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Theorem wilthlem1 20322
Description: The only elements that are equal to their own inverses in the multiplicative group of nonzero elements in  ZZ 
/  P ZZ are  1 and  -u 1  ==  P  -  1. (Note that from prmdiveq 12870,  ( N ^ ( P  - 
2 ) )  mod 
P is the modular inverse of  N in  ZZ  /  P ZZ. (Contributed by Mario Carneiro, 24-Jan-2015.)
Assertion
Ref Expression
wilthlem1  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( N  =  ( ( N ^ ( P  - 
2 ) )  mod 
P )  <->  ( N  =  1  \/  N  =  ( P  - 
1 ) ) ) )

Proof of Theorem wilthlem1
StepHypRef Expression
1 elfzelz 10814 . . . . . . . . . 10  |-  ( N  e.  ( 1 ... ( P  -  1 ) )  ->  N  e.  ZZ )
21adantl 452 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  N  e.  ZZ )
3 peano2zm 10078 . . . . . . . . 9  |-  ( N  e.  ZZ  ->  ( N  -  1 )  e.  ZZ )
42, 3syl 15 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( N  -  1 )  e.  ZZ )
54zcnd 10134 . . . . . . 7  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( N  -  1 )  e.  CC )
62peano2zd 10136 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( N  +  1 )  e.  ZZ )
76zcnd 10134 . . . . . . 7  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( N  +  1 )  e.  CC )
85, 7mulcomd 8872 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  (
( N  -  1 )  x.  ( N  +  1 ) )  =  ( ( N  +  1 )  x.  ( N  -  1 ) ) )
92zcnd 10134 . . . . . . 7  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  N  e.  CC )
10 ax-1cn 8811 . . . . . . 7  |-  1  e.  CC
11 subsq 11226 . . . . . . 7  |-  ( ( N  e.  CC  /\  1  e.  CC )  ->  ( ( N ^
2 )  -  (
1 ^ 2 ) )  =  ( ( N  +  1 )  x.  ( N  - 
1 ) ) )
129, 10, 11sylancl 643 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  (
( N ^ 2 )  -  ( 1 ^ 2 ) )  =  ( ( N  +  1 )  x.  ( N  -  1 ) ) )
139sqvald 11258 . . . . . . 7  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( N ^ 2 )  =  ( N  x.  N
) )
14 sq1 11214 . . . . . . . 8  |-  ( 1 ^ 2 )  =  1
1514a1i 10 . . . . . . 7  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  (
1 ^ 2 )  =  1 )
1613, 15oveq12d 5892 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  (
( N ^ 2 )  -  ( 1 ^ 2 ) )  =  ( ( N  x.  N )  - 
1 ) )
178, 12, 163eqtr2d 2334 . . . . 5  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  (
( N  -  1 )  x.  ( N  +  1 ) )  =  ( ( N  x.  N )  - 
1 ) )
1817breq2d 4051 . . . 4  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( P  ||  ( ( N  -  1 )  x.  ( N  +  1 ) )  <->  P  ||  (
( N  x.  N
)  -  1 ) ) )
19 1e0p1 10168 . . . . . . . 8  |-  1  =  ( 0  +  1 )
2019oveq1i 5884 . . . . . . 7  |-  ( 1 ... ( P  - 
1 ) )  =  ( ( 0  +  1 ) ... ( P  -  1 ) )
21 0z 10051 . . . . . . . 8  |-  0  e.  ZZ
22 fzp1ss 10853 . . . . . . . 8  |-  ( 0  e.  ZZ  ->  (
( 0  +  1 ) ... ( P  -  1 ) ) 
C_  ( 0 ... ( P  -  1 ) ) )
2321, 22ax-mp 8 . . . . . . 7  |-  ( ( 0  +  1 ) ... ( P  - 
1 ) )  C_  ( 0 ... ( P  -  1 ) )
2420, 23eqsstri 3221 . . . . . 6  |-  ( 1 ... ( P  - 
1 ) )  C_  ( 0 ... ( P  -  1 ) )
25 simpr 447 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  N  e.  ( 1 ... ( P  -  1 ) ) )
2624, 25sseldi 3191 . . . . 5  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  N  e.  ( 0 ... ( P  -  1 ) ) )
