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Theorem irrapxlem1 26239
Description: Lemma for irrapx1 26245. Divides the unit interval into  B half-open sections and using the pigeonhole principle fphpdo 26232 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
StepHypRef Expression
1 fzssuz 10763 . . . 4  |-  ( 0 ... B )  C_  ( ZZ>= `  0 )
2 uzssz 10179 . . . . 5  |-  ( ZZ>= ` 
0 )  C_  ZZ
3 zssre 9963 . . . . 5  |-  ZZ  C_  RR
42, 3sstri 3130 . . . 4  |-  ( ZZ>= ` 
0 )  C_  RR
51, 4sstri 3130 . . 3  |-  ( 0 ... B )  C_  RR
65a1i 12 . 2  |-  ( ( A  e.  RR+  /\  B  e.  NN )  ->  (
0 ... B )  C_  RR )
7 ovex 5782 . . 3  |-  ( 0 ... ( B  - 
1 ) )  e. 
_V
87a1i 12 . 2  |-  ( ( A  e.  RR+  /\  B  e.  NN )  ->  (
0 ... ( B  - 
1 ) )  e. 
_V )
9 nnm1nn0 9937 . . . . 5  |-  ( B  e.  NN  ->  ( B  -  1 )  e.  NN0 )
109adantl 454 . . . 4  |-  ( ( A  e.  RR+  /\  B  e.  NN )  ->  ( B  -  1 )  e.  NN0 )
11 nn0uz 10194 . . . 4  |-  NN0  =  ( ZZ>= `  0 )
1210, 11syl6eleq 2346 . . 3  |-  ( ( A  e.  RR+  /\  B  e.  NN )  ->  ( B  -  1 )  e.  ( ZZ>= `  0
) )
13 nnz 9977 . . . 4  |-  ( B  e.  NN  ->  B  e.  ZZ )
1413adantl 454 . . 3  |-  ( ( A  e.  RR+  /\  B  e.  NN )  ->  B  e.  ZZ )
15 nnre 9686 . . . . 5  |-  ( B  e.  NN  ->  B  e.  RR )
1615adantl 454 . . . 4  |-  ( ( A  e.  RR+  /\  B  e.  NN )  ->  B  e.  RR )
1716ltm1d 9622 . . 3  |-  ( ( A  e.  RR+  /\  B  e.  NN )  ->  ( B  -  1 )  <  B )
18 fzsdom2 11312 . . 3  |-  ( ( ( ( B  - 
1 )  e.  (
ZZ>= `  0 )  /\  B  e.  ZZ )  /\  ( B  -  1 )  <  B )  ->  ( 0 ... ( B  -  1 ) )  ~<  (
0 ... B ) )
1912, 14, 17, 18syl21anc 1186 . 2  |-  ( ( A  e.  RR+  /\  B  e.  NN )  ->  (
0 ... ( B  - 
1 ) )  ~< 
( 0 ... B
) )
2015ad2antlr 710 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  B  e.  RR )
21 rpre 10292 . . . . . . . . 9  |-  ( A  e.  RR+  ->  A  e.  RR )
2221ad2antrr 709 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  A  e.  RR )
23 elfzelz 10729 . . . . . . . . . 10  |-  ( a  e.  ( 0 ... B )  ->  a  e.  ZZ )
2423zred 10049 . . . . . . . . 9  |-  ( a  e.  ( 0 ... B )  ->  a  e.  RR )
2524adantl 454 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  a  e.  RR )
2622, 25remulcld 8796 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( A  x.  a )  e.  RR )
27 1rp 10290 . . . . . . 7  |-  1  e.  RR+
28 modcl 10907 . . . . . . 7  |-  ( ( ( A  x.  a
)  e.  RR  /\  1  e.  RR+ )  -> 
( ( A  x.  a )  mod  1
)  e.  RR )
2926, 27, 28sylancl 646 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( ( A  x.  a )  mod  1 )  e.  RR )
3020, 29remulcld 8796 . . . . 5  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( B  x.  ( ( A  x.  a )  mod  1
) )  e.  RR )
3130flcld 10861 . . . 4  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( |_ `  ( B  x.  (
( A  x.  a
)  mod  1 ) ) )  e.  ZZ )
3220recnd 8794 . . . . . . . . 9  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  B  e.  CC )
3332mul01d 8944 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( B  x.  0 )  =  0 )
34 modge0 10911 . . . . . . . . . 10  |-  ( ( ( A  x.  a
)  e.  RR  /\  1  e.  RR+ )  -> 
0  <_  ( ( A  x.  a )  mod  1 ) )
3526, 27, 34sylancl 646 . . . . . . . . 9  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  0  <_  ( ( A  x.  a
)  mod  1 ) )
36 0re 8771 . . . . . . . . . . 11  |-  0  e.  RR
3736a1i 12 . . . . . . . . . 10  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  0  e.  RR )
38 nngt0 9708 . . . . . . . . . . 11  |-  ( B  e.  NN  ->  0  <  B )
