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Theorem modmul1 11018
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 10991 . . . . . . . 8  |-  ( ( A  e.  RR  /\  D  e.  RR+ )  -> 
( A  mod  D
)  =  ( A  -  ( D  x.  ( |_ `  ( A  /  D ) ) ) ) )
2 modval 10991 . . . . . . . 8  |-  ( ( B  e.  RR  /\  D  e.  RR+ )  -> 
( B  mod  D
)  =  ( B  -  ( D  x.  ( |_ `  ( B  /  D ) ) ) ) )
31, 2eqeqan12d 2311 . . . . . . 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 804 . . . . . 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 696 . . . . 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 5881 . . . . 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 219 . . . 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 10378 . . . . . . . . . . 11  |-  ( D  e.  RR+  ->  D  e.  CC )
98ad2antll 709 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  D  e.  CC )
10 zcn 10045 . . . . . . . . . . 11  |-  ( C  e.  ZZ  ->  C  e.  CC )
1110ad2antrl 708 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  C  e.  CC )
12 rerpdivcl 10397 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  D  e.  RR+ )  -> 
( A  /  D
)  e.  RR )
1312flcld 10946 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  D  e.  RR+ )  -> 
( |_ `  ( A  /  D ) )  e.  ZZ )
1413zcnd 10134 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  D  e.  RR+ )  -> 
( |_ `  ( A  /  D ) )  e.  CC )
1514adantrl 696 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  ( |_ `  ( A  /  D
) )  e.  CC )
169, 11, 15mulassd 8874 . . . . . . . . 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 9038 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  ( ( D  x.  C )  x.  ( |_ `  ( A  /  D ) ) )  =  ( ( D  x.  ( |_
`  ( A  /  D ) ) )  x.  C ) )
1816, 17eqtr3d 2330 . . . . . . . 8  |-  ( ( A  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  ( D  x.  ( C  x.  ( |_ `  ( A  /  D ) ) ) )  =  ( ( D  x.  ( |_
`  ( A  /  D ) ) )  x.  C ) )
1918oveq2d 5890 . . . . . . 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 8843 . . . . . . . . 9  |-  ( A  e.  RR  ->  A  e.  CC )
2120adantr 451 . . . . . . . 8  |-  ( ( A  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  A  e.  CC )
228adantl 452 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  D  e.  RR+ )  ->  D  e.  CC )
2322, 14mulcld 8871 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  D  e.  RR+ )  -> 
( D  x.  ( |_ `  ( A  /  D ) ) )  e.  CC )
2423adantrl 696 . . . . . . . 8  |-  ( ( A  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  ( D  x.  ( |_ `  ( A  /  D ) ) )  e.  CC )
2521, 24, 11subdird 9252 . . . . . . 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 2331 . . . . . 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 695 . . . . 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 709 . . . . . . . . . 10  |-  ( ( B  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  D  e.  CC )
2910ad2antrl 708 . . . . . . . . . 10  |-  ( ( B  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  C  e.  CC )
30 rerpdivcl 10397 . . . . . . . . . . . . 13  |-  ( ( B  e.  RR  /\  D  e.  RR+ )  -> 
( B  /  D
)  e.  RR )
3130flcld 10946 . . . . . . . . . . . 12  |-  ( ( B  e.  RR  /\  D  e.  RR+ )  -> 
( |_ `  ( B  /  D ) )  e.  ZZ )
3231zcnd 10134 . . . . . . . . . . 11  |-  ( ( B  e.  RR  /\  D  e.  RR+ )  -> 
( |_ `  ( B  /  D ) )  e.  CC )
3332adantrl 696 . . . . . . . . . 10  |-  ( ( B  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  ( |_ `  ( B  /  D
) )  e.  CC )
3428, 29, 33mulassd 8874 . . . . . . . . 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 9038 . . . . . . . . 9  |-  ( ( B  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  ( ( D  x.  C )  x.  ( |_ `  ( B  /  D ) ) )  =  ( ( D  x.  ( |_
`  ( B  /  D ) ) )  x.  C ) )
3634, 35eqtr3d 2330 . . . . . . . 8  |-  ( ( B  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  ( D  x.  ( C  x.  ( |_ `  ( B  /  D ) ) ) )  =  ( ( D  x.  ( |_
