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Theorem irrapxlem3 26909
Description: Lemma for irrapx1 26913. 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 26908 . 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 9814 . . . . . . . . 9  |-  ( 1  -  1 )  =  0
3 elfzelz 10798 . . . . . . . . . . . . 13  |-  ( a  e.  ( 0 ... B )  ->  a  e.  ZZ )
43ad2antrl 708 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  -> 
a  e.  ZZ )
54zred 10117 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  -> 
a  e.  RR )
6 elfzelz 10798 . . . . . . . . . . . . 13  |-  ( b  e.  ( 0 ... B )  ->  b  e.  ZZ )
76ad2antll 709 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  -> 
b  e.  ZZ )
87zred 10117 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  -> 
b  e.  RR )
95, 8posdifd 9359 . . . . . . . . . 10  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  -> 
( a  <  b  <->  0  <  ( b  -  a ) ) )
109biimpa 470 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
0  <  ( b  -  a ) )
112, 10syl5eqbr 4056 . . . . . . . 8  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( 1  -  1 )  <  ( b  -  a ) )
12 1z 10053 . . . . . . . . 9  |-  1  e.  ZZ
13 simplrr 737 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
b  e.  ( 0 ... B ) )
1413, 6syl 15 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
b  e.  ZZ )
15 simplrl 736 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
a  e.  ( 0 ... B ) )
1615, 3syl 15 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
a  e.  ZZ )
1714, 16zsubcld 10122 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( b  -  a
)  e.  ZZ )
18 zlem1lt 10069 . . . . . . . . 9  |-  ( ( 1  e.  ZZ  /\  ( b  -  a
)  e.  ZZ )  ->  ( 1  <_ 
( b  -  a
)  <->  ( 1  -  1 )  <  (
b  -  a ) ) )
1912, 17, 18sylancr 644 . . . . . . . 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 223 . . . . . . 7  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
1  <_  ( b  -  a ) )
2114zred 10117 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
b  e.  RR )
2216zred 10117 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
a  e.  RR )
2321, 22resubcld 9211 . . . . . . . 8  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( b  -  a
)  e.  RR )
24 0re 8838 . . . . . . . . . 10  |-  0  e.  RR
2524a1i 10 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
0  e.  RR )
2621, 25resubcld 9211 . . . . . . . 8  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( b  -  0 )  e.  RR )
27 simpllr 735 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  ->  B  e.  NN )
2827nnred 9761 . . . . . . . 8  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  ->  B  e.  RR )
29 elfzle1 10799 . . . . . . . . . 10  |-  ( a  e.  ( 0 ... B )  ->  0  <_  a )
3015, 29syl 15 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
0  <_  a )
3125, 22, 21, 30lesub2dd 9389 . . . . . . . 8  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( b  -  a
)  <_  ( b  -  0 ) )
3221recnd 8861 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
b  e.  CC )
3332subid1d 9146 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( b  -  0 )  =  b )
34 elfzle2 10800 . . . . . . . . . 10  |-  ( b  e.  ( 0 ... B )  ->  b  <_  B )
3513, 34syl 15 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
b  <_  B )
3633, 35eqbrtrd 4043 . . . . . . . 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 8973 . . . . . . 7  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( b  -  a
)  <_  B )
3812a1i 10 . . . . . . . 8  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
1  e.  ZZ )
3927nnzd 10116 . . . . . . . 8  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  ->  B  e.  ZZ )
40 elfz 10788 . . . . . . . 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 1182 . . . . . . 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 888 . . . . . 6  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( b  -  a
)  e.  ( 1 ... B ) )
4342adantrr 697 . . . . 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 10360 . . . . . . . . . 10  |-  ( A  e.  RR+  ->  A  e.  RR )
4544ad3antrrr 710 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  ->  A  e.  RR )
4645, 22remulcld 8863 . . . . . . . 8  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( A  x.  a
)  e.  RR )
4745, 21remulcld 8863 . . . . . . . 8  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( A  x.  b
)  e.  RR )
48 simpr 447 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
a  <  b )
4922, 21, 48ltled 8967 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
a  <_  b )
50 rpgt0 10365 . . . . . . . . . . 11  |-  ( A  e.  RR+  ->  0  < 
A )
5150ad3antrrr 710 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
0  <  A )
52 lemul2 9609 . . . . . . . . . 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 1186 . . . . . . . . 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 201 . . . . . . . 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 10943 . . . . . . . 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 1182 . . . . . . 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 10259 . . . . . . 7  |-  ( ( |_ `  ( A  x.  b ) )  e.  ( ZZ>= `  ( |_ `  ( A  x.  a ) ) )  ->  ( ( |_
