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Theorem irrapxlem1 26918
Description: Lemma for irrapx1 26924. Divides the unit interval into  B half-open sections and using the pigeonhole principle fphpdo 26911 finds two multiples of  A in the same section mod 1. (Contributed by Stefan O'Rear, 12-Sep-2014.)
Assertion
Ref Expression
irrapxlem1  |-  ( ( A  e.  RR+  /\  B  e.  NN )  ->  E. x  e.  ( 0 ... B
) E. y  e.  ( 0 ... B
) ( x  < 
y  /\  ( |_ `  ( B  x.  (
( A  x.  x
)  mod  1 ) ) )  =  ( |_ `  ( B  x.  ( ( A  x.  y )  mod  1 ) ) ) ) )
Distinct variable groups:    x, A, y    x, B, y

Proof of Theorem irrapxlem1
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 fzssuz 10834 . . . 4  |-  ( 0 ... B )  C_  ( ZZ>= `  0 )
2 uzssz 10249 . . . . 5  |-  ( ZZ>= ` 
0 )  C_  ZZ
3 zssre 10033 . . . . 5  |-  ZZ  C_  RR
42, 3sstri 3190 . . . 4  |-  ( ZZ>= ` 
0 )  C_  RR
51, 4sstri 3190 . . 3  |-  ( 0 ... B )  C_  RR
65a1i 10 . 2  |-  ( ( A  e.  RR+  /\  B  e.  NN )  ->  (
0 ... B )  C_  RR )
7 ovex 5885 . . 3  |-  ( 0 ... ( B  - 
1 ) )  e. 
_V
87a1i 10 . 2  |-  ( ( A  e.  RR+  /\  B  e.  NN )  ->  (
0 ... ( B  - 
1 ) )  e. 
_V )
9 nnm1nn0 10007 . . . . 5  |-  ( B  e.  NN  ->  ( B  -  1 )  e.  NN0 )
109adantl 452 . . . 4  |-  ( ( A  e.  RR+  /\  B  e.  NN )  ->  ( B  -  1 )  e.  NN0 )
11 nn0uz 10264 . . . 4  |-  NN0  =  ( ZZ>= `  0 )
1210, 11syl6eleq 2375 . . 3  |-  ( ( A  e.  RR+  /\  B  e.  NN )  ->  ( B  -  1 )  e.  ( ZZ>= `  0
) )
13 nnz 10047 . . . 4  |-  ( B  e.  NN  ->  B  e.  ZZ )
1413adantl 452 . . 3  |-  ( ( A  e.  RR+  /\  B  e.  NN )  ->  B  e.  ZZ )
15 nnre 9755 . . . . 5  |-  ( B  e.  NN  ->  B  e.  RR )
1615adantl 452 . . . 4  |-  ( ( A  e.  RR+  /\  B  e.  NN )  ->  B  e.  RR )
1716ltm1d 9691 . . 3  |-  ( ( A  e.  RR+  /\  B  e.  NN )  ->  ( B  -  1 )  <  B )
18 fzsdom2 11384 . . 3  |-  ( ( ( ( B  - 
1 )  e.  (
ZZ>= `  0 )  /\  B  e.  ZZ )  /\  ( B  -  1 )  <  B )  ->  ( 0 ... ( B  -  1 ) )  ~<  (
0 ... B ) )
1912, 14, 17, 18syl21anc 1181 . 2  |-  ( ( A  e.  RR+  /\  B  e.  NN )  ->  (
0 ... ( B  - 
1 ) )  ~< 
( 0 ... B
) )
2015ad2antlr 707 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  B  e.  RR )
21 rpre 10362 . . . . . . . . 9  |-  ( A  e.  RR+  ->  A  e.  RR )
2221ad2antrr 706 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  A  e.  RR )
23 elfzelz 10800 . . . . . . . . . 10  |-  ( a  e.  ( 0 ... B )  ->  a  e.  ZZ )
2423zred 10119 . . . . . . . . 9  |-  ( a  e.  ( 0 ... B )  ->  a  e.  RR )
2524adantl 452 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  a  e.  RR )
2622, 25remulcld 8865 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( A  x.  a )  e.  RR )
27 1rp 10360 . . . . . . 7  |-  1  e.  RR+
28 modcl 10978 . . . . . . 7  |-  ( ( ( A  x.  a
)  e.  RR  /\  1  e.  RR+ )  -> 
( ( A  x.  a )  mod  1
)  e.  RR )
2926, 27, 28sylancl 643 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( ( A  x.  a )  mod  1 )  e.  RR )
3020, 29remulcld 8865 . . . . 5  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( B  x.  ( ( A  x.  a )  mod  1
) )  e.  RR )
3130flcld 10932 . . . 4  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( |_ `  ( B  x.  (
( A  x.  a
)  mod  1 ) ) )  e.  ZZ )
