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Theorem efif1olem2 20437
Description: Lemma for efif1o 20440. (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 10061 . . . . . . 7  |-  2  e.  RR
3 pire 20364 . . . . . . 7  |-  pi  e.  RR
42, 3remulcli 9096 . . . . . 6  |-  ( 2  x.  pi )  e.  RR
5 readdcl 9065 . . . . . 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 9357 . . . . . 6  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( A  -  z
)  e.  RR )
8 2pos 10074 . . . . . . . 8  |-  0  <  2
9 pipos 20365 . . . . . . . 8  |-  0  <  pi
102, 3, 8, 9mulgt0ii 9198 . . . . . . 7  |-  0  <  ( 2  x.  pi )
114, 10elrpii 10607 . . . . . 6  |-  ( 2  x.  pi )  e.  RR+
12 modcl 11245 . . . . . 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 9457 . . . 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 11250 . . . . . . 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 9221 . . . . 5  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( A  +  ( ( A  -  z
)  mod  ( 2  x.  pi ) ) )  <  ( A  +  ( 2  x.  pi ) ) )
191, 13, 6ltaddsubd 9618 . . . . 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 11249 . . . . . 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 9610 . . . . 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 9122 . . . . . 6  |-  ( A  e.  RR  ->  A  e.  RR* )
2625adantr 452 . . . . 5  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  A  e.  RR* )
27 elioc2 10965 . . . . 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 2526 . 2  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( ( A  +  ( 2  x.  pi ) )  -  (
( A  -  z
)  mod  ( 2  x.  pi ) ) )  e.  D )
32 modval 11244 . . . . . . . . . 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 6089 . . . . . . . 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 9106 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( A  +  ( 2  x.  pi ) )  e.  CC )
367recnd 9106 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( A  -  z
)  e.  CC )
374, 10gt0ne0ii 9555 . . . . . . . . . . . . . . 15  |-  ( 2  x.  pi )  =/=  0
38 redivcl 9725 . . . . . . . . . . . . . . 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 11199 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( |_ `  (
( A  -  z
)  /  ( 2  x.  pi ) ) )  e.  ZZ )
4241zred 10367 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( |_ `  (
( A  -  z
)  /  ( 2  x.  pi ) ) )  e.  RR )
43 remulcl 9067 . . . . . . . . . . 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 9106 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( ( 2  x.  pi )  x.  ( |_ `  ( ( A  -  z )  / 
( 2  x.  pi ) ) ) )  e.  CC )
4635, 36, 45subsubd 9431 . . . . . . . 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 9106 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  A  e.  CC )
484recni 9094 . . . . . . . . . . 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 9106 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  z  e.  CC )
5247, 49, 51pnncand 9442 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( ( A  +  ( 2  x.  pi ) )  -  ( A  -  z )
)  =  ( ( 2  x.  pi )  +  z ) )
5352oveq1d 6088 . . . . . . . 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 2471 . . . . . . 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 6089 . . . . . 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 9064 . . . . . . . 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 9434 . . . . . 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 9419 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  z  e.  RR )  -> 
-u ( ( ( 2  x.  pi )  +  z )  -  z )  =  ( z  -  ( ( 2  x.  pi )  +  z ) ) )
6049, 51pncand 9404 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( ( ( 2  x.  pi )  +  z )  -  z
)  =  ( 2  x.  pi ) )
6160negeqd 9292 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  z  e.  RR )  -> 
-u ( ( ( 2  x.  pi )  +  z )  -  z )  =  -u ( 2  x.  pi ) )
6259, 61eqtr3d 2469 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( z  -  (
( 2  x.  pi )  +  z )
)  =  -u (
2  x.  pi ) )
