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Theorem irrapxlem3 26878
Description: Lemma for irrapx1 26882. By subtraction, there is a multiple very close to an integer. (Contributed by Stefan O'Rear, 13-Sep-2014.)
Assertion
Ref Expression
irrapxlem3  |-  ( ( A  e.  RR+  /\  B  e.  NN )  ->  E. x  e.  ( 1 ... B
) E. y  e. 
NN0  ( abs `  (
( A  x.  x
)  -  y ) )  <  ( 1  /  B ) )
Distinct variable groups:    x, A, y    x, B, y

Proof of Theorem irrapxlem3
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 irrapxlem2 26877 . 2  |-  ( ( A  e.  RR+  /\  B  e.  NN )  ->  E. a  e.  ( 0 ... B
) E. b  e.  ( 0 ... B
) ( a  < 
b  /\  ( abs `  ( ( ( A  x.  a )  mod  1 )  -  (
( A  x.  b
)  mod  1 ) ) )  <  (
1  /  B ) ) )
2 1m1e0 10060 . . . . . . . . 9  |-  ( 1  -  1 )  =  0
3 elfzelz 11051 . . . . . . . . . . . . 13  |-  ( a  e.  ( 0 ... B )  ->  a  e.  ZZ )
43ad2antrl 709 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  -> 
a  e.  ZZ )
54zred 10367 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  -> 
a  e.  RR )
6 elfzelz 11051 . . . . . . . . . . . . 13  |-  ( b  e.  ( 0 ... B )  ->  b  e.  ZZ )
76ad2antll 710 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  -> 
b  e.  ZZ )
87zred 10367 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  -> 
b  e.  RR )
95, 8posdifd 9605 . . . . . . . . . 10  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  -> 
( a  <  b  <->  0  <  ( b  -  a ) ) )
109biimpa 471 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
0  <  ( b  -  a ) )
112, 10syl5eqbr 4237 . . . . . . . 8  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( 1  -  1 )  <  ( b  -  a ) )
12 1z 10303 . . . . . . . . 9  |-  1  e.  ZZ
13 simplrr 738 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
b  e.  ( 0 ... B ) )
1413, 6syl 16 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
b  e.  ZZ )
15 simplrl 737 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
a  e.  ( 0 ... B ) )
1615, 3syl 16 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
a  e.  ZZ )
1714, 16zsubcld 10372 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( b  -  a
)  e.  ZZ )
18 zlem1lt 10319 . . . . . . . . 9  |-  ( ( 1  e.  ZZ  /\  ( b  -  a
)  e.  ZZ )  ->  ( 1  <_ 
( b  -  a
)  <->  ( 1  -  1 )  <  (
b  -  a ) ) )
1912, 17, 18sylancr 645 . . . . . . . 8  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( 1  <_  (
b  -  a )  <-> 
( 1  -  1 )  <  ( b  -  a ) ) )
2011, 19mpbird 224 . . . . . . 7  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
1  <_  ( b  -  a ) )
2114zred 10367 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
b  e.  RR )
2216zred 10367 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
a  e.  RR )
2321, 22resubcld 9457 . . . . . . . 8  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( b  -  a
)  e.  RR )
24 0re 9083 . . . . . . . . . 10  |-  0  e.  RR
2524a1i 11 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
0  e.  RR )
2621, 25resubcld 9457 . . . . . . . 8  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( b  -  0 )  e.  RR )
27 simpllr 736 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  ->  B  e.  NN )
