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Theorem angval 20099
 Description: Define the angle function, which takes two complex numbers, treated as vectors from the origin, and returns the angle between them, in the range . To convert from the geometry notation, , the measure of the angle with legs , where is more counterclockwise for positive angles, is represented by . (Contributed by Mario Carneiro, 23-Sep-2014.)
Hypothesis
Ref Expression
ang.1
Assertion
Ref Expression
angval
Distinct variable groups:   ,,   ,,
Allowed substitution hints:   (,)

Proof of Theorem angval
StepHypRef Expression
1 eldifsn 3749 . 2
2 eldifsn 3749 . 2
3 oveq12 5867 . . . . . 6
43ancoms 439 . . . . 5
54fveq2d 5529 . . . 4
65fveq2d 5529 . . 3
7 ang.1 . . 3
8 fvex 5539 . . 3
96, 7, 8ovmpt2a 5978 . 2
101, 2, 9syl2anbr 466 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 358   wceq 1623   wcel 1684   wne 2446   cdif 3149  csn 3640  cfv 5255  (class class class)co 5858   cmpt2 5860  cc 8735  cc0 8737   cdiv 9423  cim 11583  clog 19912 This theorem is referenced by:  angcan  20100  angvald  20102  ang180lem4  20110  lawcos  20114  isosctrlem3  20120 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863
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