Step |
Hyp |
Ref |
Expression |
1 |
|
efif1olem1.1 |
|- D = ( A (,] ( A + ( 2 x. _pi ) ) ) |
2 |
|
simpl |
|- ( ( A e. RR /\ z e. RR ) -> A e. RR ) |
3 |
|
2re |
|- 2 e. RR |
4 |
|
pire |
|- _pi e. RR |
5 |
3 4
|
remulcli |
|- ( 2 x. _pi ) e. RR |
6 |
|
readdcl |
|- ( ( A e. RR /\ ( 2 x. _pi ) e. RR ) -> ( A + ( 2 x. _pi ) ) e. RR ) |
7 |
2 5 6
|
sylancl |
|- ( ( A e. RR /\ z e. RR ) -> ( A + ( 2 x. _pi ) ) e. RR ) |
8 |
|
resubcl |
|- ( ( A e. RR /\ z e. RR ) -> ( A - z ) e. RR ) |
9 |
|
2pos |
|- 0 < 2 |
10 |
|
pipos |
|- 0 < _pi |
11 |
3 4 9 10
|
mulgt0ii |
|- 0 < ( 2 x. _pi ) |
12 |
5 11
|
elrpii |
|- ( 2 x. _pi ) e. RR+ |
13 |
|
modcl |
|- ( ( ( A - z ) e. RR /\ ( 2 x. _pi ) e. RR+ ) -> ( ( A - z ) mod ( 2 x. _pi ) ) e. RR ) |
14 |
8 12 13
|
sylancl |
|- ( ( A e. RR /\ z e. RR ) -> ( ( A - z ) mod ( 2 x. _pi ) ) e. RR ) |
15 |
7 14
|
resubcld |
|- ( ( A e. RR /\ z e. RR ) -> ( ( A + ( 2 x. _pi ) ) - ( ( A - z ) mod ( 2 x. _pi ) ) ) e. RR ) |
16 |
5
|
a1i |
|- ( ( A e. RR /\ z e. RR ) -> ( 2 x. _pi ) e. RR ) |
17 |
|
modlt |
|- ( ( ( A - z ) e. RR /\ ( 2 x. _pi ) e. RR+ ) -> ( ( A - z ) mod ( 2 x. _pi ) ) < ( 2 x. _pi ) ) |
18 |
8 12 17
|
sylancl |
|- ( ( A e. RR /\ z e. RR ) -> ( ( A - z ) mod ( 2 x. _pi ) ) < ( 2 x. _pi ) ) |
19 |
14 16 2 18
|
ltadd2dd |
|- ( ( A e. RR /\ z e. RR ) -> ( A + ( ( A - z ) mod ( 2 x. _pi ) ) ) < ( A + ( 2 x. _pi ) ) ) |
20 |
2 14 7
|
ltaddsubd |
|- ( ( A e. RR /\ z e. RR ) -> ( ( A + ( ( A - z ) mod ( 2 x. _pi ) ) ) < ( A + ( 2 x. _pi ) ) <-> A < ( ( A + ( 2 x. _pi ) ) - ( ( A - z ) mod ( 2 x. _pi ) ) ) ) ) |
21 |
19 20
|
mpbid |
|- ( ( A e. RR /\ z e. RR ) -> A < ( ( A + ( 2 x. _pi ) ) - ( ( A - z ) mod ( 2 x. _pi ) ) ) ) |
22 |
|
modge0 |
|- ( ( ( A - z ) e. RR /\ ( 2 x. _pi ) e. RR+ ) -> 0 <_ ( ( A - z ) mod ( 2 x. _pi ) ) ) |
23 |
8 12 22
|
sylancl |
|- ( ( A e. RR /\ z e. RR ) -> 0 <_ ( ( A - z ) mod ( 2 x. _pi ) ) ) |
24 |
7 14
|
subge02d |
|- ( ( A e. RR /\ z e. RR ) -> ( 0 <_ ( ( A - z ) mod ( 2 x. _pi ) ) <-> ( ( A + ( 2 x. _pi ) ) - ( ( A - z ) mod ( 2 x. _pi ) ) ) <_ ( A + ( 2 x. _pi ) ) ) ) |
25 |
23 24
|
mpbid |
|- ( ( A e. RR /\ z e. RR ) -> ( ( A + ( 2 x. _pi ) ) - ( ( A - z ) mod ( 2 x. _pi ) ) ) <_ ( A + ( 2 x. _pi ) ) ) |
26 |
