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Theorem efif1olem2 22131
Description: Lemma for efif1o 22134. (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 457 . . . . . 6  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  A  e.  RR )
2 2re 10501 . . . . . . 7  |-  2  e.  RR
3 pire 22053 . . . . . . 7  |-  pi  e.  RR
42, 3remulcli 9510 . . . . . 6  |-  ( 2  x.  pi )  e.  RR
5 readdcl 9475 . . . . . 6  |-  ( ( A  e.  RR  /\  ( 2  x.  pi )  e.  RR )  ->  ( A  +  ( 2  x.  pi ) )  e.  RR )
61, 4, 5sylancl 662 . . . . 5  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( A  +  ( 2  x.  pi ) )  e.  RR )
7 resubcl 9783 . . . . . 6  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( A  -  z
)  e.  RR )
8 2pos 10523 . . . . . . . 8  |-  0  <  2
9 pipos 22055 . . . . . . . 8  |-  0  <  pi
102, 3, 8, 9mulgt0ii 9617 . . . . . . 7  |-  0  <  ( 2  x.  pi )
114, 10elrpii 11104 . . . . . 6  |-  ( 2  x.  pi )  e.  RR+
12 modcl 11828 . . . . . 6  |-  ( ( ( A  -  z
)  e.  RR  /\  ( 2  x.  pi )  e.  RR+ )  -> 
( ( A  -  z )  mod  (
2  x.  pi ) )  e.  RR )
137, 11, 12sylancl 662 . . . . 5  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( ( A  -  z )  mod  (
2  x.  pi ) )  e.  RR )
146, 13resubcld 9886 . . . 4  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( ( A  +  ( 2  x.  pi ) )  -  (
( A  -  z
)  mod  ( 2  x.  pi ) ) )  e.  RR )
154a1i 11 . . . . . 6  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( 2  x.  pi )  e.  RR )
16 modlt 11834 . . . . . . 7  |-  ( ( ( A  -  z
)  e.  RR  /\  ( 2  x.  pi )  e.  RR+ )  -> 
( ( A  -  z )  mod  (
2  x.  pi ) )  <  ( 2  x.  pi ) )
177, 11, 16sylancl 662 . . . . . 6  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( ( A  -  z )  mod  (
2  x.  pi ) )  <  ( 2  x.  pi ) )
1813, 15, 1, 17ltadd2dd 9640 . . . . 5  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( A  +  ( ( A  -  z
)  mod  ( 2  x.  pi ) ) )  <  ( A  +  ( 2  x.  pi ) ) )
191, 13, 6ltaddsubd 10049 . . . . 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 210 . . . 4  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  A  <  ( ( A  +  ( 2  x.  pi ) )  -  ( ( A  -  z )  mod  ( 2  x.  pi ) ) ) )
21 modge0 11833 . . . . . 6  |-  ( ( ( A  -  z
)  e.  RR  /\  ( 2  x.  pi )  e.  RR+ )  -> 
0  <_  ( ( A  -  z )  mod  ( 2  x.  pi ) ) )
227, 11, 21sylancl 662 . . . . 5  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  0  <_  ( ( A  -  z )  mod  ( 2  x.  pi ) ) )
236, 13subge02d 10041 . . . . 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 210 . . . 4  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( ( A  +  ( 2  x.  pi ) )  -  (
( A  -  z
)  mod  ( 2  x.  pi ) ) )  <_  ( A  +  ( 2  x.  pi ) ) )
25 rexr 9539 . . . . . 6  |-  ( A  e.  RR  ->  A  e.  RR* )
2625adantr 465 . . . . 5  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  A  e.  RR* )
27 elioc2 11468 . . . . 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 661 . . . 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 1171 . . 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 2553 . 2  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( ( A  +  ( 2  x.  pi ) )  -  (
( A  -  z
)  mod  ( 2  x.  pi ) ) )  e.  D )
32 modval 11826 . . . . . . . . . 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 662 . . . . . . . . 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 6215 . . . . . . . 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 9522 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( A  +  ( 2  x.  pi ) )  e.  CC )
367recnd 9522 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( A  -  z
)  e.  CC )
374, 10gt0ne0ii 9986 . . . . . . . . . . . . . . 15  |-  ( 2  x.  pi )  =/=  0
38 redivcl 10160 . . . . . . . . . . . . . . 15  |-  ( ( ( A  -  z
)  e.  RR  /\  ( 2  x.  pi )  e.  RR  /\  (
2  x.  pi )  =/=  0 )  -> 
( ( A  -  z )  /  (
2  x.  pi ) )  e.  RR )
394, 37, 38mp3an23 1307 . . . . . . . . . . . . . 14  |-  ( ( A  -  z )  e.  RR  ->  (
( A  -  z
)  /  ( 2  x.  pi ) )  e.  RR )
407, 39syl 16 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( ( A  -  z )  /  (
2  x.  pi ) )  e.  RR )
4140flcld 11764 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( |_ `  (
