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Theorem irrapxlem3 31002
Description: Lemma for irrapx1 31006. 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 31001 . 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 10600 . . . . . . . . 9  |-  ( 1  -  1 )  =  0
3 elfzelz 11691 . . . . . . . . . . . . 13  |-  ( a  e.  ( 0 ... B )  ->  a  e.  ZZ )
43ad2antrl 725 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  -> 
a  e.  ZZ )
54zred 10965 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  -> 
a  e.  RR )
6 elfzelz 11691 . . . . . . . . . . . . 13  |-  ( b  e.  ( 0 ... B )  ->  b  e.  ZZ )
76ad2antll 726 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  -> 
b  e.  ZZ )
87zred 10965 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  -> 
b  e.  RR )
95, 8posdifd 10135 . . . . . . . . . 10  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  -> 
( a  <  b  <->  0  <  ( b  -  a ) ) )
109biimpa 482 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
0  <  ( b  -  a ) )
112, 10syl5eqbr 4472 . . . . . . . 8  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( 1  -  1 )  <  ( b  -  a ) )
12 1z 10890 . . . . . . . . 9  |-  1  e.  ZZ
13 simplrr 760 . . . . . . . . . . 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 759 . . . . . . . . . . 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 10970 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( b  -  a
)  e.  ZZ )
18 zlem1lt 10911 . . . . . . . . 9  |-  ( ( 1  e.  ZZ  /\  ( b  -  a
)  e.  ZZ )  ->  ( 1  <_ 
( b  -  a
)  <->  ( 1  -  1 )  <  (
b  -  a ) ) )
1912, 17, 18sylancr 661 . . . . . . . 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 232 . . . . . . 7  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
1  <_  ( b  -  a ) )
2114zred 10965 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
b  e.  RR )
2216zred 10965 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
a  e.  RR )
2321, 22resubcld 9983 . . . . . . . 8  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( b  -  a
)  e.  RR )
24 0red 9586 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
0  e.  RR )
2521, 24resubcld 9983 . . . . . . . 8  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( b  -  0 )  e.  RR )
26 simpllr 758 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  ->  B  e.  NN )
2726nnred 10546 . . . . . . . 8  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  ->  B  e.  RR )
28 elfzle1 11692 . . . . . . . . . 10  |-  ( a  e.  ( 0 ... B )  ->  0  <_  a )
2915, 28syl 16 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
0  <_  a )
3024, 22, 21, 29lesub2dd 10165 . . . . . . . 8  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( b  -  a
)  <_  ( b  -  0 ) )
3121recnd 9611 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
b  e.  CC )
3231subid1d 9911 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( b  -  0 )  =  b )
33 elfzle2 11693 . . . . . . . . . 10  |-  ( b  e.  ( 0 ... B )  ->  b  <_  B )
3413, 33syl 16 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
b  <_  B )
3532, 34eqbrtrd 4459 . . . . . . . 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 9728 . . . . . . 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 10964 . . . . . . . 8  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  ->  B  e.  ZZ )
39 elfz 11681 . . . . . . . 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 1226 . . . . . . 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 920 . . . . . 6  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( b  -  a
)  e.  ( 1 ... B ) )
4241adantrr 714 . . . . 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 11227 . . . . . . . . . 10  |-  ( A  e.  RR+  ->  A  e.  RR )
4443ad3antrrr 727 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  ->  A  e.  RR )
4544, 22remulcld 9613 . . . . . . . 8  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( A  x.  a
)  e.  RR )
4644, 21remulcld 9613 . . . . . . . 8  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( A  x.  b
)  e.  RR )
47 simpr 459 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
a  <  b )
4822, 21, 47ltled 9722 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
a  <_  b )
49 rpgt0 11232 . . . . . . . . . . 11  |-  ( A  e.  RR+  ->  0  < 
A )
5049ad3antrrr 727 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
0  <  A )
51 lemul2 10391 . . . . . . . . . 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 1230 . . . . . . . . 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 210 . . . . . . . 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 11930 . . . . . . . 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 1226 . . . . . . 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 11113 . . . . . . 7  |-  ( ( |_ `  ( A  x.  b ) )  e.  ( ZZ>= `  ( |_ `  ( A  x.  a ) ) )  ->  ( ( |_
