Theorem angvald 24534

Description: The (signed) angle between two vectors is the argument of their quotient. Deduction form of angval 24531. (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

Step	Hyp	Ref	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 ) ) ) )
6	5	angval 24531	. 2 |- ( ( ( X e. CC /\ X =/= 0 ) /\ ( Y e. CC /\ Y =/= 0 ) ) -> ( X F Y ) = ( Im ` ( log ` ( Y / X ) ) ) )
7	1, 2, 3, 4, 6	syl22anc 1327	1 |- ( ph -> ( X F Y ) = ( Im ` ( log ` ( Y / X ) ) ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   =/= wne 2794   \ cdif 3571   {csn 4177   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   CCcc 9934   0cc0 9936   / cdiv 10684   Imcim 13838   logclog 24301

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655

This theorem is referenced by:  angcld  24535  angrteqvd  24536  cosangneg2d  24537  ang180lem4  24542  lawcos  24546  isosctrlem3  24550  angpieqvdlem2  24556