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Theorem irrapxlem1 26822
Description: Lemma for irrapx1 26828. Divides the unit interval into  B half-open sections and using the pigeonhole principle fphpdo 26815 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 11082 . . . 4  |-  ( 0 ... B )  C_  ( ZZ>= `  0 )
2 uzssz 10494 . . . . 5  |-  ( ZZ>= ` 
0 )  C_  ZZ
3 zssre 10278 . . . . 5  |-  ZZ  C_  RR
42, 3sstri 3349 . . . 4  |-  ( ZZ>= ` 
0 )  C_  RR
51, 4sstri 3349 . . 3  |-  ( 0 ... B )  C_  RR
65a1i 11 . 2  |-  ( ( A  e.  RR+  /\  B  e.  NN )  ->  (
0 ... B )  C_  RR )
7 ovex 6097 . . 3  |-  ( 0 ... ( B  - 
1 ) )  e. 
_V
87a1i 11 . 2  |-  ( ( A  e.  RR+  /\  B  e.  NN )  ->  (
0 ... ( B  - 
1 ) )  e. 
_V )
9 nnm1nn0 10250 . . . . 5  |-  ( B  e.  NN  ->  ( B  -  1 )  e.  NN0 )
109adantl 453 . . . 4  |-  ( ( A  e.  RR+  /\  B  e.  NN )  ->  ( B  -  1 )  e.  NN0 )
11 nn0uz 10509 . . . 4  |-  NN0  =  ( ZZ>= `  0 )
1210, 11syl6eleq 2525 . . 3  |-  ( ( A  e.  RR+  /\  B  e.  NN )  ->  ( B  -  1 )  e.  ( ZZ>= `  0
) )
13 nnz 10292 . . . 4  |-  ( B  e.  NN  ->  B  e.  ZZ )
1413adantl 453 . . 3  |-  ( ( A  e.  RR+  /\  B  e.  NN )  ->  B  e.  ZZ )
15 nnre 9996 . . . . 5  |-  ( B  e.  NN  ->  B  e.  RR )
1615adantl 453 . . . 4  |-  ( ( A  e.  RR+  /\  B  e.  NN )  ->  B  e.  RR )
1716ltm1d 9932 . . 3  |-  ( ( A  e.  RR+  /\  B  e.  NN )  ->  ( B  -  1 )  <  B )
18 fzsdom2 11681 . . 3  |-  ( ( ( ( B  - 
1 )  e.  (
ZZ>= `  0 )  /\  B  e.  ZZ )  /\  ( B  -  1 )  <  B )  ->  ( 0 ... ( B  -  1 ) )  ~<  (
0 ... B ) )
1912, 14, 17, 18syl21anc 1183 . 2  |-  ( ( A  e.  RR+  /\  B  e.  NN )  ->  (
0 ... ( B  - 
1 ) )  ~< 
( 0 ... B
) )
2015ad2antlr 708 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  B  e.  RR )
21 rpre 10607 . . . . . . . . 9  |-  ( A  e.  RR+  ->  A  e.  RR )
2221ad2antrr 707 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  A  e.  RR )
23 elfzelz 11048 . . . . . . . . . 10  |-  ( a  e.  ( 0 ... B )  ->  a  e.  ZZ )
2423zred 10364 . . . . . . . . 9  |-  ( a  e.  ( 0 ... B )  ->  a  e.  RR )
2524adantl 453 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  a  e.  RR )
2622, 25remulcld 9105 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( A  x.  a )  e.  RR )
27 1rp 10605 . . . . . . 7  |-  1  e.  RR+
28 modcl 11241 . . . . . . 7  |-  ( ( ( A  x.  a
)  e.  RR  /\  1  e.  RR+ )  -> 
( ( A  x.  a )  mod  1
)  e.  RR )
2926, 27, 28sylancl 644 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( ( A  x.  a )  mod  1 )  e.  RR )
3020, 29remulcld 9105 . . . . 5  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( B  x.  ( ( A  x.  a )  mod  1
) )  e.  RR )
3130flcld 11195 . . . 4  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( |_ `  ( B  x.  (
( A  x.  a
)  mod  1 ) ) )  e.  ZZ )
3220recnd 9103 . . . . . . . . 9  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  B  e.  CC )
3332mul01d 9254 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( B  x.  0 )  =  0 )
34 modge0 11245 . . . . . . . . . 10  |-  ( ( ( A  x.  a
)  e.  RR  /\  1  e.  RR+ )  -> 
0  <_  ( ( A  x.  a )  mod  1 ) )
3526, 27, 34sylancl 644 . . . . . . . . 9  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  0  <_  ( ( A  x.  a
)  mod  1 ) )
36 0re 9080 . . . . . . . . . . 11  |-  0  e.  RR
3736a1i 11 . . . . . . . . . 10  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  0  e.  RR )