2726biantrurd 494 . . . 4  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( P  ||  ( ( N  x.  N )  - 
1 )  <->  ( N  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( N  x.  N )  -  1 ) ) ) )
2818, 27bitrd 244 . . 3  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( P  ||  ( ( N  -  1 )  x.  ( N  +  1 ) )  <->  ( N  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( N  x.  N )  -  1 ) ) ) )
29 simpl 443 . . . 4  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  P  e.  Prime )
30 euclemma 12803 . . . 4  |-  ( ( P  e.  Prime  /\  ( N  -  1 )  e.  ZZ  /\  ( N  +  1 )  e.  ZZ )  -> 
( P  ||  (
( N  -  1 )  x.  ( N  +  1 ) )  <-> 
( P  ||  ( N  -  1 )  \/  P  ||  ( N  +  1 ) ) ) )
3129, 4, 6, 30syl3anc 1182 . . 3  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( P  ||  ( ( N  -  1 )  x.  ( N  +  1 ) )  <->  ( P  ||  ( N  -  1 )  \/  P  ||  ( N  +  1
) ) ) )
32 prmnn 12777 . . . . 5  |-  ( P  e.  Prime  ->  P  e.  NN )
33 fzm1ndvds 12596 . . . . 5  |-  ( ( P  e.  NN  /\  N  e.  ( 1 ... ( P  - 
1 ) ) )  ->  -.  P  ||  N
)
3432, 33sylan 457 . . . 4  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  -.  P  ||  N )
35 eqid 2296 . . . . 5  |-  ( ( N ^ ( P  -  2 ) )  mod  P )  =  ( ( N ^
( P  -  2 ) )  mod  P
)
3635prmdiveq 12870 . . . 4  |-  ( ( P  e.  Prime  /\  N  e.  ZZ  /\  -.  P  ||  N )  ->  (
( N  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  (
( N  x.  N
)  -  1 ) )  <->  N  =  (
( N ^ ( P  -  2 ) )  mod  P ) ) )
3729, 2, 34, 36syl3anc 1182 . . 3  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  (
( N  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  (
( N  x.  N
)  -  1 ) )  <->  N  =  (
( N ^ ( P  -  2 ) )  mod  P ) ) )
3828, 31, 373bitr3rd 275 . 2  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( N  =  ( ( N ^ ( P  - 
2 ) )  mod 
P )  <->  ( P  ||  ( N  -  1 )  \/  P  ||  ( N  +  1
) ) ) )
3929, 32syl 15 . . . . 5  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  P  e.  NN )
40 1z 10069 . . . . . 6  |-  1  e.  ZZ
4140a1i 10 . . . . 5  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  1  e.  ZZ )
42 moddvds 12554 . . . . 5  |-  ( ( P  e.  NN  /\  N  e.  ZZ  /\  1  e.  ZZ )  ->  (
( N  mod  P
)  =  ( 1  mod  P )  <->  P  ||  ( N  -  1 ) ) )
4339, 2, 41, 42syl3anc 1182 . . . 4  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  (
( N  mod  P
)  =  ( 1  mod  P )  <->  P  ||  ( N  -  1 ) ) )
44 elfznn 10835 . . . . . . . 8  |-  ( N  e.  ( 1 ... ( P  -  1 ) )  ->  N  e.  NN )
4544adantl 452 . . . . . . 7  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  N  e.  NN )
4645nnred 9777 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  N  e.  RR )
4739nnrpd 10405 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  P  e.  RR+ )
4845nnnn0d 10034 . . . . . . 7  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  N  e.  NN0 )
4948nn0ge0d 10037 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  0  <_  N )
50 elfzle2 10816 . . . . . . . 8  |-  ( N  e.  ( 1 ... ( P  -  1 ) )  ->  N  <_  ( P  -  1 ) )
5150adantl 452 . . . . . . 7  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  N  <_  ( P  -  1 ) )
52 prmz 12778 . . . . . . . 8  |-  ( P  e.  Prime  ->  P  e.  ZZ )
