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Theorem efif1olem2 20313
Description: Lemma for efif1o 20316. (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 444 . . . . . 6  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  A  e.  RR )
2 2re 10002 . . . . . . 7  |-  2  e.  RR
3 pire 20240 . . . . . . 7  |-  pi  e.  RR
42, 3remulcli 9038 . . . . . 6  |-  ( 2  x.  pi )  e.  RR
5 readdcl 9007 . . . . . 6  |-  ( ( A  e.  RR  /\  ( 2  x.  pi )  e.  RR )  ->  ( A  +  ( 2  x.  pi ) )  e.  RR )
61, 4, 5sylancl 644 . . . . 5  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( A  +  ( 2  x.  pi ) )  e.  RR )
7 resubcl 9298 . . . . . 6  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( A  -  z
)  e.  RR )
8 2pos 10015 . . . . . . . 8  |-  0  <  2
9 pipos 20241 . . . . . . . 8  |-  0  <  pi
102, 3, 8, 9mulgt0ii 9139 . . . . . . 7  |-  0  <  ( 2  x.  pi )
114, 10elrpii 10548 . . . . . 6  |-  ( 2  x.  pi )  e.  RR+
12 modcl 11181 . . . . . 6  |-  ( ( ( A  -  z
)  e.  RR  /\  ( 2  x.  pi )  e.  RR+ )  -> 
( ( A  -  z )  mod  (
2  x.  pi ) )  e.  RR )
137, 11, 12sylancl 644 . . . . 5  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( ( A  -  z )  mod  (
2  x.  pi ) )  e.  RR )
146, 13resubcld 9398 . . . 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 11186 . . . . . . 7  |-  ( ( ( A  -  z
)  e.  RR  /\  ( 2  x.  pi )  e.  RR+ )  -> 
( ( A  -  z )  mod  (
2  x.  pi ) )  <  ( 2  x.  pi ) )
177, 11, 16sylancl 644 . . . . . 6  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( ( A  -  z )  mod  (
2  x.  pi ) )  <  ( 2  x.  pi ) )
1813, 15, 1, 17ltadd2dd 9162 . . . . 5  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( A  +  ( ( A  -  z
)  mod  ( 2  x.  pi ) ) )  <  ( A  +  ( 2  x.  pi ) ) )
191, 13, 6ltaddsubd 9559 . . . . 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 202 . . . 4  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  A  <  ( ( A  +  ( 2  x.  pi ) )  -  ( ( A  -  z )  mod  ( 2  x.  pi ) ) ) )
21 modge0 11185 . . . . . 6  |-  ( ( ( A  -  z
)  e.  RR  /\  ( 2  x.  pi )  e.  RR+ )  -> 
0  <_  ( ( A  -  z )  mod  ( 2  x.  pi ) ) )
227, 11, 21sylancl 644 . . . . 5  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  0  <_  ( ( A  -  z )  mod  ( 2  x.  pi ) ) )
236, 13subge02d 9551 . . . . 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 202 . . . 4  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( ( A  +  ( 2  x.  pi ) )  -  (
( A  -  z
)  mod  ( 2  x.  pi ) ) )  <_  ( A  +  ( 2  x.  pi ) ) )
25 rexr 9064 . . . . . 6  |-  ( A  e.  RR  ->  A  e.  RR* )
2625adantr 452 . . . . 5  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  A  e.  RR* )
27 elioc2 10906 . . . . 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 643 . . . 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 1137 . . 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 2479 . 2  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( ( A  +  ( 2  x.  pi ) )  -  (
( A  -  z
)  mod  ( 2  x.  pi ) ) )  e.  D )
32 modval 11180 . . . . . . . . . 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 644 . . . . . . . . 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 6037 . . . . . . . 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 9048 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( A  +  ( 2  x.  pi ) )  e.  CC )
367recnd 9048 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( A  -  z
)  e.  CC )
374, 10gt0ne0ii 9496 . . . . . . . . . . . . . . 15  |-  ( 2  x.  pi )  =/=  0
38 redivcl 9666 . . . . . . . . . . . . . . 15  |-  ( ( ( A  -  z
)  e.  RR  /\  ( 2  x.  pi )  e.  RR  /\  (
2  x.  pi )  =/=  0 )  -> 
( ( A  -  z )  /  (
2  x.  pi ) )  e.  RR )
