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Theorem modmul1 11757
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 11715 . . . . . . . 8  |-  ( ( A  e.  RR  /\  D  e.  RR+ )  -> 
( A  mod  D
)  =  ( A  -  ( D  x.  ( |_ `  ( A  /  D ) ) ) ) )
2 modval 11715 . . . . . . . 8  |-  ( ( B  e.  RR  /\  D  e.  RR+ )  -> 
( B  mod  D
)  =  ( B  -  ( D  x.  ( |_ `  ( B  /  D ) ) ) ) )
31, 2eqeqan12d 2458 . . . . . . 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 827 . . . . . 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 715 . . . . 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 6103 . . . . 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 228 . . . 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 11004 . . . . . . . . . . 11  |-  ( D  e.  RR+  ->  D  e.  CC )
98ad2antll 728 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  D  e.  CC )
10 zcn 10656 . . . . . . . . . . 11  |-  ( C  e.  ZZ  ->  C  e.  CC )
1110ad2antrl 727 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  C  e.  CC )
12 rerpdivcl 11023 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  D  e.  RR+ )  -> 
( A  /  D
)  e.  RR )
1312flcld 11653 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  D  e.  RR+ )  -> 
( |_ `  ( A  /  D ) )  e.  ZZ )
1413zcnd 10753 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  D  e.  RR+ )  -> 
( |_ `  ( A  /  D ) )  e.  CC )
1514adantrl 715 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  ( |_ `  ( A  /  D
) )  e.  CC )
169, 11, 15mulassd 9414 . . . . . . . . 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 9584 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  ( ( D  x.  C )  x.  ( |_ `  ( A  /  D ) ) )  =  ( ( D  x.  ( |_
`  ( A  /  D ) ) )  x.  C ) )
1816, 17eqtr3d 2477 . . . . . . . 8  |-  ( ( A  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  ( D  x.  ( C  x.  ( |_ `  ( A  /  D ) ) ) )  =  ( ( D  x.  ( |_
`  ( A  /  D ) ) )  x.  C ) )
1918oveq2d 6112 . . . . . . 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 9377 . . . . . . . . 9  |-  ( A  e.  RR  ->  A  e.  CC )
2120adantr 465 . . . . . . . 8  |-  ( ( A  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  A  e.  CC )
228adantl 466 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  D  e.  RR+ )  ->  D  e.  CC )
2322, 14mulcld 9411 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  D  e.  RR+ )  -> 
( D  x.  ( |_ `  ( A  /  D ) ) )  e.  CC )
2423adantrl 715 . . . . . . . 8  |-  ( ( A  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  ( D  x.  ( |_ `  ( A  /  D ) ) )  e.  CC )
2521, 24, 11subdird 9806 . . . . . . 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 2478 . . . . . 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 714 . . . . 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 728 . . . . . . . . . 10  |-  ( ( B  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  D  e.  CC )
2910ad2antrl 727 . . . . . . . . . 10  |-  ( ( B  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  C  e.  CC )
30 rerpdivcl 11023 . . . . . . . . . . . . 13  |-  ( ( B  e.  RR  /\  D  e.  RR+ )  -> 
( B  /  D
)  e.  RR )
3130flcld 11653 . . . . . . . . . . . 12  |-  ( ( B  e.  RR  /\  D  e.  RR+ )  -> 
( |_ `  ( B  /  D ) )  e.  ZZ )
3231zcnd 10753 . . . . . . . . . . 11  |-  ( ( B  e.  RR  /\  D  e.  RR+ )  -> 
( |_ `  ( B  /  D ) )  e.  CC )
3332adantrl 715 . . . . . . . . . 10  |-  ( ( B  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  ( |_ `  ( B  /  D
) )  e.  CC )
3428, 29, 33mulassd 9414 . . . . . . . . 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 9584 . . . . . . . . 9  |-  ( ( B  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  ( ( D  x.  C )  x.  ( |_ `  ( B  /  D ) ) )  =  ( ( D  x.  ( |_
`  ( B  /  D ) ) )  x.  C ) )
3634, 35eqtr3d 2477 . . . . . . . 8  |-  ( ( B  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  ( D  x.  ( C  x.  ( |_ `  ( B  /  D ) ) ) )  =  ( ( D  x.  ( |_
`  ( B  /  D ) ) )  x.  C ) )