`  ( B  /  D ) ) )  x.  C ) )
3736oveq2d 5890 . . . . . . 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 8843 . . . . . . . . 9  |-  ( B  e.  RR  ->  B  e.  CC )
3938adantr 451 . . . . . . . 8  |-  ( ( B  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  B  e.  CC )
408adantl 452 . . . . . . . . . 10  |-  ( ( B  e.  RR  /\  D  e.  RR+ )  ->  D  e.  CC )
4140, 32mulcld 8871 . . . . . . . . 9  |-  ( ( B  e.  RR  /\  D  e.  RR+ )  -> 
( D  x.  ( |_ `  ( B  /  D ) ) )  e.  CC )
4241adantrl 696 . . . . . . . 8  |-  ( ( B  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  ( D  x.  ( |_ `  ( B  /  D ) ) )  e.  CC )
4339, 42, 29subdird 9252 . . . . . . 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 2331 . . . . . 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 694 . . . . 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 2310 . . . 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 225 . . 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 5881 . . . 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 10044 . . . . . . . . 9  |-  ( C  e.  ZZ  ->  C  e.  RR )
50 remulcl 8838 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  C  e.  RR )  ->  ( A  x.  C
)  e.  RR )
5149, 50sylan2 460 . . . . . . . 8  |-  ( ( A  e.  RR  /\  C  e.  ZZ )  ->  ( A  x.  C
)  e.  RR )
5251adantrr 697 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  ( A  x.  C )  e.  RR )
53 simprr 733 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  D  e.  RR+ )
54 simprl 732 . . . . . . . 8  |-  ( ( A  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  C  e.  ZZ )
5513adantrl 696 . . . . . . . 8  |-  ( ( A  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  ( |_ `  ( A  /  D
) )  e.  ZZ )
5654, 55zmulcld 10139 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  ( C  x.  ( |_ `  ( A  /  D ) ) )  e.  ZZ )
57 modcyc2 11016 . . . . . . 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 1182 . . . . . 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 695 . . . . 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 8838 . . . . . . . . 9  |-  ( ( B  e.  RR  /\  C  e.  RR )  ->  ( B  x.  C
)  e.  RR )
6149, 60sylan2 460 . . . . . . . 8  |-  ( ( B  e.  RR  /\  C  e.  ZZ )  ->  ( B  x.  C
)  e.  RR )
6261adantrr 697 . . . . . . 7  |-  ( ( B  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  ( B  x.  C )  e.  RR )
63 simprr 733 . . . . . . 7  |-  ( ( B  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  D  e.  RR+ )
64 simprl 732 . . . . . . . 8  |-  ( ( B  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  C  e.  ZZ )
6531adantrl 696 . . . . . . . 8  |-  ( ( B  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  ( |_ `  ( B  /  D
) )  e.  ZZ )
6664, 65zmulcld 10139 . . . . . . 7  |-  ( ( B  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  ( C  x.  ( |_ `  ( B  /  D ) ) )  e.  ZZ )
67 modcyc2 11016 . . . . . . 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 1182 . . . . . 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 694 . . . . 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 2310 . . . 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 210 . . 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 40 . 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 1148 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 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   ` cfv 5271  (class class class)co 5874   CCcc 8751   RRcr 8752    x. cmul 8758    - cmin 9053    / cdiv 9439   ZZcz 10040   RR+crp 10370   |_cfl 10940    mod cmo 10989
This theorem is referenced by:  modmul12d  11019  modnegd  11020  eulerthlem2  12866  fermltl  12868  odzdvds  12876  wilthlem2  20323  lgsdir2lem4  20581  lgsdirprm  20584  pellexlem6  27022
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-fl 10941  df-mod 10990
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