394, 37, 38mp3an23 1271 . . . . . . . . . . . . . 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 11135 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( |_ `  (
( A  -  z
)  /  ( 2  x.  pi ) ) )  e.  ZZ )
4241zred 10308 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( |_ `  (
( A  -  z
)  /  ( 2  x.  pi ) ) )  e.  RR )
43 remulcl 9009 . . . . . . . . . . 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 645 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( ( 2  x.  pi )  x.  ( |_ `  ( ( A  -  z )  / 
( 2  x.  pi ) ) ) )  e.  RR )
4544recnd 9048 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( ( 2  x.  pi )  x.  ( |_ `  ( ( A  -  z )  / 
( 2  x.  pi ) ) ) )  e.  CC )
4635, 36, 45subsubd 9372 . . . . . . . 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 9048 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  A  e.  CC )
484recni 9036 . . . . . . . . . . 11  |-  ( 2  x.  pi )  e.  CC
4948a1i 11 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( 2  x.  pi )  e.  CC )
50 simpr 448 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  z  e.  RR )
5150recnd 9048 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  z  e.  CC )
5247, 49, 51pnncand 9383 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( ( A  +  ( 2  x.  pi ) )  -  ( A  -  z )
)  =  ( ( 2  x.  pi )  +  z ) )
5352oveq1d 6036 . . . . . . . 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 2424 . . . . . . 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 6037 . . . . . 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 9006 . . . . . . . 8  |-  ( ( ( 2  x.  pi )  e.  CC  /\  z  e.  CC )  ->  (
( 2  x.  pi )  +  z )  e.  CC )
5748, 51, 56sylancr 645 . . . . . . 7  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( ( 2  x.  pi )  +  z )  e.  CC )
5851, 57, 45subsub4d 9375 . . . . . 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 9360 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  z  e.  RR )  -> 
-u ( ( ( 2  x.  pi )  +  z )  -  z )  =  ( z  -  ( ( 2  x.  pi )  +  z ) ) )
6049, 51pncand 9345 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( ( ( 2  x.  pi )  +  z )  -  z
)  =  ( 2  x.  pi ) )
6160negeqd 9233 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  z  e.  RR )  -> 
-u ( ( ( 2  x.  pi )  +  z )  -  z )  =  -u ( 2  x.  pi ) )
6259, 61eqtr3d 2422 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( z  -  (
( 2  x.  pi )  +  z )
)  =  -u (
2  x.  pi ) )
63 neg1cn 10000 . . . . . . . . . 10  |-  -u 1  e.  CC
6448mulm1i 9411 . . . . . . . . . 10  |-  ( -u
1  x.  ( 2  x.  pi ) )  =  -u ( 2  x.  pi )
6563, 48, 64mulcomli 9031 . . . . . . . . 9  |-  ( ( 2  x.  pi )  x.  -u 1 )  = 
-u ( 2  x.  pi )
6662, 65syl6eqr 2438 . . . . . . . 8  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( z  -  (
( 2  x.  pi )  +  z )
)  =  ( ( 2  x.  pi )  x.  -u 1 ) )
6766oveq1d 6036 . . . . . . 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 10309 . . . . . . . 8  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( |_ `  (
( A  -  z
)  /  ( 2  x.  pi ) ) )  e.  CC )
7049, 68, 69subdid 9422 . . . . . . 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 2423 . . . . . 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 2426 . . . . 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 6036 . . . 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 10244 . . . . . . . 8  |-  1  e.  ZZ
75 znegcl 10246 . . . . . . . 8  |-  ( 1  e.  ZZ  ->  -u 1  e.  ZZ )
7674, 75ax-mp 8 . . . . . . 7  |-  -u 1  e.  ZZ
77 zsubcl 10252 . . . . . . 7  |-  ( (
-u 1  e.  ZZ  /\  ( |_ `  (
( A  -  z
)  /  ( 2  x.  pi ) ) )  e.  ZZ )  ->  ( -u 1  -  ( |_ `  ( ( A  -  z )  /  (
2  x.  pi ) ) ) )  e.  ZZ )
7876, 41, 77sylancr 645 . . . . . 6  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( -u 1  -  ( |_ `  (
