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Theorem irrapxlem1 30589
Description: Lemma for irrapx1 30595. Divides the unit interval into  B half-open sections and using the pigeonhole principle fphpdo 30582 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 11725 . . . 4  |-  ( 0 ... B )  C_  ( ZZ>= `  0 )
2 uzssz 11102 . . . . 5  |-  ( ZZ>= ` 
0 )  C_  ZZ
3 zssre 10872 . . . . 5  |-  ZZ  C_  RR
42, 3sstri 3513 . . . 4  |-  ( ZZ>= ` 
0 )  C_  RR
51, 4sstri 3513 . . 3  |-  ( 0 ... B )  C_  RR
65a1i 11 . 2  |-  ( ( A  e.  RR+  /\  B  e.  NN )  ->  (
0 ... B )  C_  RR )
7 ovex 6310 . . 3  |-  ( 0 ... ( B  - 
1 ) )  e. 
_V
87a1i 11 . 2  |-  ( ( A  e.  RR+  /\  B  e.  NN )  ->  (
0 ... ( B  - 
1 ) )  e. 
_V )
9 nnm1nn0 10838 . . . . 5  |-  ( B  e.  NN  ->  ( B  -  1 )  e.  NN0 )
109adantl 466 . . . 4  |-  ( ( A  e.  RR+  /\  B  e.  NN )  ->  ( B  -  1 )  e.  NN0 )
11 nn0uz 11117 . . . 4  |-  NN0  =  ( ZZ>= `  0 )
1210, 11syl6eleq 2565 . . 3  |-  ( ( A  e.  RR+  /\  B  e.  NN )  ->  ( B  -  1 )  e.  ( ZZ>= `  0
) )
13 nnz 10887 . . . 4  |-  ( B  e.  NN  ->  B  e.  ZZ )
1413adantl 466 . . 3  |-  ( ( A  e.  RR+  /\  B  e.  NN )  ->  B  e.  ZZ )
15 nnre 10544 . . . . 5  |-  ( B  e.  NN  ->  B  e.  RR )
1615adantl 466 . . . 4  |-  ( ( A  e.  RR+  /\  B  e.  NN )  ->  B  e.  RR )
1716ltm1d 10479 . . 3  |-  ( ( A  e.  RR+  /\  B  e.  NN )  ->  ( B  -  1 )  <  B )
18 fzsdom2 12452 . . 3  |-  ( ( ( ( B  - 
1 )  e.  (
ZZ>= `  0 )  /\  B  e.  ZZ )  /\  ( B  -  1 )  <  B )  ->  ( 0 ... ( B  -  1 ) )  ~<  (
0 ... B ) )
1912, 14, 17, 18syl21anc 1227 . 2  |-  ( ( A  e.  RR+  /\  B  e.  NN )  ->  (
0 ... ( B  - 
1 ) )  ~< 
( 0 ... B
) )
2015ad2antlr 726 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  B  e.  RR )
21 rpre 11227 . . . . . . . . 9  |-  ( A  e.  RR+  ->  A  e.  RR )
2221ad2antrr 725 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  A  e.  RR )
23 elfzelz 11689 . . . . . . . . . 10  |-  ( a  e.  ( 0 ... B )  ->  a  e.  ZZ )
2423zred 10967 . . . . . . . . 9  |-  ( a  e.  ( 0 ... B )  ->  a  e.  RR )
2524adantl 466 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  a  e.  RR )
2622, 25remulcld 9625 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( A  x.  a )  e.  RR )
27 1rp 11225 . . . . . . 7  |-  1  e.  RR+
28 modcl 11969 . . . . . . 7  |-  ( ( ( A  x.  a
)  e.  RR  /\  1  e.  RR+ )  -> 
( ( A  x.  a )  mod  1
)  e.  RR )
2926, 27, 28sylancl 662 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( ( A  x.  a )  mod  1 )  e.  RR )
3020, 29remulcld 9625 . . . . 5  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( B  x.  ( ( A  x.  a )  mod  1
) )  e.  RR )
3130flcld 11904 . . . 4  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( |_ `  ( B  x.  (
( A  x.  a
)  mod  1 ) ) )  e.  ZZ )
3220recnd 9623 . . . . . . . . 9  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  B  e.  CC )
3332mul01d 9779 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( B  x.  0 )  =  0 )
34 modge0 11974 . . . . . . . . . 10  |-  ( ( ( A  x.  a
)  e.  RR  /\  1  e.  RR+ )  -> 
0  <_  ( ( A  x.  a )  mod  1 ) )
