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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  |-  F  =  ( x  e.  ( CC  \  { 0 } ) ,  y  e.  ( CC  \  { 0 } ) 
|->  ( Im `  ( log `  ( y  /  x ) ) ) )
angvald.1  |-  ( ph  ->  X  e.  CC )
angvald.2  |-  ( ph  ->  X  =/=  0 )
angvald.3  |-  ( ph  ->  Y  e.  CC )
angvald.4  |-  ( ph  ->  Y  =/=  0 )
Assertion
Ref Expression
angvald  |-  ( ph  ->  ( X F Y )  =  ( Im
`  ( log `  ( Y  /  X ) ) ) )
Distinct variable groups:    x, y, X    x, Y, y
Allowed substitution hints:    ph( x, y)    F( x, y)

Proof of Theorem angvald
StepHypRef Expression
1 angvald.1 . 2  |-  ( ph  ->  X  e.  CC )
2 angvald.2 . 2  |-  ( ph  ->  X  =/=  0 )
3 angvald.3 . 2  |-  ( ph  ->  Y  e.  CC )
4 angvald.4 . 2  |-  ( ph  ->  Y  =/=  0 )
5 ang.1 . . 3  |-  F  =  ( x  e.  ( CC  \  { 0 } ) ,  y  e.  ( CC  \  { 0 } ) 
|->  ( Im `  ( log `  ( y  /  x ) ) ) )
65angval 22999 . 2  |-  ( ( ( X  e.  CC  /\  X  =/=  0 )  /\  ( Y  e.  CC  /\  Y  =/=  0 ) )  -> 
( X F Y )  =  ( Im
`  ( log `  ( Y  /  X ) ) ) )
71, 2, 3, 4, 6syl22anc 1229 1  |-  ( ph  ->  ( X F Y )  =  ( Im
`  ( log `  ( Y  /  X ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767    =/= wne 2662    \ cdif 3478   {csn 4033   ` cfv 5594  (class class class)co 6295    |-> cmpt2 6297   CCcc 9502   0cc0 9504    / cdiv 10218   Imcim 12911   logclog 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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