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Theorem modmul1 11279
Description: Multiplication property of the modulo operation. Note that the multiplier  C must be an integer. (Contributed by NM, 12-Nov-2008.)
Assertion
Ref Expression
modmul1  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  ZZ  /\  D  e.  RR+ )  /\  ( A  mod  D )  =  ( B  mod  D
) )  ->  (
( A  x.  C
)  mod  D )  =  ( ( B  x.  C )  mod 
D ) )

Proof of Theorem modmul1
StepHypRef Expression
1 modval 11252 . . . . . . . 8  |-  ( ( A  e.  RR  /\  D  e.  RR+ )  -> 
( A  mod  D
)  =  ( A  -  ( D  x.  ( |_ `  ( A  /  D ) ) ) ) )
2 modval 11252 . . . . . . . 8  |-  ( ( B  e.  RR  /\  D  e.  RR+ )  -> 
( B  mod  D
)  =  ( B  -  ( D  x.  ( |_ `  ( B  /  D ) ) ) ) )
31, 2eqeqan12d 2451 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  D  e.  RR+ )  /\  ( B  e.  RR  /\  D  e.  RR+ )
)  ->  ( ( A  mod  D )  =  ( B  mod  D
)  <->  ( A  -  ( D  x.  ( |_ `  ( A  /  D ) ) ) )  =  ( B  -  ( D  x.  ( |_ `  ( B  /  D ) ) ) ) ) )
43anandirs 805 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  D  e.  RR+ )  ->  ( ( A  mod  D )  =  ( B  mod  D
)  <->  ( A  -  ( D  x.  ( |_ `  ( A  /  D ) ) ) )  =  ( B  -  ( D  x.  ( |_ `  ( B  /  D ) ) ) ) ) )
54adantrl 697 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  ZZ  /\  D  e.  RR+ ) )  ->  (
( A  mod  D
)  =  ( B  mod  D )  <->  ( A  -  ( D  x.  ( |_ `  ( A  /  D ) ) ) )  =  ( B  -  ( D  x.  ( |_ `  ( B  /  D
) ) ) ) ) )
6 oveq1 6088 . . . . 5  |-  ( ( A  -  ( D  x.  ( |_ `  ( A  /  D
) ) ) )  =  ( B  -  ( D  x.  ( |_ `  ( B  /  D ) ) ) )  ->  ( ( A  -  ( D  x.  ( |_ `  ( A  /  D ) ) ) )  x.  C
)  =  ( ( B  -  ( D  x.  ( |_ `  ( B  /  D
) ) ) )  x.  C ) )
75, 6syl6bi 220 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  ZZ  /\  D  e.  RR+ ) )  ->  (
( A  mod  D
)  =  ( B  mod  D )  -> 
( ( A  -  ( D  x.  ( |_ `  ( A  /  D ) ) ) )  x.  C )  =  ( ( B  -  ( D  x.  ( |_ `  ( B  /  D ) ) ) )  x.  C
) ) )
8 rpcn 10620 . . . . . . . . . . 11  |-  ( D  e.  RR+  ->  D  e.  CC )
98ad2antll 710 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  D  e.  CC )
10 zcn 10287 . . . . . . . . . . 11  |-  ( C  e.  ZZ  ->  C  e.  CC )
1110ad2antrl 709 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  C  e.  CC )
12 rerpdivcl 10639 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  D  e.  RR+ )  -> 
( A  /  D
)  e.  RR )
1312flcld 11207 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  D  e.  RR+ )  -> 
( |_ `  ( A  /  D ) )  e.  ZZ )
1413zcnd 10376 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  D  e.  RR+ )  -> 
( |_ `  ( A  /  D ) )  e.  CC )
1514adantrl 697 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  ( |_ `  ( A  /  D
) )  e.  CC )
169, 11, 15mulassd 9111 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  ( ( D  x.  C )  x.  ( |_ `  ( A  /  D ) ) )  =  ( D  x.  ( C  x.  ( |_ `  ( A  /  D ) ) ) ) )
179, 11, 15mul32d 9276 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  ( ( D  x.  C )  x.  ( |_ `  ( A  /  D ) ) )  =  ( ( D  x.  ( |_
`  ( A  /  D ) ) )  x.  C ) )
1816, 17eqtr3d 2470 . . . . . . . 8  |-  ( ( A  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  ( D  x.  ( C  x.  ( |_ `  ( A  /  D ) ) ) )  =  ( ( D  x.  ( |_
`  ( A  /  D ) ) )  x.  C ) )
