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Theorem wilthlem1 20233
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 12781,  ( 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 10729 . . . . . . . . . 10  |-  ( N  e.  ( 1 ... ( P  -  1 ) )  ->  N  e.  ZZ )
21adantl 454 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  N  e.  ZZ )
3 peano2zm 9994 . . . . . . . . 9  |-  ( N  e.  ZZ  ->  ( N  -  1 )  e.  ZZ )
42, 3syl 17 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( N  -  1 )  e.  ZZ )
54zcnd 10050 . . . . . . 7  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( N  -  1 )  e.  CC )
62peano2zd 10052 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( N  +  1 )  e.  ZZ )
76zcnd 10050 . . . . . . 7  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( N  +  1 )  e.  CC )
85, 7mulcomd 8789 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  (
( N  -  1 )  x.  ( N  +  1 ) )  =  ( ( N  +  1 )  x.  ( N  -  1 ) ) )
92zcnd 10050 . . . . . . 7  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  N  e.  CC )
10 ax-1cn 8728 . . . . . . 7  |-  1  e.  CC
11 subsq 11141 . . . . . . 7  |-  ( ( N  e.  CC  /\  1  e.  CC )  ->  ( ( N ^
2 )  -  (
1 ^ 2 ) )  =  ( ( N  +  1 )  x.  ( N  - 
1 ) ) )
129, 10, 11sylancl 646 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  (
( N ^ 2 )  -  ( 1 ^ 2 ) )  =  ( ( N  +  1 )  x.  ( N  -  1 ) ) )
139sqvald 11173 . . . . . . 7  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( N ^ 2 )  =  ( N  x.  N
) )
14 sq1 11129 . . . . . . . 8  |-  ( 1 ^ 2 )  =  1
1514a1i 12 . . . . . . 7  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  (
1 ^ 2 )  =  1 )
1613, 15oveq12d 5775 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  (
( N ^ 2 )  -  ( 1 ^ 2 ) )  =  ( ( N  x.  N )  - 
1 ) )
178, 12, 163eqtr2d 2294 . . . . 5  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  (
( N  -  1 )  x.  ( N  +  1 ) )  =  ( ( N  x.  N )  - 
1 ) )
1817breq2d 3975 . . . 4  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( P  ||  ( ( N  -  1 )  x.  ( N  +  1 ) )  <->  P  ||  (
( N  x.  N
)  -  1 ) ) )
19 1e0p1 10084 . . . . . . . 8  |-  1  =  ( 0  +  1 )
2019oveq1i 5767 . . . . . . 7  |-  ( 1 ... ( P  - 
1 ) )  =  ( ( 0  +  1 ) ... ( P  -  1 ) )
21 0z 9967 . . . . . . . 8  |-  0  e.  ZZ
22 fzp1ss 10768 . . . . . . . 8  |-  ( 0  e.  ZZ  ->  (
( 0  +  1 ) ... ( P  -  1 ) ) 
C_  ( 0 ... ( P  -  1 ) ) )
2321, 22ax-mp 10 . . . . . . 7  |-  ( ( 0  +  1 ) ... ( P  - 
1 ) )  C_  ( 0 ... ( P  -  1 ) )
2420, 23eqsstri 3150 . . . . . 6  |-  ( 1 ... ( P  - 
1 ) )  C_  ( 0 ... ( P  -  1 ) )
25 simpr 449 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  N  e.  ( 1 ... ( P  -  1 ) ) )
2624, 25sseldi 3120 . . . . 5  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  N  e.  ( 0 ... ( P  -  1 ) ) )
2726biantrurd 496 . . . 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 246 . . 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 445 . . . 4  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  P  e.  Prime )
30 euclemma 12714 . . . 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 1187 . . 3  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( P  ||  ( ( N  -  1 )  x.  ( N  +  1 ) )  <->  ( P  ||  ( N  -  1 )  \/  P  ||  ( N  +  1
) ) ) )
32 prmnn 12689 . . . . 5  |-  ( P  e.  Prime  ->  P  e.  NN )
33 fzm1ndvds 12507 . . . . 5  |-  ( ( P  e.  NN  /\  N  e.  ( 1 ... ( P  - 
1 ) ) )  ->  -.  P  ||  N
)
3432, 33sylan 459 . . . 4  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  -.  P  ||  N )
35 eqid 2256 . . . . 5  |-  ( ( N ^ ( P  -  2 ) )  mod  P )  =  ( ( N ^
( P  -  2 ) )  mod  P
)
3635prmdiveq 12781 . . . 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 1187 . . 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 277 . 2  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( N  =  ( ( N ^ ( P  - 
2 ) )  mod 
P )  <->  ( P  ||  ( N  -  1 )  \/  P  ||  ( N  +  1
) ) ) )
3929, 32syl 17 . . . . 5  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  P  e.  NN )
40 1z 9985 . . . . . 6  |-  1  e.  ZZ
4140a1i 12 . . . . 5  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  1  e.  ZZ )
