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Theorem wilthlem1 20719
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 13103,  ( 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 10992 . . . . . . . . . 10  |-  ( N  e.  ( 1 ... ( P  -  1 ) )  ->  N  e.  ZZ )
21adantl 453 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  N  e.  ZZ )
3 peano2zm 10253 . . . . . . . . 9  |-  ( N  e.  ZZ  ->  ( N  -  1 )  e.  ZZ )
42, 3syl 16 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( N  -  1 )  e.  ZZ )
54zcnd 10309 . . . . . . 7  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( N  -  1 )  e.  CC )
62peano2zd 10311 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( N  +  1 )  e.  ZZ )
76zcnd 10309 . . . . . . 7  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( N  +  1 )  e.  CC )
85, 7mulcomd 9043 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  (
( N  -  1 )  x.  ( N  +  1 ) )  =  ( ( N  +  1 )  x.  ( N  -  1 ) ) )
92zcnd 10309 . . . . . . 7  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  N  e.  CC )
10 ax-1cn 8982 . . . . . . 7  |-  1  e.  CC
11 subsq 11416 . . . . . . 7  |-  ( ( N  e.  CC  /\  1  e.  CC )  ->  ( ( N ^
2 )  -  (
1 ^ 2 ) )  =  ( ( N  +  1 )  x.  ( N  - 
1 ) ) )
129, 10, 11sylancl 644 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  (
( N ^ 2 )  -  ( 1 ^ 2 ) )  =  ( ( N  +  1 )  x.  ( N  -  1 ) ) )
139sqvald 11448 . . . . . . 7  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( N ^ 2 )  =  ( N  x.  N
) )
14 sq1 11404 . . . . . . . 8  |-  ( 1 ^ 2 )  =  1
1514a1i 11 . . . . . . 7  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  (
1 ^ 2 )  =  1 )
1613, 15oveq12d 6039 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  (
( N ^ 2 )  -  ( 1 ^ 2 ) )  =  ( ( N  x.  N )  - 
1 ) )
178, 12, 163eqtr2d 2426 . . . . 5  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  (
( N  -  1 )  x.  ( N  +  1 ) )  =  ( ( N  x.  N )  - 
1 ) )
1817breq2d 4166 . . . 4  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( P  ||  ( ( N  -  1 )  x.  ( N  +  1 ) )  <->  P  ||  (
( N  x.  N
)  -  1 ) ) )
19 1e0p1 10343 . . . . . . . 8  |-  1  =  ( 0  +  1 )
2019oveq1i 6031 . . . . . . 7  |-  ( 1 ... ( P  - 
1 ) )  =  ( ( 0  +  1 ) ... ( P  -  1 ) )
21 0z 10226 . . . . . . . 8  |-  0  e.  ZZ
22 fzp1ss 11031 . . . . . . . 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 3322 . . . . . 6  |-  ( 1 ... ( P  - 
1 ) )  C_  ( 0 ... ( P  -  1 ) )
25 simpr 448 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  N  e.  ( 1 ... ( P  -  1 ) ) )
2624, 25sseldi 3290 . . . . 5  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  N  e.  ( 0 ... ( P  -  1 ) ) )
2726biantrurd 495 . . . 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 245 . . 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 444 . . . 4  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  P  e.  Prime )
30 euclemma 13036 . . . 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 1184 . . 3  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( P  ||  ( ( N  -  1 )  x.  ( N  +  1 ) )  <->  ( P  ||  ( N  -  1 )  \/  P  ||  ( N  +  1
) ) ) )
32 prmnn 13010 . . . . 5  |-  ( P  e.  Prime  ->  P  e.  NN )
33 fzm1ndvds 12829 . . . . 5  |-  ( ( P  e.  NN  /\  N  e.  ( 1 ... ( P  - 
1 ) ) )  ->  -.  P  ||  N
)
3432, 33sylan 458 . . . 4  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  -.  P  ||  N )
35 eqid 2388 . . . . 5  |-  ( ( N ^ ( P  -  2 ) )  mod  P )  =  ( ( N ^
( P  -  2 ) )  mod  P
)
3635prmdiveq 13103 . . . 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 1184 . . 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 276 . 2  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( N  =  ( ( N ^ ( P  - 
2 ) )  mod 
P )  <->  ( P  ||  ( N  -  1 )  \/  P  ||  ( N  +  1
) ) ) )
