Theorem ang180lem3 24541

Description: Lemma for ang180 24544. Since ang180lem1 24539 shows that N is an integer and ang180lem2 24540 shows that N is strictly between -u 2 and 1, it follows that N e. { -u 1 , 0 }, and these two cases correspond to the two possible values for T. (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
ang180lem3	|- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> T e. { -u ( _i x. pi ) , ( _i x. pi ) } )

Distinct variable group:   x, y, A

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

Proof of Theorem ang180lem3

Step	Hyp	Ref	Expression
1		2cn 11091	. . . . . . . . . 10 |- 2 e. CC
2		picn 24211	. . . . . . . . . 10 |- pi e. CC
3	1, 2	mulcli 10045	. . . . . . . . 9 |- ( 2 x. pi ) e. CC
4		2ne0 11113	. . . . . . . . 9 |- 2 =/= 0
5	3, 1, 4	divreci 10770	. . . . . . . 8 |- ( ( 2 x. pi ) / 2 ) = ( ( 2 x. pi ) x. ( 1 / 2 ) )
6	2, 1, 4	divcan3i 10771	. . . . . . . 8 |- ( ( 2 x. pi ) / 2 ) = pi
7	5, 6	eqtr3i 2646	. . . . . . 7 |- ( ( 2 x. pi ) x. ( 1 / 2 ) ) = pi
8		ang180lem1.3	. . . . . . . . . 10 |- N = ( ( ( T / _i ) / ( 2 x. pi ) ) - ( 1 / 2 ) )
9		ang.1	. . . . . . . . . . . . . . . 16 |- F = ( x e. ( CC \ { 0 } ) , y e. ( CC \ { 0 } ) |-> ( Im ` ( log ` ( y / x ) ) ) )
10		ang180lem1.2	. . . . . . . . . . . . . . . 16 |- T = ( ( ( log ` ( 1 / ( 1 - A ) ) ) + ( log ` ( ( A - 1 ) / A ) ) ) + ( log ` A ) )
11	9, 10, 8	ang180lem2 24540	. . . . . . . . . . . . . . 15 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( -u 2 < N /\ N < 1 ) )
12	11	simprd 479	. . . . . . . . . . . . . 14 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> N < 1 )
13		1e0p1 11552	. . . . . . . . . . . . . 14 |- 1 = ( 0 + 1 )
14	12, 13	syl6breq 4694	. . . . . . . . . . . . 13 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> N < ( 0 + 1 ) )
15	9, 10, 8	ang180lem1 24539	. . . . . . . . . . . . . . 15 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( N e. ZZ /\ ( T / _i ) e. RR ) )
16	15	simpld 475	. . . . . . . . . . . . . 14 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> N e. ZZ )
17		0z 11388	. . . . . . . . . . . . . 14 |- 0 e. ZZ
18		zleltp1 11428	. . . . . . . . . . . . . 14 |- ( ( N e. ZZ /\ 0 e. ZZ ) -> ( N <_ 0 <-> N < ( 0 + 1 ) ) )
19	16, 17, 18	sylancl 694	. . . . . . . . . . . . 13 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( N <_ 0 <-> N < ( 0 + 1 ) ) )
20	14, 19	mpbird 247	. . . . . . . . . . . 12 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> N <_ 0 )
21	20	adantr 481	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) /\ -u 1 < N ) -> N <_ 0 )
22		zlem1lt 11429	. . . . . . . . . . . . . 14 |- ( ( 0 e. ZZ /\ N e. ZZ ) -> ( 0 <_ N <-> ( 0 - 1 ) < N ) )
23	17, 16, 22	sylancr 695	. . . . . . . . . . . . 13 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( 0 <_ N <-> ( 0 - 1 ) < N ) )
24		df-neg 10269	. . . . . . . . . . . . . 14 |- -u 1 = ( 0 - 1 )
25	24	breq1i 4660	. . . . . . . . . . . . 13 |- ( -u 1 < N <-> ( 0 - 1 ) < N )
26	23, 25	syl6bbr 278	. . . . . . . . . . . 12 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( 0 <_ N <-> -u 1 < N ) )