`  ( A  x.  b ) )  -  ( |_ `  ( A  x.  a ) ) )  e.  NN0 )
5856, 57syl 15 . . . . . 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 697 . . . . 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 8861 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  ->  A  e.  CC )
6122recnd 8861 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
a  e.  CC )
6260, 32, 61subdid 9235 . . . . . . . . . . . 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 5873 . . . . . . . . . . 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 8861 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( A  x.  b
)  e.  CC )
6546recnd 8861 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( A  x.  a
)  e.  CC )
6647flcld 10930 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( |_ `  ( A  x.  b )
)  e.  ZZ )
6766zcnd 10118 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( |_ `  ( A  x.  b )
)  e.  CC )
6846flcld 10930 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( |_ `  ( A  x.  a )
)  e.  ZZ )
6968zcnd 10118 . . . . . . . . . . . 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 9206 . . . . . . . . . . 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 10984 . . . . . . . . . . . . . 14  |-  ( ( A  x.  b )  e.  RR  ->  (
( A  x.  b
)  mod  1 )  =  ( ( A  x.  b )  -  ( |_ `  ( A  x.  b ) ) ) )
7247, 71syl 15 . . . . . . . . . . . . 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 2288 . . . . . . . . . . . 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 10984 . . . . . . . . . . . . . 14  |-  ( ( A  x.  a )  e.  RR  ->  (
( A  x.  a
)  mod  1 )  =  ( ( A  x.  a )  -  ( |_ `  ( A  x.  a ) ) ) )
7546, 74syl 15 . . . . . . . . . . . . 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 2288 . . . . . . . . . . . 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 5876 . . . . . . . . . . 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 2319 . . . . . . . . . 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 5529 . . . . . . . . 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 10358 . . . . . . . . . . . . 13  |-  1  e.  RR+
8180a1i 10 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
1  e.  RR+ )
8247, 81modcld 10977 . . . . . . . . . . 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 8861 . . . . . . . . . 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 10977 . . . . . . . . . . 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 8861 . . . . . . . . . 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 11935 . . . . . . . . 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 2316 . . . . . . . 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 4033 . . . . . . 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 198 . . . . . 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 602 . . . . 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 5866 . . . . . . . . 9  |-  ( x  =  ( b  -  a )  ->  ( A  x.  x )  =  ( A  x.  ( b  -  a
) ) )
9291oveq1d 5873 . . . . . . . 8  |-  ( x  =  ( b  -  a )  ->  (
( A  x.  x
)  -  y )  =  ( ( A  x.  ( b  -  a ) )  -  y ) )
9392fveq2d 5529 . . . . . . 7  |-  ( x  =  ( b  -  a )  ->  ( abs `  ( ( A  x.  x )  -  y ) )  =  ( abs `  (
( A  x.  (
b  -  a ) )  -  y ) ) )
9493breq1d 4033 . . . . . 6  |-  ( x  =  ( b  -  a )  ->  (
( abs `  (
( A  x.  x
)  -  y ) )  <  ( 1  /  B )  <->  ( abs `  ( ( A  x.  ( b  -  a
) )  -  y
) )  <  (
1  /  B ) ) )
95 oveq2 5866 . . . . . . . 8  |-  ( y  =  ( ( |_
`  ( A  x.  b ) )  -  ( |_ `  ( A  x.  a ) ) )  ->  ( ( A  x.  ( b  -  a ) )  -  y )  =  ( ( A  x.  ( b  -  a
) )  -  (
( |_ `  ( A  x.  b )
)  -  ( |_
`  ( A  x.  a ) ) ) ) )
9695fveq2d 5529 . . . . . . 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 4033 . . . . . 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 2892 . . . . 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 1182 . . . 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 423 . . 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 2674 . 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 14 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 176    /\ wa 358    = wceq 1623    e. wcel 1684   E.wrex 2544   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   RRcr 8736   0cc0 8737   1c1 8738    x. cmul 8742    < clt 8867    <_ cle 8868    - cmin 9037    / cdiv 9423   NNcn 9746   NN0cn0 9965   ZZcz 10024   ZZ>=cuz 10230   RR+crp 10354   ...cfz 10782   |_cfl 10924    mod cmo 10973   abscabs 11719
This theorem is referenced by:  irrapxlem4  26910
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-ico 10662  df-fz 10783  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721
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