38 nngt0 10018 . . . . . . . . . . 11  |-  ( B  e.  NN  ->  0  <  B )
3938ad2antlr 708 . . . . . . . . . 10  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  0  <  B )
40 lemul2 9852 . . . . . . . . . 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 1188 . . . . . . . . 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 202 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( B  x.  0 )  <_  ( B  x.  ( ( A  x.  a )  mod  1 ) ) )
4333, 42eqbrtrrd 4226 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  0  <_  ( B  x.  ( ( A  x.  a )  mod  1 ) ) )
4437, 30lenltd 9208 . . . . . . 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 202 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  -.  ( B  x.  ( ( A  x.  a )  mod  1 ) )  <  0 )
46 0z 10282 . . . . . . 7  |-  0  e.  ZZ
47 fllt 11203 . . . . . . 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 644 . . . . . 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 292 . . . . 5  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  -.  ( |_ `  ( B  x.  ( ( A  x.  a )  mod  1
) ) )  <  0 )
5031zred 10364 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( |_ `  ( B  x.  (
( A  x.  a
)  mod  1 ) ) )  e.  RR )
5137, 50lenltd 9208 . . . . 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 224 . . . 4  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  0  <_  ( |_ `  ( B  x.  ( ( A  x.  a )  mod  1 ) ) ) )
53 elnn0z 10283 . . . 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 646 . . 3  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( |_ `  ( B  x.  (
( A  x.  a
)  mod  1 ) ) )  e.  NN0 )
559ad2antlr 708 . . 3  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( B  -  1 )  e. 
NN0 )
56 flle 11196 . . . . . . 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 16 . . . . . 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 11246 . . . . . . . . 9  |-  ( ( ( A  x.  a
)  e.  RR  /\  1  e.  RR+ )  -> 
( ( A  x.  a )  mod  1
)  <  1 )
5926, 27, 58sylancl 644 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( ( A  x.  a )  mod  1 )  <  1
)
60 1re 9079 . . . . . . . . . 10  |-  1  e.  RR
6160a1i 11 . . . . . . . . 9  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  1  e.  RR )
62 ltmul2 9850 . . . . . . . . 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 1188 . . . . . . . 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 202 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( B  x.  ( ( A  x.  a )  mod  1
) )  <  ( B  x.  1 ) )
6532mulid1d 9094 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( B  x.  1 )  =  B )
6664, 65breqtrd 4228 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( B  x.  ( ( A  x.  a )  mod  1
) )  <  B
)
6750, 30, 20, 57, 66lelttrd 9217 . . . . 5  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( |_ `  ( B  x.  (
( A  x.  a
)  mod  1 ) ) )  <  B
)
68 nncn 9997 . . . . . . 7  |-  ( B  e.  NN  ->  B  e.  CC )
69 ax-1cn 9037 . . . . . . 7  |-  1  e.  CC
70 npcan 9303 . . . . . . 7  |-  ( ( B  e.  CC  /\  1  e.  CC )  ->  ( ( B  - 
1 )  +  1 )  =  B )
7168, 69, 70sylancl 644 . . . . . 6  |-  ( B  e.  NN  ->  (
( B  -  1 )  +  1 )  =  B )
7271ad2antlr 708 . . . . 5  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( ( B  -  1 )  +  1 )  =  B )