27	26	biimpar 502	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) /\ -u 1 < N ) -> 0 <_ N )
28	16	zred 11482	. . . . . . . . . . . . 13 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> N e. RR )
29	28	adantr 481	. . . . . . . . . . . 12 |- ( ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) /\ -u 1 < N ) -> N e. RR )
30		0re 10040	. . . . . . . . . . . 12 |- 0 e. RR
31		letri3 10123	. . . . . . . . . . . 12 |- ( ( N e. RR /\ 0 e. RR ) -> ( N = 0 <-> ( N <_ 0 /\ 0 <_ N ) ) )
32	29, 30, 31	sylancl 694	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) /\ -u 1 < N ) -> ( N = 0 <-> ( N <_ 0 /\ 0 <_ N ) ) )
33	21, 27, 32	mpbir2and 957	. . . . . . . . . 10 |- ( ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) /\ -u 1 < N ) -> N = 0 )
34	8, 33	syl5eqr 2670	. . . . . . . . 9 |- ( ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) /\ -u 1 < N ) -> ( ( ( T / _i ) / ( 2 x. pi ) ) - ( 1 / 2 ) ) = 0 )
35		ax-1cn 9994	. . . . . . . . . . . . . . . . . . 19 |- 1 e. CC
36		simp1 1061	. . . . . . . . . . . . . . . . . . 19 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> A e. CC )
37		subcl 10280	. . . . . . . . . . . . . . . . . . 19 |- ( ( 1 e. CC /\ A e. CC ) -> ( 1 - A ) e. CC )
38	35, 36, 37	sylancr 695	. . . . . . . . . . . . . . . . . 18 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( 1 - A ) e. CC )
39		simp3 1063	. . . . . . . . . . . . . . . . . . . 20 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> A =/= 1 )
40	39	necomd 2849	. . . . . . . . . . . . . . . . . . 19 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> 1 =/= A )
41		subeq0 10307	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( 1 e. CC /\ A e. CC ) -> ( ( 1 - A ) = 0 <-> 1 = A ) )
42	35, 36, 41	sylancr 695	. . . . . . . . . . . . . . . . . . . 20 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( ( 1 - A ) = 0 <-> 1 = A ) )
43	42	necon3bid 2838	. . . . . . . . . . . . . . . . . . 19 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( ( 1 - A ) =/= 0 <-> 1 =/= A ) )
44	40, 43	mpbird 247	. . . . . . . . . . . . . . . . . 18 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( 1 - A ) =/= 0 )
45	38, 44	reccld 10794	. . . . . . . . . . . . . . . . 17 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( 1 / ( 1 - A ) ) e. CC )
46	38, 44	recne0d 10795	. . . . . . . . . . . . . . . . 17 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( 1 / ( 1 - A ) ) =/= 0 )
47	45, 46	logcld 24317	. . . . . . . . . . . . . . . 16 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( log ` ( 1 / ( 1 - A ) ) ) e. CC )
48		subcl 10280	. . . . . . . . . . . . . . . . . . 19 |- ( ( A e. CC /\ 1 e. CC ) -> ( A - 1 ) e. CC )
49	36, 35, 48	sylancl 694	. . . . . . . . . . . . . . . . . 18 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( A - 1 ) e. CC )
50		simp2 1062	. . . . . . . . . . . . . . . . . 18 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> A =/= 0 )
