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Theorem wilthlem1 22381
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 13853,  ( 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 11445 . . . . . . . . . 10  |-  ( N  e.  ( 1 ... ( P  -  1 ) )  ->  N  e.  ZZ )
21adantl 466 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  N  e.  ZZ )
3 peano2zm 10680 . . . . . . . . 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 10740 . . . . . . 7  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( N  -  1 )  e.  CC )
62peano2zd 10742 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( N  +  1 )  e.  ZZ )
76zcnd 10740 . . . . . . 7  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( N  +  1 )  e.  CC )
85, 7mulcomd 9399 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  (
( N  -  1 )  x.  ( N  +  1 ) )  =  ( ( N  +  1 )  x.  ( N  -  1 ) ) )
92zcnd 10740 . . . . . . 7  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  N  e.  CC )
10 ax-1cn 9332 . . . . . . 7  |-  1  e.  CC
11 subsq 11965 . . . . . . 7  |-  ( ( N  e.  CC  /\  1  e.  CC )  ->  ( ( N ^
2 )  -  (
1 ^ 2 ) )  =  ( ( N  +  1 )  x.  ( N  - 
1 ) ) )
129, 10, 11sylancl 662 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  (
( N ^ 2 )  -  ( 1 ^ 2 ) )  =  ( ( N  +  1 )  x.  ( N  -  1 ) ) )
139sqvald 11997 . . . . . . 7  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( N ^ 2 )  =  ( N  x.  N
) )
14 sq1 11952 . . . . . . . 8  |-  ( 1 ^ 2 )  =  1
1514a1i 11 . . . . . . 7  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  (
1 ^ 2 )  =  1 )
1613, 15oveq12d 6104 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  (
( N ^ 2 )  -  ( 1 ^ 2 ) )  =  ( ( N  x.  N )  - 
1 ) )
178, 12, 163eqtr2d 2476 . . . . 5  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  (
( N  -  1 )  x.  ( N  +  1 ) )  =  ( ( N  x.  N )  - 
1 ) )
1817breq2d 4299 . . . 4  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( P  ||  ( ( N  -  1 )  x.  ( N  +  1 ) )  <->  P  ||  (
( N  x.  N
)  -  1 ) ) )
19 1e0p1 10775 . . . . . . . 8  |-  1  =  ( 0  +  1 )
2019oveq1i 6096 . . . . . . 7  |-  ( 1 ... ( P  - 
1 ) )  =  ( ( 0  +  1 ) ... ( P  -  1 ) )
21 0z 10649 . . . . . . . 8  |-  0  e.  ZZ
22 fzp1ss 11498 . . . . . . . 8  |-  ( 0  e.  ZZ  ->  (
( 0  +  1 ) ... ( P  -  1 ) ) 
C_  ( 0 ... ( P  -  1 ) ) )
2321, 22ax-mp 5 . . . . . . 7  |-  ( ( 0  +  1 ) ... ( P  - 
1 ) )  C_  ( 0 ... ( P  -  1 ) )
2420, 23eqsstri 3381 . . . . . 6  |-  ( 1 ... ( P  - 
1 ) )  C_  ( 0 ... ( P  -  1 ) )
25 simpr 461 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  N  e.  ( 1 ... ( P  -  1 ) ) )
2624, 25sseldi 3349 . . . . 5  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  N  e.  ( 0 ... ( P  -  1 ) ) )
2726biantrurd 508 . . . 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 253 . . 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 457 . . . 4  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  P  e.  Prime )
30 euclemma 13786 . . . 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 1218 . . 3  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( P  ||  ( ( N  -  1 )  x.  ( N  +  1 ) )  <->  ( P  ||  ( N  -  1 )  \/  P  ||  ( N  +  1
) ) ) )
32 prmnn 13758 . . . . 5  |-  ( P  e.  Prime  ->  P  e.  NN )
33 fzm1ndvds 13577 . . . . 5  |-  ( ( P  e.  NN  /\  N  e.  ( 1 ... ( P  - 
1 ) ) )  ->  -.  P  ||  N
)
3432, 33sylan 471 . . . 4  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  -.  P  ||  N )
35 eqid 2438 . . . . 5  |-  ( ( N ^ ( P  -  2 ) )  mod  P )  =  ( ( N ^
( P  -  2 ) )  mod  P
)
3635prmdiveq 13853 . . . 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 1218 . . 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 284 . 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 1zzd 10669 . . . . 5  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  1  e.  ZZ )
41 moddvds 13534 . . . . 5  |-  ( ( P  e.  NN  /\  N  e.  ZZ  /\  1  e.  ZZ )  ->  (
