Theorem lawcoslem1 24545

Description: Lemma for lawcos 24546. 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

Step	Hyp	Ref	Expression
1		lawcoslem1.1	. . 3 |- ( ph -> U e. CC )
2		lawcoslem1.2	. . 3 |- ( ph -> V e. CC )
3		sqabssub 14023	. . 3 |- ( ( U e. CC /\ V e. CC ) -> ( ( abs ` ( U - V ) ) ^ 2 ) = ( ( ( ( abs ` U ) ^ 2 ) + ( ( abs ` V ) ^ 2 ) ) - ( 2 x. ( Re ` ( U x. ( * ` V ) ) ) ) ) )
4	1, 2, 3	syl2anc 693	. 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 )
6	1, 2, 5	absdivd 14194	. . . . . . . 8 |- ( ph -> ( abs ` ( U / V ) ) = ( ( abs ` U ) / ( abs ` V ) ) )
7	6	oveq2d 6666	. . . . . . 7 |- ( ph -> ( ( Re ` ( U / V ) ) / ( abs ` ( U / V ) ) ) = ( ( Re ` ( U / V ) ) / ( ( abs ` U ) / ( abs ` V ) ) ) )
8	7	oveq2d 6666	. . . . . 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 ) ) ) ) )
9	1	abscld 14175	. . . . . . . . 9 |- ( ph -> ( abs ` U ) e. RR )
10	2	abscld 14175	. . . . . . . . 9 |- ( ph -> ( abs ` V ) e. RR )
11	9, 10	remulcld 10070	. . . . . . . 8 |- ( ph -> ( ( abs ` U ) x. ( abs ` V ) ) e. RR )
12	11	recnd 10068	. . . . . . 7 |- ( ph -> ( ( abs ` U ) x. ( abs ` V ) ) e. CC )
13	1, 2, 5	divcld 10801	. . . . . . . . 9 |- ( ph -> ( U / V ) e. CC )
14	13	recld 13934	. . . . . . . 8 |- ( ph -> ( Re ` ( U / V ) ) e. RR )
15	14	recnd 10068	. . . . . . 7 |- ( ph -> ( Re ` ( U / V ) ) e. CC )
16	9	recnd 10068	. . . . . . . 8 |- ( ph -> ( abs ` U ) e. CC )
17	10	recnd 10068	. . . . . . . 8 |- ( ph -> ( abs ` V ) e. CC )
18	2, 5	absne0d 14186	. . . . . . . 8 |- ( ph -> ( abs ` V ) =/= 0 )
19	16, 17, 18	divcld 10801	. . . . . . 7 |- ( ph -> ( ( abs ` U ) / ( abs ` V ) ) e. CC )
20		lawcoslem1.3	. . . . . . . . 9 |- ( ph -> U =/= 0 )
21	1, 20	absne0d 14186	. . . . . . . 8 |- ( ph -> ( abs ` U ) =/= 0 )
22	16, 17, 21, 18	divne0d 10817	. . . . . . 7 |- ( ph -> ( ( abs ` U ) / ( abs ` V ) ) =/= 0 )
23	12, 15, 19, 22	div12d 10837	. . . . . 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 ) ) ) ) )
24	8, 23	eqtrd 2656	. . . . 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 ) ) ) ) )
25	12, 16, 17, 21, 18	divdiv2d 10833	. . . . . . 7 |- ( ph -> ( ( ( abs ` U ) x. ( abs ` V ) ) / ( ( abs ` U ) / ( abs ` V ) ) ) = ( ( ( ( abs ` U ) x. ( abs ` V ) ) x. ( abs ` V ) ) / ( abs ` U ) ) )