1918oveq2d 6097 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  ( ( A  x.  C )  -  ( D  x.  ( C  x.  ( |_ `  ( A  /  D ) ) ) ) )  =  ( ( A  x.  C
)  -  ( ( D  x.  ( |_
`  ( A  /  D ) ) )  x.  C ) ) )
20 recn 9080 . . . . . . . . 9  |-  ( A  e.  RR  ->  A  e.  CC )
2120adantr 452 . . . . . . . 8  |-  ( ( A  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  A  e.  CC )
228adantl 453 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  D  e.  RR+ )  ->  D  e.  CC )
2322, 14mulcld 9108 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  D  e.  RR+ )  -> 
( D  x.  ( |_ `  ( A  /  D ) ) )  e.  CC )
2423adantrl 697 . . . . . . . 8  |-  ( ( A  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  ( D  x.  ( |_ `  ( A  /  D ) ) )  e.  CC )
2521, 24, 11subdird 9490 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  ( ( A  -  ( D  x.  ( |_ `  ( A  /  D ) ) ) )  x.  C
)  =  ( ( A  x.  C )  -  ( ( D  x.  ( |_ `  ( A  /  D
) ) )  x.  C ) ) )
2619, 25eqtr4d 2471 . . . . . 6  |-  ( ( A  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  ( ( A  x.  C )  -  ( D  x.  ( C  x.  ( |_ `  ( A  /  D ) ) ) ) )  =  ( ( A  -  ( D  x.  ( |_ `  ( A  /  D
) ) ) )  x.  C ) )
2726adantlr 696 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  ZZ  /\  D  e.  RR+ ) )  ->  (
( A  x.  C
)  -  ( D  x.  ( C  x.  ( |_ `  ( A  /  D ) ) ) ) )  =  ( ( A  -  ( D  x.  ( |_ `  ( A  /  D ) ) ) )  x.  C ) )
288ad2antll 710 . . . . . . . . . 10  |-  ( ( B  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  D  e.  CC )
2910ad2antrl 709 . . . . . . . . . 10  |-  ( ( B  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  C  e.  CC )
30 rerpdivcl 10639 . . . . . . . . . . . . 13  |-  ( ( B  e.  RR  /\  D  e.  RR+ )  -> 
( B  /  D
)  e.  RR )
3130flcld 11207 . . . . . . . . . . . 12  |-  ( ( B  e.  RR  /\  D  e.  RR+ )  -> 
( |_ `  ( B  /  D ) )  e.  ZZ )
3231zcnd 10376 . . . . . . . . . . 11  |-  ( ( B  e.  RR  /\  D  e.  RR+ )  -> 
( |_ `  ( B  /  D ) )  e.  CC )
3332adantrl 697 . . . . . . . . . 10  |-  ( ( B  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  ( |_ `  ( B  /  D
) )  e.  CC )
3428, 29, 33mulassd 9111 . . . . . . . . 9  |-  ( ( B  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  ( ( D  x.  C )  x.  ( |_ `  ( B  /  D ) ) )  =  ( D  x.  ( C  x.  ( |_ `  ( B  /  D ) ) ) ) )
3528, 29, 33mul32d 9276 . . . . . . . . 9  |-  ( ( B  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  ( ( D  x.  C )  x.  ( |_ `  ( B  /  D ) ) )  =  ( ( D  x.  ( |_
`  ( B  /  D ) ) )  x.  C ) )
3634, 35eqtr3d 2470 . . . . . . . 8  |-  ( ( B  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  ( D  x.  ( C  x.  ( |_ `  ( B  /  D ) ) ) )  =  ( ( D  x.  ( |_
`  ( B  /  D ) ) )  x.  C ) )
3736oveq2d 6097 . . . . . . 7  |-  ( ( B  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  ( ( B  x.  C )  -  ( D  x.  ( C  x.  ( |_ `  ( B  /  D ) ) ) ) )  =  ( ( B  x.  C
)  -  ( ( D  x.  ( |_
`  ( B  /  D ) ) )  x.  C ) ) )
38 recn 9080 . . . . . . . . 9  |-  ( B  e.  RR  ->  B  e.  CC )
3938adantr 452 . . . . . . . 8  |-  ( ( B  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  B  e.  CC )
408adantl 453 . . . . . . . . . 10  |-  ( ( B  e.  RR  /\  D  e.  RR+ )  ->  D  e.  CC )
4140, 32mulcld 9108 . . . . . . . . 9  |-  ( ( B  e.  RR  /\  D  e.  RR+ )  -> 
