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Theorem angvald 20118
Description: The (signed) angle between two vectors is the argument of their quotient. Deduction form of angval 20115. (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 20115 . 2  |-  ( ( ( X  e.  CC  /\  X  =/=  0 )  /\  ( Y  e.  CC  /\  Y  =/=  0 ) )  -> 
( X F Y )  =  ( Im
`  ( log `  ( Y  /  X ) ) ) )
71, 2, 3, 4, 6syl22anc 1183 1  |-  ( ph  ->  ( X F Y )  =  ( Im
`  ( log `  ( Y  /  X ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696    =/= wne 2459    \ cdif 3162   {csn 3653   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   CCcc 8751   0cc0 8753    / cdiv 9439   Imcim 11599   logclog 19928
This theorem is referenced by:  angcld  20119  angrteqvd  20120  cosangneg2d  20121  angpieqvdlem2  20142
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879
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