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Theorem efif1olem2 19905
Description: Lemma for efif1o 19908. (Contributed by Mario Carneiro, 13-May-2014.)
Hypothesis
Ref Expression
efif1olem1.1  |-  D  =  ( A (,] ( A  +  ( 2  x.  pi ) ) )
Assertion
Ref Expression
efif1olem2  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  E. y  e.  D  ( ( z  -  y )  /  (
2  x.  pi ) )  e.  ZZ )
Distinct variable groups:    y, z    y, A    y, D
Allowed substitution hints:    A( z)    D( z)

Proof of Theorem efif1olem2
StepHypRef Expression
1 simpl 443 . . . . . 6  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  A  e.  RR )
2 2re 9815 . . . . . . 7  |-  2  e.  RR
3 pire 19832 . . . . . . 7  |-  pi  e.  RR
42, 3remulcli 8851 . . . . . 6  |-  ( 2  x.  pi )  e.  RR
5 readdcl 8820 . . . . . 6  |-  ( ( A  e.  RR  /\  ( 2  x.  pi )  e.  RR )  ->  ( A  +  ( 2  x.  pi ) )  e.  RR )
61, 4, 5sylancl 643 . . . . 5  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( A  +  ( 2  x.  pi ) )  e.  RR )
7 resubcl 9111 . . . . . 6  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( A  -  z
)  e.  RR )
8 2pos 9828 . . . . . . . 8  |-  0  <  2
9 pipos 19833 . . . . . . . 8  |-  0  <  pi
102, 3, 8, 9mulgt0ii 8952 . . . . . . 7  |-  0  <  ( 2  x.  pi )
114, 10elrpii 10357 . . . . . 6  |-  ( 2  x.  pi )  e.  RR+
12 modcl 10976 . . . . . 6  |-  ( ( ( A  -  z
)  e.  RR  /\  ( 2  x.  pi )  e.  RR+ )  -> 
( ( A  -  z )  mod  (
2  x.  pi ) )  e.  RR )
137, 11, 12sylancl 643 . . . . 5  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( ( A  -  z )  mod  (
2  x.  pi ) )  e.  RR )
146, 13resubcld 9211 . . . 4  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( ( A  +  ( 2  x.  pi ) )  -  (
( A  -  z
)  mod  ( 2  x.  pi ) ) )  e.  RR )
154a1i 10 . . . . . 6  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( 2  x.  pi )  e.  RR )
16 modlt 10981 . . . . . . 7  |-  ( ( ( A  -  z
)  e.  RR  /\  ( 2  x.  pi )  e.  RR+ )  -> 
( ( A  -  z )  mod  (
2  x.  pi ) )  <  ( 2  x.  pi ) )
177, 11, 16sylancl 643 . . . . . 6  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( ( A  -  z )  mod  (
2  x.  pi ) )  <  ( 2  x.  pi ) )
1813, 15, 1, 17ltadd2dd 8975 . . . . 5  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( A  +  ( ( A  -  z
)  mod  ( 2  x.  pi ) ) )  <  ( A  +  ( 2  x.  pi ) ) )
191, 13, 6ltaddsubd 9372 . . . . 5  |-  ( ( 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 ) ) ) ) )
2018, 19mpbid 201 . . . 4  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  A  <  ( ( A  +  ( 2  x.  pi ) )  -  ( ( A  -  z )  mod  ( 2  x.  pi ) ) ) )
21 modge0 10980 . . . . . 6  |-  ( ( ( A  -  z
)  e.  RR  /\  ( 2  x.  pi )  e.  RR+ )  -> 
0  <_  ( ( A  -  z )  mod  ( 2  x.  pi ) ) )
227, 11, 21sylancl 643 . . . . 5  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  0  <_  ( ( A  -  z )  mod  ( 2  x.  pi ) ) )
236, 13subge02d 9364 . . . . 5  |-  ( ( 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 ) ) ) )
2422, 23mpbid 201 . . . 4  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( ( A  +  ( 2  x.  pi ) )  -  (
( A  -  z
)  mod  ( 2  x.  pi ) ) )  <_  ( A  +  ( 2  x.  pi ) ) )
25 rexr 8877 . . . . . 6  |-  ( A  e.  RR  ->  A  e.  RR* )
2625adantr 451 . . . . 5  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  A  e.  RR* )
27 elioc2 10713 . . . . 5  |-  ( ( 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 ) ) ) ) )
2826, 6, 27syl2anc 642 . . . 4  |-  ( ( 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 ) ) ) ) )
2914, 20, 24, 28mpbir3and 1135 . . 3  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( ( A  +  ( 2  x.  pi ) )  -  (
( A  -  z
)  mod  ( 2  x.  pi ) ) )  e.  ( A (,] ( A  +  ( 2  x.  pi ) ) ) )
30 efif1olem1.1 . . 3  |-  D  =  ( A (,] ( A  +  ( 2  x.  pi ) ) )
3129, 30syl6eleqr 2374 . 2  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( ( A  +  ( 2  x.  pi ) )  -  (
( A  -  z
)  mod  ( 2  x.  pi ) ) )  e.  D )
32 modval 10975 . . . . . . . . . 10  |-  ( ( ( 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 ) ) ) ) ) )