|
rexr |
|- ( A e. RR -> A e. RR* ) |
27 |
|
elioc2 |
|- ( ( A e. RR* /\ ( A + ( 2 x. _pi ) ) e. RR ) -> ( ( ( A + ( 2 x. _pi ) ) - ( ( A - z ) mod ( 2 x. _pi ) ) ) e. ( A (,] ( A + ( 2 x. _pi ) ) ) <-> ( ( ( A + ( 2 x. _pi ) ) - ( ( A - z ) mod ( 2 x. _pi ) ) ) e. RR /\ A < ( ( A + ( 2 x. _pi ) ) - ( ( A - z ) mod ( 2 x. _pi ) ) ) /\ ( ( A + ( 2 x. _pi ) ) - ( ( A - z ) mod ( 2 x. _pi ) ) ) <_ ( A + ( 2 x. _pi ) ) ) ) ) |
28 |
26 7 27
|
syl2an2r |
|- ( ( A e. RR /\ z e. RR ) -> ( ( ( A + ( 2 x. _pi ) ) - ( ( A - z ) mod ( 2 x. _pi ) ) ) e. ( A (,] ( A + ( 2 x. _pi ) ) ) <-> ( ( ( A + ( 2 x. _pi ) ) - ( ( A - z ) mod ( 2 x. _pi ) ) ) e. RR /\ A < ( ( A + ( 2 x. _pi ) ) - ( ( A - z ) mod ( 2 x. _pi ) ) ) /\ ( ( A + ( 2 x. _pi ) ) - ( ( A - z ) mod ( 2 x. _pi ) ) ) <_ ( A + ( 2 x. _pi ) ) ) ) ) |
29 |
15 21 25 28
|
mpbir3and |
|- ( ( A e. RR /\ z e. RR ) -> ( ( A + ( 2 x. _pi ) ) - ( ( A - z ) mod ( 2 x. _pi ) ) ) e. ( A (,] ( A + ( 2 x. _pi ) ) ) ) |
30 |
29 1
|
eleqtrrdi |
|- ( ( A e. RR /\ z e. RR ) -> ( ( A + ( 2 x. _pi ) ) - ( ( A - z ) mod ( 2 x. _pi ) ) ) e. D ) |
31 |
|
modval |
|- ( ( ( A - z ) e. RR /\ ( 2 x. _pi ) e. RR+ ) -> ( ( A - z ) mod ( 2 x. _pi ) ) = ( ( A - z ) - ( ( 2 x. _pi ) x. ( |_ ` ( ( A - z ) / ( 2 x. _pi ) ) ) ) ) ) |
32 |
8 12 31
|
sylancl |
|- ( ( A e. RR /\ z e. RR ) -> ( ( A - z ) mod ( 2 x. _pi ) ) = ( ( A - z ) - ( ( 2 x. _pi ) x. ( |_ ` ( ( A - z ) / ( 2 x. _pi ) ) ) ) ) ) |
33 |
32
|
oveq2d |
|- ( ( A e. RR /\ z e. RR ) -> ( ( A + ( 2 x. _pi ) ) - ( ( A - z ) mod ( 2 x. _pi ) ) ) = ( ( A + ( 2 x. _pi ) ) - ( ( A - z ) - ( ( 2 x. _pi ) x. ( |_ ` ( ( A - z ) / ( 2 x. _pi ) ) ) ) ) ) ) |
34 |
7
|
recnd |
|- ( ( A e. RR /\ z e. RR ) -> ( A + ( 2 x. _pi ) ) e. CC ) |
35 |
8
|
recnd |
|- ( ( A e. RR /\ z e. RR ) -> ( A - z ) e. CC ) |
36 |
5 11
|
gt0ne0ii |
|- ( 2 x. _pi ) =/= 0 |
37 |
|
redivcl |
|- ( ( ( A - z ) e. RR /\ ( 2 x. _pi ) e. RR /\ ( 2 x. _pi ) =/= 0 ) -> ( ( A - z ) / ( 2 x. _pi ) ) e. RR ) |
38 |
5 36 37
|
mp3an23 |
|- ( ( A - z ) e. RR -> ( ( A - z ) / ( 2 x. _pi ) ) e. RR ) |
39 |
8 38
|
syl |
|- ( ( A e. RR /\ z e. RR ) -> ( ( A - z ) / ( 2 x. _pi ) ) e. RR ) |
40 |
39
|
flcld |
|- ( ( A e. RR /\ z e. RR ) -> ( |_ ` ( ( A - z ) / ( 2 x. _pi ) ) ) e. ZZ ) |
