Theorem atandm 23071

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 3491	. . 3 |- ( A e. ( CC \ { -u _i , _i } ) <-> ( A e. CC /\ -. A e. { -u _i , _i } ) )
2		elprg 4049	. . . . . 6 |- ( A e. CC -> ( A e. { -u _i , _i } <-> ( A = -u _i \/ A = _i ) ) )
3	2	notbid 294	. . . . 5 |- ( A e. CC -> ( -. A e. { -u _i , _i } <-> -. ( A = -u _i \/ A = _i ) ) )
4		neanior 2792	. . . . 5 |- ( ( A =/= -u _i /\ A =/= _i ) <-> -. ( A = -u _i \/ A = _i ) )
5	3, 4	syl6bbr 263	. . . 4 |- ( A e. CC -> ( -. A e. { -u _i , _i } <-> ( A =/= -u _i /\ A =/= _i ) ) )
6	5	pm5.32i 637	. . 3 |- ( ( A e. CC /\ -. A e. { -u _i , _i } ) <-> ( A e. CC /\ ( A =/= -u _i /\ A =/= _i ) ) )
7	1, 6	bitri 249	. 2 |- ( A e. ( CC \ { -u _i , _i } ) <-> ( A e. CC /\ ( A =/= -u _i /\ A =/= _i ) ) )
8		ovex 6320	. . . 4 |- ( ( _i / 2 ) x. ( ( log ` ( 1 - ( _i x. x ) ) ) - ( log ` ( 1 + ( _i x. x ) ) ) ) ) e. _V
9		df-atan 23062	. . . 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 5716	. . 3 |- dom arctan = ( CC \ { -u _i , _i } )
11	10	eleq2i 2545	. 2 |- ( A e. dom arctan <-> A e. ( CC \ { -u _i , _i } ) )
12		3anass 977	. 2 |- ( ( A e. CC /\ A =/= -u _i /\ A =/= _i ) <-> ( A e. CC /\ ( A =/= -u _i /\ A =/= _i ) ) )
13	7, 11, 12	3bitr4i 277	1 |- ( A e. dom arctan <-> ( A e. CC /\ A =/= -u _i /\ A =/= _i ) )

Syntax hints:   -. wn 3   <-> wb 184   \/ wo 368   /\ wa 369   /\ w3a 973   = wceq 1379   e. wcel 1767   =/= wne 2662   \ cdif 3478   {cpr 4035   dom cdm 5005   ` cfv 5594  (class class class)co 6295   CCcc 9502   1c1 9505   _ici 9506   + caddc 9507   x. cmul 9509   - cmin 9817   -ucneg 9818   / cdiv 10218   2c2 10597   logclog 22806  arctancatan 23059

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-mpt 4513  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-fn 5597  df-fv 5602  df-ov 6298  df-atan 23062

This theorem is referenced by:  atandm2  23072  atandm3  23073  atancj  23105  2efiatan  23113  tanatan  23114  dvatan  23130