3220recnd 8863 . . . . . . . . 9  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  B  e.  CC )
3332mul01d 9013 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( B  x.  0 )  =  0 )
34 modge0 10982 . . . . . . . . . 10  |-  ( ( ( A  x.  a
)  e.  RR  /\  1  e.  RR+ )  -> 
0  <_  ( ( A  x.  a )  mod  1 ) )
3526, 27, 34sylancl 643 . . . . . . . . 9  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  0  <_  ( ( A  x.  a
)  mod  1 ) )
36 0re 8840 . . . . . . . . . . 11  |-  0  e.  RR
3736a1i 10 . . . . . . . . . 10  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  0  e.  RR )
38 nngt0 9777 . . . . . . . . . . 11  |-  ( B  e.  NN  ->  0  <  B )
3938ad2antlr 707 . . . . . . . . . 10  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  0  <  B )
40 lemul2 9611 . . . . . . . . . 10  |-  ( ( 0  e.  RR  /\  ( ( A  x.  a )  mod  1
)  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  ->  ( 0  <_  ( ( A  x.  a )  mod  1 )  <->  ( B  x.  0 )  <_  ( B  x.  ( ( A  x.  a )  mod  1 ) ) ) )
4137, 29, 20, 39, 40syl112anc 1186 . . . . . . . . 9  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( 0  <_  ( ( A  x.  a )  mod  1 )  <->  ( B  x.  0 )  <_  ( B  x.  ( ( A  x.  a )  mod  1 ) ) ) )
4235, 41mpbid 201 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( B  x.  0 )  <_  ( B  x.  ( ( A  x.  a )  mod  1 ) ) )
4333, 42eqbrtrrd 4047 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  0  <_  ( B  x.  ( ( A  x.  a )  mod  1 ) ) )
4437, 30lenltd 8967 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( 0  <_  ( B  x.  ( ( A  x.  a )  mod  1
) )  <->  -.  ( B  x.  ( ( A  x.  a )  mod  1 ) )  <  0 ) )
4543, 44mpbid 201 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  -.  ( B  x.  ( ( A  x.  a )  mod  1 ) )  <  0 )
46 0z 10037 . . . . . . 7  |-  0  e.  ZZ
47 fllt 10940 . . . . . . 7  |-  ( ( ( B  x.  (
( A  x.  a
)  mod  1 ) )  e.  RR  /\  0  e.  ZZ )  ->  ( ( B  x.  ( ( A  x.  a )  mod  1
) )  <  0  <->  ( |_ `  ( B  x.  ( ( A  x.  a )  mod  1 ) ) )  <  0 ) )
4830, 46, 47sylancl 643 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( ( B  x.  ( ( A  x.  a )  mod  1 ) )  <  0  <->  ( |_ `  ( B  x.  (
( A  x.  a
)  mod  1 ) ) )  <  0
) )
4945, 48mtbid 291 . . . . 5  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  -.  ( |_ `  ( B  x.  ( ( A  x.  a )  mod  1
) ) )  <  0 )
5031zred 10119 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( |_ `  ( B  x.  (
( A  x.  a
)  mod  1 ) ) )  e.  RR )
5137, 50lenltd 8967 . . . . 5  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( 0  <_  ( |_ `  ( B  x.  (
( A  x.  a
)  mod  1 ) ) )  <->  -.  ( |_ `  ( B  x.  ( ( A  x.  a )  mod  1
) ) )  <  0 ) )
5249, 51mpbird 223 . . . 4  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  0  <_  ( |_ `  ( B  x.  ( ( A  x.  a )  mod  1 ) ) ) )
53 elnn0z 10038 . . . 4  |-  ( ( |_ `  ( B  x.  ( ( A  x.  a )  mod  1 ) ) )  e.  NN0  <->  ( ( |_
`  ( B  x.  ( ( A  x.  a )  mod  1
) ) )  e.  ZZ  /\  0  <_ 
( |_ `  ( B  x.  ( ( A  x.  a )  mod  1 ) ) ) ) )
5431, 52, 53sylanbrc 645 . . 3  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( |_ `  ( B  x.  (
( A  x.  a
)  mod  1 ) ) )  e.  NN0 )
559ad2antlr 707 . . 3  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( B  -  1 )  e. 
