Theorem ang180lem1 24539

Description: Lemma for ang180 24544. Show that the "revolution number" N is an integer, using efeq1 24275 to show that since the product of the three arguments A , 1 / ( 1 - A ) , ( A - 1 ) / A is -u 1, the sum of the logarithms must be an integer multiple of 2 pi _i away from pi _i = log ( -u 1 ). (Contributed by Mario Carneiro, 23-Sep-2014.)

Hypotheses

Ref	Expression
ang.1	|- F = ( x e. ( CC \ { 0 } ) , y e. ( CC \ { 0 } ) |-> ( Im ` ( log ` ( y / x ) ) ) )
ang180lem1.2	|- T = ( ( ( log ` ( 1 / ( 1 - A ) ) ) + ( log ` ( ( A - 1 ) / A ) ) ) + ( log ` A ) )
ang180lem1.3	|- N = ( ( ( T / _i ) / ( 2 x. pi ) ) - ( 1 / 2 ) )

Assertion

Ref	Expression
ang180lem1	|- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( N e. ZZ /\ ( T / _i ) e. RR ) )

Distinct variable group:   x, y, A

Allowed substitution hints:   T( x, y)   F( x, y)   N( x, y)

Proof of Theorem ang180lem1

Step	Hyp	Ref	Expression
1		picn 24211	. . . . . . 7 |- pi e. CC
2		2re 11090	. . . . . . . . . 10 |- 2 e. RR
3		pire 24210	. . . . . . . . . 10 |- pi e. RR
4	2, 3	remulcli 10054	. . . . . . . . 9 |- ( 2 x. pi ) e. RR
5	4	recni 10052	. . . . . . . 8 |- ( 2 x. pi ) e. CC
6		2pos 11112	. . . . . . . . . 10 |- 0 < 2
7		pipos 24212	. . . . . . . . . 10 |- 0 < pi
8	2, 3, 6, 7	mulgt0ii 10170	. . . . . . . . 9 |- 0 < ( 2 x. pi )
9	4, 8	gt0ne0ii 10564	. . . . . . . 8 |- ( 2 x. pi ) =/= 0
10	5, 9	pm3.2i 471	. . . . . . 7 |- ( ( 2 x. pi ) e. CC /\ ( 2 x. pi ) =/= 0 )
11		ax-icn 9995	. . . . . . . 8 |- _i e. CC
12		ine0 10465	. . . . . . . 8 |- _i =/= 0
13	11, 12	pm3.2i 471	. . . . . . 7 |- ( _i e. CC /\ _i =/= 0 )
14		divcan5 10727	. . . . . . 7 |- ( ( pi e. CC /\ ( ( 2 x. pi ) e. CC /\ ( 2 x. pi ) =/= 0 ) /\ ( _i e. CC /\ _i =/= 0 ) ) -> ( ( _i x. pi ) / ( _i x. ( 2 x. pi ) ) ) = ( pi / ( 2 x. pi ) ) )
15	1, 10, 13, 14	mp3an 1424	. . . . . 6 |- ( ( _i x. pi ) / ( _i x. ( 2 x. pi ) ) ) = ( pi / ( 2 x. pi ) )
16	3, 7	gt0ne0ii 10564	. . . . . . 7 |- pi =/= 0
17		recdiv 10731	. . . . . . 7 |- ( ( ( ( 2 x. pi ) e. CC /\ ( 2 x. pi ) =/= 0 ) /\ ( pi e. CC /\ pi =/= 0 ) ) -> ( 1 / ( ( 2 x. pi ) / pi ) ) = ( pi / ( 2 x. pi ) ) )
18	5, 9, 1, 16, 17	mp4an 709	. . . . . 6 |- ( 1 / ( ( 2 x. pi ) / pi ) ) = ( pi / ( 2 x. pi ) )
19	2	recni 10052	. . . . . . . 8 |- 2 e. CC
20	19, 1, 16	divcan4i 10772	. . . . . . 7 |- ( ( 2 x. pi ) / pi ) = 2
21	20	oveq2i 6661	. . . . . 6 |- ( 1 / ( ( 2 x. pi ) / pi ) ) = ( 1 / 2 )
22	15, 18, 21	3eqtr2i 2650	. . . . 5 |- ( ( _i x. pi ) / ( _i x. ( 2 x. pi ) ) ) = ( 1 / 2 )
23	22	oveq2i 6661	. . . 4 |- ( ( T / ( _i x. ( 2 x. pi ) ) ) - ( ( _i x. pi ) / ( _i x. ( 2 x. pi ) ) ) ) = ( ( T / ( _i x. ( 2 x. pi ) ) ) - ( 1 / 2 ) )
24		ang180lem1.2	. . . . . 6 |- T = ( ( ( log ` ( 1 / ( 1 - A ) ) ) + ( log ` ( ( A - 1 ) / A ) ) ) + ( log ` A ) )
25		ax-1cn 9994	. . . . . . . . . . 11 |- 1 e. CC
26		simp1 1061	. . . . . . . . . . 11 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> A e. CC )
27		subcl 10280	. . . . . . . . . . 11 |- ( ( 1 e. CC /\ A e. CC ) -> ( 1 - A ) e. CC )