3736oveq2d 6112 . . . . . . 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 9377 . . . . . . . . 9  |-  ( B  e.  RR  ->  B  e.  CC )
3938adantr 465 . . . . . . . 8  |-  ( ( B  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  B  e.  CC )
408adantl 466 . . . . . . . . . 10  |-  ( ( B  e.  RR  /\  D  e.  RR+ )  ->  D  e.  CC )
4140, 32mulcld 9411 . . . . . . . . 9  |-  ( ( B  e.  RR  /\  D  e.  RR+ )  -> 
( D  x.  ( |_ `  ( B  /  D ) ) )  e.  CC )
4241adantrl 715 . . . . . . . 8  |-  ( ( B  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  ( D  x.  ( |_ `  ( B  /  D ) ) )  e.  CC )
4339, 42, 29subdird 9806 . . . . . . 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 2478 . . . . . 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 713 . . . . 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 2457 . . . 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 234 . . 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 6103 . . . 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 10655 . . . . . . . . 9  |-  ( C  e.  ZZ  ->  C  e.  RR )
50 remulcl 9372 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  C  e.  RR )  ->  ( A  x.  C
)  e.  RR )
5149, 50sylan2 474 . . . . . . . 8  |-  ( ( A  e.  RR  /\  C  e.  ZZ )  ->  ( A  x.  C
)  e.  RR )
5251adantrr 716 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  ( A  x.  C )  e.  RR )
53 simprr 756 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  D  e.  RR+ )
54 simprl 755 . . . . . . . 8  |-  ( ( A  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  C  e.  ZZ )
5513adantrl 715 . . . . . . . 8  |-  ( ( A  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  ( |_ `  ( A  /  D
) )  e.  ZZ )
5654, 55zmulcld 10758 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  ( C  x.  ( |_ `  ( A  /  D ) ) )  e.  ZZ )
57 modcyc2 11749 . . . . . . 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 1218 . . . . . 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 714 . . . . 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 9372 . . . . . . . . 9  |-  ( ( B  e.  RR  /\  C  e.  RR )  ->  ( B  x.  C
)  e.  RR )
6149, 60sylan2 474 . . . . . . . 8  |-  ( ( B  e.  RR  /\  C  e.  ZZ )  ->  ( B  x.  C
)  e.  RR )
6261adantrr 716 . . . . . . 7  |-  ( ( B  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  ( B  x.  C )  e.  RR )
63 simprr 756 . . . . . . 7  |-  ( ( B  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  D  e.  RR+ )
64 simprl 755 . . . . . . . 8  |-  ( ( B  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  C  e.  ZZ )
6531adantrl 715 . . . . . . . 8  |-  ( ( B  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  ( |_ `  ( B  /  D
) )  e.  ZZ )
6664, 65zmulcld 10758 . . . . . . 7  |-  ( ( B  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  ( C  x.  ( |_ `  ( B  /  D ) ) )  e.  ZZ )
67 modcyc2 11749 . . . . . . 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 1218 . . . . . 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 713 . . . . 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 2457 . . . 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 219 . . 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 44 . 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 1184 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   ` cfv 5423  (class class class)co 6096   CCcc 9285   RRcr 9286    x. cmul 9292    - cmin 9600    / cdiv 9998   ZZcz 10651   RR+crp 10996   |_cfl 11645    mod cmo 11713
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-pre-sup 9365
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-recs 6837  df-rdg 6871  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-sup 7696  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-nn 10328  df-n0 10585  df-z 10652  df-uz 10867  df-rp 10997  df-fl 11647  df-mod 11714
This theorem is referenced by:  modmul12d  11758  modnegd  11759  modmulmod  11769  eulerthlem2  13862  fermltl  13864  odzdvds  13872  wilthlem2  22412  lgsdir2lem4  22670  lgsdirprm  22673  pellexlem6  29180
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