53 zltlem1 10086 . . . . . . . 8  |-  ( ( N  e.  ZZ  /\  P  e.  ZZ )  ->  ( N  <  P  <->  N  <_  ( P  - 
1 ) ) )
541, 52, 53syl2anr 464 . . . . . . 7  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( N  <  P  <->  N  <_  ( P  -  1 ) ) )
5551, 54mpbird 223 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  N  <  P )
56 modid 11009 . . . . . 6  |-  ( ( ( N  e.  RR  /\  P  e.  RR+ )  /\  ( 0  <_  N  /\  N  <  P ) )  ->  ( N  mod  P )  =  N )
5746, 47, 49, 55, 56syl22anc 1183 . . . . 5  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( N  mod  P )  =  N )
5839nnred 9777 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  P  e.  RR )
59 prmuz2 12792 . . . . . . . 8  |-  ( P  e.  Prime  ->  P  e.  ( ZZ>= `  2 )
)
6029, 59syl 15 . . . . . . 7  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  P  e.  ( ZZ>= `  2 )
)
61 eluz2b2 10306 . . . . . . . 8  |-  ( P  e.  ( ZZ>= `  2
)  <->  ( P  e.  NN  /\  1  < 
P ) )
6261simprbi 450 . . . . . . 7  |-  ( P  e.  ( ZZ>= `  2
)  ->  1  <  P )
6360, 62syl 15 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  1  <  P )
64 1mod 11012 . . . . . 6  |-  ( ( P  e.  RR  /\  1  <  P )  -> 
( 1  mod  P
)  =  1 )
6558, 63, 64syl2anc 642 . . . . 5  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  (
1  mod  P )  =  1 )
6657, 65eqeq12d 2310 . . . 4  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  (
( N  mod  P
)  =  ( 1  mod  P )  <->  N  = 
1 ) )
6743, 66bitr3d 246 . . 3  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( P  ||  ( N  - 
1 )  <->  N  = 
1 ) )
6841znegcld 10135 . . . . 5  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  -u 1  e.  ZZ )
69 moddvds 12554 . . . . 5  |-  ( ( P  e.  NN  /\  N  e.  ZZ  /\  -u 1  e.  ZZ )  ->  (
( N  mod  P
)  =  ( -u
1  mod  P )  <->  P 
||  ( N  -  -u 1 ) ) )
7039, 2, 68, 69syl3anc 1182 . . . 4  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  (
( N  mod  P
)  =  ( -u
1  mod  P )  <->  P 
||  ( N  -  -u 1 ) ) )
7139nncnd 9778 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  P  e.  CC )
7271mulid2d 8869 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  (
1  x.  P )  =  P )
7372oveq2d 5890 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( -u 1  +  ( 1  x.  P ) )  =  ( -u 1  +  P ) )
74 neg1cn 9829 . . . . . . . . 9  |-  -u 1  e.  CC
75 addcom 9014 . . . . . . . . 9  |-  ( (
-u 1  e.  CC  /\  P  e.  CC )  ->  ( -u 1  +  P )  =  ( P  +  -u 1
) )
7674, 71, 75sylancr 644 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( -u 1  +  P )  =  ( P  +  -u 1 ) )
77 negsub 9111 . . . . . . . . 9  |-  ( ( P  e.  CC  /\  1  e.  CC )  ->  ( P  +  -u
1 )  =  ( P  -  1 ) )
7871, 10, 77sylancl 643 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( P  +  -u 1 )  =  ( P  - 
1 ) )
7973, 76, 783eqtrd 2332 . . . . . . 7  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( -u 1  +  ( 1  x.  P ) )  =  ( P  - 
1 ) )
8079oveq1d 5889 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  (
( -u 1  +  ( 1  x.  P ) )  mod  P )  =  ( ( P  -  1 )  mod 
P ) )
81 1re 8853 . . . . . . . . 9  |-  1  e.  RR
8281renegcli 9124 . . . . . . . 8  |-  -u 1  e.  RR
8382a1i 10 . . . . . . 7  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  -u 1  e.  RR )
84 modcyc 11015 . . . . . . 7  |-  ( (