41 |
40
|
zred |
|- ( ( A e. RR /\ z e. RR ) -> ( |_ ` ( ( A - z ) / ( 2 x. _pi ) ) ) e. RR ) |
42 |
|
remulcl |
|- ( ( ( 2 x. _pi ) e. RR /\ ( |_ ` ( ( A - z ) / ( 2 x. _pi ) ) ) e. RR ) -> ( ( 2 x. _pi ) x. ( |_ ` ( ( A - z ) / ( 2 x. _pi ) ) ) ) e. RR ) |
43 |
5 41 42
|
sylancr |
|- ( ( A e. RR /\ z e. RR ) -> ( ( 2 x. _pi ) x. ( |_ ` ( ( A - z ) / ( 2 x. _pi ) ) ) ) e. RR ) |
44 |
43
|
recnd |
|- ( ( A e. RR /\ z e. RR ) -> ( ( 2 x. _pi ) x. ( |_ ` ( ( A - z ) / ( 2 x. _pi ) ) ) ) e. CC ) |
45 |
34 35 44
|
subsubd |
|- ( ( A e. RR /\ z e. RR ) -> ( ( A + ( 2 x. _pi ) ) - ( ( A - z ) - ( ( 2 x. _pi ) x. ( |_ ` ( ( A - z ) / ( 2 x. _pi ) ) ) ) ) ) = ( ( ( A + ( 2 x. _pi ) ) - ( A - z ) ) + ( ( 2 x. _pi ) x. ( |_ ` ( ( A - z ) / ( 2 x. _pi ) ) ) ) ) ) |
46 |
2
|
recnd |
|- ( ( A e. RR /\ z e. RR ) -> A e. CC ) |
47 |
5
|
recni |
|- ( 2 x. _pi ) e. CC |
48 |
47
|
a1i |
|- ( ( A e. RR /\ z e. RR ) -> ( 2 x. _pi ) e. CC ) |
49 |
|
simpr |
|- ( ( A e. RR /\ z e. RR ) -> z e. RR ) |
50 |
49
|
recnd |
|- ( ( A e. RR /\ z e. RR ) -> z e. CC ) |
51 |
46 48 50
|
pnncand |
|- ( ( A e. RR /\ z e. RR ) -> ( ( A + ( 2 x. _pi ) ) - ( A - z ) ) = ( ( 2 x. _pi ) + z ) ) |
52 |
51
|
oveq1d |
|- ( ( A e. RR /\ z e. RR ) -> ( ( ( A + ( 2 x. _pi ) ) - ( A - z ) ) + ( ( 2 x. _pi ) x. ( |_ ` ( ( A - z ) / ( 2 x. _pi ) ) ) ) ) = ( ( ( 2 x. _pi ) + z ) + ( ( 2 x. _pi ) x. ( |_ ` ( ( A - z ) / ( 2 x. _pi ) ) ) ) ) ) |
53 |
33 45 52
|
3eqtrd |
|- ( ( A e. RR /\ z e. RR ) -> ( ( A + ( 2 x. _pi ) ) - ( ( A - z ) mod ( 2 x. _pi ) ) ) = ( ( ( 2 x. _pi ) + z ) + ( ( 2 x. _pi ) x. ( |_ ` ( ( A - z ) / ( 2 x. _pi ) ) ) ) ) ) |
54 |
53
|
oveq2d |
|- ( ( A e. RR /\ z e. RR ) -> ( z - ( ( A + ( 2 x. _pi ) ) - ( ( A - z ) mod ( 2 x. _pi ) ) ) ) = ( z - ( ( ( 2 x. _pi ) + z ) + ( ( 2 x. _pi ) x. ( |_ ` ( ( A - z ) / ( 2 x. _pi ) ) ) ) ) ) ) |
55 |
|
addcl |
|- ( ( ( 2 x. _pi ) e. CC /\ z e. CC ) -> ( ( 2 x. _pi ) + z ) e. CC ) |
56 |
47 50 55
|
sylancr |
|- ( ( A e. RR /\ z e. RR ) -> ( ( 2 x. _pi ) + z ) e. CC ) |
57 |
50 56 44
|
subsub4d |
|- ( ( A e. RR /\ z e. RR ) -> ( ( z - ( ( 2 x. _pi ) + z ) ) - ( ( 2 x. _pi ) x. ( |_ ` ( ( A - z ) / ( 2 x. _pi ) ) ) ) ) = ( z - ( ( ( 2 x. _pi ) + z ) + ( ( 2 x. _pi ) x. ( |_ ` ( ( A - z ) / ( 2 x. _pi ) ) ) ) ) ) ) |