3938ad2antlr 710 . . . . . . . . . 10  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  0  <  B )
40 lemul2 9542 . . . . . . . . . 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 1191 . . . . . . . . 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 203 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( B  x.  0 )  <_  ( B  x.  ( ( A  x.  a )  mod  1 ) ) )
4333, 42eqbrtrrd 3985 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  0  <_  ( B  x.  ( ( A  x.  a )  mod  1 ) ) )
4437, 30lenltd 8898 . . . . . . 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 203 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  -.  ( B  x.  ( ( A  x.  a )  mod  1 ) )  <  0 )
46 0z 9967 . . . . . . 7  |-  0  e.  ZZ
47 fllt 10869 . . . . . . 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 646 . . . . . 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 293 . . . . 5  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  -.  ( |_ `  ( B  x.  ( ( A  x.  a )  mod  1
) ) )  <  0 )
5031zred 10049 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( |_ `  ( B  x.  (
( A  x.  a
)  mod  1 ) ) )  e.  RR )
5137, 50lenltd 8898 . . . . 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 225 . . . 4  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  0  <_  ( |_ `  ( B  x.  ( ( A  x.  a )  mod  1 ) ) ) )
53 elnn0z 9968 . . . 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 648 . . 3  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( |_ `  ( B  x.  (
( A  x.  a
)  mod  1 ) ) )  e.  NN0 )
559ad2antlr 710 . . 3  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( B  -  1 )  e. 
NN0 )
56 flle 10862 . . . . . . 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 17 . . . . . 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 10912 . . . . . . . . 9  |-  ( ( ( A  x.  a
)  e.  RR  /\  1  e.  RR+ )  -> 
( ( A  x.  a )  mod  1
)  <  1 )
5926, 27, 58sylancl 646 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( ( A  x.  a )  mod  1 )  <  1
)
60 1re 8770 . . . . . . . . . 10  |-  1  e.  RR
6160a1i 12 . . . . . . . . 9  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  1  e.  RR )
62 ltmul2 9540 . . . . . . . . 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 1191 . . . . . . . 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 203 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( B  x.  ( ( A  x.  a )  mod  1
) )  <  ( B  x.  1 ) )
6532mulid1d 8785 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( B  x.  1 )  =  B )
6664, 65breqtrd 3987 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( B  x.  ( ( A  x.  a )  mod  1
) )  <  B
)
6750, 30, 20, 57, 66lelttrd 8907 . . . . 5  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( |_ `  ( B  x.  (
( A  x.  a
)  mod  1 ) ) )  <  B
)
68 nncn 9687 . . . . . . 7  |-  ( B  e.  NN  ->  B  e.  CC )
69 ax-1cn 8728 . . . . . . 7  |-  1  e.  CC
70 npcan 8993 . . . . . . 7  |-  ( ( B  e.  CC  /\  1  e.  CC )  ->  ( ( B  - 
1 )  +  1 )  =  B )
7168, 69, 70sylancl 646 . . . . . 6  |-  ( B  e.  NN  ->  (
( B  -  1 )  +  1 )  =  B )
7271ad2antlr 710 . . . . 5  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( ( B  -  1 )  +  1 )  =  B )
7367, 72breqtrrd 3989 . . . 4  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( |_ `  ( B  x.  (
( A  x.  a
)  mod  1 ) ) )  <  (
( B  -  1 )  +  1 ) )
7413ad2antlr 710 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  B  e.  ZZ )