-u 1  e.  RR  /\  P  e.  RR+  /\  1  e.  ZZ )  ->  (
( -u 1  +  ( 1  x.  P ) )  mod  P )  =  ( -u 1  mod  P ) )
8583, 47, 41, 84syl3anc 1182 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  (
( -u 1  +  ( 1  x.  P ) )  mod  P )  =  ( -u 1  mod  P ) )
86 peano2rem 9129 . . . . . . . 8  |-  ( P  e.  RR  ->  ( P  -  1 )  e.  RR )
8758, 86syl 15 . . . . . . 7  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( P  -  1 )  e.  RR )
88 nnm1nn0 10021 . . . . . . . . 9  |-  ( P  e.  NN  ->  ( P  -  1 )  e.  NN0 )
8939, 88syl 15 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( P  -  1 )  e.  NN0 )
9089nn0ge0d 10037 . . . . . . 7  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  0  <_  ( P  -  1 ) )
9158ltm1d 9705 . . . . . . 7  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( P  -  1 )  <  P )
92 modid 11009 . . . . . . 7  |-  ( ( ( ( P  - 
1 )  e.  RR  /\  P  e.  RR+ )  /\  ( 0  <_  ( P  -  1 )  /\  ( P  - 
1 )  <  P
) )  ->  (
( P  -  1 )  mod  P )  =  ( P  - 
1 ) )
9387, 47, 90, 91, 92syl22anc 1183 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  (
( P  -  1 )  mod  P )  =  ( P  - 
1 ) )
9480, 85, 933eqtr3d 2336 . . . . 5  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( -u 1  mod  P )  =  ( P  - 
1 ) )
9557, 94eqeq12d 2310 . . . 4  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  (
( N  mod  P
)  =  ( -u
1  mod  P )  <->  N  =  ( P  - 
1 ) ) )
96 subneg 9112 . . . . . 6  |-  ( ( N  e.  CC  /\  1  e.  CC )  ->  ( N  -  -u 1
)  =  ( N  +  1 ) )
979, 10, 96sylancl 643 . . . . 5  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( N  -  -u 1 )  =  ( N  + 
1 ) )
9897breq2d 4051 . . . 4  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( P  ||  ( N  -  -u 1 )  <->  P  ||  ( N  +  1 ) ) )
9970, 95, 983bitr3rd 275 . . 3  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( P  ||  ( N  + 
1 )  <->  N  =  ( P  -  1
) ) )
10067, 99orbi12d 690 . 2  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  (
( P  ||  ( N  -  1 )  \/  P  ||  ( N  +  1 ) )  <->  ( N  =  1  \/  N  =  ( P  -  1 ) ) ) )
10138, 100bitrd 244 1  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( N  =  ( ( N ^ ( P  - 
2 ) )  mod 
P )  <->  ( N  =  1  \/  N  =  ( P  - 
1 ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1632    e. wcel 1696    C_ wss 3165   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758    < clt 8883    <_ cle 8884    - cmin 9053   -ucneg 9054   NNcn 9762   2c2 9811   NN0cn0 9981   ZZcz 10040   ZZ>=cuz 10246   RR+crp 10370   ...cfz 10798    mod cmo 10989   ^cexp 11120    || cdivides 12547   Primecprime 12774
This theorem is referenced by:  wilthlem2  20323
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-rep 4147  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-int 3879  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-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-card 7588  df-cda 7810  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-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-fz 10799  df-fzo 10887  df-fl 10941  df-mod 10990  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-dvds 12548  df-gcd 12702  df-prm 12775  df-phi 12850
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