58 |
56 50
|
negsubdi2d |
|- ( ( A e. RR /\ z e. RR ) -> -u ( ( ( 2 x. _pi ) + z ) - z ) = ( z - ( ( 2 x. _pi ) + z ) ) ) |
59 |
48 50
|
pncand |
|- ( ( A e. RR /\ z e. RR ) -> ( ( ( 2 x. _pi ) + z ) - z ) = ( 2 x. _pi ) ) |
60 |
59
|
negeqd |
|- ( ( A e. RR /\ z e. RR ) -> -u ( ( ( 2 x. _pi ) + z ) - z ) = -u ( 2 x. _pi ) ) |
61 |
58 60
|
eqtr3d |
|- ( ( A e. RR /\ z e. RR ) -> ( z - ( ( 2 x. _pi ) + z ) ) = -u ( 2 x. _pi ) ) |
62 |
|
neg1cn |
|- -u 1 e. CC |
63 |
47
|
mulm1i |
|- ( -u 1 x. ( 2 x. _pi ) ) = -u ( 2 x. _pi ) |
64 |
62 47 63
|
mulcomli |
|- ( ( 2 x. _pi ) x. -u 1 ) = -u ( 2 x. _pi ) |
65 |
61 64
|
eqtr4di |
|- ( ( A e. RR /\ z e. RR ) -> ( z - ( ( 2 x. _pi ) + z ) ) = ( ( 2 x. _pi ) x. -u 1 ) ) |
66 |
65
|
oveq1d |
|- ( ( A e. RR /\ z e. RR ) -> ( ( z - ( ( 2 x. _pi ) + z ) ) - ( ( 2 x. _pi ) x. ( |_ ` ( ( A - z ) / ( 2 x. _pi ) ) ) ) ) = ( ( ( 2 x. _pi ) x. -u 1 ) - ( ( 2 x. _pi ) x. ( |_ ` ( ( A - z ) / ( 2 x. _pi ) ) ) ) ) ) |
67 |
62
|
a1i |
|- ( ( A e. RR /\ z e. RR ) -> -u 1 e. CC ) |
68 |
40
|
zcnd |
|- ( ( A e. RR /\ z e. RR ) -> ( |_ ` ( ( A - z ) / ( 2 x. _pi ) ) ) e. CC ) |
69 |
48 67 68
|
subdid |
|- ( ( A e. RR /\ z e. RR ) -> ( ( 2 x. _pi ) x. ( -u 1 - ( |_ ` ( ( A - z ) / ( 2 x. _pi ) ) ) ) ) = ( ( ( 2 x. _pi ) x. -u 1 ) - ( ( 2 x. _pi ) x. ( |_ ` ( ( A - z ) / ( 2 x. _pi ) ) ) ) ) ) |
70 |
66 69
|
eqtr4d |
|- ( ( A e. RR /\ z e. RR ) -> ( ( z - ( ( 2 x. _pi ) + z ) ) - ( ( 2 x. _pi ) x. ( |_ ` ( ( A - z ) / ( 2 x. _pi ) ) ) ) ) = ( ( 2 x. _pi ) x. ( -u 1 - ( |_ ` ( ( A - z ) / ( 2 x. _pi ) ) ) ) ) ) |
71 |
54 57 70
|
3eqtr2d |
|- ( ( A e. RR /\ z e. RR ) -> ( z - ( ( A + ( 2 x. _pi ) ) - ( ( A - z ) mod ( 2 x. _pi ) ) ) ) = ( ( 2 x. _pi ) x. ( -u 1 - ( |_ ` ( ( A - z ) / ( 2 x. _pi ) ) ) ) ) ) |
72 |
71
|
oveq1d |
|- ( ( A e. RR /\ z e. RR ) -> ( ( z - ( ( A + ( 2 x. _pi ) ) - ( ( A - z ) mod ( 2 x. _pi ) ) ) ) / ( 2 x. _pi ) ) = ( ( ( 2 x. _pi ) x. ( -u 1 - ( |_ ` ( ( A - z ) / ( 2 x. _pi ) ) ) ) ) / ( 2 x. _pi ) ) ) |
73 |
|
neg1z |
|- -u 1 e. ZZ |
74 |
|
zsubcl |
|- ( ( -u 1 e. ZZ /\ ( |_ ` ( ( A - z ) / ( 2 x. _pi ) ) ) e. ZZ ) -> ( -u 1 - ( |_ ` ( ( A - z ) / ( 2 x. _pi ) ) ) ) e. ZZ ) |
