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Theorem lawcoslem1 23346
Description: Lemma for lawcos 23347. 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 13198 . . 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 659 . 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 13368 . . . . . . . 8  |-  ( ph  ->  ( abs `  ( U  /  V ) )  =  ( ( abs `  U )  /  ( abs `  V ) ) )
76oveq2d 6286 . . . . . . 7  |-  ( ph  ->  ( ( Re `  ( U  /  V
) )  /  ( abs `  ( U  /  V ) ) )  =  ( ( Re
`  ( U  /  V ) )  / 
( ( abs `  U
)  /  ( abs `  V ) ) ) )
87oveq2d 6286 . . . . . 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 13349 . . . . . . . . 9  |-  ( ph  ->  ( abs `  U
)  e.  RR )
102abscld 13349 . . . . . . . . 9  |-  ( ph  ->  ( abs `  V
)  e.  RR )
119, 10remulcld 9613 . . . . . . . 8  |-  ( ph  ->  ( ( abs `  U
)  x.  ( abs `  V ) )  e.  RR )
1211recnd 9611 . . . . . . 7  |-  ( ph  ->  ( ( abs `  U
)  x.  ( abs `  V ) )  e.  CC )
131, 2, 5divcld 10316 . . . . . . . . 9  |-  ( ph  ->  ( U  /  V
)  e.  CC )
1413recld 13109 . . . . . . . 8  |-  ( ph  ->  ( Re `  ( U  /  V ) )  e.  RR )
1514recnd 9611 . . . . . . 7  |-  ( ph  ->  ( Re `  ( U  /  V ) )  e.  CC )
169recnd 9611 . . . . . . . 8  |-  ( ph  ->  ( abs `  U
)  e.  CC )
1710recnd 9611 . . . . . . . 8  |-  ( ph  ->  ( abs `  V
)  e.  CC )
182, 5absne0d 13360 . . . . . . . 8  |-  ( ph  ->  ( abs `  V
)  =/=  0 )
1916, 17, 18divcld 10316 . . . . . . 7  |-  ( ph  ->  ( ( abs `  U
)  /  ( abs `  V ) )  e.  CC )
20 lawcoslem1.3 . . . . . . . . 9  |-  ( ph  ->  U  =/=  0 )
211, 20absne0d 13360 . . . . . . . 8  |-  ( ph  ->  ( abs `  U
)  =/=  0 )
2216, 17, 21, 18divne0d 10332 . . . . . . 7  |-  ( ph  ->  ( ( abs `  U
)  /  ( abs `  V ) )  =/=  0 )
2312, 15, 19, 22div12d 10352 . . . . . 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 2495 . . . . 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 10348 . . . . . . 7  |-  ( ph  ->  ( ( ( abs `  U )  x.  ( abs `  V ) )  /  ( ( abs `  U )  /  ( abs `  V ) ) )  =  ( ( ( ( abs `  U
)  x.  ( abs `  V ) )  x.  ( abs `  V
) )  /  ( abs `  U ) ) )
2617sqvald 12289 . . . . . . . . . 10  |-  ( ph  ->  ( ( abs `  V
) ^ 2 )  =  ( ( abs `  V )  x.  ( abs `  V ) ) )
2726oveq1d 6285 . . . . . . . . 9  |-  ( ph  ->  ( ( ( abs `  V ) ^ 2 )  x.  ( abs `  U ) )  =  ( ( ( abs `  V )  x.  ( abs `  V ) )  x.  ( abs `  U
) ) )
2816, 17, 17mul31d 9780 . . . . . . . . 9  |-  ( ph  ->  ( ( ( abs `  U )  x.  ( abs `  V ) )  x.  ( abs `  V
) )  =  ( ( ( abs `  V
)  x.  ( abs `  V ) )  x.  ( abs `  U
) ) )
2927, 28eqtr4d 2498 . . . . . . . 8  |-  ( ph  ->  ( ( ( abs `  V ) ^ 2 )  x.  ( abs `  U ) )  =  ( ( ( abs `  U )  x.  ( abs `  V ) )  x.  ( abs `  V
) ) )
3029oveq1d 6285 . . . . . . 7  |-  ( ph  ->  ( ( ( ( abs `  V ) ^ 2 )  x.  ( abs `  U
) )  /  ( abs `  U ) )  =  ( ( ( ( abs `  U
)  x.  ( abs `  V ) )  x.  ( abs `  V
) )  /  ( abs `  U ) ) )
3117sqcld 12290 . . . . . . . 8  |-  ( ph  ->  ( ( abs `  V
) ^ 2 )  e.  CC )
3231, 16, 21divcan4d 10322 . . . . . . 7  |-  ( ph  ->  ( ( ( ( abs `  V ) ^ 2 )  x.  ( abs `  U
) )  /  ( abs `  U ) )  =  ( ( abs `  V ) ^ 2 ) )
3325, 30, 323eqtr2rd 2502 . . . . . 6  |-  ( ph  ->  ( ( abs `  V
) ^ 2 )  =  ( ( ( abs `  U )  x.  ( abs `  V
) )  /  (
( abs `  U
)  /  ( abs `  V ) ) ) )
3433oveq2d 6286 . . . . 5  |-  ( ph  ->  ( ( Re `  ( U  /  V
) )  x.  (
( abs `  V
) ^ 2 ) )  =  ( ( Re `  ( U  /  V ) )  x.  ( ( ( abs `  U )  x.  ( abs `  V
) )  /  (
( abs `  U
)  /  ( abs `  V ) ) ) ) )