28	25, 26, 27	sylancr 695	. . . . . . . . . 10 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( 1 - A ) e. CC )
29		simp3 1063	. . . . . . . . . . . 12 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> A =/= 1 )
30	29	necomd 2849	. . . . . . . . . . 11 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> 1 =/= A )
31		subeq0 10307	. . . . . . . . . . . . 13 |- ( ( 1 e. CC /\ A e. CC ) -> ( ( 1 - A ) = 0 <-> 1 = A ) )
32	25, 26, 31	sylancr 695	. . . . . . . . . . . 12 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( ( 1 - A ) = 0 <-> 1 = A ) )
33	32	necon3bid 2838	. . . . . . . . . . 11 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( ( 1 - A ) =/= 0 <-> 1 =/= A ) )
34	30, 33	mpbird 247	. . . . . . . . . 10 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( 1 - A ) =/= 0 )
35	28, 34	reccld 10794	. . . . . . . . 9 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( 1 / ( 1 - A ) ) e. CC )
36	28, 34	recne0d 10795	. . . . . . . . 9 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( 1 / ( 1 - A ) ) =/= 0 )
37	35, 36	logcld 24317	. . . . . . . 8 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( log ` ( 1 / ( 1 - A ) ) ) e. CC )
38		subcl 10280	. . . . . . . . . . 11 |- ( ( A e. CC /\ 1 e. CC ) -> ( A - 1 ) e. CC )
39	26, 25, 38	sylancl 694	. . . . . . . . . 10 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( A - 1 ) e. CC )
40		simp2 1062	. . . . . . . . . 10 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> A =/= 0 )
41	39, 26, 40	divcld 10801	. . . . . . . . 9 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( ( A - 1 ) / A ) e. CC )
42		subeq0 10307	. . . . . . . . . . . . 13 |- ( ( A e. CC /\ 1 e. CC ) -> ( ( A - 1 ) = 0 <-> A = 1 ) )
43	26, 25, 42	sylancl 694	. . . . . . . . . . . 12 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( ( A - 1 ) = 0 <-> A = 1 ) )
44	43	necon3bid 2838	. . . . . . . . . . 11 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( ( A - 1 ) =/= 0 <-> A =/= 1 ) )
45	29, 44	mpbird 247	. . . . . . . . . 10 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( A - 1 ) =/= 0 )
46	39, 26, 45, 40	divne0d 10817	. . . . . . . . 9 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( ( A - 1 ) / A ) =/= 0 )
47	41, 46	logcld 24317	. . . . . . . 8 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( log ` ( ( A - 1 ) / A ) ) e. CC )
48	37, 47	addcld 10059	. . . . . . 7 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( ( log ` ( 1 / ( 1 - A ) ) ) + ( log ` ( ( A - 1 ) / A ) ) ) e. CC )
49	26, 40	logcld 24317	. . . . . . 7 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( log ` A ) e. CC )
50	48, 49	addcld 10059	. . . . . 6 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( ( ( log ` ( 1 / ( 1 - A ) ) ) + ( log ` ( ( A - 1 ) / A ) ) ) + ( log ` A ) ) e. CC )
51	24, 50	syl5eqel 2705	. . . . 5 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> T e. CC )
52	11, 1	mulcli 10045	. . . . . 6 |- ( _i x. pi ) e. CC
53	52	a1i 11	. . . . 5 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( _i x. pi ) e. CC )