2827nnred 10007 . . . . . . . 8  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  ->  B  e.  RR )
29 elfzle1 11052 . . . . . . . . . 10  |-  ( a  e.  ( 0 ... B )  ->  0  <_  a )
3015, 29syl 16 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
0  <_  a )
3125, 22, 21, 30lesub2dd 9635 . . . . . . . 8  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( b  -  a
)  <_  ( b  -  0 ) )
3221recnd 9106 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
b  e.  CC )
3332subid1d 9392 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( b  -  0 )  =  b )
34 elfzle2 11053 . . . . . . . . . 10  |-  ( b  e.  ( 0 ... B )  ->  b  <_  B )
3513, 34syl 16 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
b  <_  B )
3633, 35eqbrtrd 4224 . . . . . . . 8  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( b  -  0 )  <_  B )
3723, 26, 28, 31, 36letrd 9219 . . . . . . 7  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( b  -  a
)  <_  B )
3812a1i 11 . . . . . . . 8  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
1  e.  ZZ )
3927nnzd 10366 . . . . . . . 8  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  ->  B  e.  ZZ )
40 elfz 11041 . . . . . . . 8  |-  ( ( ( b  -  a
)  e.  ZZ  /\  1  e.  ZZ  /\  B  e.  ZZ )  ->  (
( b  -  a
)  e.  ( 1 ... B )  <->  ( 1  <_  ( b  -  a )  /\  (
b  -  a )  <_  B ) ) )
4117, 38, 39, 40syl3anc 1184 . . . . . . 7  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( ( b  -  a )  e.  ( 1 ... B )  <-> 
( 1  <_  (
b  -  a )  /\  ( b  -  a )  <_  B
) ) )
4220, 37, 41mpbir2and 889 . . . . . 6  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( b  -  a
)  e.  ( 1 ... B ) )
4342adantrr 698 . . . . 5  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  ( a  <  b  /\  ( abs `  (
( ( A  x.  a )  mod  1
)  -  ( ( A  x.  b )  mod  1 ) ) )  <  ( 1  /  B ) ) )  ->  ( b  -  a )  e.  ( 1 ... B
) )
44 rpre 10610 . . . . . . . . . 10  |-  ( A  e.  RR+  ->  A  e.  RR )
4544ad3antrrr 711 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  ->  A  e.  RR )
4645, 22remulcld 9108 . . . . . . . 8  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( A  x.  a
)  e.  RR )
4745, 21remulcld 9108 . . . . . . . 8  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( A  x.  b
)  e.  RR )
48 simpr 448 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
a  <  b )
4922, 21, 48ltled 9213 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
a  <_  b )
50 rpgt0 10615 . . . . . . . . . . 11  |-  ( A  e.  RR+  ->  0  < 
A )
5150ad3antrrr 711 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
0  <  A )
52 lemul2 9855 . . . . . . . . . 10  |-  ( ( a  e.  RR  /\  b  e.  RR  /\  ( A  e.  RR  /\  0  <  A ) )  -> 
( a  <_  b  <->  ( A  x.  a )  <_  ( A  x.  b ) ) )
5322, 21, 45, 51, 52syl112anc 1188 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( a  <_  b  <->  ( A  x.  a )  <_  ( A  x.  b ) ) )
5449, 53mpbid 202 . . . . . . . 8  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( A  x.  a
)  <_  ( A  x.  b ) )
55 flword2 11212 . . . . . . . 8  |-  ( ( ( A  x.  a
)  e.  RR  /\  ( A  x.  b
)  e.  RR  /\  ( A  x.  a
)  <_  ( A  x.  b ) )  -> 
( |_ `  ( A  x.  b )
