Theorem atandm 23323

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 3399	. . 3 |- ( A e. ( CC \ { -u _i , _i } ) <-> ( A e. CC /\ -. A e. { -u _i , _i } ) )
2		elprg 3960	. . . . . 6 |- ( A e. CC -> ( A e. { -u _i , _i } <-> ( A = -u _i \/ A = _i ) ) )
3	2	notbid 292	. . . . 5 |- ( A e. CC -> ( -. A e. { -u _i , _i } <-> -. ( A = -u _i \/ A = _i ) ) )
4		neanior 2707	. . . . 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 635	. . 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 6224	. . . 4 |- ( ( _i / 2 ) x. ( ( log ` ( 1 - ( _i x. x ) ) ) - ( log ` ( 1 + ( _i x. x ) ) ) ) ) e. _V
9		df-atan 23314	. . . 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 5618	. . 3 |- dom arctan = ( CC \ { -u _i , _i } )
11	10	eleq2i 2460	. 2 |- ( A e. dom arctan <-> A e. ( CC \ { -u _i , _i } ) )
12		3anass 975	. 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 366   /\ wa 367   /\ w3a 971   = wceq 1399   e. wcel 1826   =/= wne 2577   \ cdif 3386   {cpr 3946   dom cdm 4913   ` cfv 5496  (class class class)co 6196   CCcc 9401   1c1 9404   _ici 9405   + caddc 9406   x. cmul 9408   - cmin 9718   -ucneg 9719   / cdiv 10123   2c2 10502   logclog 23027  arctancatan 23311

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pr 4601

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-sbc 3253  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-iota 5460  df-fun 5498  df-fn 5499  df-fv 5504  df-ov 6199  df-atan 23314

This theorem is referenced by:  atandm2  23324  atandm3  23325  atancj  23357  2efiatan  23365  tanatan  23366  dvatan  23382