337, 11, 32sylancl 643 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( ( A  -  z )  mod  (
2  x.  pi ) )  =  ( ( A  -  z )  -  ( ( 2  x.  pi )  x.  ( |_ `  (
( A  -  z
)  /  ( 2  x.  pi ) ) ) ) ) )
3433oveq2d 5874 . . . . . . . 8  |-  ( ( 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 ) ) ) ) ) ) )
356recnd 8861 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( A  +  ( 2  x.  pi ) )  e.  CC )
367recnd 8861 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( A  -  z
)  e.  CC )
374, 10gt0ne0ii 9309 . . . . . . . . . . . . . . 15  |-  ( 2  x.  pi )  =/=  0
38 redivcl 9479 . . . . . . . . . . . . . . 15  |-  ( ( ( A  -  z
)  e.  RR  /\  ( 2  x.  pi )  e.  RR  /\  (
2  x.  pi )  =/=  0 )  -> 
( ( A  -  z )  /  (
2  x.  pi ) )  e.  RR )
394, 37, 38mp3an23 1269 . . . . . . . . . . . . . 14  |-  ( ( A  -  z )  e.  RR  ->  (
( A  -  z
)  /  ( 2  x.  pi ) )  e.  RR )
407, 39syl 15 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( ( A  -  z )  /  (
2  x.  pi ) )  e.  RR )
4140flcld 10930 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( |_ `  (
( A  -  z
)  /  ( 2  x.  pi ) ) )  e.  ZZ )
4241zred 10117 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( |_ `  (
( A  -  z
)  /  ( 2  x.  pi ) ) )  e.  RR )
43 remulcl 8822 . . . . . . . . . . 11  |-  ( ( ( 2  x.  pi )  e.  RR  /\  ( |_ `  ( ( A  -  z )  / 
( 2  x.  pi ) ) )  e.  RR )  ->  (
( 2  x.  pi )  x.  ( |_ `  ( ( A  -  z )  /  (
2  x.  pi ) ) ) )  e.  RR )
444, 42, 43sylancr 644 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( ( 2  x.  pi )  x.  ( |_ `  ( ( A  -  z )  / 
( 2  x.  pi ) ) ) )  e.  RR )
4544recnd 8861 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( ( 2  x.  pi )  x.  ( |_ `  ( ( A  -  z )  / 
( 2  x.  pi ) ) ) )  e.  CC )
4635, 36, 45subsubd 9185 . . . . . . . 8  |-  ( ( 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 ) ) ) ) ) )
471recnd 8861 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  A  e.  CC )
484recni 8849 . . . . . . . . . . 11  |-  ( 2  x.  pi )  e.  CC
4948a1i 10 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( 2  x.  pi )  e.  CC )
50 simpr 447 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  z  e.  RR )
5150recnd 8861 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  z  e.  CC )
5247, 49, 51pnncand 9196 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( ( A  +  ( 2  x.  pi ) )  -  ( A  -  z )
)  =  ( ( 2  x.  pi )  +  z ) )
5352oveq1d 5873 . . . . . . . 8  |-  ( ( 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 ) ) ) ) ) )
5434, 46, 533eqtrd 2319 . . . . . . 7  |-  ( ( 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 ) ) ) ) ) )
5554oveq2d 5874 . . . . . 6  |-  ( ( 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 ) ) ) ) ) ) )
56 addcl 8819 . . . . . . . 8  |-  ( ( ( 2  x.  pi )  e.  CC  /\  z  e.  CC )  ->  (
( 2  x.  pi )  +  z )  e.  CC )
5748, 51, 56sylancr 644 . . . . . . 7  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( ( 2  x.  pi )  +  z )  e.  CC )
5851, 57, 45subsub4d 9188 . . . . . 6  |-  ( ( 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 ) ) ) ) ) ) )
5957, 51negsubdi2d 9173 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  z  e.  RR )  -> 
-u ( ( ( 2  x.  pi )  +  z )  -  z )  =  ( z  -  ( ( 2  x.  pi )  +  z ) ) )
6049, 51pncand 9158 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( ( ( 2  x.  pi )  +  z )  -  z
)  =  ( 2  x.  pi ) )
6160negeqd 9046 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  z  e.  RR )  -> 
-u ( ( ( 2  x.  pi )  +  z )  -  z )  =  -u ( 2  x.  pi ) )
6259, 61eqtr3d 2317 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( z  -  (
( 2  x.  pi )  +  z )
)  =  -u (
2  x.  pi ) )
63 neg1cn 9813 . . . . . . . . . 10  |-  -u 1  e.  CC
6448mulm1i 9224 . . . . . . . . . 10  |-  ( -u
1  x.  ( 2  x.  pi ) )  =  -u ( 2  x.  pi )
6563, 48, 64mulcomli 8844 . . . . . . . . 9  |-  ( ( 2  x.  pi )  x.  -u 1 )  = 
-u ( 2  x.  pi )