)  e.  ( ZZ>= `  ( |_ `  ( A  x.  a ) ) ) )
5646, 47, 54, 55syl3anc 1184 . . . . . . 7  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( |_ `  ( A  x.  b )
)  e.  ( ZZ>= `  ( |_ `  ( A  x.  a ) ) ) )
57 uznn0sub 10509 . . . . . . 7  |-  ( ( |_ `  ( A  x.  b ) )  e.  ( ZZ>= `  ( |_ `  ( A  x.  a ) ) )  ->  ( ( |_
`  ( A  x.  b ) )  -  ( |_ `  ( A  x.  a ) ) )  e.  NN0 )
5856, 57syl 16 . . . . . 6  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( ( |_ `  ( A  x.  b
) )  -  ( |_ `  ( A  x.  a ) ) )  e.  NN0 )
5958adantrr 698 . . . . 5  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  ( a  <  b  /\  ( abs `  (
( ( A  x.  a )  mod  1
)  -  ( ( A  x.  b )  mod  1 ) ) )  <  ( 1  /  B ) ) )  ->  ( ( |_ `  ( A  x.  b ) )  -  ( |_ `  ( A  x.  a ) ) )  e.  NN0 )
6045recnd 9106 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  ->  A  e.  CC )
6122recnd 9106 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
a  e.  CC )
6260, 32, 61subdid 9481 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( A  x.  (
b  -  a ) )  =  ( ( A  x.  b )  -  ( A  x.  a ) ) )
6362oveq1d 6088 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( ( A  x.  ( b  -  a
) )  -  (
( |_ `  ( A  x.  b )
)  -  ( |_
`  ( A  x.  a ) ) ) )  =  ( ( ( A  x.  b
)  -  ( A  x.  a ) )  -  ( ( |_
`  ( A  x.  b ) )  -  ( |_ `  ( A  x.  a ) ) ) ) )
6447recnd 9106 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( A  x.  b
)  e.  CC )
6546recnd 9106 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( A  x.  a
)  e.  CC )
6647flcld 11199 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( |_ `  ( A  x.  b )
)  e.  ZZ )
6766zcnd 10368 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( |_ `  ( A  x.  b )
)  e.  CC )
6846flcld 11199 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( |_ `  ( A  x.  a )
)  e.  ZZ )
6968zcnd 10368 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( |_ `  ( A  x.  a )
)  e.  CC )
7064, 65, 67, 69sub4d 9452 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( ( ( A  x.  b )  -  ( A  x.  a
) )  -  (
( |_ `  ( A  x.  b )
)  -  ( |_
`  ( A  x.  a ) ) ) )  =  ( ( ( A  x.  b
)  -  ( |_
`  ( A  x.  b ) ) )  -  ( ( A  x.  a )  -  ( |_ `  ( A  x.  a ) ) ) ) )
71 modfrac 11253 . . . . . . . . . . . . . 14  |-  ( ( A  x.  b )  e.  RR  ->  (
( A  x.  b
)  mod  1 )  =  ( ( A  x.  b )  -  ( |_ `  ( A  x.  b ) ) ) )
7247, 71syl 16 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( ( A  x.  b )  mod  1
)  =  ( ( A  x.  b )  -  ( |_ `  ( A  x.  b
) ) ) )
7372eqcomd 2440 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( ( A  x.  b )  -  ( |_ `  ( A  x.  b ) ) )  =  ( ( A  x.  b )  mod  1 ) )
74 modfrac 11253 . . . . . . . . . . . . . 14  |-  ( ( A  x.  a )  e.  RR  ->  (
( A  x.  a
)  mod  1 )  =  ( ( A  x.  a )  -  ( |_ `  ( A  x.  a ) ) ) )
7546, 74syl 16 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( ( A  x.  a )  mod  1
)  =  ( ( A  x.  a )  -  ( |_ `  ( A  x.  a
) ) ) )
7675eqcomd 2440 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( ( A  x.  a )  -  ( |_ `  ( A  x.  a ) ) )  =  ( ( A  x.  a )  mod  1 ) )