3929, 32syl 16 . . . . 5  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  P  e.  NN )
40 1z 10244 . . . . . 6  |-  1  e.  ZZ
4140a1i 11 . . . . 5  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  1  e.  ZZ )
42 moddvds 12787 . . . . 5  |-  ( ( P  e.  NN  /\  N  e.  ZZ  /\  1  e.  ZZ )  ->  (
( N  mod  P
)  =  ( 1  mod  P )  <->  P  ||  ( N  -  1 ) ) )
4339, 2, 41, 42syl3anc 1184 . . . 4  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  (
( N  mod  P
)  =  ( 1  mod  P )  <->  P  ||  ( N  -  1 ) ) )
44 elfznn 11013 . . . . . . . 8  |-  ( N  e.  ( 1 ... ( P  -  1 ) )  ->  N  e.  NN )
4544adantl 453 . . . . . . 7  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  N  e.  NN )
4645nnred 9948 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  N  e.  RR )
4739nnrpd 10580 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  P  e.  RR+ )
4845nnnn0d 10207 . . . . . . 7  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  N  e.  NN0 )
4948nn0ge0d 10210 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  0  <_  N )
50 elfzle2 10994 . . . . . . . 8  |-  ( N  e.  ( 1 ... ( P  -  1 ) )  ->  N  <_  ( P  -  1 ) )
5150adantl 453 . . . . . . 7  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  N  <_  ( P  -  1 ) )
52 prmz 13011 . . . . . . . 8  |-  ( P  e.  Prime  ->  P  e.  ZZ )
53 zltlem1 10261 . . . . . . . 8  |-  ( ( N  e.  ZZ  /\  P  e.  ZZ )  ->  ( N  <  P  <->  N  <_  ( P  - 
1 ) ) )
541, 52, 53syl2anr 465 . . . . . . 7  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( N  <  P  <->  N  <_  ( P  -  1 ) ) )
5551, 54mpbird 224 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  N  <  P )
56 modid 11198 . . . . . 6  |-  ( ( ( N  e.  RR  /\  P  e.  RR+ )  /\  ( 0  <_  N  /\  N  <  P ) )  ->  ( N  mod  P )  =  N )
5746, 47, 49, 55, 56syl22anc 1185 . . . . 5  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( N  mod  P )  =  N )
5839nnred 9948 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  P  e.  RR )
59 prmuz2 13025 . . . . . . . 8  |-  ( P  e.  Prime  ->  P  e.  ( ZZ>= `  2 )
)
6029, 59syl 16 . . . . . . 7  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  P  e.  ( ZZ>= `  2 )
)
61 eluz2b2 10481 . . . . . . . 8  |-  ( P  e.  ( ZZ>= `  2
)  <->  ( P  e.  NN  /\  1  < 
P ) )
6261simprbi 451 . . . . . . 7  |-  ( P  e.  ( ZZ>= `  2
)  ->  1  <  P )
6360, 62syl 16 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  1  <  P )
64 1mod 11201 . . . . . 6  |-  ( ( P  e.  RR  /\  1  <  P )  -> 
( 1  mod  P
)  =  1 )
6558, 63, 64syl2anc 643 . . . . 5  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  (
1  mod  P )  =  1 )
6657, 65eqeq12d 2402 . . . 4  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  (
( N  mod  P
)  =  ( 1  mod  P )  <->  N  = 
1 ) )
6743, 66bitr3d 247 . . 3  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( P  ||  ( N  - 
1 )  <->  N  = 
1 ) )
6841znegcld 10310 . . . . 5  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  -u 1  e.  ZZ )
69 moddvds 12787 . . . . 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 1184 . . . 4  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  (
( N  mod  P
)  =  ( -u
1  mod  P )  <->  P 
||  ( N  -  -u 1 ) ) )
7139nncnd 9949 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  P  e.  CC )
7271mulid2d 9040 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  (
1  x.  P )  =  P )
7372oveq2d 6037 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( -u 1  +  ( 1  x.  P ) )  =  ( -u 1  +  P ) )
74 neg1cn 10000 . . . . . . . . 9  |-  -u 1  e.  CC
75 addcom 9185 . . . . . . . . 9  |-  ( (
-u 1  e.  CC  /\  P  e.  CC )  ->  ( -u 1  +  P )  =  ( P  +  -u 1
) )
7674, 71, 75sylancr 645 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( -u 1  +  P )  =  ( P  +  -u 1 ) )
77 negsub 9282 . . . . . . . . 9  |-  ( ( P  e.  CC  /\  1  e.  CC )  ->  ( P  +  -u
