MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  efif1olem2 Structured version   Unicode version

Theorem efif1olem2 21974
Description: Lemma for efif1o 21977. (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 10383 . . . . . . 7  |-  2  e.  RR
3 pire 21896 . . . . . . 7  |-  pi  e.  RR
42, 3remulcli 9392 . . . . . 6  |-  ( 2  x.  pi )  e.  RR
5 readdcl 9357 . . . . . 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 9665 . . . . . 6  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( A  -  z
)  e.  RR )
8 2pos 10405 . . . . . . . 8  |-  0  <  2
9 pipos 21898 . . . . . . . 8  |-  0  <  pi
102, 3, 8, 9mulgt0ii 9499 . . . . . . 7  |-  0  <  ( 2  x.  pi )
114, 10elrpii 10986 . . . . . 6  |-  ( 2  x.  pi )  e.  RR+
12 modcl 11704 . . . . . 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 9768 . . . 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 11710 . . . . . . 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 9522 . . . . 5  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( A  +  ( ( A  -  z
)  mod  ( 2  x.  pi ) ) )  <  ( A  +  ( 2  x.  pi ) ) )
191, 13, 6ltaddsubd 9931 . . . . 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 11709 . . . . . 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 9923 . . . . 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 9421 . . . . . 6  |-  ( A  e.  RR  ->  A  e.  RR* )
2625adantr 465 . . . . 5  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  A  e.  RR* )
27 elioc2 11350 . . . . 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 2529 . 2  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( ( A  +  ( 2  x.  pi ) )  -  (
( A  -  z
)  mod  ( 2  x.  pi ) ) )  e.  D )
32 modval 11702 . . . . . . . . . 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 6102 . . . . . . . 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 9404 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( A  +  ( 2  x.  pi ) )  e.  CC )
367recnd 9404 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( A  -  z
)  e.  CC )
374, 10gt0ne0ii 9868 . . . . . . . . . . . . . . 15  |-  ( 2  x.  pi )  =/=  0
38 redivcl 10042 . . . . . . . . . . . . . . 15  |-  ( ( ( A  -  z
)  e.  RR  /\  ( 2  x.  pi )  e.  RR  /\  (
2  x.  pi )  =/=  0 )  -> 
( ( A  -  z )  /  (
2  x.  pi ) )  e.  RR )
394, 37, 38mp3an23 1306 . . . . . . . . . . . . . 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 11640 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( |_ `  (
( A  -  z
)  /  ( 2  x.  pi ) ) )  e.  ZZ )
4241zred 10739 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( |_ `  (
( A  -  z
)  /  ( 2  x.  pi ) ) )  e.  RR )
43 remulcl 9359 . . . . . . . . . . 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 9404 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( ( 2  x.  pi )  x.  ( |_ `  ( ( A  -  z )  / 
( 2  x.  pi ) ) ) )  e.  CC )
4635, 36, 45subsubd 9739 . . . . . . . 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 9404 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  A  e.  CC )
484recni 9390 . . . . . . . . . . 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 9404 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  z  e.  CC )
5247, 49, 51pnncand 9750 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( ( A  +  ( 2  x.  pi ) )  -  ( A  -  z )
)  =  ( ( 2  x.  pi )  +  z ) )
5352oveq1d 6101 . . . . . . . 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 2474 . . . . . . 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 6102 . . . . . 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 9356 . . . . . . . 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 9742 . . . . . 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 9727 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  z  e.  RR )  -> 
-u ( ( ( 2  x.  pi )  +  z )  -  z )  =  ( z  -  ( ( 2  x.  pi )  +  z ) ) )
6049, 51pncand 9712 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( ( ( 2  x.  pi )  +  z )  -  z
)  =  ( 2  x.  pi ) )
6160negeqd 9596 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  z  e.  RR )  -> 
-u ( ( ( 2  x.  pi )  +  z )  -  z )  =  -u ( 2  x.  pi ) )
6259, 61eqtr3d 2472 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( z  -  (
( 2  x.  pi )  +  z )
)  =  -u (
2  x.  pi ) )