( A  -  z
)  /  ( 2  x.  pi ) ) )  e.  ZZ )
4241zred 10857 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( |_ `  (
( A  -  z
)  /  ( 2  x.  pi ) ) )  e.  RR )
43 remulcl 9477 . . . . . . . . . . 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 663 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( ( 2  x.  pi )  x.  ( |_ `  ( ( A  -  z )  / 
( 2  x.  pi ) ) ) )  e.  RR )
4544recnd 9522 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( ( 2  x.  pi )  x.  ( |_ `  ( ( A  -  z )  / 
( 2  x.  pi ) ) ) )  e.  CC )
4635, 36, 45subsubd 9857 . . . . . . . 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 9522 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  A  e.  CC )
484recni 9508 . . . . . . . . . . 11  |-  ( 2  x.  pi )  e.  CC
4948a1i 11 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( 2  x.  pi )  e.  CC )
50 simpr 461 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  z  e.  RR )
5150recnd 9522 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  z  e.  CC )
5247, 49, 51pnncand 9868 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( ( A  +  ( 2  x.  pi ) )  -  ( A  -  z )
)  =  ( ( 2  x.  pi )  +  z ) )
5352oveq1d 6214 . . . . . . . 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 2499 . . . . . . 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 6215 . . . . . 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 9474 . . . . . . . 8  |-  ( ( ( 2  x.  pi )  e.  CC  /\  z  e.  CC )  ->  (
( 2  x.  pi )  +  z )  e.  CC )
5748, 51, 56sylancr 663 . . . . . . 7  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( ( 2  x.  pi )  +  z )  e.  CC )
5851, 57, 45subsub4d 9860 . . . . . 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 9845 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  z  e.  RR )  -> 
-u ( ( ( 2  x.  pi )  +  z )  -  z )  =  ( z  -  ( ( 2  x.  pi )  +  z ) ) )
6049, 51pncand 9830 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( ( ( 2  x.  pi )  +  z )  -  z
)  =  ( 2  x.  pi ) )
6160negeqd 9714 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  z  e.  RR )  -> 
-u ( ( ( 2  x.  pi )  +  z )  -  z )  =  -u ( 2  x.  pi ) )
6259, 61eqtr3d 2497 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( z  -  (
( 2  x.  pi )  +  z )
)  =  -u (
2  x.  pi ) )
63 neg1cn 10535 . . . . . . . . . 10  |-  -u 1  e.  CC
6448mulm1i 9899 . . . . . . . . . 10  |-  ( -u
1  x.  ( 2  x.  pi ) )  =  -u ( 2  x.  pi )
6563, 48, 64mulcomli 9503 . . . . . . . . 9  |-  ( ( 2  x.  pi )  x.  -u 1 )  = 
-u ( 2  x.  pi )
6662, 65syl6eqr 2513 . . . . . . . 8  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( z  -  (
( 2  x.  pi )  +  z )
)  =  ( ( 2  x.  pi )  x.  -u 1 ) )
6766oveq1d 6214 . . . . . . 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 11 . . . . . . . 8  |-  ( ( A  e.  RR  /\  z  e.  RR )  -> 
-u 1  e.  CC )
6941zcnd 10858 . . . . . . . 8  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( |_ `  (
( A  -  z
)  /  ( 2  x.  pi ) ) )  e.  CC )
7049, 68, 69subdid 9910 . . . . . . 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 2498 . . . . . 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 2501 . . . . 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 6214 . . . 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 neg1z 10791 . . . . . . 7  |-  -u 1  e.  ZZ
75 zsubcl 10797 . . . . . . 7  |-  ( (
-u 1  e.  ZZ  /\  ( |_ `  (
( A  -  z
)  /  ( 2  x.  pi ) ) )  e.  ZZ )  ->  ( -u 1  -  ( |_ `  ( ( A  -  z )  /  (
2  x.  pi ) ) ) )  e.  ZZ )
7674, 41, 75sylancr 663 . . . . . 6  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( -u 1  -  ( |_ `  (
( A  -  z
)  /  ( 2  x.  pi ) ) ) )  e.  ZZ )
7776zcnd 10858 . . . . 5  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( -u 1  -  ( |_ `  (
( A  -  z
)  /  ( 2  x.  pi ) ) ) )  e.  CC )
78 divcan3 10128 . . . . . 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 ) ) ) ) )
7948, 37, 78mp3an23 1307 . . . . 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 ) ) ) ) )
8077, 79syl 16 . . . 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 ) ) ) ) )
8173, 80eqtrd 2495 . . 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 ) ) ) ) )