3526, 27, 34sylancl 662 . . . . . . . . 9  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  0  <_  ( ( A  x.  a
)  mod  1 ) )
36 0re 9597 . . . . . . . . . . 11  |-  0  e.  RR
3736a1i 11 . . . . . . . . . 10  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  0  e.  RR )
38 nngt0 10566 . . . . . . . . . . 11  |-  ( B  e.  NN  ->  0  <  B )
3938ad2antlr 726 . . . . . . . . . 10  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  0  <  B )
40 lemul2 10396 . . . . . . . . . 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 1232 . . . . . . . . 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 210 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( B  x.  0 )  <_  ( B  x.  ( ( A  x.  a )  mod  1 ) ) )
4333, 42eqbrtrrd 4469 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  0  <_  ( B  x.  ( ( A  x.  a )  mod  1 ) ) )
4437, 30lenltd 9731 . . . . . . 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 210 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  -.  ( B  x.  ( ( A  x.  a )  mod  1 ) )  <  0 )
46 0z 10876 . . . . . . 7  |-  0  e.  ZZ
47 fllt 11912 . . . . . . 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 662 . . . . . 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 300 . . . . 5  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  -.  ( |_ `  ( B  x.  ( ( A  x.  a )  mod  1
) ) )  <  0 )
5031zred 10967 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( |_ `  ( B  x.  (
( A  x.  a
)  mod  1 ) ) )  e.  RR )
5137, 50lenltd 9731 . . . . 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 232 . . . 4  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  0  <_  ( |_ `  ( B  x.  ( ( A  x.  a )  mod  1 ) ) ) )
53 elnn0z 10878 . . . 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 664 . . 3  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( |_ `  ( B  x.  (
( A  x.  a
)  mod  1 ) ) )  e.  NN0 )
559ad2antlr 726 . . 3  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( B  -  1 )  e. 
NN0 )
56 flle 11905 . . . . . . 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 11975 . . . . . . . . 9  |-  ( ( ( A  x.  a
)  e.  RR  /\  1  e.  RR+ )  -> 
( ( A  x.  a )  mod  1
)  <  1 )
5926, 27, 58sylancl 662 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( ( A  x.  a )  mod  1 )  <  1
)
60 1re 9596 . . . . . . . . . 10  |-  1  e.  RR
6160a1i 11 . . . . . . . . 9  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  1  e.  RR )
62 ltmul2 10394 . . . . . . . . 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 1232 . . . . . . . 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 210 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( B  x.  ( ( A  x.  a )  mod  1
) )  <  ( B  x.  1 ) )
6532mulid1d 9614 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( B  x.  1 )  =  B )
6664, 65breqtrd 4471 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( B  x.  ( ( A  x.  a )  mod  1
) )  <  B
)
6750, 30, 20, 57, 66lelttrd 9740 . . . . 5  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( |_ `  ( B  x.  (
( A  x.  a
)  mod  1 ) ) )  <  B
)
68 nncn 10545 . . . . . . 7  |-  ( B  e.  NN  ->  B  e.  CC )
69 ax-1cn 9551 . . . . . . 7  |-  1  e.  CC