7773, 76oveq12d 6091 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( ( ( A  x.  b )  -  ( |_ `  ( A  x.  b ) ) )  -  ( ( A  x.  a )  -  ( |_ `  ( A  x.  a
) ) ) )  =  ( ( ( A  x.  b )  mod  1 )  -  ( ( A  x.  a )  mod  1
) ) )
7863, 70, 773eqtrd 2471 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( ( A  x.  ( b  -  a
) )  -  (
( |_ `  ( A  x.  b )
)  -  ( |_
`  ( A  x.  a ) ) ) )  =  ( ( ( A  x.  b
)  mod  1 )  -  ( ( A  x.  a )  mod  1 ) ) )
7978fveq2d 5724 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( abs `  (
( A  x.  (
b  -  a ) )  -  ( ( |_ `  ( A  x.  b ) )  -  ( |_ `  ( A  x.  a
) ) ) ) )  =  ( abs `  ( ( ( A  x.  b )  mod  1 )  -  (
( A  x.  a
)  mod  1 ) ) ) )
80 1rp 10608 . . . . . . . . . . . . 13  |-  1  e.  RR+
8180a1i 11 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
1  e.  RR+ )
8247, 81modcld 11246 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( ( A  x.  b )  mod  1
)  e.  RR )
8382recnd 9106 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( ( A  x.  b )  mod  1
)  e.  CC )
8446, 81modcld 11246 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( ( A  x.  a )  mod  1
)  e.  RR )
8584recnd 9106 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( ( A  x.  a )  mod  1
)  e.  CC )
8683, 85abssubd 12247 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( abs `  (
( ( A  x.  b )  mod  1
)  -  ( ( A  x.  a )  mod  1 ) ) )  =  ( abs `  ( ( ( A  x.  a )  mod  1 )  -  (
( A  x.  b
)  mod  1 ) ) ) )
8779, 86eqtr2d 2468 . . . . . . . 8  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( abs `  (
( ( A  x.  a )  mod  1
)  -  ( ( A  x.  b )  mod  1 ) ) )  =  ( abs `  ( ( A  x.  ( b  -  a
) )  -  (
( |_ `  ( A  x.  b )
)  -  ( |_
`  ( A  x.  a ) ) ) ) ) )
8887breq1d 4214 . . . . . . 7  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( ( abs `  (
( ( A  x.  a )  mod  1
)  -  ( ( A  x.  b )  mod  1 ) ) )  <  ( 1  /  B )  <->  ( abs `  ( ( A  x.  ( b  -  a
) )  -  (
( |_ `  ( A  x.  b )
)  -  ( |_
`  ( A  x.  a ) ) ) ) )  <  (
1  /  B ) ) )
8988biimpd 199 . . . . . 6  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( ( abs `  (
( ( A  x.  a )  mod  1
)  -  ( ( A  x.  b )  mod  1 ) ) )  <  ( 1  /  B )  -> 
( abs `  (
( A  x.  (
b  -  a ) )  -  ( ( |_ `  ( A  x.  b ) )  -  ( |_ `  ( A  x.  a
) ) ) ) )  <  ( 1  /  B ) ) )
9089impr 603 . . . . 5  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  ( a  <  b  /\  ( abs `  (
( ( A  x.  a )  mod  1
)  -  ( ( A  x.  b )  mod  1 ) ) )  <  ( 1  /  B ) ) )  ->  ( abs `  ( ( A  x.  ( b  -  a
) )  -  (
( |_ `  ( A  x.  b )
)  -  ( |_
`  ( A  x.  a ) ) ) ) )  <  (
1  /  B ) )
91 oveq2 6081 . . . . . . . . 9  |-  ( x  =  ( b  -  a )  ->  ( A  x.  x )  =  ( A  x.  ( b  -  a
) ) )
9291oveq1d 6088 . . . . . . . 8  |-  ( x  =  ( b  -  a )  ->  (
( A  x.  x
)  -  y )  =  ( ( A  x.  ( b  -  a ) )  -  y ) )
9392fveq2d 5724 . . . . . . 7  |-  ( x  =  ( b  -  a )  ->  ( abs `  ( ( A  x.  x )  -  y ) )  =  ( abs `  (
( A  x.  (
b  -  a ) )  -  y ) ) )
9493breq1d 4214 . . . . . 6  |-  ( x  =  ( b  -  a )  ->  (
( abs `  (
( A  x.  x
)  -  y ) )  <  ( 1  /  B )  <->  ( abs `  ( ( A  x.  ( b  -  a
) )  -  y
) )  <  (
1  /  B ) ) )
95 oveq2 6081 . . . . . . . 8  |-  ( y  =  ( ( |_
