph  ->  ( ( ( abs `  U )  x.  ( abs `  V ) )  /  ( ( abs `  U )  /  ( abs `  V ) ) )  =  ( ( ( ( abs `  U
)  x.  ( abs `  V ) )  x.  ( abs `  V
) )  /  ( abs `  U ) ) )
2617sqvald 11258 . . . . . . . . . 10  |-  ( ph  ->  ( ( abs `  V
) ^ 2 )  =  ( ( abs `  V )  x.  ( abs `  V ) ) )
2726oveq1d 5889 . . . . . . . . 9  |-  ( ph  ->  ( ( ( abs `  V ) ^ 2 )  x.  ( abs `  U ) )  =  ( ( ( abs `  V )  x.  ( abs `  V ) )  x.  ( abs `  U
) ) )
2816, 17, 17mul31d 9039 . . . . . . . . 9  |-  ( ph  ->  ( ( ( abs `  U )  x.  ( abs `  V ) )  x.  ( abs `  V
) )  =  ( ( ( abs `  V
)  x.  ( abs `  V ) )  x.  ( abs `  U
) ) )
2927, 28eqtr4d 2331 . . . . . . . 8  |-  ( ph  ->  ( ( ( abs `  V ) ^ 2 )  x.  ( abs `  U ) )  =  ( ( ( abs `  U )  x.  ( abs `  V ) )  x.  ( abs `  V
) ) )
3029oveq1d 5889 . . . . . . 7  |-  ( ph  ->  ( ( ( ( abs `  V ) ^ 2 )  x.  ( abs `  U
) )  /  ( abs `  U ) )  =  ( ( ( ( abs `  U
)  x.  ( abs `  V ) )  x.  ( abs `  V
) )  /  ( abs `  U ) ) )
3117sqcld 11259 . . . . . . . 8  |-  ( ph  ->  ( ( abs `  V
) ^ 2 )  e.  CC )
3231, 16, 21divcan4d 9558 . . . . . . 7  |-  ( ph  ->  ( ( ( ( abs `  V ) ^ 2 )  x.  ( abs `  U
) )  /  ( abs `  U ) )  =  ( ( abs `  V ) ^ 2 ) )
3325, 30, 323eqtr2rd 2335 . . . . . 6  |-  ( ph  ->  ( ( abs `  V
) ^ 2 )  =  ( ( ( abs `  U )  x.  ( abs `  V
) )  /  (
( abs `  U
)  /  ( abs `  V ) ) ) )
3433oveq2d 5890 . . . . 5  |-  ( ph  ->  ( ( Re `  ( U  /  V
) )  x.  (
( abs `  V
) ^ 2 ) )  =  ( ( Re `  ( U  /  V ) )  x.  ( ( ( abs `  U )  x.  ( abs `  V
) )  /  (
( abs `  U
)  /  ( abs `  V ) ) ) ) )
3515, 31mulcomd 8872 . . . . . . 7  |-  ( ph  ->  ( ( Re `  ( U  /  V
) )  x.  (
( abs `  V
) ^ 2 ) )  =  ( ( ( abs `  V
) ^ 2 )  x.  ( Re `  ( U  /  V
) ) ) )
3610resqcld 11287 . . . . . . . 8  |-  ( ph  ->  ( ( abs `  V
) ^ 2 )  e.  RR )
3736, 13remul2d 11728 . . . . . . 7  |-  ( ph  ->  ( Re `  (
( ( abs `  V
) ^ 2 )  x.  ( U  /  V ) ) )  =  ( ( ( abs `  V ) ^ 2 )  x.  ( Re `  ( U  /  V ) ) ) )
3835, 37eqtr4d 2331 . . . . . 6  |-  ( ph  ->  ( ( Re `  ( U  /  V
) )  x.  (
( abs `  V
) ^ 2 ) )  =  ( Re
`  ( ( ( abs `  V ) ^ 2 )  x.  ( U  /  V
) ) ) )
391, 31, 2, 5div12d 9588 . . . . . . . 8  |-  ( ph  ->  ( U  x.  (
( ( abs `  V
) ^ 2 )  /  V ) )  =  ( ( ( abs `  V ) ^ 2 )  x.  ( U  /  V
) ) )
4031, 2, 5divrecd 9555 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( abs `  V ) ^ 2 )  /  V )  =  ( ( ( abs `  V ) ^ 2 )  x.  ( 1  /  V
) ) )
41 recval 11822 . . . . . . . . . . . . 13  |-  ( ( V  e.  CC  /\  V  =/=  0 )  -> 
( 1  /  V
)  =  ( ( * `  V )  /  ( ( abs `  V ) ^ 2 ) ) )
422, 5, 41syl2anc 642 . . . . . . . . . . . 12  |-  ( ph  ->  ( 1  /  V