51	49, 36, 50	divcld 10801	. . . . . . . . . . . . . . . . 17 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( ( A - 1 ) / A ) e. CC )
52		subeq0 10307	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( A e. CC /\ 1 e. CC ) -> ( ( A - 1 ) = 0 <-> A = 1 ) )
53	36, 35, 52	sylancl 694	. . . . . . . . . . . . . . . . . . . 20 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( ( A - 1 ) = 0 <-> A = 1 ) )
54	53	necon3bid 2838	. . . . . . . . . . . . . . . . . . 19 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( ( A - 1 ) =/= 0 <-> A =/= 1 ) )
55	39, 54	mpbird 247	. . . . . . . . . . . . . . . . . 18 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( A - 1 ) =/= 0 )
56	49, 36, 55, 50	divne0d 10817	. . . . . . . . . . . . . . . . 17 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( ( A - 1 ) / A ) =/= 0 )
57	51, 56	logcld 24317	. . . . . . . . . . . . . . . 16 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( log ` ( ( A - 1 ) / A ) ) e. CC )
58	47, 57	addcld 10059	. . . . . . . . . . . . . . 15 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( ( log ` ( 1 / ( 1 - A ) ) ) + ( log ` ( ( A - 1 ) / A ) ) ) e. CC )
59		logcl 24315	. . . . . . . . . . . . . . . 16 |- ( ( A e. CC /\ A =/= 0 ) -> ( log ` A ) e. CC )
60	59	3adant3 1081	. . . . . . . . . . . . . . 15 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( log ` A ) e. CC )
61	58, 60	addcld 10059	. . . . . . . . . . . . . 14 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( ( ( log ` ( 1 / ( 1 - A ) ) ) + ( log ` ( ( A - 1 ) / A ) ) ) + ( log ` A ) ) e. CC )
62	10, 61	syl5eqel 2705	. . . . . . . . . . . . 13 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> T e. CC )
63		ax-icn 9995	. . . . . . . . . . . . . 14 |- _i e. CC
64	63	a1i 11	. . . . . . . . . . . . 13 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> _i e. CC )
65		ine0 10465	. . . . . . . . . . . . . 14 |- _i =/= 0
66	65	a1i 11	. . . . . . . . . . . . 13 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> _i =/= 0 )
67	62, 64, 66	divcld 10801	. . . . . . . . . . . 12 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( T / _i ) e. CC )
68	3	a1i 11	. . . . . . . . . . . 12 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( 2 x. pi ) e. CC )
69		pire 24210	. . . . . . . . . . . . . . 15 |- pi e. RR
70		pipos 24212	. . . . . . . . . . . . . . 15 |- 0 < pi
71	69, 70	gt0ne0ii 10564	. . . . . . . . . . . . . 14 |- pi =/= 0
72	1, 2, 4, 71	mulne0i 10670	. . . . . . . . . . . . 13 |- ( 2 x. pi ) =/= 0
73	72	a1i 11	. . . . . . . . . . . 12 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( 2 x. pi ) =/= 0 )
74	67, 68, 73	divcld 10801	. . . . . . . . . . 11 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( ( T / _i ) / ( 2 x. pi ) ) e. CC )
75	74	adantr 481	. . . . . . . . . 10 |- ( ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) /\ -u 1 < N ) -> ( ( T / _i ) / ( 2 x. pi ) ) e. CC )
76		halfcn 11247	. . . . . . . . . 10 |- ( 1 / 2 ) e. CC
77		subeq0 10307	. . . . . . . . . 10 |- ( ( ( ( T / _i ) / ( 2 x. pi ) ) e. CC /\ ( 1 / 2 ) e. CC ) -> ( ( ( ( T / _i ) / ( 2 x. pi ) ) - ( 1 / 2 ) ) = 0 <-> ( ( T / _i ) / ( 2 x. pi ) ) = ( 1 / 2 ) ) )