( N  mod  P
)  =  ( 1  mod  P )  <->  P  ||  ( N  -  1 ) ) )
4239, 2, 40, 41syl3anc 1218 . . . 4  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  (
( N  mod  P
)  =  ( 1  mod  P )  <->  P  ||  ( N  -  1 ) ) )
43 elfznn 11470 . . . . . . . 8  |-  ( N  e.  ( 1 ... ( P  -  1 ) )  ->  N  e.  NN )
4443adantl 466 . . . . . . 7  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  N  e.  NN )
4544nnred 10329 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  N  e.  RR )
4639nnrpd 11018 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  P  e.  RR+ )
4744nnnn0d 10628 . . . . . . 7  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  N  e.  NN0 )
4847nn0ge0d 10631 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  0  <_  N )
49 elfzle2 11447 . . . . . . . 8  |-  ( N  e.  ( 1 ... ( P  -  1 ) )  ->  N  <_  ( P  -  1 ) )
5049adantl 466 . . . . . . 7  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  N  <_  ( P  -  1 ) )
51 prmz 13759 . . . . . . . 8  |-  ( P  e.  Prime  ->  P  e.  ZZ )
52 zltlem1 10689 . . . . . . . 8  |-  ( ( N  e.  ZZ  /\  P  e.  ZZ )  ->  ( N  <  P  <->  N  <_  ( P  - 
1 ) ) )
531, 51, 52syl2anr 478 . . . . . . 7  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( N  <  P  <->  N  <_  ( P  -  1 ) ) )
5450, 53mpbird 232 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  N  <  P )
55 modid 11724 . . . . . 6  |-  ( ( ( N  e.  RR  /\  P  e.  RR+ )  /\  ( 0  <_  N  /\  N  <  P ) )  ->  ( N  mod  P )  =  N )
5645, 46, 48, 54, 55syl22anc 1219 . . . . 5  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( N  mod  P )  =  N )
5739nnred 10329 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  P  e.  RR )
58 prmuz2 13773 . . . . . . . 8  |-  ( P  e.  Prime  ->  P  e.  ( ZZ>= `  2 )
)
5929, 58syl 16 . . . . . . 7  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  P  e.  ( ZZ>= `  2 )
)
60 eluz2b2 10919 . . . . . . . 8  |-  ( P  e.  ( ZZ>= `  2
)  <->  ( P  e.  NN  /\  1  < 
P ) )
6160simprbi 464 . . . . . . 7  |-  ( P  e.  ( ZZ>= `  2
)  ->  1  <  P )
6259, 61syl 16 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  1  <  P )
63 1mod 11732 . . . . . 6  |-  ( ( P  e.  RR  /\  1  <  P )  -> 
( 1  mod  P
)  =  1 )
6457, 62, 63syl2anc 661 . . . . 5  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  (
1  mod  P )  =  1 )
6556, 64eqeq12d 2452 . . . 4  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  (
( N  mod  P
)  =  ( 1  mod  P )  <->  N  = 
1 ) )
6642, 65bitr3d 255 . . 3  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( P  ||  ( N  - 
1 )  <->  N  = 
1 ) )
6740znegcld 10741 . . . . 5  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  -u 1  e.  ZZ )
68 moddvds 13534 . . . . 5  |-  ( ( P  e.  NN  /\  N  e.  ZZ  /\  -u 1  e.  ZZ )  ->  (
( N  mod  P
)  =  ( -u
1  mod  P )  <->  P 
||  ( N  -  -u 1 ) ) )
6939, 2, 67, 68syl3anc 1218 . . . 4  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  (
( N  mod  P
)  =  ( -u
1  mod  P )  <->  P 
||  ( N  -  -u 1 ) ) )
7039nncnd 10330 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  P  e.  CC )
7170mulid2d 9396 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  (
1  x.  P )  =  P )
7271oveq2d 6102 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( -u 1  +  ( 1  x.  P ) )  =  ( -u 1  +  P ) )
73 neg1cn 10417 . . . . . . . . 9  |-  -u 1  e.  CC
74 addcom 9547 . . . . . . . . 9  |-  ( (
-u 1  e.  CC  /\  P  e.  CC )  ->  ( -u 1  +  P )  =  ( P  +  -u 1
) )
7573, 70, 74sylancr 663 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( -u 1  +  P )  =  ( P  +  -u 1 ) )
76 negsub 9649 . . . . . . . . 9  |-  ( ( P  e.  CC  /\  1  e.  CC )  ->  ( P  +  -u
1 )  =  ( P  -  1 ) )
7770, 10, 76sylancl 662 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( P  +  -u 1 )  =  ( P  - 
1 ) )