63 neg1cn 10417 . . . . . . . . . 10  |-  -u 1  e.  CC
6448mulm1i 9781 . . . . . . . . . 10  |-  ( -u
1  x.  ( 2  x.  pi ) )  =  -u ( 2  x.  pi )
6563, 48, 64mulcomli 9385 . . . . . . . . 9  |-  ( ( 2  x.  pi )  x.  -u 1 )  = 
-u ( 2  x.  pi )
6662, 65syl6eqr 2488 . . . . . . . 8  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( z  -  (
( 2  x.  pi )  +  z )
)  =  ( ( 2  x.  pi )  x.  -u 1 ) )
6766oveq1d 6101 . . . . . . 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 10740 . . . . . . . 8  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( |_ `  (
( A  -  z
)  /  ( 2  x.  pi ) ) )  e.  CC )
7049, 68, 69subdid 9792 . . . . . . 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 2473 . . . . . 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 2476 . . . . 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 6101 . . . 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 10673 . . . . . . 7  |-  -u 1  e.  ZZ
75 zsubcl 10679 . . . . . . 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 10740 . . . . 5  |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  ( -u 1  -  ( |_ `  (
( A  -  z
)  /  ( 2  x.  pi ) ) ) )  e.  CC )
78 divcan3 10010 . . . . . 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 1306 . . . . 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 2470 . . 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 2512 . 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 6094 . . . . 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 6101 . . . 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 2504 . . 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 3068 . 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 1369    e. wcel 1756    =/= wne 2601   E.wrex 2711   class class class wbr 4287   ` cfv 5413  (class class class)co 6086   CCcc 9272   RRcr 9273   0cc0 9274   1c1 9275    + caddc 9277    x. cmul 9279   RR*cxr 9409    < clt 9410    <_ cle 9411    - cmin 9587   -ucneg 9588    / cdiv 9985   2c2 10363   ZZcz 10638   RR+crp 10983   (,]cioc 11293   |_cfl 11632    mod cmo 11700   picpi 13344
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352  ax-addf 9353  ax-mulf 9354
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-iin 4169  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-of 6315  df-om 6472  df-1st 6572  df-2nd 6573  df-supp 6686  df-recs 6824  df-rdg 6858  df-1o 6912  df-2o 6913  df-oadd 6916  df-er 7093  df-map 7208  df-pm 7209  df-ixp 7256  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-fsupp 7613  df-fi 7653  df-sup 7683  df-oi 7716  df-card 8101  df-cda 8329  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-7 10377  df-8 10378  df-9 10379  df-10 10380  df-n0 10572  df-z 10639  df-dec 10748  df-uz 10854  df-q 10946  df-rp 10984  df-xneg 11081  df-xadd 11082  df-xmul 11083  df-ioo 11296  df-ioc 11297  df-ico 11298  df-icc 11299  df-fz 11430  df-fzo 11541  df-fl 11634  df-mod 11701  df-seq 11799  df-exp 11858  df-fac 12044  df-bc 12071  df-hash 12096  df-shft 12548  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-limsup 12941  df-clim 12958  df-rlim 12959  df-sum 13156  df-ef 13345  df-sin 13347  df-cos 13348  df-pi 13350  df-struct 14168  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-ress 14173  df-plusg 14243  df-mulr 14244  df-starv 14245  df-sca 14246  df-vsca 14247  df-ip 14248  df-tset 14249  df-ple 14250  df-ds 14252  df-unif 14253  df-hom 14254  df-cco 14255  df-rest 14353  df-topn 14354  df-0g 14372  df-gsum 14373  df-topgen 14374  df-pt 14375  df-prds 14378  df-xrs 14432  df-qtop 14437  df-imas 14438  df-xps 14440  df-mre 14516  df-mrc 14517  df-acs 14519  df-mnd 15407  df-submnd 15457  df-mulg 15539  df-cntz 15826  df-cmn 16270  df-psmet 17784  df-xmet 17785  df-met 17786  df-bl 17787  df-mopn 17788  df-fbas 17789  df-fg 17790  df-cnfld 17794  df-top 18478  df-bases 18480  df-topon 18481  df-topsp 18482  df-cld 18598  df-ntr 18599  df-cls 18600  df-nei 18677  df-lp 18715  df-perf 18716  df-cn 18806  df-cnp 18807  df-haus 18894  df-tx 19110  df-hmeo 19303  df-fil 19394  df-fm 19486  df-flim 19487  df-flf 19488  df-xms 19870  df-ms 19871  df-tms 19872  df-cncf 20429  df-limc 21316  df-dv 21317
This theorem is referenced by:  efif1o  21977  eff1o  21980
  Copyright terms: Public domain W3C validator