75 |
73 40 74
|
sylancr |
|- ( ( A e. RR /\ z e. RR ) -> ( -u 1 - ( |_ ` ( ( A - z ) / ( 2 x. _pi ) ) ) ) e. ZZ ) |
76 |
75
|
zcnd |
|- ( ( A e. RR /\ z e. RR ) -> ( -u 1 - ( |_ ` ( ( A - z ) / ( 2 x. _pi ) ) ) ) e. CC ) |
77 |
|
divcan3 |
|- ( ( ( -u 1 - ( |_ ` ( ( A - z ) / ( 2 x. _pi ) ) ) ) e. CC /\ ( 2 x. _pi ) e. CC /\ ( 2 x. _pi ) =/= 0 ) -> ( ( ( 2 x. _pi ) x. ( -u 1 - ( |_ ` ( ( A - z ) / ( 2 x. _pi ) ) ) ) ) / ( 2 x. _pi ) ) = ( -u 1 - ( |_ ` ( ( A - z ) / ( 2 x. _pi ) ) ) ) ) |
78 |
47 36 77
|
mp3an23 |
|- ( ( -u 1 - ( |_ ` ( ( A - z ) / ( 2 x. _pi ) ) ) ) e. CC -> ( ( ( 2 x. _pi ) x. ( -u 1 - ( |_ ` ( ( A - z ) / ( 2 x. _pi ) ) ) ) ) / ( 2 x. _pi ) ) = ( -u 1 - ( |_ ` ( ( A - z ) / ( 2 x. _pi ) ) ) ) ) |
79 |
76 78
|
syl |
|- ( ( A e. RR /\ z e. RR ) -> ( ( ( 2 x. _pi ) x. ( -u 1 - ( |_ ` ( ( A - z ) / ( 2 x. _pi ) ) ) ) ) / ( 2 x. _pi ) ) = ( -u 1 - ( |_ ` ( ( A - z ) / ( 2 x. _pi ) ) ) ) ) |
80 |
72 79
|
eqtrd |
|- ( ( A e. RR /\ z e. RR ) -> ( ( z - ( ( A + ( 2 x. _pi ) ) - ( ( A - z ) mod ( 2 x. _pi ) ) ) ) / ( 2 x. _pi ) ) = ( -u 1 - ( |_ ` ( ( A - z ) / ( 2 x. _pi ) ) ) ) ) |
81 |
80 75
|
eqeltrd |
|- ( ( A e. RR /\ z e. RR ) -> ( ( z - ( ( A + ( 2 x. _pi ) ) - ( ( A - z ) mod ( 2 x. _pi ) ) ) ) / ( 2 x. _pi ) ) e. ZZ ) |
82 |
|
oveq2 |
|- ( y = ( ( A + ( 2 x. _pi ) ) - ( ( A - z ) mod ( 2 x. _pi ) ) ) -> ( z - y ) = ( z - ( ( A + ( 2 x. _pi ) ) - ( ( A - z ) mod ( 2 x. _pi ) ) ) ) ) |
83 |
82
|
oveq1d |
|- ( y = ( ( A + ( 2 x. _pi ) ) - ( ( A - z ) mod ( 2 x. _pi ) ) ) -> ( ( z - y ) / ( 2 x. _pi ) ) = ( ( z - ( ( A + ( 2 x. _pi ) ) - ( ( A - z ) mod ( 2 x. _pi ) ) ) ) / ( 2 x. _pi ) ) ) |
84 |
83
|
eleq1d |
|- ( y = ( ( A + ( 2 x. _pi ) ) - ( ( A - z ) mod ( 2 x. _pi ) ) ) -> ( ( ( z - y ) / ( 2 x. _pi ) ) e. ZZ <-> ( ( z - ( ( A + ( 2 x. _pi ) ) - ( ( A - z ) mod ( 2 x. _pi ) ) ) ) / ( 2 x. _pi ) ) e. ZZ ) ) |
85 |
84
|
rspcev |
|- ( ( ( ( A + ( 2 x. _pi ) ) - ( ( A - z ) mod ( 2 x. _pi ) ) ) e. D /\ ( ( z - ( ( A + ( 2 x. _pi ) ) - ( ( A - z ) mod ( 2 x. _pi ) ) ) ) / ( 2 x. _pi ) ) e. ZZ ) -> E. y e. D ( ( z - y ) / ( 2 x. _pi ) ) e. ZZ ) |
86 |
30 81 85
|
syl2anc |
|- ( ( A e. RR /\ z e. RR ) -> E. y e. D ( ( z - y ) / ( 2 x. _pi ) ) e. ZZ ) |