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Theorem modmul1 11002
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 10975 . . . . . . . 8  |-  ( ( A  e.  RR  /\  D  e.  RR+ )  -> 
( A  mod  D
)  =  ( A  -  ( D  x.  ( |_ `  ( A  /  D ) ) ) ) )
2 modval 10975 . . . . . . . 8  |-  ( ( B  e.  RR  /\  D  e.  RR+ )  -> 
( B  mod  D
)  =  ( B  -  ( D  x.  ( |_ `  ( B  /  D ) ) ) ) )
31, 2eqeqan12d 2298 . . . . . . 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 5865 . . . . 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 10362 . . . . . . . . . . 11  |-  ( D  e.  RR+  ->  D  e.  CC )
98ad2antll 709 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  D  e.  CC )
10 zcn 10029 . . . . . . . . . . 11  |-  ( C  e.  ZZ  ->  C  e.  CC )
1110ad2antrl 708 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  C  e.  CC )
12 rerpdivcl 10381 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  D  e.  RR+ )  -> 
( A  /  D
)  e.  RR )
1312flcld 10930 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  D  e.  RR+ )  -> 
( |_ `  ( A  /  D ) )  e.  ZZ )
1413zcnd 10118 . . . . . . . . . . 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 8858 . . . . . . . . 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 9022 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  ( ( D  x.  C )  x.  ( |_ `  ( A  /  D ) ) )  =  ( ( D  x.  ( |_
`  ( A  /  D ) ) )  x.  C ) )
1816, 17eqtr3d 2317 . . . . . . . 8  |-  ( ( A  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  ( D  x.  ( C  x.  ( |_ `  ( A  /  D ) ) ) )  =  ( ( D  x.  ( |_
`  ( A  /  D ) ) )  x.  C ) )
1918oveq2d 5874 . . . . . . 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 8827 . . . . . . . . 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 8855 . . . . . . . . 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 9236 . . . . . . 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 2318 . . . . . 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 10381 . . . . . . . . . . . . 13  |-  ( ( B  e.  RR  /\  D  e.  RR+ )  -> 
( B  /  D
)  e.  RR )
3130flcld 10930 . . . . . . . . . . . 12  |-  ( ( B  e.  RR  /\  D  e.  RR+ )  -> 
( |_ `  ( B  /  D ) )  e.  ZZ )
3231zcnd 10118 . . . . . . . . . . 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 8858 . . . . . . . . 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 9022 . . . . . . . . 9  |-  ( ( B  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  ( ( D  x.  C )  x.  ( |_ `  ( B  /  D ) ) )  =  ( ( D  x.  ( |_
`  ( B  /  D ) ) )  x.  C ) )
3634, 35eqtr3d 2317 . . . . . . . 8  |-  ( ( B  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  ( D  x.  ( C  x.  ( |_ `  ( B  /  D ) ) ) )  =  ( ( D  x.  ( |_
`  ( B  /  D ) ) )  x.  C ) )
3736oveq2d 5874 . . . . . . 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 8827 . . . . . . . . 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 8855 . . . . . . . . 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 9236 . . . . . . 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 2318 . . . . . 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 2297 . . . 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 5865 . . . 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 10028 . . . . . . . . 9  |-  ( C  e.  ZZ  ->  C  e.  RR )
50 remulcl 8822 . . . . . . . . 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 10123 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  ( C  x.  ( |_ `  ( A  /  D ) ) )  e.  ZZ )
57 modcyc2 11000 . . . . . . 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 8822 . . . . . . . . 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 10123 . . . . . . 7  |-  ( ( B  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  ( C  x.  ( |_ `  ( B  /  D ) ) )  e.  ZZ )
67 modcyc2 11000 . . . . . . 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 2297 . . . 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 1623    e. wcel 1684   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736    x. cmul 8742    - cmin 9037    / cdiv 9423   ZZcz 10024   RR+crp 10354   |_cfl 10924    mod cmo 10973
This theorem is referenced by:  modmul12d  11003  modnegd  11004  eulerthlem2  12850  fermltl  12852  odzdvds  12860  wilthlem2  20307  lgsdir2lem4  20565  lgsdirprm  20568  pellexlem6  26919
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fl 10925  df-mod 10974
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