78	75, 76, 77	sylancl 694	. . . . . . . . 9 |- ( ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) /\ -u 1 < N ) -> ( ( ( ( T / _i ) / ( 2 x. pi ) ) - ( 1 / 2 ) ) = 0 <-> ( ( T / _i ) / ( 2 x. pi ) ) = ( 1 / 2 ) ) )
79	34, 78	mpbid 222	. . . . . . . 8 |- ( ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) /\ -u 1 < N ) -> ( ( T / _i ) / ( 2 x. pi ) ) = ( 1 / 2 ) )
80	67	adantr 481	. . . . . . . . 9 |- ( ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) /\ -u 1 < N ) -> ( T / _i ) e. CC )
81	3	a1i 11	. . . . . . . . 9 |- ( ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) /\ -u 1 < N ) -> ( 2 x. pi ) e. CC )
82	76	a1i 11	. . . . . . . . 9 |- ( ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) /\ -u 1 < N ) -> ( 1 / 2 ) e. CC )
83	72	a1i 11	. . . . . . . . 9 |- ( ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) /\ -u 1 < N ) -> ( 2 x. pi ) =/= 0 )
84	80, 81, 82, 83	divmuld 10823	. . . . . . . 8 |- ( ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) /\ -u 1 < N ) -> ( ( ( T / _i ) / ( 2 x. pi ) ) = ( 1 / 2 ) <-> ( ( 2 x. pi ) x. ( 1 / 2 ) ) = ( T / _i ) ) )
85	79, 84	mpbid 222	. . . . . . 7 |- ( ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) /\ -u 1 < N ) -> ( ( 2 x. pi ) x. ( 1 / 2 ) ) = ( T / _i ) )
86	7, 85	syl5reqr 2671	. . . . . 6 |- ( ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) /\ -u 1 < N ) -> ( T / _i ) = pi )
87	62	adantr 481	. . . . . . 7 |- ( ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) /\ -u 1 < N ) -> T e. CC )
88	63	a1i 11	. . . . . . 7 |- ( ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) /\ -u 1 < N ) -> _i e. CC )
89	2	a1i 11	. . . . . . 7 |- ( ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) /\ -u 1 < N ) -> pi e. CC )
90	65	a1i 11	. . . . . . 7 |- ( ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) /\ -u 1 < N ) -> _i =/= 0 )
91	87, 88, 89, 90	divmuld 10823	. . . . . 6 |- ( ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) /\ -u 1 < N ) -> ( ( T / _i ) = pi <-> ( _i x. pi ) = T ) )
92	86, 91	mpbid 222	. . . . 5 |- ( ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) /\ -u 1 < N ) -> ( _i x. pi ) = T )
93	92	eqcomd 2628	. . . 4 |- ( ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) /\ -u 1 < N ) -> T = ( _i x. pi ) )
94	93	olcd 408	. . 3 |- ( ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) /\ -u 1 < N ) -> ( T = -u ( _i x. pi ) \/ T = ( _i x. pi ) ) )
95	2, 63	mulneg1i 10476	. . . . . . 7 |- ( -u pi x. _i ) = -u ( pi x. _i )
96	2, 63	mulcomi 10046	. . . . . . . 8 |- ( pi x. _i ) = ( _i x. pi )
97	96	negeqi 10274	. . . . . . 7 |- -u ( pi x. _i ) = -u ( _i x. pi )
98	95, 97	eqtri 2644	. . . . . 6 |- ( -u pi x. _i ) = -u ( _i x. pi )
99	76, 3	mulneg1i 10476	. . . . . . . . . 10 |- ( -u ( 1 / 2 ) x. ( 2 x. pi ) ) = -u ( ( 1 / 2 ) x. ( 2 x. pi ) )
100	35, 1, 4	divcan1i 10769	. . . . . . . . . . . . 13 |- ( ( 1 / 2 ) x. 2 ) = 1
101	100	oveq1i 6660	. . . . . . . . . . . 12 |- ( ( ( 1 / 2 ) x. 2 ) x. pi ) = ( 1 x. pi )