6662, 65syl6eqr 2333 . . . . . . . 8  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( z  -  (
( 2  x.  pi )  +  z )
)  =  ( ( 2  x.  pi )  x.  -u 1 ) )
6766oveq1d 5873 . . . . . . 7  |-  ( ( 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 ) ) ) ) ) )
6863a1i 10 . . . . . . . 8  |-  ( ( A  e.  RR  /\  z  e.  RR )  -> 
-u 1  e.  CC )
6941zcnd 10118 . . . . . . . 8  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( |_ `  (
( A  -  z
)  /  ( 2  x.  pi ) ) )  e.  CC )
7049, 68, 69subdid 9235 . . . . . . 7  |-  ( ( 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 ) ) ) ) ) )
7167, 70eqtr4d 2318 . . . . . 6  |-  ( ( 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 ) ) ) ) ) )
7255, 58, 713eqtr2d 2321 . . . . 5  |-  ( ( 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 ) ) ) ) ) )
7372oveq1d 5873 . . . 4  |-  ( ( 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 ) ) )
74 1z 10053 . . . . . . . 8  |-  1  e.  ZZ
75 znegcl 10055 . . . . . . . 8  |-  ( 1  e.  ZZ  ->  -u 1  e.  ZZ )
7674, 75ax-mp 8 . . . . . . 7  |-  -u 1  e.  ZZ
77 zsubcl 10061 . . . . . . 7  |-  ( (
-u 1  e.  ZZ  /\  ( |_ `  (
( A  -  z
)  /  ( 2  x.  pi ) ) )  e.  ZZ )  ->  ( -u 1  -  ( |_ `  ( ( A  -  z )  /  (
2  x.  pi ) ) ) )  e.  ZZ )
7876, 41, 77sylancr 644 . . . . . 6  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( -u 1  -  ( |_ `  (
( A  -  z
)  /  ( 2  x.  pi ) ) ) )  e.  ZZ )
7978zcnd 10118 . . . . 5  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( -u 1  -  ( |_ `  (
( A  -  z
)  /  ( 2  x.  pi ) ) ) )  e.  CC )
80 divcan3 9448 . . . . . 6  |-  ( ( ( -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 ) ) ) ) )
8148, 37, 80mp3an23 1269 . . . . 5  |-  ( (
-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 ) ) ) ) )
8279, 81syl 15 . . . 4  |-  ( ( 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 ) ) ) ) )
8373, 82eqtrd 2315 . . 3  |-  ( ( 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 ) ) ) ) )
8483, 78eqeltrd 2357 . 2  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( ( z  -  ( ( A  +  ( 2  x.  pi ) )  -  (
( A  -  z
)  mod  ( 2  x.  pi ) ) ) )  /  (
2  x.  pi ) )  e.  ZZ )
85 oveq2 5866 . . . . 5  |-  ( y  =  ( ( A  +  ( 2  x.  pi ) )  -  ( ( A  -  z )  mod  (
2  x.  pi ) ) )  ->  (
z  -  y )  =  ( z  -  ( ( A  +  ( 2  x.  pi ) )  -  (
( A  -  z
)  mod  ( 2  x.  pi ) ) ) ) )
8685oveq1d 5873 . . . 4  |-  ( 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 ) ) )
8786eleq1d 2349 . . 3  |-  ( 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 ) )
8887rspcev 2884 . 2  |-  ( ( ( ( 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 )
8931, 84, 88syl2anc 642 1  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  E. y  e.  D  ( ( z  -  y )  /  (
2  x.  pi ) )  e.  ZZ )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742   RR*cxr 8866    < clt 8867    <_ cle 8868    - cmin 9037   -ucneg 9038    / cdiv 9423   2c2 9795   ZZcz 10024   RR+crp 10354   (,]cioc 10657   |_cfl 10924    mod cmo 10973   picpi 12348
This theorem is referenced by:  efif1o  19908  eff1o  19911
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-ioc 10661  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-fac 11289  df-bc 11316  df-hash 11338  df-shft 11562  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-limsup 11945  df-clim 11962  df-rlim 11963  df-sum 12159  df-ef 12349  df-sin 12351  df-cos 12352  df-pi 12354  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-pt 13345  df-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  df-qtop 13410  df-imas 13411  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-mulg 14492  df-cntz 14793  df-cmn 15091  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-lp 16868  df-perf 16869  df-cn 16957  df-cnp 16958  df-haus 17043  df-tx 17257  df-hmeo 17446  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633  df-flim 17634  df-flf 17635  df-xms 17885  df-ms 17886  df-tms 17887  df-cncf 18382  df-limc 19216  df-dv 19217
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