54	11, 5	mulcli 10045	. . . . . 6 |- ( _i x. ( 2 x. pi ) ) e. CC
55	54	a1i 11	. . . . 5 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( _i x. ( 2 x. pi ) ) e. CC )
56	11, 5, 12, 9	mulne0i 10670	. . . . . 6 |- ( _i x. ( 2 x. pi ) ) =/= 0
57	56	a1i 11	. . . . 5 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( _i x. ( 2 x. pi ) ) =/= 0 )
58	51, 53, 55, 57	divsubdird 10840	. . . 4 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( ( T - ( _i x. pi ) ) / ( _i x. ( 2 x. pi ) ) ) = ( ( T / ( _i x. ( 2 x. pi ) ) ) - ( ( _i x. pi ) / ( _i x. ( 2 x. pi ) ) ) ) )
59		ang180lem1.3	. . . . 5 |- N = ( ( ( T / _i ) / ( 2 x. pi ) ) - ( 1 / 2 ) )
60	13	a1i 11	. . . . . . 7 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( _i e. CC /\ _i =/= 0 ) )
61	10	a1i 11	. . . . . . 7 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( ( 2 x. pi ) e. CC /\ ( 2 x. pi ) =/= 0 ) )
62		divdiv1 10736	. . . . . . 7 |- ( ( T e. CC /\ ( _i e. CC /\ _i =/= 0 ) /\ ( ( 2 x. pi ) e. CC /\ ( 2 x. pi ) =/= 0 ) ) -> ( ( T / _i ) / ( 2 x. pi ) ) = ( T / ( _i x. ( 2 x. pi ) ) ) )
63	51, 60, 61, 62	syl3anc 1326	. . . . . 6 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( ( T / _i ) / ( 2 x. pi ) ) = ( T / ( _i x. ( 2 x. pi ) ) ) )
64	63	oveq1d 6665	. . . . 5 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( ( ( T / _i ) / ( 2 x. pi ) ) - ( 1 / 2 ) ) = ( ( T / ( _i x. ( 2 x. pi ) ) ) - ( 1 / 2 ) ) )
65	59, 64	syl5eq 2668	. . . 4 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> N = ( ( T / ( _i x. ( 2 x. pi ) ) ) - ( 1 / 2 ) ) )
66	23, 58, 65	3eqtr4a 2682	. . 3 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( ( T - ( _i x. pi ) ) / ( _i x. ( 2 x. pi ) ) ) = N )
67		efsub 14830	. . . . . 6 |- ( ( T e. CC /\ ( _i x. pi ) e. CC ) -> ( exp ` ( T - ( _i x. pi ) ) ) = ( ( exp ` T ) / ( exp ` ( _i x. pi ) ) ) )
68	51, 52, 67	sylancl 694	. . . . 5 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( exp ` ( T - ( _i x. pi ) ) ) = ( ( exp ` T ) / ( exp ` ( _i x. pi ) ) ) )
69		efipi 24225	. . . . . . 7 |- ( exp ` ( _i x. pi ) ) = -u 1
70	69	oveq2i 6661	. . . . . 6 |- ( ( exp ` T ) / ( exp ` ( _i x. pi ) ) ) = ( ( exp ` T ) / -u 1 )
71	24	fveq2i 6194	. . . . . . . . 9 |- ( exp ` T ) = ( exp ` ( ( ( log ` ( 1 / ( 1 - A ) ) ) + ( log ` ( ( A - 1 ) / A ) ) ) + ( log ` A ) ) )
72		efadd 14824	. . . . . . . . . . 11 |- ( ( ( ( log ` ( 1 / ( 1 - A ) ) ) + ( log ` ( ( A - 1 ) / A ) ) ) e. CC /\ ( log ` A ) e. CC ) -> ( exp ` ( ( ( log ` ( 1 / ( 1 - A ) ) ) + ( log ` ( ( A - 1 ) / A ) ) ) + ( log ` A ) ) ) = ( ( exp ` ( ( log ` ( 1 / ( 1 - A ) ) ) + ( log ` ( ( A - 1 ) / A ) ) ) ) x. ( exp ` ( log ` A ) ) ) )
73	48, 49, 72	syl2anc 693	. . . . . . . . . 10 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( exp ` ( ( ( log ` ( 1 / ( 1 - A ) ) ) + ( log ` ( ( A - 1 ) / A ) ) ) + ( log ` A ) ) ) = ( ( exp ` ( ( log ` ( 1 / ( 1 - A ) ) ) + ( log ` ( ( A - 1 ) / A ) ) ) ) x. ( exp ` ( log ` A ) ) ) )
74		efadd 14824	. . . . . . . . . . . . 13 |- ( ( ( log ` ( 1 / ( 1 - A ) ) ) e. CC /\ ( log ` ( ( A - 1 ) / A ) ) e. CC ) -> ( exp ` ( ( log ` ( 1 / ( 1 - A ) ) ) + ( log ` ( ( A - 1 ) / A ) ) ) ) = ( ( exp ` ( log ` ( 1 / ( 1 - A ) ) ) ) x. ( exp ` ( log ` ( ( A - 1 ) / A ) ) ) ) )