42 moddvds 12465 . . . . 5  |-  ( ( P  e.  NN  /\  N  e.  ZZ  /\  1  e.  ZZ )  ->  (
( N  mod  P
)  =  ( 1  mod  P )  <->  P  ||  ( N  -  1 ) ) )
4339, 2, 41, 42syl3anc 1187 . . . 4  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  (
( N  mod  P
)  =  ( 1  mod  P )  <->  P  ||  ( N  -  1 ) ) )
44 elfznn 10750 . . . . . . . 8  |-  ( N  e.  ( 1 ... ( P  -  1 ) )  ->  N  e.  NN )
4544adantl 454 . . . . . . 7  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  N  e.  NN )
4645nnred 9694 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  N  e.  RR )
4739nnrpd 10321 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  P  e.  RR+ )
4845nnnn0d 9950 . . . . . . 7  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  N  e.  NN0 )
4948nn0ge0d 9953 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  0  <_  N )
50 elfzle2 10731 . . . . . . . 8  |-  ( N  e.  ( 1 ... ( P  -  1 ) )  ->  N  <_  ( P  -  1 ) )
5150adantl 454 . . . . . . 7  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  N  <_  ( P  -  1 ) )
52 prmz 12688 . . . . . . . 8  |-  ( P  e.  Prime  ->  P  e.  ZZ )
53 zltlem1 10002 . . . . . . . 8  |-  ( ( N  e.  ZZ  /\  P  e.  ZZ )  ->  ( N  <  P  <->  N  <_  ( P  - 
1 ) ) )
541, 52, 53syl2anr 466 . . . . . . 7  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( N  <  P  <->  N  <_  ( P  -  1 ) ) )
5551, 54mpbird 225 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  N  <  P )
56 modid 10924 . . . . . 6  |-  ( ( ( N  e.  RR  /\  P  e.  RR+ )  /\  ( 0  <_  N  /\  N  <  P ) )  ->  ( N  mod  P )  =  N )
5746, 47, 49, 55, 56syl22anc 1188 . . . . 5  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( N  mod  P )  =  N )
5839nnred 9694 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  P  e.  RR )
59 prmuz2 12703 . . . . . . . 8  |-  ( P  e.  Prime  ->  P  e.  ( ZZ>= `  2 )
)
6029, 59syl 17 . . . . . . 7  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  P  e.  ( ZZ>= `  2 )
)
61 eluz2b2 10222 . . . . . . . 8  |-  ( P  e.  ( ZZ>= `  2
)  <->  ( P  e.  NN  /\  1  < 
P ) )
6261simprbi 452 . . . . . . 7  |-  ( P  e.  ( ZZ>= `  2
)  ->  1  <  P )
6360, 62syl 17 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  1  <  P )
64 1mod 10927 . . . . . 6  |-  ( ( P  e.  RR  /\  1  <  P )  -> 
( 1  mod  P
)  =  1 )
6558, 63, 64syl2anc 645 . . . . 5  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  (
1  mod  P )  =  1 )
6657, 65eqeq12d 2270 . . . 4  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  (
( N  mod  P
)  =  ( 1  mod  P )  <->  N  = 
1 ) )
6743, 66bitr3d 248 . . 3  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( P  ||  ( N  - 
1 )  <->  N  = 
1 ) )
6841znegcld 10051 . . . . 5  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  -u 1  e.  ZZ )
69 moddvds 12465 . . . . 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 1187 . . . 4  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  (
( N  mod  P
)  =  ( -u
1  mod  P )  <->  P 
||  ( N  -  -u 1 ) ) )
7139nncnd 9695 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  P  e.  CC )
7271mulid2d 8786 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  (
1  x.  P )  =  P )
7372oveq2d 5773 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( -u 1  +  ( 1  x.  P ) )  =  ( -u 1  +  P ) )
74 neg1cn 9746 . . . . . . . . 9  |-  -u 1  e.  CC
75 addcom 8931 . . . . . . . . 9  |-  ( (
-u 1  e.  CC  /\  P  e.  CC )  ->  ( -u 1  +  P )  =  ( P  +  -u 1
) )
7674, 71, 75sylancr 647 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( -u 1  +  P )  =  ( P  +  -u 1 ) )
77 negsub 9028 . . . . . . . . 9  |-  ( ( P  e.  CC  /\  1  e.  CC )  ->  ( P  +  -u
1 )  =  ( P  -  1 ) )
7871, 10, 77sylancl 646 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( P  +  -u 1 )  =  ( P  - 
1 ) )
7973, 76, 783eqtrd 2292 . . . . . . 7  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( -u 1  +  ( 1  x.  P ) )  =  ( P  - 
1 ) )
8079oveq1d 5772 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  (
( -u 1  +  ( 1  x.  P ) )  mod  P )  =  ( ( P  -  1 )  mod 
P ) )
81 1re 8770 . . . . . . . . 9  |-  1  e.  RR
