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Theorem angvald 20648
 Description: The (signed) angle between two vectors is the argument of their quotient. Deduction form of angval 20645. (Contributed by David Moews, 28-Feb-2017.)
Hypotheses
Ref Expression
ang.1
angvald.1
angvald.2
angvald.3
angvald.4
Assertion
Ref Expression
angvald
Distinct variable groups:   ,,   ,,
Allowed substitution hints:   (,)   (,)

Proof of Theorem angvald
StepHypRef Expression
1 angvald.1 . 2
2 angvald.2 . 2
3 angvald.3 . 2
4 angvald.4 . 2
5 ang.1 . . 3
65angval 20645 . 2
71, 2, 3, 4, 6syl22anc 1186 1
 Colors of variables: wff set class Syntax hints:   wi 4   wceq 1653   wcel 1726   wne 2601   cdif 3319  csn 3816  cfv 5456  (class class class)co 6083   cmpt2 6085  cc 8990  cc0 8992   cdiv 9679  cim 11905  clog 20454 This theorem is referenced by:  angcld  20649  angrteqvd  20650  cosangneg2d  20651  ang180lem4  20656  lawcos  20660  isosctrlem3  20666  angpieqvdlem2  20672 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-iota 5420  df-fun 5458  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088
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