75	37, 47, 74	syl2anc 693	. . . . . . . . . . . 12 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( exp ` ( ( log ` ( 1 / ( 1 - A ) ) ) + ( log ` ( ( A - 1 ) / A ) ) ) ) = ( ( exp ` ( log ` ( 1 / ( 1 - A ) ) ) ) x. ( exp ` ( log ` ( ( A - 1 ) / A ) ) ) ) )
76		eflog 24323	. . . . . . . . . . . . . 14 |- ( ( ( 1 / ( 1 - A ) ) e. CC /\ ( 1 / ( 1 - A ) ) =/= 0 ) -> ( exp ` ( log ` ( 1 / ( 1 - A ) ) ) ) = ( 1 / ( 1 - A ) ) )
77	35, 36, 76	syl2anc 693	. . . . . . . . . . . . 13 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( exp ` ( log ` ( 1 / ( 1 - A ) ) ) ) = ( 1 / ( 1 - A ) ) )
78		eflog 24323	. . . . . . . . . . . . . 14 |- ( ( ( ( A - 1 ) / A ) e. CC /\ ( ( A - 1 ) / A ) =/= 0 ) -> ( exp ` ( log ` ( ( A - 1 ) / A ) ) ) = ( ( A - 1 ) / A ) )
79	41, 46, 78	syl2anc 693	. . . . . . . . . . . . 13 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( exp ` ( log ` ( ( A - 1 ) / A ) ) ) = ( ( A - 1 ) / A ) )
80	77, 79	oveq12d 6668	. . . . . . . . . . . 12 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( ( exp ` ( log ` ( 1 / ( 1 - A ) ) ) ) x. ( exp ` ( log ` ( ( A - 1 ) / A ) ) ) ) = ( ( 1 / ( 1 - A ) ) x. ( ( A - 1 ) / A ) ) )
81	35, 41	mulcomd 10061	. . . . . . . . . . . . 13 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( ( 1 / ( 1 - A ) ) x. ( ( A - 1 ) / A ) ) = ( ( ( A - 1 ) / A ) x. ( 1 / ( 1 - A ) ) ) )
82	25	a1i 11	. . . . . . . . . . . . . . . 16 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> 1 e. CC )
83	82, 28, 34	div2negd 10816	. . . . . . . . . . . . . . 15 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( -u 1 / -u ( 1 - A ) ) = ( 1 / ( 1 - A ) ) )
84		negsubdi2 10340	. . . . . . . . . . . . . . . . 17 |- ( ( 1 e. CC /\ A e. CC ) -> -u ( 1 - A ) = ( A - 1 ) )
85	25, 26, 84	sylancr 695	. . . . . . . . . . . . . . . 16 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> -u ( 1 - A ) = ( A - 1 ) )
86	85	oveq2d 6666	. . . . . . . . . . . . . . 15 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( -u 1 / -u ( 1 - A ) ) = ( -u 1 / ( A - 1 ) ) )
87	83, 86	eqtr3d 2658	. . . . . . . . . . . . . 14 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( 1 / ( 1 - A ) ) = ( -u 1 / ( A - 1 ) ) )
88	87	oveq2d 6666	. . . . . . . . . . . . 13 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( ( ( A - 1 ) / A ) x. ( 1 / ( 1 - A ) ) ) = ( ( ( A - 1 ) / A ) x. ( -u 1 / ( A - 1 ) ) ) )
89		neg1cn 11124	. . . . . . . . . . . . . . 15 |- -u 1 e. CC
90	89	a1i 11	. . . . . . . . . . . . . 14 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> -u 1 e. CC )
91	90, 39, 26, 45, 40	dmdcand 10830	. . . . . . . . . . . . 13 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( ( ( A - 1 ) / A ) x. ( -u 1 / ( A - 1 ) ) ) = ( -u 1 / A ) )
92	81, 88, 91	3eqtrd 2660	. . . . . . . . . . . 12 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( ( 1 / ( 1 - A ) ) x. ( ( A - 1 ) / A ) ) = ( -u 1 / A ) )