8281renegcli 9041 . . . . . . . 8  |-  -u 1  e.  RR
8382a1i 12 . . . . . . 7  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  -u 1  e.  RR )
84 modcyc 10930 . . . . . . 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 1187 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  (
( -u 1  +  ( 1  x.  P ) )  mod  P )  =  ( -u 1  mod  P ) )
86 peano2rem 9046 . . . . . . . 8  |-  ( P  e.  RR  ->  ( P  -  1 )  e.  RR )
8758, 86syl 17 . . . . . . 7  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( P  -  1 )  e.  RR )
88 nnm1nn0 9937 . . . . . . . . 9  |-  ( P  e.  NN  ->  ( P  -  1 )  e.  NN0 )
8939, 88syl 17 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( P  -  1 )  e.  NN0 )
9089nn0ge0d 9953 . . . . . . 7  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  0  <_  ( P  -  1 ) )
9158ltm1d 9622 . . . . . . 7  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( P  -  1 )  <  P )
92 modid 10924 . . . . . . 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 1188 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  (
( P  -  1 )  mod  P )  =  ( P  - 
1 ) )
9480, 85, 933eqtr3d 2296 . . . . 5  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( -u 1  mod  P )  =  ( P  - 
1 ) )
9557, 94eqeq12d 2270 . . . 4  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  (
( N  mod  P
)  =  ( -u
1  mod  P )  <->  N  =  ( P  - 
1 ) ) )
96 subneg 9029 . . . . . 6  |-  ( ( N  e.  CC  /\  1  e.  CC )  ->  ( N  -  -u 1
)  =  ( N  +  1 ) )
979, 10, 96sylancl 646 . . . . 5  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( N  -  -u 1 )  =  ( N  + 
1 ) )
9897breq2d 3975 . . . 4  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( P  ||  ( N  -  -u 1 )  <->  P  ||  ( N  +  1 ) ) )
9970, 95, 983bitr3rd 277 . . 3  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( P  ||  ( N  + 
1 )  <->  N  =  ( P  -  1
) ) )
10067, 99orbi12d 693 . 2  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  (
( P  ||  ( N  -  1 )  \/  P  ||  ( N  +  1 ) )  <->  ( N  =  1  \/  N  =  ( P  -  1 ) ) ) )
10138, 100bitrd 246 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 5    -> wi 6    <-> wb 178    \/ wo 359    /\ wa 360    = wceq 1619    e. wcel 1621    C_ wss 3094   class class class wbr 3963   ` cfv 4638  (class class class)co 5757   CCcc 8668   RRcr 8669   0cc0 8670   1c1 8671    + caddc 8673    x. cmul 8675    < clt 8800    <_ cle 8801    - cmin 8970   -ucneg 8971   NNcn 9679   2c2 9728   NN0cn0 9897   ZZcz 9956   ZZ>=cuz 10162   RR+crp 10286   ...cfz 10713    mod cmo 10904   ^cexp 11035    || cdivides 12458   Primecprime 12685
This theorem is referenced by:  wilthlem2  20234
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4071  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449  ax-cnex 8726  ax-resscn 8727  ax-1cn 8728  ax-icn 8729  ax-addcl 8730  ax-addrcl 8731  ax-mulcl 8732  ax-mulrcl 8733  ax-mulcom 8734  ax-addass 8735  ax-mulass 8736  ax-distr 8737  ax-i2m1 8738  ax-1ne0 8739  ax-1rid 8740  ax-rnegex 8741  ax-rrecex 8742  ax-cnre 8743  ax-pre-lttri 8744  ax-pre-lttrn 8745  ax-pre-ltadd 8746  ax-pre-mulgt0 8747  ax-pre-sup 8748
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-int 3804  df-iun 3848  df-br 3964  df-opab 4018  df-mpt 4019  df-tr 4054  df-eprel 4242  df-id 4246  df-po 4251  df-so 4252  df-fr 4289  df-we 4291  df-ord 4332  df-on 4333  df-lim 4334  df-suc 4335  df-om 4594  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-1st 6021  df-2nd 6022  df-iota 6190  df-riota 6237  df-recs 6321  df-rdg 6356  df-1o 6412  df-2o 6413  df-oadd 6416  df-er 6593  df-map 6707  df-en 6797  df-dom 6798  df-sdom 6799  df-fin 6800  df-sup 7127  df-card 7505  df-cda 7727  df-pnf 8802  df-mnf 8803  df-xr 8804  df-ltxr 8805  df-le 8806  df-sub 8972  df-neg 8973  df-div 9357  df-n 9680  df-2 9737  df-3 9738  df-n0 9898  df-z 9957  df-uz 10163  df-rp 10287  df-fz 10714  df-fzo 10802  df-fl 10856  df-mod 10905  df-seq 10978  df-exp 11036  df-hash 11269  df-cj 11514  df-re 11515  df-im 11516  df-sqr 11650  df-abs 11651  df-divides 12459  df-gcd 12613  df-prime 12686  df-phi 12761
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