75 1z 9985 . . . . . 6  |-  1  e.  ZZ
76 zsubcl 9993 . . . . . 6  |-  ( ( B  e.  ZZ  /\  1  e.  ZZ )  ->  ( B  -  1 )  e.  ZZ )
7774, 75, 76sylancl 646 . . . . 5  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( B  -  1 )  e.  ZZ )
78 zleltp1 10000 . . . . 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 645 . . . 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 225 . . 3  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( |_ `  ( B  x.  (
( A  x.  a
)  mod  1 ) ) )  <_  ( B  -  1 ) )
81 elfz2nn0 10752 . . 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 1141 . 2  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( |_ `  ( B  x.  (
( A  x.  a
)  mod  1 ) ) )  e.  ( 0 ... ( B  -  1 ) ) )
83 oveq2 5765 . . . . 5  |-  ( a  =  x  ->  ( A  x.  a )  =  ( A  x.  x ) )
8483oveq1d 5772 . . . 4  |-  ( a  =  x  ->  (
( A  x.  a
)  mod  1 )  =  ( ( A  x.  x )  mod  1 ) )
8584oveq2d 5773 . . 3  |-  ( a  =  x  ->  ( B  x.  ( ( A  x.  a )  mod  1 ) )  =  ( B  x.  (
( A  x.  x
)  mod  1 ) ) )
8685fveq2d 5427 . 2  |-  ( a  =  x  ->  ( |_ `  ( B  x.  ( ( A  x.  a )  mod  1
) ) )  =  ( |_ `  ( B  x.  ( ( A  x.  x )  mod  1 ) ) ) )
87 oveq2 5765 . . . . 5  |-  ( a  =  y  ->  ( A  x.  a )  =  ( A  x.  y ) )
8887oveq1d 5772 . . . 4  |-  ( a  =  y  ->  (
( A  x.  a
)  mod  1 )  =  ( ( A  x.  y )  mod  1 ) )
8988oveq2d 5773 . . 3  |-  ( a  =  y  ->  ( B  x.  ( ( A  x.  a )  mod  1 ) )  =  ( B  x.  (
( A  x.  y
)  mod  1 ) ) )
9089fveq2d 5427 . 2  |-  ( a  =  y  ->  ( |_ `  ( B  x.  ( ( A  x.  a )  mod  1
) ) )  =  ( |_ `  ( B  x.  ( ( A  x.  y )  mod  1 ) ) ) )
916, 8, 19, 82, 86, 90fphpdo 26232 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 5    -> wi 6    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621   E.wrex 2517   _Vcvv 2740    C_ wss 3094   class class class wbr 3963   ` cfv 4638  (class class class)co 5757    ~< csdm 6795   CCcc 8668   RRcr 8669   0cc0 8670   1c1 8671    + caddc 8673    x. cmul 8675    < clt 8800    <_ cle 8801    - cmin 8970   NNcn 9679   NN0cn0 9897   ZZcz 9956   ZZ>=cuz 10162   RR+crp 10286   ...cfz 10713   |_cfl 10855    mod cmo 10904
This theorem is referenced by:  irrapxlem2  26240
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4071  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449  ax-cnex 8726  ax-resscn 8727  ax-1cn 8728  ax-icn 8729  ax-addcl 8730  ax-addrcl 8731  ax-mulcl 8732  ax-mulrcl 8733  ax-mulcom 8734  ax-addass 8735  ax-mulass 8736  ax-distr 8737  ax-i2m1 8738  ax-1ne0 8739  ax-1rid 8740  ax-rnegex 8741  ax-rrecex 8742  ax-cnre 8743  ax-pre-lttri 8744  ax-pre-lttrn 8745  ax-pre-ltadd 8746  ax-pre-mulgt0 8747  ax-pre-sup 8748
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-int 3804  df-iun 3848  df-br 3964  df-opab 4018  df-mpt 4019  df-tr 4054  df-eprel 4242  df-id 4246  df-po 4251  df-so 4252  df-fr 4289  df-we 4291  df-ord 4332  df-on 4333  df-lim 4334  df-suc 4335  df-om 4594  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-1st 6021  df-2nd 6022  df-iota 6190  df-riota 6237  df-recs 6321  df-rdg 6356  df-1o 6412  df-oadd 6416  df-er 6593  df-en 6797  df-dom 6798  df-sdom 6799  df-fin 6800  df-sup 7127  df-card 7505  df-pnf 8802  df-mnf 8803  df-xr 8804  df-ltxr 8805  df-le 8806  df-sub 8972  df-neg 8973  df-div 9357  df-n 9680  df-n0 9898  df-z 9957  df-uz 10163  df-rp 10287  df-fz 10714  df-fl 10856  df-mod 10905  df-hash 11269
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