8281, 76eqeltrd 2542 . 2  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( ( z  -  ( ( A  +  ( 2  x.  pi ) )  -  (
( A  -  z
)  mod  ( 2  x.  pi ) ) ) )  /  (
2  x.  pi ) )  e.  ZZ )
83 oveq2 6207 . . . . 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 ) ) ) ) )
8483oveq1d 6214 . . . 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 ) ) )
8584eleq1d 2523 . . 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 ) )
8685rspcev 3177 . 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 )
8731, 82, 86syl2anc 661 1  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  E. y  e.  D  ( ( z  -  y )  /  (
2  x.  pi ) )  e.  ZZ )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2647   E.wrex 2799   class class class wbr 4399   ` cfv 5525  (class class class)co 6199   CCcc 9390   RRcr 9391   0cc0 9392   1c1 9393    + caddc 9395    x. cmul 9397   RR*cxr 9527    < clt 9528    <_ cle 9529    - cmin 9705   -ucneg 9706    / cdiv 10103   2c2 10481   ZZcz 10756   RR+crp 11101   (,]cioc 11411   |_cfl 11756    mod cmo 11824   picpi 13469
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-inf2 7957  ax-cnex 9448  ax-resscn 9449  ax-1cn 9450  ax-icn 9451  ax-addcl 9452  ax-addrcl 9453  ax-mulcl 9454  ax-mulrcl 9455  ax-mulcom 9456  ax-addass 9457  ax-mulass 9458  ax-distr 9459  ax-i2m1 9460  ax-1ne0 9461  ax-1rid 9462  ax-rnegex 9463  ax-rrecex 9464  ax-cnre 9465  ax-pre-lttri 9466  ax-pre-lttrn 9467  ax-pre-ltadd 9468  ax-pre-mulgt0 9469  ax-pre-sup 9470  ax-addf 9471  ax-mulf 9472
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-reu 2805  df-rmo 2806  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-uni 4199  df-int 4236  df-iun 4280  df-iin 4281  df-br 4400  df-opab 4458  df-mpt 4459  df-tr 4493  df-eprel 4739  df-id 4743  df-po 4748  df-so 4749  df-fr 4786  df-se 4787  df-we 4788  df-ord 4829  df-on 4830  df-lim 4831  df-suc 4832  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-isom 5534  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-of 6429  df-om 6586  df-1st 6686  df-2nd 6687  df-supp 6800  df-recs 6941  df-rdg 6975  df-1o 7029  df-2o 7030  df-oadd 7033  df-er 7210  df-map 7325  df-pm 7326  df-ixp 7373  df-en 7420  df-dom 7421  df-sdom 7422  df-fin 7423  df-fsupp 7731  df-fi 7771  df-sup 7801  df-oi 7834  df-card 8219  df-cda 8447  df-pnf 9530  df-mnf 9531  df-xr 9532  df-ltxr 9533  df-le 9534  df-sub 9707  df-neg 9708  df-div 10104  df-nn 10433  df-2 10490  df-3 10491  df-4 10492  df-5 10493  df-6 10494  df-7 10495  df-8 10496  df-9 10497  df-10 10498  df-n0 10690  df-z 10757  df-dec 10866  df-uz 10972  df-q 11064  df-rp 11102  df-xneg 11199  df-xadd 11200  df-xmul 11201  df-ioo 11414  df-ioc 11415  df-ico 11416  df-icc 11417  df-fz 11554  df-fzo 11665  df-fl 11758  df-mod 11825  df-seq 11923  df-exp 11982  df-fac 12168  df-bc 12195  df-hash 12220  df-shft 12673  df-cj 12705  df-re 12706  df-im 12707  df-sqr 12841  df-abs 12842  df-limsup 13066  df-clim 13083  df-rlim 13084  df-sum 13281  df-ef 13470  df-sin 13472  df-cos 13473  df-pi 13475  df-struct 14293  df-ndx 14294  df-slot 14295  df-base 14296  df-sets 14297  df-ress 14298  df-plusg 14369  df-mulr 14370  df-starv 14371  df-sca 14372  df-vsca 14373  df-ip 14374  df-tset 14375  df-ple 14376  df-ds 14378  df-unif 14379  df-hom 14380  df-cco 14381  df-rest 14479  df-topn 14480  df-0g 14498  df-gsum 14499  df-topgen 14500  df-pt 14501  df-prds 14504  df-xrs 14558  df-qtop 14563  df-imas 14564  df-xps 14566  df-mre 14642  df-mrc 14643  df-acs 14645  df-mnd 15533  df-submnd 15583  df-mulg 15666  df-cntz 15953  df-cmn 16399  df-psmet 17933  df-xmet 17934  df-met 17935  df-bl 17936  df-mopn 17937  df-fbas 17938  df-fg 17939  df-cnfld 17943  df-top 18634  df-bases 18636  df-topon 18637  df-topsp 18638  df-cld 18754  df-ntr 18755  df-cls 18756  df-nei 18833  df-lp 18871  df-perf 18872  df-cn 18962  df-cnp 18963  df-haus 19050  df-tx 19266  df-hmeo 19459  df-fil 19550  df-fm 19642  df-flim 19643  df-flf 19644  df-xms 20026  df-ms 20027  df-tms 20028  df-cncf 20585  df-limc 21473  df-dv 21474
This theorem is referenced by:  efif1o  22134  eff1o  22137
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