Theorem atandm 23851

Description: Since the property is a little lengthy, we abbreviate A e. CC /\ A =/= -u _i /\ A =/= _i as A e. dom arctan. This is the necessary precondition for the definition of arctan to make sense. (Contributed by Mario Carneiro, 31-Mar-2015.)

Assertion

Ref	Expression
atandm	|- ( A e. dom arctan <-> ( A e. CC /\ A =/= -u _i /\ A =/= _i ) )

Proof of Theorem atandm

Step	Hyp	Ref	Expression
1		eldif 3426	. . 3 |- ( A e. ( CC \ { -u _i , _i } ) <-> ( A e. CC /\ -. A e. { -u _i , _i } ) )
2		elprg 3996	. . . . . 6 |- ( A e. CC -> ( A e. { -u _i , _i } <-> ( A = -u _i \/ A = _i ) ) )
3	2	notbid 300	. . . . 5 |- ( A e. CC -> ( -. A e. { -u _i , _i } <-> -. ( A = -u _i \/ A = _i ) ) )
4		neanior 2728	. . . . 5 |- ( ( A =/= -u _i /\ A =/= _i ) <-> -. ( A = -u _i \/ A = _i ) )
5	3, 4	syl6bbr 271	. . . 4 |- ( A e. CC -> ( -. A e. { -u _i , _i } <-> ( A =/= -u _i /\ A =/= _i ) ) )
6	5	pm5.32i 647	. . 3 |- ( ( A e. CC /\ -. A e. { -u _i , _i } ) <-> ( A e. CC /\ ( A =/= -u _i /\ A =/= _i ) ) )
7	1, 6	bitri 257	. 2 |- ( A e. ( CC \ { -u _i , _i } ) <-> ( A e. CC /\ ( A =/= -u _i /\ A =/= _i ) ) )
8		ovex 6343	. . . 4 |- ( ( _i / 2 ) x. ( ( log ` ( 1 - ( _i x. x ) ) ) - ( log ` ( 1 + ( _i x. x ) ) ) ) ) e. _V
9		df-atan 23842	. . . 4 |- arctan = ( x e. ( CC \ { -u _i , _i } ) |-> ( ( _i / 2 ) x. ( ( log ` ( 1 - ( _i x. x ) ) ) - ( log ` ( 1 + ( _i x. x ) ) ) ) ) )
10	8, 9	dmmpti 5729	. . 3 |- dom arctan = ( CC \ { -u _i , _i } )
11	10	eleq2i 2532	. 2 |- ( A e. dom arctan <-> A e. ( CC \ { -u _i , _i } ) )
12		3anass 995	. 2 |- ( ( A e. CC /\ A =/= -u _i /\ A =/= _i ) <-> ( A e. CC /\ ( A =/= -u _i /\ A =/= _i ) ) )
13	7, 11, 12	3bitr4i 285	1 |- ( A e. dom arctan <-> ( A e. CC /\ A =/= -u _i /\ A =/= _i ) )

Syntax hints:   -. wn 3   <-> wb 189   \/ wo 374   /\ wa 375   /\ w3a 991   = wceq 1455   e. wcel 1898   =/= wne 2633   \ cdif 3413   {cpr 3982   dom cdm 4853   ` cfv 5601  (class class class)co 6315   CCcc 9563   1c1 9566   _ici 9567   + caddc 9568   x. cmul 9570   - cmin 9886   -ucneg 9887   / cdiv 10297   2c2 10687   logclog 23553  arctancatan 23839

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4539  ax-nul 4548  ax-pr 4653

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-sbc 3280  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-br 4417  df-opab 4476  df-mpt 4477  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-iota 5565  df-fun 5603  df-fn 5604  df-fv 5609  df-ov 6318  df-atan 23842

This theorem is referenced by:  atandm2  23852  atandm3  23853  atancj  23885  2efiatan  23893  tanatan  23894  dvatan  23910