93	75, 80, 92	3eqtrd 2660	. . . . . . . . . . 11 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( exp ` ( ( log ` ( 1 / ( 1 - A ) ) ) + ( log ` ( ( A - 1 ) / A ) ) ) ) = ( -u 1 / A ) )
94		eflog 24323	. . . . . . . . . . . 12 |- ( ( A e. CC /\ A =/= 0 ) -> ( exp ` ( log ` A ) ) = A )
95	26, 40, 94	syl2anc 693	. . . . . . . . . . 11 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( exp ` ( log ` A ) ) = A )
96	93, 95	oveq12d 6668	. . . . . . . . . 10 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( ( exp ` ( ( log ` ( 1 / ( 1 - A ) ) ) + ( log ` ( ( A - 1 ) / A ) ) ) ) x. ( exp ` ( log ` A ) ) ) = ( ( -u 1 / A ) x. A ) )
97	90, 26, 40	divcan1d 10802	. . . . . . . . . 10 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( ( -u 1 / A ) x. A ) = -u 1 )
98	73, 96, 97	3eqtrd 2660	. . . . . . . . 9 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( exp ` ( ( ( log ` ( 1 / ( 1 - A ) ) ) + ( log ` ( ( A - 1 ) / A ) ) ) + ( log ` A ) ) ) = -u 1 )
99	71, 98	syl5eq 2668	. . . . . . . 8 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( exp ` T ) = -u 1 )
100	99	oveq1d 6665	. . . . . . 7 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( ( exp ` T ) / -u 1 ) = ( -u 1 / -u 1 ) )
101		neg1ne0 11126	. . . . . . . 8 |- -u 1 =/= 0
102	89, 101	dividi 10758	. . . . . . 7 |- ( -u 1 / -u 1 ) = 1
103	100, 102	syl6eq 2672	. . . . . 6 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( ( exp ` T ) / -u 1 ) = 1 )
104	70, 103	syl5eq 2668	. . . . 5 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( ( exp ` T ) / ( exp ` ( _i x. pi ) ) ) = 1 )
105	68, 104	eqtrd 2656	. . . 4 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( exp ` ( T - ( _i x. pi ) ) ) = 1 )
106		subcl 10280	. . . . . 6 |- ( ( T e. CC /\ ( _i x. pi ) e. CC ) -> ( T - ( _i x. pi ) ) e. CC )
107	51, 52, 106	sylancl 694	. . . . 5 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( T - ( _i x. pi ) ) e. CC )
108		efeq1 24275	. . . . 5 |- ( ( T - ( _i x. pi ) ) e. CC -> ( ( exp ` ( T - ( _i x. pi ) ) ) = 1 <-> ( ( T - ( _i x. pi ) ) / ( _i x. ( 2 x. pi ) ) ) e. ZZ ) )
109	107, 108	syl 17	. . . 4 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( ( exp ` ( T - ( _i x. pi ) ) ) = 1 <-> ( ( T - ( _i x. pi ) ) / ( _i x. ( 2 x. pi ) ) ) e. ZZ ) )
110	105, 109	mpbid 222	. . 3 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( ( T - ( _i x. pi ) ) / ( _i x. ( 2 x. pi ) ) ) e. ZZ )
111	66, 110	eqeltrrd 2702	. 2 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> N e. ZZ )
112	11	a1i 11	. . . . 5 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> _i e. CC )
113	12	a1i 11	. . . . 5 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> _i =/= 0 )
114	51, 112, 113	divcld 10801	. . . 4 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( T / _i ) e. CC )
115	5	a1i 11	. . . 4 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( 2 x. pi ) e. CC )
116	9	a1i 11	. . . 4 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( 2 x. pi ) =/= 0 )
117	114, 115, 116	divcan1d 10802	. . 3 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( ( ( T / _i ) / ( 2 x. pi ) ) x. ( 2 x. pi ) ) = ( T / _i ) )