)  =  ( ( * `  V )  /  ( ( abs `  V ) ^ 2 ) ) )
4342oveq2d 5890 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( abs `  V ) ^ 2 )  x.  ( 1  /  V ) )  =  ( ( ( abs `  V ) ^ 2 )  x.  ( ( * `  V )  /  (
( abs `  V
) ^ 2 ) ) ) )
442cjcld 11697 . . . . . . . . . . . 12  |-  ( ph  ->  ( * `  V
)  e.  CC )
45 sqne0 11186 . . . . . . . . . . . . . 14  |-  ( ( abs `  V )  e.  CC  ->  (
( ( abs `  V
) ^ 2 )  =/=  0  <->  ( abs `  V )  =/=  0
) )
4617, 45syl 15 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( ( abs `  V ) ^ 2 )  =/=  0  <->  ( abs `  V )  =/=  0 ) )
4718, 46mpbird 223 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( abs `  V
) ^ 2 )  =/=  0 )
4844, 31, 47divcan2d 9554 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( abs `  V ) ^ 2 )  x.  ( ( * `  V )  /  ( ( abs `  V ) ^ 2 ) ) )  =  ( * `  V
) )
4943, 48eqtrd 2328 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( abs `  V ) ^ 2 )  x.  ( 1  /  V ) )  =  ( * `  V ) )
5040, 49eqtrd 2328 . . . . . . . . 9  |-  ( ph  ->  ( ( ( abs `  V ) ^ 2 )  /  V )  =  ( * `  V ) )
5150oveq2d 5890 . . . . . . . 8  |-  ( ph  ->  ( U  x.  (
( ( abs `  V
) ^ 2 )  /  V ) )  =  ( U  x.  ( * `  V
) ) )
5239, 51eqtr3d 2330 . . . . . . 7  |-  ( ph  ->  ( ( ( abs `  V ) ^ 2 )  x.  ( U  /  V ) )  =  ( U  x.  ( * `  V
) ) )
5352fveq2d 5545 . . . . . 6  |-  ( ph  ->  ( Re `  (
( ( abs `  V
) ^ 2 )  x.  ( U  /  V ) ) )  =  ( Re `  ( U  x.  (
* `  V )
) ) )
5438, 53eqtrd 2328 . . . . 5  |-  ( ph  ->  ( ( Re `  ( U  /  V
) )  x.  (
( abs `  V
) ^ 2 ) )  =  ( Re
`  ( U  x.  ( * `  V
) ) ) )
5524, 34, 543eqtr2rd 2335 . . . 4  |-  ( ph  ->  ( Re `  ( U  x.  ( * `  V ) ) )  =  ( ( ( abs `  U )  x.  ( abs `  V
) )  x.  (
( Re `  ( U  /  V ) )  /  ( abs `  ( U  /  V ) ) ) ) )
5655oveq2d 5890 . . 3  |-  ( ph  ->  ( 2  x.  (
Re `  ( U  x.  ( * `  V
) ) ) )  =  ( 2  x.  ( ( ( abs `  U )  x.  ( abs `  V ) )  x.  ( ( Re
`  ( U  /  V ) )  / 
( abs `  ( U  /  V ) ) ) ) ) )
5756oveq2d 5890 . 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 2328 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 176    = wceq 1632    e. wcel 1696    =/= wne 2459   ` cfv 5271  (class class class)co 5874   CCcc 8751   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758    - cmin 9053    / cdiv 9439   2c2 9811   ^cexp 11120   *ccj 11597   Recre 11598   abscabs 11735
This theorem is referenced by:  lawcos  20130
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-seq 11063  df-exp 11121  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737
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