3515, 31mulcomd 9606 . . . . . . 7  |-  ( ph  ->  ( ( Re `  ( U  /  V
) )  x.  (
( abs `  V
) ^ 2 ) )  =  ( ( ( abs `  V
) ^ 2 )  x.  ( Re `  ( U  /  V
) ) ) )
3610resqcld 12318 . . . . . . . 8  |-  ( ph  ->  ( ( abs `  V
) ^ 2 )  e.  RR )
3736, 13remul2d 13142 . . . . . . 7  |-  ( ph  ->  ( Re `  (
( ( abs `  V
) ^ 2 )  x.  ( U  /  V ) ) )  =  ( ( ( abs `  V ) ^ 2 )  x.  ( Re `  ( U  /  V ) ) ) )
3835, 37eqtr4d 2498 . . . . . 6  |-  ( ph  ->  ( ( Re `  ( U  /  V
) )  x.  (
( abs `  V
) ^ 2 ) )  =  ( Re
`  ( ( ( abs `  V ) ^ 2 )  x.  ( U  /  V
) ) ) )
391, 31, 2, 5div12d 10352 . . . . . . . 8  |-  ( ph  ->  ( U  x.  (
( ( abs `  V
) ^ 2 )  /  V ) )  =  ( ( ( abs `  V ) ^ 2 )  x.  ( U  /  V
) ) )
4031, 2, 5divrecd 10319 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( abs `  V ) ^ 2 )  /  V )  =  ( ( ( abs `  V ) ^ 2 )  x.  ( 1  /  V
) ) )
41 recval 13237 . . . . . . . . . . . . 13  |-  ( ( V  e.  CC  /\  V  =/=  0 )  -> 
( 1  /  V
)  =  ( ( * `  V )  /  ( ( abs `  V ) ^ 2 ) ) )
422, 5, 41syl2anc 659 . . . . . . . . . . . 12  |-  ( ph  ->  ( 1  /  V
)  =  ( ( * `  V )  /  ( ( abs `  V ) ^ 2 ) ) )
4342oveq2d 6286 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( abs `  V ) ^ 2 )  x.  ( 1  /  V ) )  =  ( ( ( abs `  V ) ^ 2 )  x.  ( ( * `  V )  /  (
( abs `  V
) ^ 2 ) ) ) )
442cjcld 13111 . . . . . . . . . . . 12  |-  ( ph  ->  ( * `  V
)  e.  CC )
45 sqne0 12216 . . . . . . . . . . . . . 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 10318 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( abs `  V ) ^ 2 )  x.  ( ( * `  V )  /  ( ( abs `  V ) ^ 2 ) ) )  =  ( * `  V
) )
4943, 48eqtrd 2495 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( abs `  V ) ^ 2 )  x.  ( 1  /  V ) )  =  ( * `  V ) )
5040, 49eqtrd 2495 . . . . . . . . 9  |-  ( ph  ->  ( ( ( abs `  V ) ^ 2 )  /  V )  =  ( * `  V ) )
5150oveq2d 6286 . . . . . . . 8  |-  ( ph  ->  ( U  x.  (
( ( abs `  V
) ^ 2 )  /  V ) )  =  ( U  x.  ( * `  V
) ) )
5239, 51eqtr3d 2497 . . . . . . 7  |-  ( ph  ->  ( ( ( abs `  V ) ^ 2 )  x.  ( U  /  V ) )  =  ( U  x.  ( * `  V
) ) )
5352fveq2d 5852 . . . . . 6  |-  ( ph  ->  ( Re `  (
( ( abs `  V
) ^ 2 )  x.  ( U  /  V ) ) )  =  ( Re `  ( U  x.  (
* `  V )
) ) )
5438, 53eqtrd 2495 . . . . 5  |-  ( ph  ->  ( ( Re `  ( U  /  V
) )  x.  (
( abs `  V
) ^ 2 ) )  =  ( Re
`  ( U  x.  ( * `  V
) ) ) )
5524, 34, 543eqtr2rd 2502 . . . 4  |-  ( ph  ->  ( Re `  ( U  x.  ( * `  V ) ) )  =  ( ( ( abs `  U )  x.  ( abs `  V
) )  x.  (
( Re `  ( U  /  V ) )  /  ( abs `  ( U  /  V ) ) ) ) )
5655oveq2d 6286 . . 3  |-  ( ph  ->  ( 2  x.  (
Re `  ( U  x.  ( * `  V
) ) ) )  =  ( 2  x.  ( ( ( abs `  U )  x.  ( abs `  V ) )  x.  ( ( Re
`  ( U  /  V ) )  / 
( abs `  ( U  /  V ) ) ) ) ) )
5756oveq2d 6286 . 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 2495 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 1398    e. wcel 1823    =/= wne 2649   ` cfv 5570  (class class class)co 6270   CCcc 9479   0cc0 9481   1c1 9482    + caddc 9484    x. cmul 9486    - cmin 9796    / cdiv 10202   2c2 10581   ^cexp 12148   *ccj 13011   Recre 13012   abscabs 13149
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-2nd 6774  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-sup 7893  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11083  df-rp 11222  df-seq 12090  df-exp 12149  df-cj 13014  df-re 13015  df-im 13016  df-sqrt 13150  df-abs 13151
This theorem is referenced by:  lawcos  23347
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