Theorem atandm2 24604

Description: This form of atandm 24603 is a bit more useful for showing that the logarithms in df-atan 24594 are well-defined. (Contributed by Mario Carneiro, 31-Mar-2015.)

Assertion

Ref	Expression
atandm2	|- ( A e. dom arctan <-> ( A e. CC /\ ( 1 - ( _i x. A ) ) =/= 0 /\ ( 1 + ( _i x. A ) ) =/= 0 ) )

Proof of Theorem atandm2

Step	Hyp	Ref	Expression
1		atandm 24603	. 2 |- ( A e. dom arctan <-> ( A e. CC /\ A =/= -u _i /\ A =/= _i ) )
2		3anass 1042	. . 3 |- ( ( A e. CC /\ ( 1 - ( _i x. A ) ) =/= 0 /\ ( 1 + ( _i x. A ) ) =/= 0 ) <-> ( A e. CC /\ ( ( 1 - ( _i x. A ) ) =/= 0 /\ ( 1 + ( _i x. A ) ) =/= 0 ) ) )
3		ax-1cn 9994	. . . . . . . . . 10 |- 1 e. CC
4		ax-icn 9995	. . . . . . . . . . 11 |- _i e. CC
5		mulcl 10020	. . . . . . . . . . 11 |- ( ( _i e. CC /\ A e. CC ) -> ( _i x. A ) e. CC )
6	4, 5	mpan 706	. . . . . . . . . 10 |- ( A e. CC -> ( _i x. A ) e. CC )
7		subeq0 10307	. . . . . . . . . 10 |- ( ( 1 e. CC /\ ( _i x. A ) e. CC ) -> ( ( 1 - ( _i x. A ) ) = 0 <-> 1 = ( _i x. A ) ) )
8	3, 6, 7	sylancr 695	. . . . . . . . 9 |- ( A e. CC -> ( ( 1 - ( _i x. A ) ) = 0 <-> 1 = ( _i x. A ) ) )
9	4, 4	mulneg2i 10477	. . . . . . . . . . . 12 |- ( _i x. -u _i ) = -u ( _i x. _i )
10		ixi 10656	. . . . . . . . . . . . 13 |- ( _i x. _i ) = -u 1
11	10	negeqi 10274	. . . . . . . . . . . 12 |- -u ( _i x. _i ) = -u -u 1
12		negneg1e1 11128	. . . . . . . . . . . 12 |- -u -u 1 = 1
13	9, 11, 12	3eqtri 2648	. . . . . . . . . . 11 |- ( _i x. -u _i ) = 1
14	13	eqeq2i 2634	. . . . . . . . . 10 |- ( ( _i x. A ) = ( _i x. -u _i ) <-> ( _i x. A ) = 1 )
15		eqcom 2629	. . . . . . . . . 10 |- ( ( _i x. A ) = 1 <-> 1 = ( _i x. A ) )
16	14, 15	bitri 264	. . . . . . . . 9 |- ( ( _i x. A ) = ( _i x. -u _i ) <-> 1 = ( _i x. A ) )
17	8, 16	syl6bbr 278	. . . . . . . 8 |- ( A e. CC -> ( ( 1 - ( _i x. A ) ) = 0 <-> ( _i x. A ) = ( _i x. -u _i ) ) )
18		id 22	. . . . . . . . 9 |- ( A e. CC -> A e. CC )
19	4	negcli 10349	. . . . . . . . . 10 |- -u _i e. CC
20	19	a1i 11	. . . . . . . . 9 |- ( A e. CC -> -u _i e. CC )
21	4	a1i 11	. . . . . . . . 9 |- ( A e. CC -> _i e. CC )
22		ine0 10465	. . . . . . . . . 10 |- _i =/= 0
23	22	a1i 11	. . . . . . . . 9 |- ( A e. CC -> _i =/= 0 )
24	18, 20, 21, 23	mulcand 10660	. . . . . . . 8 |- ( A e. CC -> ( ( _i x. A ) = ( _i x. -u _i ) <-> A = -u _i ) )
25	17, 24	bitrd 268	. . . . . . 7 |- ( A e. CC -> ( ( 1 - ( _i x. A ) ) = 0 <-> A = -u _i ) )
26	25	necon3bid 2838	. . . . . 6 |- ( A e. CC -> ( ( 1 - ( _i x. A ) ) =/= 0 <-> A =/= -u _i ) )
27		addcom 10222	. . . . . . . . . . . . 13 |- ( ( 1 e. CC /\ ( _i x. A ) e. CC ) -> ( 1 + ( _i x. A ) ) = ( ( _i x. A ) + 1 ) )
28	3, 6, 27	sylancr 695	. . . . . . . . . . . 12 |- ( A e. CC -> ( 1 + ( _i x. A ) ) = ( ( _i x. A ) + 1 ) )
29		subneg 10330	. . . . . . . . . . . . 13 |- ( ( ( _i x. A ) e. CC /\ 1 e. CC ) -> ( ( _i x. A ) - -u 1 ) = ( ( _i x. A ) + 1 ) )