`  ( A  x.  b ) )  -  ( |_ `  ( A  x.  a ) ) )  ->  ( ( A  x.  ( b  -  a ) )  -  y )  =  ( ( A  x.  ( b  -  a
) )  -  (
( |_ `  ( A  x.  b )
)  -  ( |_
`  ( A  x.  a ) ) ) ) )
9695fveq2d 5724 . . . . . . 7  |-  ( y  =  ( ( |_
`  ( A  x.  b ) )  -  ( |_ `  ( A  x.  a ) ) )  ->  ( abs `  ( ( A  x.  ( b  -  a
) )  -  y
) )  =  ( abs `  ( ( A  x.  ( b  -  a ) )  -  ( ( |_
`  ( A  x.  b ) )  -  ( |_ `  ( A  x.  a ) ) ) ) ) )
9796breq1d 4214 . . . . . 6  |-  ( y  =  ( ( |_
`  ( A  x.  b ) )  -  ( |_ `  ( A  x.  a ) ) )  ->  ( ( abs `  ( ( A  x.  ( b  -  a ) )  -  y ) )  < 
( 1  /  B
)  <->  ( abs `  (
( A  x.  (
b  -  a ) )  -  ( ( |_ `  ( A  x.  b ) )  -  ( |_ `  ( A  x.  a
) ) ) ) )  <  ( 1  /  B ) ) )
9894, 97rspc2ev 3052 . . . . 5  |-  ( ( ( b  -  a
)  e.  ( 1 ... B )  /\  ( ( |_ `  ( A  x.  b
) )  -  ( |_ `  ( A  x.  a ) ) )  e.  NN0  /\  ( abs `  ( ( A  x.  ( b  -  a ) )  -  ( ( |_ `  ( A  x.  b
) )  -  ( |_ `  ( A  x.  a ) ) ) ) )  <  (
1  /  B ) )  ->  E. x  e.  ( 1 ... B
) E. y  e. 
NN0  ( abs `  (
( A  x.  x
)  -  y ) )  <  ( 1  /  B ) )
9943, 59, 90, 98syl3anc 1184 . . . 4  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  ( a  <  b  /\  ( abs `  (
( ( A  x.  a )  mod  1
)  -  ( ( A  x.  b )  mod  1 ) ) )  <  ( 1  /  B ) ) )  ->  E. x  e.  ( 1 ... B
) E. y  e. 
NN0  ( abs `  (
( A  x.  x
)  -  y ) )  <  ( 1  /  B ) )
10099ex 424 . . 3  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  -> 
( ( a  < 
b  /\  ( abs `  ( ( ( A  x.  a )  mod  1 )  -  (
( A  x.  b
)  mod  1 ) ) )  <  (
1  /  B ) )  ->  E. x  e.  ( 1 ... B
) E. y  e. 
NN0  ( abs `  (
( A  x.  x
)  -  y ) )  <  ( 1  /  B ) ) )
101100rexlimdvva 2829 . 2  |-  ( ( A  e.  RR+  /\  B  e.  NN )  ->  ( E. a  e.  (
0 ... B ) E. b  e.  ( 0 ... B ) ( a  <  b  /\  ( abs `  ( ( ( A  x.  a
)  mod  1 )  -  ( ( A  x.  b )  mod  1 ) ) )  <  ( 1  /  B ) )  ->  E. x  e.  (
1 ... B ) E. y  e.  NN0  ( abs `  ( ( A  x.  x )  -  y ) )  < 
( 1  /  B
) ) )
1021, 101mpd 15 1  |-  ( ( A  e.  RR+  /\  B  e.  NN )  ->  E. x  e.  ( 1 ... B
) E. y  e. 
NN0  ( abs `  (
( A  x.  x
)  -  y ) )  <  ( 1  /  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   E.wrex 2698   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   RRcr 8981   0cc0 8982   1c1 8983    x. cmul 8987    < clt 9112    <_ cle 9113    - cmin 9283    / cdiv 9669   NNcn 9992   NN0cn0 10213   ZZcz 10274   ZZ>=cuz 10480   RR+crp 10604   ...cfz 11035   |_cfl 11193    mod cmo 11242   abscabs 12031
This theorem is referenced by:  irrapxlem4  26879
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-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
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-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-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-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-card 7818  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-n0 10214  df-z 10275  df-uz 10481  df-rp 10605  df-ico 10914  df-fz 11036  df-fl 11194  df-mod 11243  df-seq 11316  df-exp 11375  df-hash 11611  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033
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