1 )  =  ( P  -  1 ) )
7871, 10, 77sylancl 644 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( P  +  -u 1 )  =  ( P  - 
1 ) )
7973, 76, 783eqtrd 2424 . . . . . . 7  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( -u 1  +  ( 1  x.  P ) )  =  ( P  - 
1 ) )
8079oveq1d 6036 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  (
( -u 1  +  ( 1  x.  P ) )  mod  P )  =  ( ( P  -  1 )  mod 
P ) )
81 1re 9024 . . . . . . . . 9  |-  1  e.  RR
8281renegcli 9295 . . . . . . . 8  |-  -u 1  e.  RR
8382a1i 11 . . . . . . 7  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  -u 1  e.  RR )
84 modcyc 11204 . . . . . . 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 1184 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  (
( -u 1  +  ( 1  x.  P ) )  mod  P )  =  ( -u 1  mod  P ) )
86 peano2rem 9300 . . . . . . . 8  |-  ( P  e.  RR  ->  ( P  -  1 )  e.  RR )
8758, 86syl 16 . . . . . . 7  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( P  -  1 )  e.  RR )
88 nnm1nn0 10194 . . . . . . . . 9  |-  ( P  e.  NN  ->  ( P  -  1 )  e.  NN0 )
8939, 88syl 16 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( P  -  1 )  e.  NN0 )
9089nn0ge0d 10210 . . . . . . 7  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  0  <_  ( P  -  1 ) )
9158ltm1d 9876 . . . . . . 7  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( P  -  1 )  <  P )
92 modid 11198 . . . . . . 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 1185 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  (
( P  -  1 )  mod  P )  =  ( P  - 
1 ) )
9480, 85, 933eqtr3d 2428 . . . . 5  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( -u 1  mod  P )  =  ( P  - 
1 ) )
9557, 94eqeq12d 2402 . . . 4  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  (
( N  mod  P
)  =  ( -u
1  mod  P )  <->  N  =  ( P  - 
1 ) ) )
96 subneg 9283 . . . . . 6  |-  ( ( N  e.  CC  /\  1  e.  CC )  ->  ( N  -  -u 1
)  =  ( N  +  1 ) )
979, 10, 96sylancl 644 . . . . 5  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( N  -  -u 1 )  =  ( N  + 
1 ) )
9897breq2d 4166 . . . 4  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( P  ||  ( N  -  -u 1 )  <->  P  ||  ( N  +  1 ) ) )
9970, 95, 983bitr3rd 276 . . 3  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( P  ||  ( N  + 
1 )  <->  N  =  ( P  -  1
) ) )
10067, 99orbi12d 691 . 2  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  (
( P  ||  ( N  -  1 )  \/  P  ||  ( N  +  1 ) )  <->  ( N  =  1  \/  N  =  ( P  -  1 ) ) ) )
10138, 100bitrd 245 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 177    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1717    C_ wss 3264   class class class wbr 4154   ` cfv 5395  (class class class)co 6021   CCcc 8922   RRcr 8923   0cc0 8924   1c1 8925    + caddc 8927    x. cmul 8929    < clt 9054    <_ cle 9055    - cmin 9224   -ucneg 9225   NNcn 9933   2c2 9982   NN0cn0 10154   ZZcz 10215   ZZ>=cuz 10421   RR+crp 10545   ...cfz 10976    mod cmo 11178   ^cexp 11310    || cdivides 12780   Primecprime 13007
This theorem is referenced by:  wilthlem2  20720
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001  ax-pre-sup 9002
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-riota 6486  df-recs 6570  df-rdg 6605  df-1o 6661  df-2o 6662  df-oadd 6665  df-er 6842  df-map 6957  df-en 7047  df-dom 7048  df-sdom 7049  df-fin 7050  df-sup 7382  df-card 7760  df-cda 7982  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-div 9611  df-nn 9934  df-2 9991  df-3 9992  df-n0 10155  df-z 10216  df-uz 10422  df-rp 10546  df-fz 10977  df-fzo 11067  df-fl 11130  df-mod 11179  df-seq 11252  df-exp 11311  df-hash 11547  df-cj 11832  df-re 11833  df-im 11834  df-sqr 11968  df-abs 11969  df-dvds 12781  df-gcd 12935  df-prm 13008  df-phi 13083
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