`  ( A  x.  b ) )  -  ( |_ `  ( A  x.  a ) ) )  e.  NN0 )
5755, 56syl 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 )
5857adantrr 714 . . . . 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 9611 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  ->  A  e.  CC )
6022recnd 9611 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
a  e.  CC )
6159, 31, 60subdid 10008 . . . . . . . . . . . 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 6285 . . . . . . . . . . 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 9611 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( A  x.  b
)  e.  CC )
6445recnd 9611 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( A  x.  a
)  e.  CC )
6546flcld 11916 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( |_ `  ( A  x.  b )
)  e.  ZZ )
6665zcnd 10966 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( |_ `  ( A  x.  b )
)  e.  CC )
6745flcld 11916 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  ( a  e.  ( 0 ... B
)  /\  b  e.  ( 0 ... B
) ) )  /\  a  <  b )  -> 
( |_ `  ( A  x.  a )
)  e.  ZZ )
6867zcnd 10966 . . . . . . . . . . . 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 9971 . . . . . . . . . . 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 11992 . . . . . . . . . . . . . 14  |-  ( ( A  x.  b )  e.  RR  ->  (
( A  x.  b
)  mod  1 )  =  ( ( A  x.  b )  -  ( |_ `  ( A  x.  b ) ) ) )
7146, 70syl 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
) ) ) )
7271eqcomd 2462 . . . . . . . . . . . 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 11992 . . . . . . . . . . . . . 14  |-  ( ( A  x.  a )  e.  RR  ->  (
( A  x.  a
)  mod  1 )  =  ( ( A  x.  a )  -  ( |_ `  ( A  x.  a ) ) ) )
7445, 73syl 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
) ) ) )
7574eqcomd 2462 . . . . . . . . . . . 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 6288 . . . . . . . . . . 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 2499 . . . . . . . . . 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 5852 . . . . . . . . 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 11225 . . . . . . . . . . . . 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 11984 . . . . . . . . . . 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 9611 . . . . . . . . . 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 11984 . . . . . . . . . . 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 9611 . . . . . . . . . 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 13369 . . . . . . . . 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 2496 . . . . . . . 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 4449 . . . . . . 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 207 . . . . . 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 617 . . . . 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 6278 . . . . . . . . 9  |-  ( x  =  ( b  -  a )  ->  ( A  x.  x )  =  ( A  x.  ( b  -  a
) ) )
9190oveq1d 6285 . . . . . . . 8  |-  ( x  =  ( b  -  a )  ->  (
( A  x.  x
)  -  y )  =  ( ( A  x.  ( b  -  a ) )  -  y ) )
9291fveq2d 5852 . . . . . . 7  |-  ( x  =  ( b  -  a )  ->  ( abs `  ( ( A  x.  x )  -  y ) )  =  ( abs `  (
( A  x.  (
b  -  a ) )  -  y ) ) )
9392breq1d 4449 . . . . . 6  |-  ( x  =  ( b  -  a )  ->  (
( abs `  (
( A  x.  x
)  -  y ) )  <  ( 1  /  B )  <->  ( abs `  ( ( A  x.  ( b  -  a
) )  -  y
) )  <  (
1  /  B ) ) )
94 oveq2 6278 . . . . . . . 8  |-  ( y  =  ( ( |_
`  ( A  x.  b ) )  -  ( |_ `  ( A  x.  a ) ) )  ->  ( ( A  x.  ( b  -  a ) )  -  y )  =  ( ( A  x.  ( b  -  a
) )  -  (
( |_ `  ( A  x.  b )
)  -  ( |_
`  ( A  x.  a ) ) ) ) )
9594fveq2d 5852 . . . . . . 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 4449 . . . . . 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 3218 . . . . 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 1226 . . . 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 432 . . 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 2953 . 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 184    /\ wa 367    = wceq 1398    e. wcel 1823   E.wrex 2805   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   RRcr 9480   0cc0 9481   1c1 9482    x. cmul 9486    < clt 9617    <_ cle 9618    - cmin 9796    / cdiv 10202   NNcn 10531   NN0cn0 10791   ZZcz 10860   ZZ>=cuz 11082   RR+crp 11221   ...cfz 11675   |_cfl 11908    mod cmo 11978   abscabs 13152
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-sup 7893  df-card 8311  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11083  df-rp 11222  df-ico 11538  df-fz 11676  df-fl 11910  df-mod 11979  df-seq 12093  df-exp 12152  df-hash 12391  df-cj 13017  df-re 13018  df-im 13019  df-sqrt 13153  df-abs 13154
This theorem is referenced by:  irrapxlem4  31003
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