( D  x.  ( |_ `  ( B  /  D ) ) )  e.  CC )
4241adantrl 697 . . . . . . . 8  |-  ( ( B  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  ( D  x.  ( |_ `  ( B  /  D ) ) )  e.  CC )
4339, 42, 29subdird 9490 . . . . . . 7  |-  ( ( B  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  ( ( B  -  ( D  x.  ( |_ `  ( B  /  D ) ) ) )  x.  C
)  =  ( ( B  x.  C )  -  ( ( D  x.  ( |_ `  ( B  /  D
) ) )  x.  C ) ) )
4437, 43eqtr4d 2471 . . . . . 6  |-  ( ( B  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  ( ( B  x.  C )  -  ( D  x.  ( C  x.  ( |_ `  ( B  /  D ) ) ) ) )  =  ( ( B  -  ( D  x.  ( |_ `  ( B  /  D
) ) ) )  x.  C ) )
4544adantll 695 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  ZZ  /\  D  e.  RR+ ) )  ->  (
( B  x.  C
)  -  ( D  x.  ( C  x.  ( |_ `  ( B  /  D ) ) ) ) )  =  ( ( B  -  ( D  x.  ( |_ `  ( B  /  D ) ) ) )  x.  C ) )
4627, 45eqeq12d 2450 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  ZZ  /\  D  e.  RR+ ) )  ->  (
( ( A  x.  C )  -  ( D  x.  ( C  x.  ( |_ `  ( A  /  D ) ) ) ) )  =  ( ( B  x.  C )  -  ( D  x.  ( C  x.  ( |_ `  ( B  /  D ) ) ) ) )  <->  ( ( A  -  ( D  x.  ( |_ `  ( A  /  D ) ) ) )  x.  C
)  =  ( ( B  -  ( D  x.  ( |_ `  ( B  /  D
) ) ) )  x.  C ) ) )
477, 46sylibrd 226 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  ZZ  /\  D  e.  RR+ ) )  ->  (
( A  mod  D
)  =  ( B  mod  D )  -> 
( ( A  x.  C )  -  ( D  x.  ( C  x.  ( |_ `  ( A  /  D ) ) ) ) )  =  ( ( B  x.  C )  -  ( D  x.  ( C  x.  ( |_ `  ( B  /  D ) ) ) ) ) ) )
48 oveq1 6088 . . . 4  |-  ( ( ( A  x.  C
)  -  ( D  x.  ( C  x.  ( |_ `  ( A  /  D ) ) ) ) )  =  ( ( B  x.  C )  -  ( D  x.  ( C  x.  ( |_ `  ( B  /  D ) ) ) ) )  -> 
( ( ( A  x.  C )  -  ( D  x.  ( C  x.  ( |_ `  ( A  /  D
) ) ) ) )  mod  D )  =  ( ( ( B  x.  C )  -  ( D  x.  ( C  x.  ( |_ `  ( B  /  D ) ) ) ) )  mod  D
) )
49 zre 10286 . . . . . . . . 9  |-  ( C  e.  ZZ  ->  C  e.  RR )
50 remulcl 9075 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  C  e.  RR )  ->  ( A  x.  C
)  e.  RR )
5149, 50sylan2 461 . . . . . . . 8  |-  ( ( A  e.  RR  /\  C  e.  ZZ )  ->  ( A  x.  C
)  e.  RR )
5251adantrr 698 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  ( A  x.  C )  e.  RR )
53 simprr 734 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  D  e.  RR+ )
54 simprl 733 . . . . . . . 8  |-  ( ( A  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  C  e.  ZZ )
5513adantrl 697 . . . . . . . 8  |-  ( ( A  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  ( |_ `  ( A  /  D
) )  e.  ZZ )
5654, 55zmulcld 10381 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  ( C  x.  ( |_ `  ( A  /  D ) ) )  e.  ZZ )
57 modcyc2 11277 . . . . . . 7  |-  ( ( ( A  x.  C
)  e.  RR  /\  D  e.  RR+  /\  ( C  x.  ( |_ `  ( A  /  D
) ) )  e.  ZZ )  ->  (
( ( A  x.  C )  -  ( D  x.  ( C  x.  ( |_ `  ( A  /  D ) ) ) ) )  mod 
D )  =  ( ( A  x.  C
)  mod  D )
)
5852, 53, 56, 57syl3anc 1184 . . . . . 6  |-  ( ( A  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  ( (
( A  x.  C
)  -  ( D  x.  ( C  x.  ( |_ `  ( A  /  D ) ) ) ) )  mod 
D )  =  ( ( A  x.  C
)  mod  D )
)
5958adantlr 696 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  ZZ  /\  D  e.  RR+ ) )  ->  (
( ( A  x.  C )  -  ( D  x.  ( C  x.  ( |_ `  ( A  /  D ) ) ) ) )  mod 