102	76, 1, 2	mulassi 10049	. . . . . . . . . . . 12 |- ( ( ( 1 / 2 ) x. 2 ) x. pi ) = ( ( 1 / 2 ) x. ( 2 x. pi ) )
103	2	mulid2i 10043	. . . . . . . . . . . 12 |- ( 1 x. pi ) = pi
104	101, 102, 103	3eqtr3i 2652	. . . . . . . . . . 11 |- ( ( 1 / 2 ) x. ( 2 x. pi ) ) = pi
105	104	negeqi 10274	. . . . . . . . . 10 |- -u ( ( 1 / 2 ) x. ( 2 x. pi ) ) = -u pi
106	99, 105	eqtri 2644	. . . . . . . . 9 |- ( -u ( 1 / 2 ) x. ( 2 x. pi ) ) = -u pi
107	35, 76	negsubdii 10366	. . . . . . . . . . . . 13 |- -u ( 1 - ( 1 / 2 ) ) = ( -u 1 + ( 1 / 2 ) )
108		1mhlfehlf 11251	. . . . . . . . . . . . . 14 |- ( 1 - ( 1 / 2 ) ) = ( 1 / 2 )
109	108	negeqi 10274	. . . . . . . . . . . . 13 |- -u ( 1 - ( 1 / 2 ) ) = -u ( 1 / 2 )
110	107, 109	eqtr3i 2646	. . . . . . . . . . . 12 |- ( -u 1 + ( 1 / 2 ) ) = -u ( 1 / 2 )
111		simpr 477	. . . . . . . . . . . . . 14 |- ( ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) /\ -u 1 = N ) -> -u 1 = N )
112	111, 8	syl6eq 2672	. . . . . . . . . . . . 13 |- ( ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) /\ -u 1 = N ) -> -u 1 = ( ( ( T / _i ) / ( 2 x. pi ) ) - ( 1 / 2 ) ) )
113	112	oveq1d 6665	. . . . . . . . . . . 12 |- ( ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) /\ -u 1 = N ) -> ( -u 1 + ( 1 / 2 ) ) = ( ( ( ( T / _i ) / ( 2 x. pi ) ) - ( 1 / 2 ) ) + ( 1 / 2 ) ) )
114	110, 113	syl5eqr 2670	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) /\ -u 1 = N ) -> -u ( 1 / 2 ) = ( ( ( ( T / _i ) / ( 2 x. pi ) ) - ( 1 / 2 ) ) + ( 1 / 2 ) ) )
115		npcan 10290	. . . . . . . . . . . . 13 |- ( ( ( ( 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 ) ) )
116	74, 76, 115	sylancl 694	. . . . . . . . . . . 12 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( ( ( ( T / _i ) / ( 2 x. pi ) ) - ( 1 / 2 ) ) + ( 1 / 2 ) ) = ( ( T / _i ) / ( 2 x. pi ) ) )
117	116	adantr 481	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) /\ -u 1 = N ) -> ( ( ( ( T / _i ) / ( 2 x. pi ) ) - ( 1 / 2 ) ) + ( 1 / 2 ) ) = ( ( T / _i ) / ( 2 x. pi ) ) )
118	114, 117	eqtrd 2656	. . . . . . . . . 10 |- ( ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) /\ -u 1 = N ) -> -u ( 1 / 2 ) = ( ( T / _i ) / ( 2 x. pi ) ) )
119	118	oveq1d 6665	. . . . . . . . 9 |- ( ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) /\ -u 1 = N ) -> ( -u ( 1 / 2 ) x. ( 2 x. pi ) ) = ( ( ( T / _i ) / ( 2 x. pi ) ) x. ( 2 x. pi ) ) )
120	106, 119	syl5eqr 2670	. . . . . . . 8 |- ( ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) /\ -u 1 = N ) -> -u pi = ( ( ( T / _i ) / ( 2 x. pi ) ) x. ( 2 x. pi ) ) )
121	67, 68, 73	divcan1d 10802	. . . . . . . . 9 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( ( ( T / _i ) / ( 2 x. pi ) ) x. ( 2 x. pi ) ) = ( T / _i ) )
122	121	adantr 481	. . . . . . . 8 |- ( ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) /\ -u 1 = N ) -> ( ( ( T / _i ) / ( 2 x. pi ) ) x. ( 2 x. pi ) ) = ( T / _i ) )
123	120, 122	eqtrd 2656	. . . . . . 7 |- ( ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) /\ -u 1 = N ) -> -u pi = ( T / _i ) )
