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Theorem irrapxlem3 35739
Description: Lemma for irrapx1 35743. 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 35738 . 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 10700 . . . . . . . . 9  |-  ( 1  -  1 )  =  0
3 elfzelz 11826 . . . . . . . . . . . . 13  |-  ( a  e.  ( 0 ... B )  ->  a  e.  ZZ )
43ad2antrl 742 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  -> 
a  e.  ZZ )
54zred 11063 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  -> 
a  e.  RR )
6 elfzelz 11826 . . . . . . . . . . . . 13  |-  ( b  e.  ( 0 ... B )  ->  b  e.  ZZ )
76ad2antll 743 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  -> 
b  e.  ZZ )
87zred 11063 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  -> 
b  e.  RR )
95, 8posdifd 10221 . . . . . . . . . 10  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  -> 
( a  <  b  <->  0  <  ( b  -  a ) ) )
109biimpa 492 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
0  <  ( b  -  a ) )
112, 10syl5eqbr 4429 . . . . . . . 8  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( 1  -  1 )  <  ( b  -  a ) )
12 1z 10991 . . . . . . . . 9  |-  1  e.  ZZ
13 simplrr 779 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
b  e.  ( 0 ... B ) )
1413, 6syl 17 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
b  e.  ZZ )
15 simplrl 778 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
a  e.  ( 0 ... B ) )
1615, 3syl 17 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
a  e.  ZZ )
1714, 16zsubcld 11068 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( b  -  a
)  e.  ZZ )
18 zlem1lt 11012 . . . . . . . . 9  |-  ( ( 1  e.  ZZ  /\  ( b  -  a
)  e.  ZZ )  ->  ( 1  <_ 
( b  -  a
)  <->  ( 1  -  1 )  <  (
b  -  a ) ) )
1912, 17, 18sylancr 676 . . . . . . . 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 240 . . . . . . 7  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
1  <_  ( b  -  a ) )
2114zred 11063 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
b  e.  RR )
2216zred 11063 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
a  e.  RR )
2321, 22resubcld 10068 . . . . . . . 8  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( b  -  a
)  e.  RR )
24 0red 9662 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
0  e.  RR )
2521, 24resubcld 10068 . . . . . . . 8  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( b  -  0 )  e.  RR )
26 simpllr 777 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  ->  B  e.  NN )
2726nnred 10646 . . . . . . . 8  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  ->  B  e.  RR )
28 elfzle1 11828 . . . . . . . . . 10  |-  ( a  e.  ( 0 ... B )  ->  0  <_  a )
2915, 28syl 17 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
0  <_  a )
3024, 22, 21, 29lesub2dd 10251 . . . . . . . 8  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( b  -  a
)  <_  ( b  -  0 ) )
3121recnd 9687 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
b  e.  CC )
3231subid1d 9994 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( b  -  0 )  =  b )
33 elfzle2 11829 . . . . . . . . . 10  |-  ( b  e.  ( 0 ... B )  ->  b  <_  B )
3413, 33syl 17 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
b  <_  B )
3532, 34eqbrtrd 4416 . . . . . . . 8  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( b  -  0 )  <_  B )
3623, 25, 27, 30, 35letrd 9809 . . . . . . 7  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( b  -  a
)  <_  B )
3712a1i 11 . . . . . . . 8  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
1  e.  ZZ )
3826nnzd 11062 . . . . . . . 8  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  ->  B  e.  ZZ )
39 elfz 11816 . . . . . . . 8  |-  ( ( ( b  -  a
)  e.  ZZ  /\  1  e.  ZZ  /\  B  e.  ZZ )  ->  (
( b  -  a
)  e.  ( 1 ... B )  <->  ( 1  <_  ( b  -  a )  /\  (
b  -  a )  <_  B ) ) )
4017, 37, 38, 39syl3anc 1292 . . . . . . 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
) ) )
4120, 36, 40mpbir2and 936 . . . . . 6  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( b  -  a
)  e.  ( 1 ... B ) )
4241adantrr 731 . . . . 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
) )
43 rpre 11331 . . . . . . . . . 10  |-  ( A  e.  RR+  ->  A  e.  RR )
4443ad3antrrr 744 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  ->  A  e.  RR )
4544, 22remulcld 9689 . . . . . . . 8  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( A  x.  a
)  e.  RR )
4644, 21remulcld 9689 . . . . . . . 8  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( A  x.  b
)  e.  RR )
47 simpr 468 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
a  <  b )
4822, 21, 47ltled 9800 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
a  <_  b )
49 rpgt0 11336 . . . . . . . . . . 11  |-  ( A  e.  RR+  ->  0  < 
A )
5049ad3antrrr 744 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
0  <  A )
51 lemul2 10480 . . . . . . . . . 10  |-  ( ( a  e.  RR  /\  b  e.  RR  /\  ( A  e.  RR  /\  0  <  A ) )  -> 
( a  <_  b  <->  ( A  x.  a )  <_  ( A  x.  b ) ) )
5222, 21, 44, 50, 51syl112anc 1296 . . . . . . . . 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 ) ) )
5348, 52mpbid 215 . . . . . . . 8  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( A  x.  a
)  <_  ( A  x.  b ) )
54 flword2 12081 . . . . . . . 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 ) ) ) )
5545, 46, 53, 54syl3anc 1292 . . . . . . 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 ) ) ) )
56 uznn0sub 11214 . . . . . . 7  |-  ( ( |_ `  ( A  x.  b ) )  e.  ( ZZ>= `  ( |_ `  ( A  x.  a ) ) )  ->  ( ( |_