118	59	oveq1i 6660	. . . . . 6 |- ( N + ( 1 / 2 ) ) = ( ( ( ( T / _i ) / ( 2 x. pi ) ) - ( 1 / 2 ) ) + ( 1 / 2 ) )
119	114, 115, 116	divcld 10801	. . . . . . 7 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( ( T / _i ) / ( 2 x. pi ) ) e. CC )
120		halfre 11246	. . . . . . . 8 |- ( 1 / 2 ) e. RR
121	120	recni 10052	. . . . . . 7 |- ( 1 / 2 ) e. CC
122		npcan 10290	. . . . . . 7 |- ( ( ( ( T / _i ) / ( 2 x. pi ) ) e. CC /\ ( 1 / 2 ) e. CC ) -> ( ( ( ( T / _i ) / ( 2 x. pi ) ) - ( 1 / 2 ) ) + ( 1 / 2 ) ) = ( ( T / _i ) / ( 2 x. pi ) ) )
123	119, 121, 122	sylancl 694	. . . . . 6 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( ( ( ( T / _i ) / ( 2 x. pi ) ) - ( 1 / 2 ) ) + ( 1 / 2 ) ) = ( ( T / _i ) / ( 2 x. pi ) ) )
124	118, 123	syl5eq 2668	. . . . 5 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( N + ( 1 / 2 ) ) = ( ( T / _i ) / ( 2 x. pi ) ) )
125	111	zred 11482	. . . . . 6 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> N e. RR )
126		readdcl 10019	. . . . . 6 |- ( ( N e. RR /\ ( 1 / 2 ) e. RR ) -> ( N + ( 1 / 2 ) ) e. RR )
127	125, 120, 126	sylancl 694	. . . . 5 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( N + ( 1 / 2 ) ) e. RR )
128	124, 127	eqeltrrd 2702	. . . 4 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( ( T / _i ) / ( 2 x. pi ) ) e. RR )
129		remulcl 10021	. . . 4 |- ( ( ( ( T / _i ) / ( 2 x. pi ) ) e. RR /\ ( 2 x. pi ) e. RR ) -> ( ( ( T / _i ) / ( 2 x. pi ) ) x. ( 2 x. pi ) ) e. RR )
130	128, 4, 129	sylancl 694	. . 3 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( ( ( T / _i ) / ( 2 x. pi ) ) x. ( 2 x. pi ) ) e. RR )
131	117, 130	eqeltrrd 2702	. 2 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( T / _i ) e. RR )
132	111, 131	jca 554	1 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( N e. ZZ /\ ( T / _i ) e. RR ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   \ cdif 3571   {csn 4177   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   _ici 9938   + caddc 9939   x. cmul 9941   - cmin 10266   -ucneg 10267   / cdiv 10684   2c2 11070   ZZcz 11377   Imcim 13838   expce 14792   picpi 14797   logclog 24301

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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  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  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  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-rmo 2920  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ioc 12180  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-shft 13807  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-sum 14417  df-ef 14798  df-sin 14800  df-cos 14801  df-pi 14803  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lp 20940  df-perf 20941  df-cn 21031  df-cnp 21032  df-haus 21119  df-tx 21365  df-hmeo 21558  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-xms 22125  df-ms 22126  df-tms 22127  df-cncf 22681  df-limc 23630  df-dv 23631  df-log 24303

This theorem is referenced by:  ang180lem2  24540  ang180lem3  24541