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Theorem lawcoslem1 22972
Description: Lemma for lawcos 22973. 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 13082 . . 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 661 . 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 13252 . . . . . . . 8  |-  ( ph  ->  ( abs `  ( U  /  V ) )  =  ( ( abs `  U )  /  ( abs `  V ) ) )
76oveq2d 6301 . . . . . . 7  |-  ( ph  ->  ( ( Re `  ( U  /  V
) )  /  ( abs `  ( U  /  V ) ) )  =  ( ( Re
`  ( U  /  V ) )  / 
( ( abs `  U
)  /  ( abs `  V ) ) ) )
87oveq2d 6301 . . . . . 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 13233 . . . . . . . . 9  |-  ( ph  ->  ( abs `  U
)  e.  RR )
102abscld 13233 . . . . . . . . 9  |-  ( ph  ->  ( abs `  V
)  e.  RR )
119, 10remulcld 9625 . . . . . . . 8  |-  ( ph  ->  ( ( abs `  U
)  x.  ( abs `  V ) )  e.  RR )
1211recnd 9623 . . . . . . 7  |-  ( ph  ->  ( ( abs `  U
)  x.  ( abs `  V ) )  e.  CC )
131, 2, 5divcld 10321 . . . . . . . . 9  |-  ( ph  ->  ( U  /  V
)  e.  CC )
1413recld 12993 . . . . . . . 8  |-  ( ph  ->  ( Re `  ( U  /  V ) )  e.  RR )
1514recnd 9623 . . . . . . 7  |-  ( ph  ->  ( Re `  ( U  /  V ) )  e.  CC )
169recnd 9623 . . . . . . . 8  |-  ( ph  ->  ( abs `  U
)  e.  CC )
1710recnd 9623 . . . . . . . 8  |-  ( ph  ->  ( abs `  V
)  e.  CC )
182, 5absne0d 13244 . . . . . . . 8  |-  ( ph  ->  ( abs `  V
)  =/=  0 )
1916, 17, 18divcld 10321 . . . . . . 7  |-  ( ph  ->  ( ( abs `  U
)  /  ( abs `  V ) )  e.  CC )
20 lawcoslem1.3 . . . . . . . . 9  |-  ( ph  ->  U  =/=  0 )
211, 20absne0d 13244 . . . . . . . 8  |-  ( ph  ->  ( abs `  U
)  =/=  0 )
2216, 17, 21, 18divne0d 10337 . . . . . . 7  |-  ( ph  ->  ( ( abs `  U
)  /  ( abs `  V ) )  =/=  0 )
2312, 15, 19, 22div12d 10357 . . . . . 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 2508 . . . . 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 10353 . . . . . . 7  |-  ( ph  ->  ( ( ( abs `  U )  x.  ( abs `  V ) )  /  ( ( abs `  U )  /  ( abs `  V ) ) )  =  ( ( ( ( abs `  U
)  x.  ( abs `  V ) )  x.  ( abs `  V
) )  /  ( abs `  U ) ) )
2617sqvald 12276 . . . . . . . . . 10  |-  ( ph  ->  ( ( abs `  V
) ^ 2 )  =  ( ( abs `  V )  x.  ( abs `  V ) ) )
2726oveq1d 6300 . . . . . . . . 9  |-  ( ph  ->  ( ( ( abs `  V ) ^ 2 )  x.  ( abs `  U ) )  =  ( ( ( abs `  V )  x.  ( abs `  V ) )  x.  ( abs `  U
) ) )
2816, 17, 17mul31d 9791 . . . . . . . . 9  |-  ( ph  ->  ( ( ( abs `  U )  x.  ( abs `  V ) )  x.  ( abs `  V
) )  =  ( ( ( abs `  V
)  x.  ( abs `  V ) )  x.  ( abs `  U
) ) )
2927, 28eqtr4d 2511 . . . . . . . 8  |-  ( ph  ->  ( ( ( abs `  V ) ^ 2 )  x.  ( abs `  U ) )  =  ( ( ( abs `  U )  x.  ( abs `  V ) )  x.  ( abs `  V
) ) )
3029oveq1d 6300 . . . . . . 7  |-  ( ph  ->  ( ( ( ( abs `  V ) ^ 2 )  x.  ( abs `  U
) )  /  ( abs `  U ) )  =  ( ( ( ( abs `  U
)  x.  ( abs `  V ) )  x.  ( abs `  V
) )  /  ( abs `  U ) ) )
3117sqcld 12277 . . . . . . . 8  |-  ( ph  ->  ( ( abs `  V
) ^ 2 )  e.  CC )
3231, 16, 21divcan4d 10327 . . . . . . 7  |-  ( ph  ->  ( ( ( ( abs `  V ) ^ 2 )  x.  ( abs `  U
) )  /  ( abs `  U ) )  =  ( ( abs `  V ) ^ 2 ) )
3325, 30, 323eqtr2rd 2515 . . . . . 6  |-  ( ph  ->  ( ( abs `  V
) ^ 2 )  =  ( ( ( abs `  U )  x.  ( abs `  V
) )  /  (
( abs `  U
)  /  ( abs `  V ) ) ) )
3433oveq2d 6301 . . . . 5  |-  ( ph  ->  ( ( Re `  ( U  /  V
) )  x.  (
( abs `  V
) ^ 2 ) )  =  ( ( Re `  ( U  /  V ) )  x.  ( ( ( abs `  U )  x.  ( abs `  V
) )  /  (
( abs `  U
)  /  ( abs `  V ) ) ) ) )
3515, 31mulcomd 9618 . . . . . . 7  |-  ( ph  ->  ( ( Re `  ( U  /  V
) )  x.  (
( abs `  V
) ^ 2 ) )  =  ( ( ( abs `  V
) ^ 2 )  x.  ( Re `  ( U  /  V