26	17	sqvald 13005	. . . . . . . . . 10 |- ( ph -> ( ( abs ` V ) ^ 2 ) = ( ( abs ` V ) x. ( abs ` V ) ) )
27	26	oveq1d 6665	. . . . . . . . 9 |- ( ph -> ( ( ( abs ` V ) ^ 2 ) x. ( abs ` U ) ) = ( ( ( abs ` V ) x. ( abs ` V ) ) x. ( abs ` U ) ) )
28	16, 17, 17	mul31d 10247	. . . . . . . . 9 |- ( ph -> ( ( ( abs ` U ) x. ( abs ` V ) ) x. ( abs ` V ) ) = ( ( ( abs ` V ) x. ( abs ` V ) ) x. ( abs ` U ) ) )
29	27, 28	eqtr4d 2659	. . . . . . . 8 |- ( ph -> ( ( ( abs ` V ) ^ 2 ) x. ( abs ` U ) ) = ( ( ( abs ` U ) x. ( abs ` V ) ) x. ( abs ` V ) ) )
30	29	oveq1d 6665	. . . . . . 7 |- ( ph -> ( ( ( ( abs ` V ) ^ 2 ) x. ( abs ` U ) ) / ( abs ` U ) ) = ( ( ( ( abs ` U ) x. ( abs ` V ) ) x. ( abs ` V ) ) / ( abs ` U ) ) )
31	17	sqcld 13006	. . . . . . . 8 |- ( ph -> ( ( abs ` V ) ^ 2 ) e. CC )
32	31, 16, 21	divcan4d 10807	. . . . . . 7 |- ( ph -> ( ( ( ( abs ` V ) ^ 2 ) x. ( abs ` U ) ) / ( abs ` U ) ) = ( ( abs ` V ) ^ 2 ) )
33	25, 30, 32	3eqtr2rd 2663	. . . . . 6 |- ( ph -> ( ( abs ` V ) ^ 2 ) = ( ( ( abs ` U ) x. ( abs ` V ) ) / ( ( abs ` U ) / ( abs ` V ) ) ) )
34	33	oveq2d 6666	. . . . 5 |- ( ph -> ( ( Re ` ( U / V ) ) x. ( ( abs ` V ) ^ 2 ) ) = ( ( Re ` ( U / V ) ) x. ( ( ( abs ` U ) x. ( abs ` V ) ) / ( ( abs ` U ) / ( abs ` V ) ) ) ) )
35	15, 31	mulcomd 10061	. . . . . . 7 |- ( ph -> ( ( Re ` ( U / V ) ) x. ( ( abs ` V ) ^ 2 ) ) = ( ( ( abs ` V ) ^ 2 ) x. ( Re ` ( U / V ) ) ) )
36	10	resqcld 13035	. . . . . . . 8 |- ( ph -> ( ( abs ` V ) ^ 2 ) e. RR )
37	36, 13	remul2d 13967	. . . . . . 7 |- ( ph -> ( Re ` ( ( ( abs ` V ) ^ 2 ) x. ( U / V ) ) ) = ( ( ( abs ` V ) ^ 2 ) x. ( Re ` ( U / V ) ) ) )
38	35, 37	eqtr4d 2659	. . . . . 6 |- ( ph -> ( ( Re ` ( U / V ) ) x. ( ( abs ` V ) ^ 2 ) ) = ( Re ` ( ( ( abs ` V ) ^ 2 ) x. ( U / V ) ) ) )
39	1, 31, 2, 5	div12d 10837	. . . . . . . 8 |- ( ph -> ( U x. ( ( ( abs ` V ) ^ 2 ) / V ) ) = ( ( ( abs ` V ) ^ 2 ) x. ( U / V ) ) )
40	31, 2, 5	divrecd 10804	. . . . . . . . . 10 |- ( ph -> ( ( ( abs ` V ) ^ 2 ) / V ) = ( ( ( abs ` V ) ^ 2 ) x. ( 1 / V ) ) )
41		recval 14062	. . . . . . . . . . . . 13 |- ( ( V e. CC /\ V =/= 0 ) -> ( 1 / V ) = ( ( * ` V ) / ( ( abs ` V ) ^ 2 ) ) )
