MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  modmul1 Structured version   Unicode version

Theorem modmul1 12042
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 12000 . . . . . . . 8  |-  ( ( A  e.  RR  /\  D  e.  RR+ )  -> 
( A  mod  D
)  =  ( A  -  ( D  x.  ( |_ `  ( A  /  D ) ) ) ) )
2 modval 12000 . . . . . . . 8  |-  ( ( B  e.  RR  /\  D  e.  RR+ )  -> 
( B  mod  D
)  =  ( B  -  ( D  x.  ( |_ `  ( B  /  D ) ) ) ) )
31, 2eqeqan12d 2480 . . . . . . 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 831 . . . . . 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 6303 . . . . 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 11253 . . . . . . . . . . 11  |-  ( D  e.  RR+  ->  D  e.  CC )
98ad2antll 728 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  D  e.  CC )
10 zcn 10890 . . . . . . . . . . 11  |-  ( C  e.  ZZ  ->  C  e.  CC )
1110ad2antrl 727 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  C  e.  CC )
12 rerpdivcl 11272 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  D  e.  RR+ )  -> 
( A  /  D
)  e.  RR )
1312flcld 11937 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  D  e.  RR+ )  -> 
( |_ `  ( A  /  D ) )  e.  ZZ )
1413zcnd 10991 . . . . . . . . . . 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 9636 . . . . . . . . 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 9807 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  ( ( D  x.  C )  x.  ( |_ `  ( A  /  D ) ) )  =  ( ( D  x.  ( |_
`  ( A  /  D ) ) )  x.  C ) )
1816, 17eqtr3d 2500 . . . . . . . 8  |-  ( ( A  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  ( D  x.  ( C  x.  ( |_ `  ( A  /  D ) ) ) )  =  ( ( D  x.  ( |_
`  ( A  /  D ) ) )  x.  C ) )
1918oveq2d 6312 . . . . . . 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 9599 . . . . . . . . 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 9633 . . . . . . . . 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 10034 . . . . . . 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 2501 . . . . . 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 11272 . . . . . . . . . . . . 13  |-  ( ( B  e.  RR  /\  D  e.  RR+ )  -> 
( B  /  D
)  e.  RR )
3130flcld 11937 . . . . . . . . . . . 12  |-  ( ( B  e.  RR  /\  D  e.  RR+ )  -> 
( |_ `  ( B  /  D ) )  e.  ZZ )
3231zcnd 10991 . . . . . . . . . . 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 9636 . . . . . . . . 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 9807 . . . . . . . . 9  |-  ( ( B  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  ( ( D  x.  C )  x.  ( |_ `  ( B  /  D ) ) )  =  ( ( D  x.  ( |_
`  ( B  /  D ) ) )  x.  C ) )
3634, 35eqtr3d 2500 . . . . . . . 8  |-  ( ( B  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  ( D  x.  ( C  x.  ( |_ `  ( B  /  D ) ) ) )  =  ( ( D  x.  ( |_
`  ( B  /  D ) ) )  x.  C ) )
3736oveq2d 6312 . . . . . . 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 9599 . . . . . . . . 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 9633 . . . . . . . . 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 10034 . . . . . . 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 2501 . . . . . 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 2479 . . . 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 6303 . . . 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 10889 . . . . . . . . 9  |-  ( C  e.  ZZ  ->  C  e.  RR )
50 remulcl 9594 . . . . . . . . 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 757 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  D  e.  RR+ )
54 simprl 756 . . . . . . . 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 10996 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  ( C  x.  ( |_ `  ( A  /  D ) ) )  e.  ZZ )
57 modcyc2 12034 . . . . . . 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 1228 . . . . . 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 9594 . . . . . . . . 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 757 . . . . . . 7  |-  ( ( B  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  D  e.  RR+ )
64 simprl 756 . . . . . . . 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 10996 . . . . . . 7  |-  ( ( B  e.  RR  /\  ( C  e.  ZZ  /\  D  e.  RR+ )
)  ->  ( C  x.  ( |_ `  ( B  /  D ) ) )  e.  ZZ )
67 modcyc2 12034 . . . . . . 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 1228 . . . . . 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 2479 . . . 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 1193 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 973    = wceq 1395    e. wcel 1819   ` cfv 5594  (class class class)co 6296   CCcc 9507   RRcr 9508    x. cmul 9514    - cmin 9824    / cdiv 10227   ZZcz 10885   RR+crp 11245   |_cfl 11929    mod cmo 11998
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-recs 7060  df-rdg 7094  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-sup 7919  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-n0 10817  df-z 10886  df-uz 11107  df-rp 11246  df-fl 11931  df-mod 11999
This theorem is referenced by:  modmul12d  12043  modnegd  12044  modmulmod  12054  eulerthlem2  14323  fermltl  14325  odzdvds  14333  wilthlem2  23468  lgsdir2lem4  23726  lgsdirprm  23729  pellexlem6  30932
  Copyright terms: Public domain W3C validator