70 npcan 9830 . . . . . . 7  |-  ( ( B  e.  CC  /\  1  e.  CC )  ->  ( ( B  - 
1 )  +  1 )  =  B )
7168, 69, 70sylancl 662 . . . . . 6  |-  ( B  e.  NN  ->  (
( B  -  1 )  +  1 )  =  B )
7271ad2antlr 726 . . . . 5  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( ( B  -  1 )  +  1 )  =  B )
7367, 72breqtrrd 4473 . . . 4  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( |_ `  ( B  x.  (
( A  x.  a
)  mod  1 ) ) )  <  (
( B  -  1 )  +  1 ) )
7413ad2antlr 726 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  B  e.  ZZ )
75 1z 10895 . . . . . 6  |-  1  e.  ZZ
76 zsubcl 10906 . . . . . 6  |-  ( ( B  e.  ZZ  /\  1  e.  ZZ )  ->  ( B  -  1 )  e.  ZZ )
7774, 75, 76sylancl 662 . . . . 5  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( B  -  1 )  e.  ZZ )
78 zleltp1 10914 . . . . 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 661 . . . 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 232 . . 3  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( |_ `  ( B  x.  (
( A  x.  a
)  mod  1 ) ) )  <_  ( B  -  1 ) )
81 elfz2nn0 11769 . . 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 1180 . 2  |-  ( ( ( A  e.  RR+  /\  B  e.  NN )  /\  a  e.  ( 0 ... B ) )  ->  ( |_ `  ( B  x.  (
( A  x.  a
)  mod  1 ) ) )  e.  ( 0 ... ( B  -  1 ) ) )
83 oveq2 6293 . . . . 5  |-  ( a  =  x  ->  ( A  x.  a )  =  ( A  x.  x ) )
8483oveq1d 6300 . . . 4  |-  ( a  =  x  ->  (
( A  x.  a
)  mod  1 )  =  ( ( A  x.  x )  mod  1 ) )
8584oveq2d 6301 . . 3  |-  ( a  =  x  ->  ( B  x.  ( ( A  x.  a )  mod  1 ) )  =  ( B  x.  (
( A  x.  x
)  mod  1 ) ) )
8685fveq2d 5870 . 2  |-  ( a  =  x  ->  ( |_ `  ( B  x.  ( ( A  x.  a )  mod  1
) ) )  =  ( |_ `  ( B  x.  ( ( A  x.  x )  mod  1 ) ) ) )
87 oveq2 6293 . . . . 5  |-  ( a  =  y  ->  ( A  x.  a )  =  ( A  x.  y ) )
8887oveq1d 6300 . . . 4  |-  ( a  =  y  ->  (
( A  x.  a
)  mod  1 )  =  ( ( A  x.  y )  mod  1 ) )
8988oveq2d 6301 . . 3  |-  ( a  =  y  ->  ( B  x.  ( ( A  x.  a )  mod  1 ) )  =  ( B  x.  (
( A  x.  y
)  mod  1 ) ) )
9089fveq2d 5870 . 2  |-  ( a  =  y  ->  ( |_ `  ( B  x.  ( ( A  x.  a )  mod  1
) ) )  =  ( |_ `  ( B  x.  ( ( A  x.  y )  mod  1 ) ) ) )
916, 8, 19, 82, 86, 90fphpdo 30582 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   E.wrex 2815   _Vcvv 3113    C_ wss 3476   class class class wbr 4447   ` cfv 5588  (class class class)co 6285    ~< csdm 7516   CCcc 9491   RRcr 9492   0cc0 9493   1c1 9494    + caddc 9496    x. cmul 9498    < clt 9629    <_ cle 9630    - cmin 9806   NNcn 10537   NN0cn0 10796   ZZcz 10865   ZZ>=cuz 11083   RR+crp 11221   ...cfz 11673   |_cfl 11896    mod cmo 11965
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570  ax-pre-sup 9571
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7043  df-rdg 7077  df-1o 7131  df-oadd 7135  df-er 7312  df-en 7518  df-dom 7519  df-sdom 7520  df-fin 7521  df-sup 7902  df-card 8321  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-div 10208  df-nn 10538  df-n0 10797  df-z 10866  df-uz 11084  df-rp 11222  df-fz 11674  df-fl 11898  df-mod 11966  df-hash 12375
This theorem is referenced by:  irrapxlem2  30590
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