42	2, 5, 41	syl2anc 693	. . . . . . . . . . . 12 |- ( ph -> ( 1 / V ) = ( ( * ` V ) / ( ( abs ` V ) ^ 2 ) ) )
43	42	oveq2d 6666	. . . . . . . . . . 11 |- ( ph -> ( ( ( abs ` V ) ^ 2 ) x. ( 1 / V ) ) = ( ( ( abs ` V ) ^ 2 ) x. ( ( * ` V ) / ( ( abs ` V ) ^ 2 ) ) ) )
44	2	cjcld 13936	. . . . . . . . . . . 12 |- ( ph -> ( * ` V ) e. CC )
45		sqne0 12930	. . . . . . . . . . . . . 14 |- ( ( abs ` V ) e. CC -> ( ( ( abs ` V ) ^ 2 ) =/= 0 <-> ( abs ` V ) =/= 0 ) )
46	17, 45	syl 17	. . . . . . . . . . . . 13 |- ( ph -> ( ( ( abs ` V ) ^ 2 ) =/= 0 <-> ( abs ` V ) =/= 0 ) )
47	18, 46	mpbird 247	. . . . . . . . . . . 12 |- ( ph -> ( ( abs ` V ) ^ 2 ) =/= 0 )
48	44, 31, 47	divcan2d 10803	. . . . . . . . . . 11 |- ( ph -> ( ( ( abs ` V ) ^ 2 ) x. ( ( * ` V ) / ( ( abs ` V ) ^ 2 ) ) ) = ( * ` V ) )
49	43, 48	eqtrd 2656	. . . . . . . . . 10 |- ( ph -> ( ( ( abs ` V ) ^ 2 ) x. ( 1 / V ) ) = ( * ` V ) )
50	40, 49	eqtrd 2656	. . . . . . . . 9 |- ( ph -> ( ( ( abs ` V ) ^ 2 ) / V ) = ( * ` V ) )
51	50	oveq2d 6666	. . . . . . . 8 |- ( ph -> ( U x. ( ( ( abs ` V ) ^ 2 ) / V ) ) = ( U x. ( * ` V ) ) )
52	39, 51	eqtr3d 2658	. . . . . . 7 |- ( ph -> ( ( ( abs ` V ) ^ 2 ) x. ( U / V ) ) = ( U x. ( * ` V ) ) )
53	52	fveq2d 6195	. . . . . 6 |- ( ph -> ( Re ` ( ( ( abs ` V ) ^ 2 ) x. ( U / V ) ) ) = ( Re ` ( U x. ( * ` V ) ) ) )
54	38, 53	eqtrd 2656	. . . . 5 |- ( ph -> ( ( Re ` ( U / V ) ) x. ( ( abs ` V ) ^ 2 ) ) = ( Re ` ( U x. ( * ` V ) ) ) )
55	24, 34, 54	3eqtr2rd 2663	. . . 4 |- ( ph -> ( Re ` ( U x. ( * ` V ) ) ) = ( ( ( abs ` U ) x. ( abs ` V ) ) x. ( ( Re ` ( U / V ) ) / ( abs ` ( U / V ) ) ) ) )
56	55	oveq2d 6666	. . 3 |- ( ph -> ( 2 x. ( Re ` ( U x. ( * ` V ) ) ) ) = ( 2 x. ( ( ( abs ` U ) x. ( abs ` V ) ) x. ( ( Re ` ( U / V ) ) / ( abs ` ( U / V ) ) ) ) ) )
57	56	oveq2d 6666	. 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 ) ) ) ) ) ) )
58	4, 57	eqtrd 2656	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 ) ) ) ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   =/= wne 2794   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   - cmin 10266   / cdiv 10684   2c2 11070   ^cexp 12860   *ccj 13836   Recre 13837   abscabs 13974

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976

This theorem is referenced by:  lawcos  24546