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Theorem angvald 23002
 Description: The (signed) angle between two vectors is the argument of their quotient. Deduction form of angval 22999. (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 22999 . 2
71, 2, 3, 4, 6syl22anc 1229 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wceq 1379   wcel 1767   wne 2662   cdif 3478  csn 4033  cfv 5594  (class class class)co 6295   cmpt2 6297  cc 9502  cc0 9504   cdiv 10218  cim 12911  clog 22808 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-iota 5557  df-fun 5596  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300 This theorem is referenced by:  angcld  23003  angrteqvd  23004  cosangneg2d  23005  ang180lem4  23010  lawcos  23014  isosctrlem3  23020  angpieqvdlem2  23026
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