7367, 72breqtrrd 4230 . . . 4  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( |_ `  ( B  x.  (
( A  x.  a
)  mod  1 ) ) )  <  (
( B  -  1 )  +  1 ) )
7413ad2antlr 708 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  B  e.  ZZ )
75 1z 10300 . . . . . 6  |-  1  e.  ZZ
76 zsubcl 10308 . . . . . 6  |-  ( ( B  e.  ZZ  /\  1  e.  ZZ )  ->  ( B  -  1 )  e.  ZZ )
7774, 75, 76sylancl 644 . . . . 5  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( B  -  1 )  e.  ZZ )
78 zleltp1 10315 . . . . 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 643 . . . 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 224 . . 3  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( |_ `  ( B  x.  (
( A  x.  a
)  mod  1 ) ) )  <_  ( B  -  1 ) )
81 elfz2nn0 11071 . . 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 1138 . 2  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( |_ `  ( B  x.  (
( A  x.  a
)  mod  1 ) ) )  e.  ( 0 ... ( B  -  1 ) ) )
83 oveq2 6080 . . . . 5  |-  ( a  =  x  ->  ( A  x.  a )  =  ( A  x.  x ) )
8483oveq1d 6087 . . . 4  |-  ( a  =  x  ->  (
( A  x.  a
)  mod  1 )  =  ( ( A  x.  x )  mod  1 ) )
8584oveq2d 6088 . . 3  |-  ( a  =  x  ->  ( B  x.  ( ( A  x.  a )  mod  1 ) )  =  ( B  x.  (
( A  x.  x
)  mod  1 ) ) )
8685fveq2d 5723 . 2  |-  ( a  =  x  ->  ( |_ `  ( B  x.  ( ( A  x.  a )  mod  1
) ) )  =  ( |_ `  ( B  x.  ( ( A  x.  x )  mod  1 ) ) ) )
87 oveq2 6080 . . . . 5  |-  ( a  =  y  ->  ( A  x.  a )  =  ( A  x.  y ) )
8887oveq1d 6087 . . . 4  |-  ( a  =  y  ->  (
( A  x.  a
)  mod  1 )  =  ( ( A  x.  y )  mod  1 ) )
8988oveq2d 6088 . . 3  |-  ( a  =  y  ->  ( B  x.  ( ( A  x.  a )  mod  1 ) )  =  ( B  x.  (
( A  x.  y
)  mod  1 ) ) )
9089fveq2d 5723 . 2  |-  ( a  =  y  ->  ( |_ `  ( B  x.  ( ( A  x.  a )  mod  1
) ) )  =  ( |_ `  ( B  x.  ( ( A  x.  y )  mod  1 ) ) ) )
916, 8, 19, 82, 86, 90fphpdo 26815 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 177    /\ wa 359    = wceq 1652    e. wcel 1725   E.wrex 2698   _Vcvv 2948    C_ wss 3312   class class class wbr 4204   ` cfv 5445  (class class class)co 6072    ~< csdm 7099   CCcc 8977   RRcr 8978   0cc0 8979   1c1 8980    + caddc 8982    x. cmul 8984    < clt 9109    <_ cle 9110    - cmin 9280   NNcn 9989   NN0cn0 10210   ZZcz 10271   ZZ>=cuz 10477   RR+crp 10601   ...cfz 11032   |_cfl 11189    mod cmo 11238
This theorem is referenced by:  irrapxlem2  26823
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692  ax-cnex 9035  ax-resscn 9036  ax-1cn 9037  ax-icn 9038  ax-addcl 9039  ax-addrcl 9040  ax-mulcl 9041  ax-mulrcl 9042  ax-mulcom 9043  ax-addass 9044  ax-mulass 9045  ax-distr 9046  ax-i2m1 9047  ax-1ne0 9048  ax-1rid 9049  ax-rnegex 9050  ax-rrecex 9051  ax-cnre 9052  ax-pre-lttri 9053  ax-pre-lttrn 9054  ax-pre-ltadd 9055  ax-pre-mulgt0 9056  ax-pre-sup 9057
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4837  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-1st 6340  df-2nd 6341  df-riota 6540  df-recs 6624  df-rdg 6659  df-1o 6715  df-oadd 6719  df-er 6896  df-en 7101  df-dom 7102  df-sdom 7103  df-fin 7104  df-sup 7437  df-card 7815  df-pnf 9111  df-mnf 9112  df-xr 9113  df-ltxr 9114  df-le 9115  df-sub 9282  df-neg 9283  df-div 9667  df-nn 9990  df-n0 10211  df-z 10272  df-uz 10478  df-rp 10602  df-fz 11033  df-fl 11190  df-mod 11239  df-hash 11607
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