D )  =  ( ( A  x.  C
)  mod  D )
)
60 remulcl 9075 . . . . . . . . 9  |-  ( ( B  e.  RR  /\  C  e.  RR )  ->  ( B  x.  C
)  e.  RR )
6149, 60sylan2 461 . . . . . . . 8  |-  ( ( B  e.  RR  /\  C  e.  ZZ )  ->  ( B  x.  C
)  e.  RR )
6261adantrr 698 . . . . . . 7  |-  ( ( B  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  ( B  x.  C )  e.  RR )
63 simprr 734 . . . . . . 7  |-  ( ( B  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  D  e.  RR+ )
64 simprl 733 . . . . . . . 8  |-  ( ( B  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  C  e.  ZZ )
6531adantrl 697 . . . . . . . 8  |-  ( ( B  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  ( |_ `  ( B  /  D
) )  e.  ZZ )
6664, 65zmulcld 10381 . . . . . . 7  |-  ( ( B  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  ( C  x.  ( |_ `  ( B  /  D ) ) )  e.  ZZ )
67 modcyc2 11277 . . . . . . 7  |-  ( ( ( B  x.  C
)  e.  RR  /\  D  e.  RR+  /\  ( C  x.  ( |_ `  ( B  /  D
) ) )  e.  ZZ )  ->  (
( ( B  x.  C )  -  ( D  x.  ( C  x.  ( |_ `  ( B  /  D ) ) ) ) )  mod 
D )  =  ( ( B  x.  C
)  mod  D )
)
6862, 63, 66, 67syl3anc 1184 . . . . . 6  |-  ( ( B  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  ( (
( B  x.  C
)  -  ( D  x.  ( C  x.  ( |_ `  ( B  /  D ) ) ) ) )  mod 
D )  =  ( ( B  x.  C
)  mod  D )
)
6968adantll 695 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  ZZ  /\  D  e.  RR+ ) )  ->  (
( ( B  x.  C )  -  ( D  x.  ( C  x.  ( |_ `  ( B  /  D ) ) ) ) )  mod 
D )  =  ( ( B  x.  C
)  mod  D )
)
7059, 69eqeq12d 2450 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  ZZ  /\  D  e.  RR+ ) )  ->  (
( ( ( A  x.  C )  -  ( D  x.  ( C  x.  ( |_ `  ( A  /  D
) ) ) ) )  mod  D )  =  ( ( ( B  x.  C )  -  ( D  x.  ( C  x.  ( |_ `  ( B  /  D ) ) ) ) )  mod  D
)  <->  ( ( A  x.  C )  mod 
D )  =  ( ( B  x.  C
)  mod  D )
) )
7148, 70syl5ib 211 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  ZZ  /\  D  e.  RR+ ) )  ->  (
( ( A  x.  C )  -  ( D  x.  ( C  x.  ( |_ `  ( A  /  D ) ) ) ) )  =  ( ( B  x.  C )  -  ( D  x.  ( C  x.  ( |_ `  ( B  /  D ) ) ) ) )  -> 
( ( A  x.  C )  mod  D
)  =  ( ( B  x.  C )  mod  D ) ) )
7247, 71syld 42 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  ZZ  /\  D  e.  RR+ ) )  ->  (
( A  mod  D
)  =  ( B  mod  D )  -> 
( ( A  x.  C )  mod  D
)  =  ( ( B  x.  C )  mod  D ) ) )
73723impia 1150 1  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  ZZ  /\  D  e.  RR+ )  /\  ( A  mod  D )  =  ( B  mod  D
) )  ->  (
( A  x.  C
)  mod  D )  =  ( ( B  x.  C )  mod 
D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   ` cfv 5454  (class class class)co 6081   CCcc 8988   RRcr 8989    x. cmul 8995    - cmin 9291    / cdiv 9677   ZZcz 10282   RR+crp 10612   |_cfl 11201    mod cmo 11250
This theorem is referenced by:  modmul12d  11280  modnegd  11281  eulerthlem2  13171  fermltl  13173  odzdvds  13181  wilthlem2  20852  lgsdir2lem4  21110  lgsdirprm  21113  pellexlem6  26897  modmulmod  28157
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 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-sup 7446  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-n0 10222  df-z 10283  df-uz 10489  df-rp 10613  df-fl 11202  df-mod 11251
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