7872, 75, 773eqtrd 2474 . . . . . . 7  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( -u 1  +  ( 1  x.  P ) )  =  ( P  - 
1 ) )
7978oveq1d 6101 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  (
( -u 1  +  ( 1  x.  P ) )  mod  P )  =  ( ( P  -  1 )  mod 
P ) )
80 neg1rr 10418 . . . . . . . 8  |-  -u 1  e.  RR
8180a1i 11 . . . . . . 7  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  -u 1  e.  RR )
82 modcyc 11735 . . . . . . 7  |-  ( (
-u 1  e.  RR  /\  P  e.  RR+  /\  1  e.  ZZ )  ->  (
( -u 1  +  ( 1  x.  P ) )  mod  P )  =  ( -u 1  mod  P ) )
8381, 46, 40, 82syl3anc 1218 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  (
( -u 1  +  ( 1  x.  P ) )  mod  P )  =  ( -u 1  mod  P ) )
84 peano2rem 9667 . . . . . . . 8  |-  ( P  e.  RR  ->  ( P  -  1 )  e.  RR )
8557, 84syl 16 . . . . . . 7  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( P  -  1 )  e.  RR )
86 nnm1nn0 10613 . . . . . . . . 9  |-  ( P  e.  NN  ->  ( P  -  1 )  e.  NN0 )
8739, 86syl 16 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( P  -  1 )  e.  NN0 )
8887nn0ge0d 10631 . . . . . . 7  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  0  <_  ( P  -  1 ) )
8957ltm1d 10257 . . . . . . 7  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( P  -  1 )  <  P )
90 modid 11724 . . . . . . 7  |-  ( ( ( ( P  - 
1 )  e.  RR  /\  P  e.  RR+ )  /\  ( 0  <_  ( P  -  1 )  /\  ( P  - 
1 )  <  P
) )  ->  (
( P  -  1 )  mod  P )  =  ( P  - 
1 ) )
9185, 46, 88, 89, 90syl22anc 1219 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  (
( P  -  1 )  mod  P )  =  ( P  - 
1 ) )
9279, 83, 913eqtr3d 2478 . . . . 5  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( -u 1  mod  P )  =  ( P  - 
1 ) )
9356, 92eqeq12d 2452 . . . 4  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  (
( N  mod  P
)  =  ( -u
1  mod  P )  <->  N  =  ( P  - 
1 ) ) )
94 subneg 9650 . . . . . 6  |-  ( ( N  e.  CC  /\  1  e.  CC )  ->  ( N  -  -u 1
)  =  ( N  +  1 ) )
959, 10, 94sylancl 662 . . . . 5  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( N  -  -u 1 )  =  ( N  + 
1 ) )
9695breq2d 4299 . . . 4  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( P  ||  ( N  -  -u 1 )  <->  P  ||  ( N  +  1 ) ) )
9769, 93, 963bitr3rd 284 . . 3  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( P  ||  ( N  + 
1 )  <->  N  =  ( P  -  1
) ) )
9866, 97orbi12d 709 . 2  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  (
( P  ||  ( N  -  1 )  \/  P  ||  ( N  +  1 ) )  <->  ( N  =  1  \/  N  =  ( P  -  1 ) ) ) )
9938, 98bitrd 253 1  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( N  =  ( ( N ^ ( P  - 
2 ) )  mod 
P )  <->  ( N  =  1  \/  N  =  ( P  - 
1 ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1369    e. wcel 1756    C_ wss 3323   class class class wbr 4287   ` cfv 5413  (class class class)co 6086   CCcc 9272   RRcr 9273   0cc0 9274   1c1 9275    + caddc 9277    x. cmul 9279    < clt 9410    <_ cle 9411    - cmin 9587   -ucneg 9588   NNcn 10314   2c2 10363   NN0cn0 10571   ZZcz 10638   ZZ>=cuz 10853   RR+crp 10983   ...cfz 11429    mod cmo 11700   ^cexp 11857    || cdivides 13527   Primecprime 13755
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 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352
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 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-1o 6912  df-2o 6913  df-oadd 6916  df-er 7093  df-map 7208  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-sup 7683  df-card 8101  df-cda 8329  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-n0 10572  df-z 10639  df-uz 10854  df-rp 10984  df-fz 11430  df-fzo 11541  df-fl 11634  df-mod 11701  df-seq 11799  df-exp 11858  df-hash 12096  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-dvds 13528  df-gcd 13683  df-prm 13756  df-phi 13833
This theorem is referenced by:  wilthlem2  22382
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