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Theorem irrapxlem1 26576
Description: Lemma for irrapx1 26582. Divides the unit interval into  B half-open sections and using the pigeonhole principle fphpdo 26569 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 11025 . . . 4  |-  ( 0 ... B )  C_  ( ZZ>= `  0 )
2 uzssz 10437 . . . . 5  |-  ( ZZ>= ` 
0 )  C_  ZZ
3 zssre 10221 . . . . 5  |-  ZZ  C_  RR
42, 3sstri 3300 . . . 4  |-  ( ZZ>= ` 
0 )  C_  RR
51, 4sstri 3300 . . 3  |-  ( 0 ... B )  C_  RR
65a1i 11 . 2  |-  ( ( A  e.  RR+  /\  B  e.  NN )  ->  (
0 ... B )  C_  RR )
7 ovex 6045 . . 3  |-  ( 0 ... ( B  - 
1 ) )  e. 
_V
87a1i 11 . 2  |-  ( ( A  e.  RR+  /\  B  e.  NN )  ->  (
0 ... ( B  - 
1 ) )  e. 
_V )
9 nnm1nn0 10193 . . . . 5  |-  ( B  e.  NN  ->  ( B  -  1 )  e.  NN0 )
109adantl 453 . . . 4  |-  ( ( A  e.  RR+  /\  B  e.  NN )  ->  ( B  -  1 )  e.  NN0 )
11 nn0uz 10452 . . . 4  |-  NN0  =  ( ZZ>= `  0 )
1210, 11syl6eleq 2477 . . 3  |-  ( ( A  e.  RR+  /\  B  e.  NN )  ->  ( B  -  1 )  e.  ( ZZ>= `  0
) )
13 nnz 10235 . . . 4  |-  ( B  e.  NN  ->  B  e.  ZZ )
1413adantl 453 . . 3  |-  ( ( A  e.  RR+  /\  B  e.  NN )  ->  B  e.  ZZ )
15 nnre 9939 . . . . 5  |-  ( B  e.  NN  ->  B  e.  RR )
1615adantl 453 . . . 4  |-  ( ( A  e.  RR+  /\  B  e.  NN )  ->  B  e.  RR )
1716ltm1d 9875 . . 3  |-  ( ( A  e.  RR+  /\  B  e.  NN )  ->  ( B  -  1 )  <  B )
18 fzsdom2 11620 . . 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 10550 . . . . . . . . 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 10991 . . . . . . . . . 10  |-  ( a  e.  ( 0 ... B )  ->  a  e.  ZZ )
2423zred 10307 . . . . . . . . 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 9049 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( A  x.  a )  e.  RR )
27 1rp 10548 . . . . . . 7  |-  1  e.  RR+
28 modcl 11180 . . . . . . 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 9049 . . . . 5  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( B  x.  ( ( A  x.  a )  mod  1
) )  e.  RR )
3130flcld 11134 . . . 4  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( |_ `  ( B  x.  (
( A  x.  a
)  mod  1 ) ) )  e.  ZZ )
3220recnd 9047 . . . . . . . . 9  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  B  e.  CC )
3332mul01d 9197 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( B  x.  0 )  =  0 )
34 modge0 11184 . . . . . . . . . 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 9024 . . . . . . . . . . 11  |-  0  e.  RR
3736a1i 11 . . . . . . . . . 10  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  0  e.  RR )
38 nngt0 9961 . . . . . . . . . . 11  |-  ( B  e.  NN  ->  0  <  B )
3938ad2antlr 708 . . . . . . . . . 10  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  0  <  B )
40 lemul2 9795 . . . . . . . . . 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 4175 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  0  <_  ( B  x.  ( ( A  x.  a )  mod  1 ) ) )
4437, 30lenltd 9151 . . . . . . 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 10225 . . . . . . 7  |-  0  e.  ZZ
47 fllt 11142 . . . . . . 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 10307 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( |_ `  ( B  x.  (
( A  x.  a
)  mod  1 ) ) )  e.  RR )
5137, 50lenltd 9151 . . . . 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 10226 . . . 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 11135 . . . . . . 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 11185 . . . . . . . . 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 9023 . . . . . . . . . 10  |-  1  e.  RR
6160a1i 11 . . . . . . . . 9  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  1  e.  RR )
62 ltmul2 9793 . . . . . . . . 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 9038 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( B  x.  1 )  =  B )
6664, 65breqtrd 4177 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( B  x.  ( ( A  x.  a )  mod  1
) )  <  B
)
6750, 30, 20, 57, 66lelttrd 9160 . . . . 5  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( |_ `  ( B  x.  (
( A  x.  a
)  mod  1 ) ) )  <  B
)
68 nncn 9940 . . . . . . 7  |-  ( B  e.  NN  ->  B  e.  CC )
69 ax-1cn 8981 . . . . . . 7  |-  1  e.  CC
70 npcan 9246 . . . . . . 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 4179 . . . 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 10243 . . . . . 6  |-  1  e.  ZZ
76 zsubcl 10251 . . . . . 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 10258 . . . . 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 11014 . . 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 6028 . . . . 5  |-  ( a  =  x  ->  ( A  x.  a )  =  ( A  x.  x ) )
8483oveq1d 6035 . . . 4  |-  ( a  =  x  ->  (
( A  x.  a
)  mod  1 )  =  ( ( A  x.  x )  mod  1 ) )
8584oveq2d 6036 . . 3  |-  ( a  =  x  ->  ( B  x.  ( ( A  x.  a )  mod  1 ) )  =  ( B  x.  (
( A  x.  x
)  mod  1 ) ) )
8685fveq2d 5672 . 2  |-  ( a  =  x  ->  ( |_ `  ( B  x.  ( ( A  x.  a )  mod  1
) ) )  =  ( |_ `  ( B  x.  ( ( A  x.  x )  mod  1 ) ) ) )
87 oveq2 6028 . . . . 5  |-  ( a  =  y  ->  ( A  x.  a )  =  ( A  x.  y ) )
8887oveq1d 6035 . . . 4  |-  ( a  =  y  ->  (
( A  x.  a
)  mod  1 )  =  ( ( A  x.  y )  mod  1 ) )
8988oveq2d 6036 . . 3  |-  ( a  =  y  ->  ( B  x.  ( ( A  x.  a )  mod  1 ) )  =  ( B  x.  (
( A  x.  y
)  mod  1 ) ) )
9089fveq2d 5672 . 2  |-  ( a  =  y  ->  ( |_ `  ( B  x.  ( ( A  x.  a )  mod  1
) ) )  =  ( |_ `  ( B  x.  ( ( A  x.  y )  mod  1 ) ) ) )
916, 8, 19, 82, 86, 90fphpdo 26569 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 1649    e. wcel 1717   E.wrex 2650   _Vcvv 2899    C_ wss 3263   class class class wbr 4153   ` cfv 5394  (class class class)co 6020    ~< csdm 7044   CCcc 8921   RRcr 8922   0cc0 8923   1c1 8924    + caddc 8926    x. cmul 8928    < clt 9053    <_ cle 9054    - cmin 9223   NNcn 9932   NN0cn0 10153   ZZcz 10214   ZZ>=cuz 10420   RR+crp 10544   ...cfz 10975   |_cfl 11128    mod cmo 11177
This theorem is referenced by:  irrapxlem2  26577
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000  ax-pre-sup 9001
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-1o 6660  df-oadd 6664  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-fin 7049  df-sup 7381  df-card 7759  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-div 9610  df-nn 9933  df-n0 10154  df-z 10215  df-uz 10421  df-rp 10545  df-fz 10976  df-fl 11129  df-mod 11178  df-hash 11546
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