`  ( A  x.  b ) )  -  ( |_ `  ( A  x.  a ) ) )  e.  NN0 )
5755, 56syl 17 . . . . . 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 )
5857adantrr 731 . . . . 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 )
5944recnd 9687 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  ->  A  e.  CC )
6022recnd 9687 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
a  e.  CC )
6159, 31, 60subdid 10095 . . . . . . . . . . . 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 ) ) )
6261oveq1d 6323 . . . . . . . . . . 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 ) ) ) ) )
6346recnd 9687 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( A  x.  b
)  e.  CC )
6445recnd 9687 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( A  x.  a
)  e.  CC )
6546flcld 12067 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( |_ `  ( A  x.  b )
)  e.  ZZ )
6665zcnd 11064 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( |_ `  ( A  x.  b )
)  e.  CC )
6745flcld 12067 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( |_ `  ( A  x.  a )
)  e.  ZZ )
6867zcnd 11064 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( |_ `  ( A  x.  a )
)  e.  CC )
6963, 64, 66, 68sub4d 10054 . . . . . . . . . . 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 ) ) ) ) )
70 modfrac 12143 . . . . . . . . . . . . . 14  |-  ( ( A  x.  b )  e.  RR  ->  (
( A  x.  b
)  mod  1 )  =  ( ( A  x.  b )  -  ( |_ `  ( A  x.  b ) ) ) )
7146, 70syl 17 . . . . . . . . . . . . 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
) ) ) )
7271eqcomd 2477 . . . . . . . . . . . 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 ) )
73 modfrac 12143 . . . . . . . . . . . . . 14  |-  ( ( A  x.  a )  e.  RR  ->  (
( A  x.  a
)  mod  1 )  =  ( ( A  x.  a )  -  ( |_ `  ( A  x.  a ) ) ) )
7445, 73syl 17 . . . . . . . . . . . . 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
) ) ) )
7574eqcomd 2477 . . . . . . . . . . . 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 ) )
7672, 75oveq12d 6326 . . . . . . . . . . 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
) ) )
7762, 69, 763eqtrd 2509 . . . . . . . . . 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 ) ) )
7877fveq2d 5883 . . . . . . . . 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 ) ) ) )
79 1rp 11329 . . . . . . . . . . . . 13  |-  1  e.  RR+
8079a1i 11 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
1  e.  RR+ )
8146, 80modcld 12135 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( ( A  x.  b )  mod  1
)  e.  RR )
8281recnd 9687 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( ( A  x.  b )  mod  1
)  e.  CC )
8345, 80modcld 12135 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( ( A  x.  a )  mod  1
)  e.  RR )
8483recnd 9687 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( ( A  x.  a )  mod  1
)  e.  CC )
8582, 84abssubd 13592 . . . . . . . . 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 ) ) ) )
8678, 85eqtr2d 2506 . . . . . . . 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 ) ) ) ) ) )
8786breq1d 4405 . . . . . . 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 ) ) )
8887biimpd 212 . . . . . 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 ) ) )
8988impr 631 . . . . 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 ) )
90 oveq2 6316 . . . . . . . . 9  |-  ( x  =  ( b  -  a )  ->  ( A  x.  x )  =  ( A  x.  ( b  -  a
) ) )
9190oveq1d 6323 . . . . . . . 8  |-  ( x  =  ( b  -  a )  ->  (
( A  x.  x
)  -  y )  =  ( ( A  x.  ( b  -  a ) )  -  y ) )
9291fveq2d 5883 . . . . . . 7  |-  ( x  =  ( b  -  a )  ->  ( abs `  ( ( A  x.  x )  -  y ) )  =  ( abs `  (
( A  x.  (
b  -  a ) )  -  y ) ) )
9392breq1d 4405 . . . . . 6  |-  ( x  =  ( b  -  a )  ->  (
( abs `  (
( A  x.  x
)  -  y ) )  <  ( 1  /  B )  <->  ( abs `  ( ( A  x.  ( b  -  a
) )  -  y
) )  <  (
1  /  B ) ) )
94 oveq2 6316 . . . . . . . 8  |-  ( y  =  ( ( |_
`  ( A  x.  b ) )  -  ( |_ `  ( A  x.  a ) ) )  ->  ( ( A  x.  ( b  -  a ) )  -  y )  =  ( ( A  x.  ( b  -  a
) )  -  (
( |_ `  ( A  x.  b )
)  -  ( |_
`  ( A  x.  a ) ) ) ) )
9594fveq2d 5883 . . . . . . 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 ) ) ) ) ) )
9695breq1d 4405 . . . . . 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 ) ) )
9793, 96rspc2ev 3149 . . . . 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 ) )
9842, 58, 89, 97syl3anc 1292 . . . 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 ) )
9998ex 441 . . 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 ) ) )
10099rexlimdvva 2878 . 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
) ) )
1011, 100mpd 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 setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    = wceq 1452    e. wcel 1904   E.wrex 2757   class class class wbr 4395   ` cfv 5589  (class class class)co 6308   RRcr 9556   0cc0 9557   1c1 9558    x. cmul 9562    < clt 9693    <_ cle 9694    - cmin 9880    / cdiv 10291   NNcn 10631   NN0cn0 10893   ZZcz 10961   ZZ>=cuz 11182   RR+crp 11325   ...cfz 11810   |_cfl 12059    mod cmo 12129   abscabs 13374
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-sup 7974  df-inf 7975  df-card 8391  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-n0 10894  df-z 10962  df-uz 11183  df-rp 11326  df-ico 11666  df-fz 11811  df-fl 12061  df-mod 12130  df-seq 12252  df-exp 12311  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376
This theorem is referenced by:  irrapxlem4  35740
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