30	6, 3, 29	sylancl 694	. . . . . . . . . . . 12 |- ( A e. CC -> ( ( _i x. A ) - -u 1 ) = ( ( _i x. A ) + 1 ) )
31	28, 30	eqtr4d 2659	. . . . . . . . . . 11 |- ( A e. CC -> ( 1 + ( _i x. A ) ) = ( ( _i x. A ) - -u 1 ) )
32	31	eqeq1d 2624	. . . . . . . . . 10 |- ( A e. CC -> ( ( 1 + ( _i x. A ) ) = 0 <-> ( ( _i x. A ) - -u 1 ) = 0 ) )
33	3	negcli 10349	. . . . . . . . . . 11 |- -u 1 e. CC
34		subeq0 10307	. . . . . . . . . . 11 |- ( ( ( _i x. A ) e. CC /\ -u 1 e. CC ) -> ( ( ( _i x. A ) - -u 1 ) = 0 <-> ( _i x. A ) = -u 1 ) )
35	6, 33, 34	sylancl 694	. . . . . . . . . 10 |- ( A e. CC -> ( ( ( _i x. A ) - -u 1 ) = 0 <-> ( _i x. A ) = -u 1 ) )
36	32, 35	bitrd 268	. . . . . . . . 9 |- ( A e. CC -> ( ( 1 + ( _i x. A ) ) = 0 <-> ( _i x. A ) = -u 1 ) )
37	10	eqeq2i 2634	. . . . . . . . 9 |- ( ( _i x. A ) = ( _i x. _i ) <-> ( _i x. A ) = -u 1 )
38	36, 37	syl6bbr 278	. . . . . . . 8 |- ( A e. CC -> ( ( 1 + ( _i x. A ) ) = 0 <-> ( _i x. A ) = ( _i x. _i ) ) )
39	18, 21, 21, 23	mulcand 10660	. . . . . . . 8 |- ( A e. CC -> ( ( _i x. A ) = ( _i x. _i ) <-> A = _i ) )
40	38, 39	bitrd 268	. . . . . . 7 |- ( A e. CC -> ( ( 1 + ( _i x. A ) ) = 0 <-> A = _i ) )
41	40	necon3bid 2838	. . . . . 6 |- ( A e. CC -> ( ( 1 + ( _i x. A ) ) =/= 0 <-> A =/= _i ) )
42	26, 41	anbi12d 747	. . . . 5 |- ( A e. CC -> ( ( ( 1 - ( _i x. A ) ) =/= 0 /\ ( 1 + ( _i x. A ) ) =/= 0 ) <-> ( A =/= -u _i /\ A =/= _i ) ) )
43	42	pm5.32i 669	. . . 4 |- ( ( A e. CC /\ ( ( 1 - ( _i x. A ) ) =/= 0 /\ ( 1 + ( _i x. A ) ) =/= 0 ) ) <-> ( A e. CC /\ ( A =/= -u _i /\ A =/= _i ) ) )
44		3anass 1042	. . . 4 |- ( ( A e. CC /\ A =/= -u _i /\ A =/= _i ) <-> ( A e. CC /\ ( A =/= -u _i /\ A =/= _i ) ) )
45	43, 44	bitr4i 267	. . 3 |- ( ( A e. CC /\ ( ( 1 - ( _i x. A ) ) =/= 0 /\ ( 1 + ( _i x. A ) ) =/= 0 ) ) <-> ( A e. CC /\ A =/= -u _i /\ A =/= _i ) )
46	2, 45	bitri 264	. 2 |- ( ( A e. CC /\ ( 1 - ( _i x. A ) ) =/= 0 /\ ( 1 + ( _i x. A ) ) =/= 0 ) <-> ( A e. CC /\ A =/= -u _i /\ A =/= _i ) )
47	1, 46	bitr4i 267	1 |- ( A e. dom arctan <-> ( A e. CC /\ ( 1 - ( _i x. A ) ) =/= 0 /\ ( 1 + ( _i x. A ) ) =/= 0 ) )

Syntax hints:   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   dom cdm 5114  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   _ici 9938   + caddc 9939   x. cmul 9941   - cmin 10266   -ucneg 10267  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-8 1992  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-pow 4843  ax-pr 4906  ax-un 6949  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  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-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-atan 24594

This theorem is referenced by:  atanf  24607  atanneg  24634  atancj  24637  efiatan  24639  atanlogaddlem  24640  atanlogadd  24641  atanlogsublem  24642  atanlogsub  24643  efiatan2  24644  2efiatan  24645  atantan  24650  atanbndlem  24652  dvatan  24662  atantayl  24664