) ) ) )
3610resqcld 12305 . . . . . . . 8  |-  ( ph  ->  ( ( abs `  V
) ^ 2 )  e.  RR )
3736, 13remul2d 13026 . . . . . . 7  |-  ( ph  ->  ( Re `  (
( ( abs `  V
) ^ 2 )  x.  ( U  /  V ) ) )  =  ( ( ( abs `  V ) ^ 2 )  x.  ( Re `  ( U  /  V ) ) ) )
3835, 37eqtr4d 2511 . . . . . 6  |-  ( ph  ->  ( ( Re `  ( U  /  V
) )  x.  (
( abs `  V
) ^ 2 ) )  =  ( Re
`  ( ( ( abs `  V ) ^ 2 )  x.  ( U  /  V
) ) ) )
391, 31, 2, 5div12d 10357 . . . . . . . 8  |-  ( ph  ->  ( U  x.  (
( ( abs `  V
) ^ 2 )  /  V ) )  =  ( ( ( abs `  V ) ^ 2 )  x.  ( U  /  V
) ) )
4031, 2, 5divrecd 10324 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( abs `  V ) ^ 2 )  /  V )  =  ( ( ( abs `  V ) ^ 2 )  x.  ( 1  /  V
) ) )
41 recval 13121 . . . . . . . . . . . . 13  |-  ( ( V  e.  CC  /\  V  =/=  0 )  -> 
( 1  /  V
)  =  ( ( * `  V )  /  ( ( abs `  V ) ^ 2 ) ) )
422, 5, 41syl2anc 661 . . . . . . . . . . . 12  |-  ( ph  ->  ( 1  /  V
)  =  ( ( * `  V )  /  ( ( abs `  V ) ^ 2 ) ) )
4342oveq2d 6301 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( abs `  V ) ^ 2 )  x.  ( 1  /  V ) )  =  ( ( ( abs `  V ) ^ 2 )  x.  ( ( * `  V )  /  (
( abs `  V
) ^ 2 ) ) ) )
442cjcld 12995 . . . . . . . . . . . 12  |-  ( ph  ->  ( * `  V
)  e.  CC )
45 sqne0 12203 . . . . . . . . . . . . . 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 232 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( abs `  V
) ^ 2 )  =/=  0 )
4844, 31, 47divcan2d 10323 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( abs `  V ) ^ 2 )  x.  ( ( * `  V )  /  ( ( abs `  V ) ^ 2 ) ) )  =  ( * `  V
) )
4943, 48eqtrd 2508 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( abs `  V ) ^ 2 )  x.  ( 1  /  V ) )  =  ( * `  V ) )
5040, 49eqtrd 2508 . . . . . . . . 9  |-  ( ph  ->  ( ( ( abs `  V ) ^ 2 )  /  V )  =  ( * `  V ) )
5150oveq2d 6301 . . . . . . . 8  |-  ( ph  ->  ( U  x.  (
( ( abs `  V
) ^ 2 )  /  V ) )  =  ( U  x.  ( * `  V
) ) )
5239, 51eqtr3d 2510 . . . . . . 7  |-  ( ph  ->  ( ( ( abs `  V ) ^ 2 )  x.  ( U  /  V ) )  =  ( U  x.  ( * `  V
) ) )
5352fveq2d 5870 . . . . . 6  |-  ( ph  ->  ( Re `  (
( ( abs `  V
) ^ 2 )  x.  ( U  /  V ) ) )  =  ( Re `  ( U  x.  (
* `  V )
) ) )
5438, 53eqtrd 2508 . . . . 5  |-  ( ph  ->  ( ( Re `  ( U  /  V
) )  x.  (
( abs `  V
) ^ 2 ) )  =  ( Re
`  ( U  x.  ( * `  V
) ) ) )
5524, 34, 543eqtr2rd 2515 . . . 4  |-  ( ph  ->  ( Re `  ( U  x.  ( * `  V ) ) )  =  ( ( ( abs `  U )  x.  ( abs `  V
) )  x.  (
( Re `  ( U  /  V ) )  /  ( abs `  ( U  /  V ) ) ) ) )
5655oveq2d 6301 . . 3  |-  ( ph  ->  ( 2  x.  (
Re `  ( U  x.  ( * `  V
) ) ) )  =  ( 2  x.  ( ( ( abs `  U )  x.  ( abs `  V ) )  x.  ( ( Re
`  ( U  /  V ) )  / 
( abs `  ( U  /  V ) ) ) ) ) )
5756oveq2d 6301 . 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 2508 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1379    e. wcel 1767    =/= wne 2662   ` cfv 5588  (class class class)co 6285   CCcc 9491   0cc0 9493   1c1 9494    + caddc 9496    x. cmul 9498    - cmin 9806    / cdiv 10207   2c2 10586   ^cexp 12135   *ccj 12895   Recre 12896   abscabs 13033
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570  ax-pre-sup 9571
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-2nd 6786  df-recs 7043  df-rdg 7077  df-er 7312  df-en 7518  df-dom 7519  df-sdom 7520  df-sup 7902  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-div 10208  df-nn 10538  df-2 10595  df-3 10596  df-n0 10797  df-z 10866  df-uz 11084  df-rp 11222  df-seq 12077  df-exp 12136  df-cj 12898  df-re 12899  df-im 12900  df-sqrt 13034  df-abs 13035
This theorem is referenced by:  lawcos  22973
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