( A  -  z
)  /  ( 2  x.  pi ) ) ) )  e.  ZZ )
7978zcnd 10309 . . . . 5  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( -u 1  -  ( |_ `  (
( A  -  z
)  /  ( 2  x.  pi ) ) ) )  e.  CC )
80 divcan3 9635 . . . . . 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 1271 . . . . 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 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 ) ) ) ) )
8373, 82eqtrd 2420 . . 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 2462 . 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 6029 . . . . 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 6036 . . . 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 2454 . . 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 2996 . 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 643 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 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2551   E.wrex 2651   class class class wbr 4154   ` cfv 5395  (class class class)co 6021   CCcc 8922   RRcr 8923   0cc0 8924   1c1 8925    + caddc 8927    x. cmul 8929   RR*cxr 9053    < clt 9054    <_ cle 9055    - cmin 9224   -ucneg 9225    / cdiv 9610   2c2 9982   ZZcz 10215   RR+crp 10545   (,]cioc 10850   |_cfl 11129    mod cmo 11178   picpi 12597
This theorem is referenced by:  efif1o  20316  eff1o  20319
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-inf2 7530  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001  ax-pre-sup 9002  ax-addf 9003  ax-mulf 9004
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-iin 4039  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-se 4484  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-isom 5404  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-of 6245  df-1st 6289  df-2nd 6290  df-riota 6486  df-recs 6570  df-rdg 6605  df-1o 6661  df-2o 6662  df-oadd 6665  df-er 6842  df-map 6957  df-pm 6958  df-ixp 7001  df-en 7047  df-dom 7048  df-sdom 7049  df-fin 7050  df-fi 7352  df-sup 7382  df-oi 7413  df-card 7760  df-cda 7982  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-div 9611  df-nn 9934  df-2 9991  df-3 9992  df-4 9993  df-5 9994  df-6 9995  df-7 9996  df-8 9997  df-9 9998  df-10 9999  df-n0 10155  df-z 10216  df-dec 10316  df-uz 10422  df-q 10508  df-rp 10546  df-xneg 10643  df-xadd 10644  df-xmul 10645  df-ioo 10853  df-ioc 10854  df-ico 10855  df-icc 10856  df-fz 10977  df-fzo 11067  df-fl 11130  df-mod 11179  df-seq 11252  df-exp 11311  df-fac 11495  df-bc 11522  df-hash 11547  df-shft 11810  df-cj 11832  df-re 11833  df-im 11834  df-sqr 11968  df-abs 11969  df-limsup 12193  df-clim 12210  df-rlim 12211  df-sum 12408  df-ef 12598  df-sin 12600  df-cos 12601  df-pi 12603  df-struct 13399  df-ndx 13400  df-slot 13401  df-base 13402  df-sets 13403  df-ress 13404  df-plusg 13470  df-mulr 13471  df-starv 13472  df-sca 13473  df-vsca 13474  df-tset 13476  df-ple 13477  df-ds 13479  df-unif 13480  df-hom 13481  df-cco 13482  df-rest 13578  df-topn 13579  df-topgen 13595  df-pt 13596  df-prds 13599  df-xrs 13654  df-0g 13655  df-gsum 13656  df-qtop 13661  df-imas 13662  df-xps 13664  df-mre 13739  df-mrc 13740  df-acs 13742  df-mnd 14618  df-submnd 14667  df-mulg 14743  df-cntz 15044  df-cmn 15342  df-xmet 16620  df-met 16621  df-bl 16622  df-mopn 16623  df-fbas 16624  df-fg 16625  df-cnfld 16628  df-top 16887  df-bases 16889  df-topon 16890  df-topsp 16891  df-cld 17007  df-ntr 17008  df-cls 17009  df-nei 17086  df-lp 17124  df-perf 17125  df-cn 17214  df-cnp 17215  df-haus 17302  df-tx 17516  df-hmeo 17709  df-fil 17800  df-fm 17892  df-flim 17893  df-flf 17894  df-xms 18260  df-ms 18261  df-tms 18262  df-cncf 18780  df-limc 19621  df-dv 19622
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