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Theorem irrapxlem3 35668
Description: Lemma for irrapx1 35672. 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 35667 . 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 10678 . . . . . . . . 9  |-  ( 1  -  1 )  =  0
3 elfzelz 11800 . . . . . . . . . . . . 13  |-  ( a  e.  ( 0 ... B )  ->  a  e.  ZZ )
43ad2antrl 734 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  -> 
a  e.  ZZ )
54zred 11040 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  -> 
a  e.  RR )
6 elfzelz 11800 . . . . . . . . . . . . 13  |-  ( b  e.  ( 0 ... B )  ->  b  e.  ZZ )
76ad2antll 735 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  -> 
b  e.  ZZ )
87zred 11040 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  -> 
b  e.  RR )
95, 8posdifd 10200 . . . . . . . . . 10  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  -> 
( a  <  b  <->  0  <  ( b  -  a ) ) )
109biimpa 487 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
0  <  ( b  -  a ) )
112, 10syl5eqbr 4436 . . . . . . . 8  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( 1  -  1 )  <  ( b  -  a ) )
12 1z 10967 . . . . . . . . 9  |-  1  e.  ZZ
13 simplrr 771 . . . . . . . . . . 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 770 . . . . . . . . . . 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 11045 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( b  -  a
)  e.  ZZ )
18 zlem1lt 10988 . . . . . . . . 9  |-  ( ( 1  e.  ZZ  /\  ( b  -  a
)  e.  ZZ )  ->  ( 1  <_ 
( b  -  a
)  <->  ( 1  -  1 )  <  (
b  -  a ) ) )
1912, 17, 18sylancr 669 . . . . . . . 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 236 . . . . . . 7  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
1  <_  ( b  -  a ) )
2114zred 11040 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
b  e.  RR )
2216zred 11040 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
a  e.  RR )
2321, 22resubcld 10047 . . . . . . . 8  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( b  -  a
)  e.  RR )
24 0red 9644 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
0  e.  RR )
2521, 24resubcld 10047 . . . . . . . 8  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( b  -  0 )  e.  RR )
26 simpllr 769 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  ->  B  e.  NN )
2726nnred 10624 . . . . . . . 8  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  ->  B  e.  RR )
28 elfzle1 11802 . . . . . . . . . 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 10230 . . . . . . . 8  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( b  -  a
)  <_  ( b  -  0 ) )
3121recnd 9669 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
b  e.  CC )
3231subid1d 9975 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( b  -  0 )  =  b )
33 elfzle2 11803 . . . . . . . . . 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 4423 . . . . . . . 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 9792 . . . . . . 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 11039 . . . . . . . 8  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  ->  B  e.  ZZ )
39 elfz 11790 . . . . . . . 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 1268 . . . . . . 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 933 . . . . . 6  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( b  -  a
)  e.  ( 1 ... B ) )
4241adantrr 723 . . . . 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 11308 . . . . . . . . . 10  |-  ( A  e.  RR+  ->  A  e.  RR )
4443ad3antrrr 736 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  ->  A  e.  RR )
4544, 22remulcld 9671 . . . . . . . 8  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( A  x.  a
)  e.  RR )
4644, 21remulcld 9671 . . . . . . . 8  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( A  x.  b
)  e.  RR )
47 simpr 463 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
a  <  b )
4822, 21, 47ltled 9783 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
a  <_  b )
49 rpgt0 11313 . . . . . . . . . . 11  |-  ( A  e.  RR+  ->  0  < 
A )
5049ad3antrrr 736 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
0  <  A )
51 lemul2 10458 . . . . . . . . . 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 1272 . . . . . . . . 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 214 . . . . . . . 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 12048 . . . . . . . 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 1268 . . . . . . 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 11190 . . . . . . 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 723 . . . . 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 9669 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  ->  A  e.  CC )
6022recnd 9669 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
a  e.  CC )
6159, 31, 60subdid 10074 . . . . . . . . . . . 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 6305 . . . . . . . . . . 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 9669 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( A  x.  b
)  e.  CC )
6445recnd 9669 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( A  x.  a
)  e.  CC )
6546flcld 12034 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( |_ `  ( A  x.  b )
)  e.  ZZ )
6665zcnd 11041 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( |_ `  ( A  x.  b )
)  e.  CC )
6745flcld 12034 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( |_ `  ( A  x.  a )
)  e.  ZZ )
6867zcnd 11041 . . . . . . . . . . . 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 10035 . . . . . . . . . . 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 12110 . . . . . . . . . . . . . 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 2457 . . . . . . . . . . . 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 12110 . . . . . . . . . . . . . 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 2457 . . . . . . . . . . . 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 6308 . . . . . . . . . . 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 2489 . . . . . . . . . 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 5869 . . . . . . . . 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 11306 . . . . . . . . . . . . 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 12102 . . . . . . . . . . 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 9669 . . . . . . . . . 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 12102 . . . . . . . . . . 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 9669 . . . . . . . . . 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 13515 . . . . . . . . 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 2486 . . . . . . . 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 4412 . . . . . . 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 211 . . . . . 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 625 . . . . 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 6298 . . . . . . . . 9  |-  ( x  =  ( b  -  a )  ->  ( A  x.  x )  =  ( A  x.  ( b  -  a
) ) )
9190oveq1d 6305 . . . . . . . 8  |-  ( x  =  ( b  -  a )  ->  (
( A  x.  x
)  -  y )  =  ( ( A  x.  ( b  -  a ) )  -  y ) )
9291fveq2d 5869 . . . . . . 7  |-  ( x  =  ( b  -  a )  ->  ( abs `  ( ( A  x.  x )  -  y ) )  =  ( abs `  (
( A  x.  (
b  -  a ) )  -  y ) ) )
9392breq1d 4412 . . . . . 6  |-  ( x  =  ( b  -  a )  ->  (
( abs `  (
( A  x.  x
)  -  y ) )  <  ( 1  /  B )  <->  ( abs `  ( ( A  x.  ( b  -  a
) )  -  y
) )  <  (
1  /  B ) ) )
94 oveq2 6298 . . . . . . . 8  |-  ( y  =  ( ( |_
`  ( A  x.  b ) )  -  ( |_ `  ( A  x.  a ) ) )  ->  ( ( A  x.  ( b  -  a ) )  -  y )  =  ( ( A  x.  ( b  -  a
) )  -  (
( |_ `  ( A  x.  b )
)  -  ( |_
`  ( A  x.  a ) ) ) ) )
9594fveq2d 5869 . . . . . . 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 4412 . . . . . 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 3161 . . . . 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 1268 . . . 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 436 . . 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 2886 . 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 188    /\ wa 371    = wceq 1444    e. wcel 1887   E.wrex 2738   class class class wbr 4402   ` cfv 5582  (class class class)co 6290   RRcr 9538   0cc0 9539   1c1 9540    x. cmul 9544    < clt 9675    <_ cle 9676    - cmin 9860    / cdiv 10269   NNcn 10609   NN0cn0 10869   ZZcz 10937   ZZ>=cuz 11159   RR+crp 11302   ...cfz 11784   |_cfl 12026    mod cmo 12096   abscabs 13297
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-sup 7956  df-inf 7957  df-card 8373  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-ico 11641  df-fz 11785  df-fl 12028  df-mod 12097  df-seq 12214  df-exp 12273  df-hash 12516  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299
This theorem is referenced by:  irrapxlem4  35669
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