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Theorem angval 20047
 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 3709 . 2
2 eldifsn 3709 . 2
3 oveq12 5787 . . . . . 6
43ancoms 441 . . . . 5
54fveq2d 5448 . . . 4
65fveq2d 5448 . . 3
7 ang.1 . . 3
8 fvex 5458 . . 3
96, 7, 8ovmpt2a 5898 . 2
101, 2, 9syl2anbr 468 1
 Colors of variables: wff set class Syntax hints:   wi 6   wa 360   wceq 1619   wcel 1621   wne 2419   cdif 3110  csn 3600  cfv 4659  (class class class)co 5778   cmpt2 5780  cc 8689  cc0 8691   cdiv 9377  cim 11534  clog 19860 This theorem is referenced by:  angcan  20048  angvald  20050  ang180lem4  20058  lawcos  20062  isosctrlem3  20068 This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4101  ax-nul 4109  ax-pr 4172  ax-un 4470 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2521  df-rex 2522  df-rab 2525  df-v 2759  df-sbc 2953  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3417  df-if 3526  df-sn 3606  df-pr 3607  df-op 3609  df-uni 3788  df-br 3984  df-opab 4038  df-id 4267  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fv 4675  df-ov 5781  df-oprab 5782  df-mpt2 5783
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