Theorem atandm 22290

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 3357	. . 3 |- ( A e. ( CC \ { -u _i , _i } ) <-> ( A e. CC /\ -. A e. { -u _i , _i } ) )
2		elprg 3912	. . . . . 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 2716	. . . . 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 6135	. . . 4 |- ( ( _i / 2 ) x. ( ( log ` ( 1 - ( _i x. x ) ) ) - ( log ` ( 1 + ( _i x. x ) ) ) ) ) e. _V
9		df-atan 22281	. . . 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 5559	. . 3 |- dom arctan = ( CC \ { -u _i , _i } )
11	10	eleq2i 2507	. 2 |- ( A e. dom arctan <-> A e. ( CC \ { -u _i , _i } ) )
12		3anass 969	. 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 965   = wceq 1369   e. wcel 1756   =/= wne 2620   \ cdif 3344   {cpr 3898   dom cdm 4859   ` cfv 5437  (class class class)co 6110   CCcc 9299   1c1 9302   _ici 9303   + caddc 9304   x. cmul 9306   - cmin 9614   -ucneg 9615   / cdiv 10012   2c2 10390   logclog 22025  arctancatan 22278

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4432  ax-nul 4440  ax-pr 4550

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2739  df-rex 2740  df-rab 2743  df-v 2993  df-sbc 3206  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-nul 3657  df-if 3811  df-sn 3897  df-pr 3899  df-op 3903  df-uni 4111  df-br 4312  df-opab 4370  df-mpt 4371  df-id 4655  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-iota 5400  df-fun 5439  df-fn 5440  df-fv 5445  df-ov 6113  df-atan 22281

This theorem is referenced by:  atandm2  22291  atandm3  22292  atancj  22324  2efiatan  22332  tanatan  22333  dvatan  22349