NN0 )
56 flle 10933 . . . . . . 7  |-  ( ( B  x.  ( ( A  x.  a )  mod  1 ) )  e.  RR  ->  ( |_ `  ( B  x.  ( ( A  x.  a )  mod  1
) ) )  <_ 
( B  x.  (
( A  x.  a
)  mod  1 ) ) )
5730, 56syl 15 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( |_ `  ( B  x.  (
( A  x.  a
)  mod  1 ) ) )  <_  ( B  x.  ( ( A  x.  a )  mod  1 ) ) )
58 modlt 10983 . . . . . . . . 9  |-  ( ( ( A  x.  a
)  e.  RR  /\  1  e.  RR+ )  -> 
( ( A  x.  a )  mod  1
)  <  1 )
5926, 27, 58sylancl 643 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( ( A  x.  a )  mod  1 )  <  1
)
60 1re 8839 . . . . . . . . . 10  |-  1  e.  RR
6160a1i 10 . . . . . . . . 9  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  1  e.  RR )
62 ltmul2 9609 . . . . . . . . 9  |-  ( ( ( ( A  x.  a )  mod  1
)  e.  RR  /\  1  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  -> 
( ( ( A  x.  a )  mod  1 )  <  1  <->  ( B  x.  ( ( A  x.  a )  mod  1 ) )  <  ( B  x.  1 ) ) )
6329, 61, 20, 39, 62syl112anc 1186 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( (
( A  x.  a
)  mod  1 )  <  1  <->  ( B  x.  ( ( A  x.  a )  mod  1
) )  <  ( B  x.  1 ) ) )
6459, 63mpbid 201 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( B  x.  ( ( A  x.  a )  mod  1
) )  <  ( B  x.  1 ) )
6532mulid1d 8854 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( B  x.  1 )  =  B )
6664, 65breqtrd 4049 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( B  x.  ( ( A  x.  a )  mod  1
) )  <  B
)
6750, 30, 20, 57, 66lelttrd 8976 . . . . 5  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( |_ `  ( B  x.  (
( A  x.  a
)  mod  1 ) ) )  <  B
)
68 nncn 9756 . . . . . . 7  |-  ( B  e.  NN  ->  B  e.  CC )
69 ax-1cn 8797 . . . . . . 7  |-  1  e.  CC
70 npcan 9062 . . . . . . 7  |-  ( ( B  e.  CC  /\  1  e.  CC )  ->  ( ( B  - 
1 )  +  1 )  =  B )
7168, 69, 70sylancl 643 . . . . . 6  |-  ( B  e.  NN  ->  (
( B  -  1 )  +  1 )  =  B )
7271ad2antlr 707 . . . . 5  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( ( B  -  1 )  +  1 )  =  B )
7367, 72breqtrrd 4051 . . . 4  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( |_ `  ( B  x.  (
( A  x.  a
)  mod  1 ) ) )  <  (
( B  -  1 )  +  1 ) )
7413ad2antlr 707 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  B  e.  ZZ )
75 1z 10055 . . . . . 6  |-  1  e.  ZZ
76 zsubcl 10063 . . . . . 6  |-  ( ( B  e.  ZZ  /\  1  e.  ZZ )  ->  ( B  -  1 )  e.  ZZ )
7774, 75, 76sylancl 643 . . . . 5  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( B  -  1 )  e.  ZZ )
78 zleltp1 10070 . . . . 5  |-  ( ( ( |_ `  ( B  x.  ( ( A  x.  a )  mod  1 ) ) )  e.  ZZ  /\  ( B  -  1 )  e.  ZZ )  -> 
( ( |_ `  ( B  x.  (
( A  x.  a
)  mod  1 ) ) )  <_  ( B  -  1 )  <-> 
( |_ `  ( B  x.  ( ( A  x.  a )  mod  1 ) ) )  <  ( ( B  -  1 )  +  1 ) ) )
7931, 77, 78syl2anc 642 . . . 4  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( ( |_ `  ( B  x.  ( ( A  x.  a )  mod  1
) ) )  <_ 
( B  -  1 )  <->  ( |_ `  ( B  x.  (
( A  x.  a
)  mod  1 ) ) )  <  (
( B  -  1 )  +  1 ) ) )
8073, 79mpbird 223 . . 3  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( |_ `  ( B  x.  (
( A  x.  a
)  mod  1 ) ) )  <_  ( B  -  1 ) )
81 elfz2nn0 10823 . . 3  |-  ( ( |_ `  ( B  x.  ( ( A  x.  a )  mod  1 ) ) )  e.  ( 0 ... ( B  -  1 ) )  <->  ( ( |_ `  ( B  x.  ( ( A  x.  a )  mod  1
) ) )  e. 
