Theorem atandm 24603

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 3584	. . 3 |- ( A e. ( CC \ { -u _i , _i } ) <-> ( A e. CC /\ -. A e. { -u _i , _i } ) )
2		elprg 4196	. . . . . 6 |- ( A e. CC -> ( A e. { -u _i , _i } <-> ( A = -u _i \/ A = _i ) ) )
3	2	notbid 308	. . . . 5 |- ( A e. CC -> ( -. A e. { -u _i , _i } <-> -. ( A = -u _i \/ A = _i ) ) )
4		neanior 2886	. . . . 5 |- ( ( A =/= -u _i /\ A =/= _i ) <-> -. ( A = -u _i \/ A = _i ) )
5	3, 4	syl6bbr 278	. . . 4 |- ( A e. CC -> ( -. A e. { -u _i , _i } <-> ( A =/= -u _i /\ A =/= _i ) ) )
6	5	pm5.32i 669	. . 3 |- ( ( A e. CC /\ -. A e. { -u _i , _i } ) <-> ( A e. CC /\ ( A =/= -u _i /\ A =/= _i ) ) )
7	1, 6	bitri 264	. 2 |- ( A e. ( CC \ { -u _i , _i } ) <-> ( A e. CC /\ ( A =/= -u _i /\ A =/= _i ) ) )
8		ovex 6678	. . . 4 |- ( ( _i / 2 ) x. ( ( log ` ( 1 - ( _i x. x ) ) ) - ( log ` ( 1 + ( _i x. x ) ) ) ) ) e. _V
9		df-atan 24594	. . . 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 6023	. . 3 |- dom arctan = ( CC \ { -u _i , _i } )
11	10	eleq2i 2693	. 2 |- ( A e. dom arctan <-> A e. ( CC \ { -u _i , _i } ) )
12		3anass 1042	. 2 |- ( ( A e. CC /\ A =/= -u _i /\ A =/= _i ) <-> ( A e. CC /\ ( A =/= -u _i /\ A =/= _i ) ) )
13	7, 11, 12	3bitr4i 292	1 |- ( A e. dom arctan <-> ( A e. CC /\ A =/= -u _i /\ A =/= _i ) )

Syntax hints:   -. wn 3   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   \ cdif 3571   {cpr 4179   dom cdm 5114   ` cfv 5888  (class class class)co 6650   CCcc 9934   1c1 9937   _ici 9938   + caddc 9939   x. cmul 9941   - cmin 10266   -ucneg 10267   / cdiv 10684   2c2 11070   logclog 24301  arctancatan 24591

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-mpt 4730  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-fn 5891  df-fv 5896  df-ov 6653  df-atan 24594

This theorem is referenced by:  atandm2  24604  atandm3  24605  atancj  24637  2efiatan  24645  tanatan  24646  dvatan  24662