124	123	oveq1d 6665	. . . . . 6 |- ( ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) /\ -u 1 = N ) -> ( -u pi x. _i ) = ( ( T / _i ) x. _i ) )
125	98, 124	syl5eqr 2670	. . . . 5 |- ( ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) /\ -u 1 = N ) -> -u ( _i x. pi ) = ( ( T / _i ) x. _i ) )
126	62, 64, 66	divcan1d 10802	. . . . . 6 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( ( T / _i ) x. _i ) = T )
127	126	adantr 481	. . . . 5 |- ( ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) /\ -u 1 = N ) -> ( ( T / _i ) x. _i ) = T )
128	125, 127	eqtr2d 2657	. . . 4 |- ( ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) /\ -u 1 = N ) -> T = -u ( _i x. pi ) )
129	128	orcd 407	. . 3 |- ( ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) /\ -u 1 = N ) -> ( T = -u ( _i x. pi ) \/ T = ( _i x. pi ) ) )
130		df-2 11079	. . . . . . . 8 |- 2 = ( 1 + 1 )
131	130	negeqi 10274	. . . . . . 7 |- -u 2 = -u ( 1 + 1 )
132		negdi2 10339	. . . . . . . 8 |- ( ( 1 e. CC /\ 1 e. CC ) -> -u ( 1 + 1 ) = ( -u 1 - 1 ) )
133	35, 35, 132	mp2an 708	. . . . . . 7 |- -u ( 1 + 1 ) = ( -u 1 - 1 )
134	131, 133	eqtri 2644	. . . . . 6 |- -u 2 = ( -u 1 - 1 )
135	11	simpld 475	. . . . . 6 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> -u 2 < N )
136	134, 135	syl5eqbrr 4689	. . . . 5 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( -u 1 - 1 ) < N )
137		neg1z 11413	. . . . . 6 |- -u 1 e. ZZ
138		zlem1lt 11429	. . . . . 6 |- ( ( -u 1 e. ZZ /\ N e. ZZ ) -> ( -u 1 <_ N <-> ( -u 1 - 1 ) < N ) )
139	137, 16, 138	sylancr 695	. . . . 5 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( -u 1 <_ N <-> ( -u 1 - 1 ) < N ) )
140	136, 139	mpbird 247	. . . 4 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> -u 1 <_ N )
141		neg1rr 11125	. . . . 5 |- -u 1 e. RR
142		leloe 10124	. . . . 5 |- ( ( -u 1 e. RR /\ N e. RR ) -> ( -u 1 <_ N <-> ( -u 1 < N \/ -u 1 = N ) ) )
143	141, 28, 142	sylancr 695	. . . 4 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( -u 1 <_ N <-> ( -u 1 < N \/ -u 1 = N ) ) )
144	140, 143	mpbid 222	. . 3 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( -u 1 < N \/ -u 1 = N ) )
145	94, 129, 144	mpjaodan 827	. 2 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( T = -u ( _i x. pi ) \/ T = ( _i x. pi ) ) )
146		ovex 6678	. . . 4 |- ( ( ( log ` ( 1 / ( 1 - A ) ) ) + ( log ` ( ( A - 1 ) / A ) ) ) + ( log ` A ) ) e. _V
147	10, 146	eqeltri 2697	. . 3 |- T e. _V
148	147	elpr 4198	. 2 |- ( T e. { -u ( _i x. pi ) , ( _i x. pi ) } <-> ( T = -u ( _i x. pi ) \/ T = ( _i x. pi ) ) )
149	145, 148	sylibr 224	1 |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> T e. { -u ( _i x. pi ) , ( _i x. pi ) } )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   \ cdif 3571   {csn 4177   {cpr 4179   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   _ici 9938   + caddc 9939   x. cmul 9941   < clt 10074   <_ cle 10075   - cmin 10266   -ucneg 10267   / cdiv 10684   2c2 11070   ZZcz 11377   Imcim 13838   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:  ang180lem4  24542