63 neg1cn 10059 . . . . . . . . . 10  |-  -u 1  e.  CC
6448mulm1i 9470 . . . . . . . . . 10  |-  ( -u
1  x.  ( 2  x.  pi ) )  =  -u ( 2  x.  pi )
6563, 48, 64mulcomli 9089 . . . . . . . . 9  |-  ( ( 2  x.  pi )  x.  -u 1 )  = 
-u ( 2  x.  pi )
6662, 65syl6eqr 2485 . . . . . . . 8  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( z  -  (
( 2  x.  pi )  +  z )
)  =  ( ( 2  x.  pi )  x.  -u 1 ) )
6766oveq1d 6088 . . . . . . 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 10368 . . . . . . . 8  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( |_ `  (
( A  -  z
)  /  ( 2  x.  pi ) ) )  e.  CC )
7049, 68, 69subdid 9481 . . . . . . 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 2470 . . . . . 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 2473 . . . . 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 6088 . . . 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 10303 . . . . . . . 8  |-  1  e.  ZZ
75 znegcl 10305 . . . . . . . 8  |-  ( 1  e.  ZZ  ->  -u 1  e.  ZZ )
7674, 75ax-mp 8 . . . . . . 7  |-  -u 1  e.  ZZ
77 zsubcl 10311 . . . . . . 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 10368 . . . . 5  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( -u 1  -  ( |_ `  (
( A  -  z
)  /  ( 2  x.  pi ) ) ) )  e.  CC )
80 divcan3 9694 . . . . . 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 2467 . . 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 2509 . 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 6081 . . . . 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 6088 . . . 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 2501 . . 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 3044 . 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 1652    e. wcel 1725    =/= wne 2598   E.wrex 2698   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   CCcc 8980   RRcr 8981   0cc0 8982   1c1 8983    + caddc 8985    x. cmul 8987   RR*cxr 9111    < clt 9112    <_ cle 9113    - cmin 9283   -ucneg 9284    / cdiv 9669   2c2 10041   ZZcz 10274   RR+crp 10604   (,]cioc 10909   |_cfl 11193    mod cmo 11242   picpi 12661
This theorem is referenced by:  efif1o  20440  eff1o  20443
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060  ax-addf 9061  ax-mulf 9062
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-er 6897  df-map 7012  df-pm 7013  df-ixp 7056  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-fi 7408  df-sup 7438  df-oi 7471  df-card 7818  df-cda 8040  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-7 10055  df-8 10056  df-9 10057  df-10 10058  df-n0 10214  df-z 10275  df-dec 10375  df-uz 10481  df-q 10567  df-rp 10605  df-xneg 10702  df-xadd 10703  df-xmul 10704  df-ioo 10912  df-ioc 10913  df-ico 10914  df-icc 10915  df-fz 11036  df-fzo 11128  df-fl 11194  df-mod 11243  df-seq 11316  df-exp 11375  df-fac 11559  df-bc 11586  df-hash 11611  df-shft 11874  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-limsup 12257  df-clim 12274  df-rlim 12275  df-sum 12472  df-ef 12662  df-sin 12664  df-cos 12665  df-pi 12667  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-mulr 13535  df-starv 13536  df-sca 13537  df-vsca 13538  df-tset 13540  df-ple 13541  df-ds 13543  df-unif 13544  df-hom 13545  df-cco 13546  df-rest 13642  df-topn 13643  df-topgen 13659  df-pt 13660  df-prds 13663  df-xrs 13718  df-0g 13719  df-gsum 13720  df-qtop 13725  df-imas 13726  df-xps 13728  df-mre 13803  df-mrc 13804  df-acs 13806  df-mnd 14682  df-submnd 14731  df-mulg 14807  df-cntz 15108  df-cmn 15406  df-psmet 16686  df-xmet 16687  df-met 16688  df-bl 16689  df-mopn 16690  df-fbas 16691  df-fg 16692  df-cnfld 16696  df-top 16955  df-bases 16957  df-topon 16958  df-topsp 16959  df-cld 17075  df-ntr 17076  df-cls 17077  df-nei 17154  df-lp 17192  df-perf 17193  df-cn 17283  df-cnp 17284  df-haus 17371  df-tx 17586  df-hmeo 17779  df-fil 17870  df-fm 17962  df-flim 17963  df-flf 17964  df-xms 18342  df-ms 18343  df-tms 18344  df-cncf 18900  df-limc 19745  df-dv 19746
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