NN0  /\  ( B  -  1 )  e. 
NN0  /\  ( |_ `  ( B  x.  (
( A  x.  a
)  mod  1 ) ) )  <_  ( B  -  1 ) ) )
8254, 55, 80, 81syl3anbrc 1136 . 2  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( |_ `  ( B  x.  (
( A  x.  a
)  mod  1 ) ) )  e.  ( 0 ... ( B  -  1 ) ) )
83 oveq2 5868 . . . . 5  |-  ( a  =  x  ->  ( A  x.  a )  =  ( A  x.  x ) )
8483oveq1d 5875 . . . 4  |-  ( a  =  x  ->  (
( A  x.  a
)  mod  1 )  =  ( ( A  x.  x )  mod  1 ) )
8584oveq2d 5876 . . 3  |-  ( a  =  x  ->  ( B  x.  ( ( A  x.  a )  mod  1 ) )  =  ( B  x.  (
( A  x.  x
)  mod  1 ) ) )
8685fveq2d 5531 . 2  |-  ( a  =  x  ->  ( |_ `  ( B  x.  ( ( A  x.  a )  mod  1
) ) )  =  ( |_ `  ( B  x.  ( ( A  x.  x )  mod  1 ) ) ) )
87 oveq2 5868 . . . . 5  |-  ( a  =  y  ->  ( A  x.  a )  =  ( A  x.  y ) )
8887oveq1d 5875 . . . 4  |-  ( a  =  y  ->  (
( A  x.  a
)  mod  1 )  =  ( ( A  x.  y )  mod  1 ) )
8988oveq2d 5876 . . 3  |-  ( a  =  y  ->  ( B  x.  ( ( A  x.  a )  mod  1 ) )  =  ( B  x.  (
( A  x.  y
)  mod  1 ) ) )
9089fveq2d 5531 . 2  |-  ( a  =  y  ->  ( |_ `  ( B  x.  ( ( A  x.  a )  mod  1
) ) )  =  ( |_ `  ( B  x.  ( ( A  x.  y )  mod  1 ) ) ) )
916, 8, 19, 82, 86, 90fphpdo 26911 1  |-  ( ( A  e.  RR+  /\  B  e.  NN )  ->  E. x  e.  ( 0 ... B
) E. y  e.  ( 0 ... B
) ( x  < 
y  /\  ( |_ `  ( B  x.  (
( A  x.  x
)  mod  1 ) ) )  =  ( |_ `  ( B  x.  ( ( A  x.  y )  mod  1 ) ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1625    e. wcel 1686   E.wrex 2546   _Vcvv 2790    C_ wss 3154   class class class wbr 4025   ` cfv 5257  (class class class)co 5860    ~< csdm 6864   CCcc 8737   RRcr 8738   0cc0 8739   1c1 8740    + caddc 8742    x. cmul 8744    < clt 8869    <_ cle 8870    - cmin 9039   NNcn 9748   NN0cn0 9967   ZZcz 10026   ZZ>=cuz 10232   RR+crp 10356   ...cfz 10784   |_cfl 10926    mod cmo 10975
This theorem is referenced by:  irrapxlem2  26919
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-cnex 8795  ax-resscn 8796  ax-1cn 8797  ax-icn 8798  ax-addcl 8799  ax-addrcl 8800  ax-mulcl 8801  ax-mulrcl 8802  ax-mulcom 8803  ax-addass 8804  ax-mulass 8805  ax-distr 8806  ax-i2m1 8807  ax-1ne0 8808  ax-1rid 8809  ax-rnegex 8810  ax-rrecex 8811  ax-cnre 8812  ax-pre-lttri 8813  ax-pre-lttrn 8814  ax-pre-ltadd 8815  ax-pre-mulgt0 8816  ax-pre-sup 8817
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 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-1st 6124  df-2nd 6125  df-riota 6306  df-recs 6390  df-rdg 6425  df-1o 6481  df-oadd 6485  df-er 6662  df-en 6866  df-dom 6867  df-sdom 6868  df-fin 6869  df-sup 7196  df-card 7574  df-pnf 8871  df-mnf 8872  df-xr 8873  df-ltxr 8874  df-le 8875  df-sub 9041  df-neg 9042  df-div 9426  df-nn 9749  df-n0 9968  df-z 10027  df-uz 10233  df-rp 10357  df-fz 10785  df-fl 10927  df-mod 10976  df-hash 11340
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