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Theorem wilthlem1 24005
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 14746,  ( 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 11807 . . . . . . . . . 10  |-  ( N  e.  ( 1 ... ( P  -  1 ) )  ->  N  e.  ZZ )
21adantl 468 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  N  e.  ZZ )
3 peano2zm 10987 . . . . . . . . 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 11048 . . . . . . 7  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( N  -  1 )  e.  CC )
62peano2zd 11050 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( N  +  1 )  e.  ZZ )
76zcnd 11048 . . . . . . 7  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( N  +  1 )  e.  CC )
85, 7mulcomd 9669 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  (
( N  -  1 )  x.  ( N  +  1 ) )  =  ( ( N  +  1 )  x.  ( N  -  1 ) ) )
92zcnd 11048 . . . . . . 7  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  N  e.  CC )
10 ax-1cn 9602 . . . . . . 7  |-  1  e.  CC
11 subsq 12389 . . . . . . 7  |-  ( ( N  e.  CC  /\  1  e.  CC )  ->  ( ( N ^
2 )  -  (
1 ^ 2 ) )  =  ( ( N  +  1 )  x.  ( N  - 
1 ) ) )
129, 10, 11sylancl 669 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  (
( N ^ 2 )  -  ( 1 ^ 2 ) )  =  ( ( N  +  1 )  x.  ( N  -  1 ) ) )
139sqvald 12420 . . . . . . 7  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( N ^ 2 )  =  ( N  x.  N
) )
14 sq1 12376 . . . . . . . 8  |-  ( 1 ^ 2 )  =  1
1514a1i 11 . . . . . . 7  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  (
1 ^ 2 )  =  1 )
1613, 15oveq12d 6313 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  (
( N ^ 2 )  -  ( 1 ^ 2 ) )  =  ( ( N  x.  N )  - 
1 ) )
178, 12, 163eqtr2d 2493 . . . . 5  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  (
( N  -  1 )  x.  ( N  +  1 ) )  =  ( ( N  x.  N )  - 
1 ) )
1817breq2d 4417 . . . 4  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( P  ||  ( ( N  -  1 )  x.  ( N  +  1 ) )  <->  P  ||  (
( N  x.  N
)  -  1 ) ) )
19 1e0p1 11086 . . . . . . . 8  |-  1  =  ( 0  +  1 )
2019oveq1i 6305 . . . . . . 7  |-  ( 1 ... ( P  - 
1 ) )  =  ( ( 0  +  1 ) ... ( P  -  1 ) )
21 0z 10955 . . . . . . . 8  |-  0  e.  ZZ
22 fzp1ss 11854 . . . . . . . 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 3464 . . . . . 6  |-  ( 1 ... ( P  - 
1 ) )  C_  ( 0 ... ( P  -  1 ) )
25 simpr 463 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  N  e.  ( 1 ... ( P  -  1 ) ) )
2624, 25sseldi 3432 . . . . 5  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  N  e.  ( 0 ... ( P  -  1 ) ) )
2726biantrurd 511 . . . 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 257 . . 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 459 . . . 4  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  P  e.  Prime )
30 euclemma 14677 . . . 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 1269 . . 3  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( P  ||  ( ( N  -  1 )  x.  ( N  +  1 ) )  <->  ( P  ||  ( N  -  1 )  \/  P  ||  ( N  +  1
) ) ) )
32 prmnn 14637 . . . . 5  |-  ( P  e.  Prime  ->  P  e.  NN )
33 fzm1ndvds 14369 . . . . 5  |-  ( ( P  e.  NN  /\  N  e.  ( 1 ... ( P  - 
1 ) ) )  ->  -.  P  ||  N
)
3432, 33sylan 474 . . . 4  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  -.  P  ||  N )
35 eqid 2453 . . . . 5  |-  ( ( N ^ ( P  -  2 ) )  mod  P )  =  ( ( N ^
( P  -  2 ) )  mod  P
)
3635prmdiveq 14746 . . . 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 1269 . . 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 288 . 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 1zzd 10975 . . . . 5  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  1  e.  ZZ )
41 moddvds 14324 . . . . 5  |-  ( ( P  e.  NN  /\  N  e.  ZZ  /\  1  e.  ZZ )  ->  (
( N  mod  P
)  =  ( 1  mod  P )  <->  P  ||  ( N  -  1 ) ) )
4239, 2, 40, 41syl3anc 1269 . . . 4  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  (
( N  mod  P
)  =  ( 1  mod  P )  <->  P  ||  ( N  -  1 ) ) )
43 elfznn 11835 . . . . . . . 8  |-  ( N  e.  ( 1 ... ( P  -  1 ) )  ->  N  e.  NN )
4443adantl 468 . . . . . . 7  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  N  e.  NN )
4544nnred 10631 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  N  e.  RR )
4639nnrpd 11346 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  P  e.  RR+ )
4744nnnn0d 10932 . . . . . . 7  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  N  e.  NN0 )
4847nn0ge0d 10935 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  0  <_  N )
49 elfzle2 11810 . . . . . . . 8  |-  ( N  e.  ( 1 ... ( P  -  1 ) )  ->  N  <_  ( P  -  1 ) )
5049adantl 468 . . . . . . 7  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  N  <_  ( P  -  1 ) )
51 prmz 14638 . . . . . . . 8  |-  ( P  e.  Prime  ->  P  e.  ZZ )
52 zltlem1 10996 . . . . . . . 8  |-  ( ( N  e.  ZZ  /\  P  e.  ZZ )  ->  ( N  <  P  <->  N  <_  ( P  - 
1 ) ) )
531, 51, 52syl2anr 481 . . . . . . 7  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( N  <  P  <->  N  <_  ( P  -  1 ) ) )
5450, 53mpbird 236 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  N  <  P )
55 modid 12128 . . . . . 6  |-  ( ( ( N  e.  RR  /\  P  e.  RR+ )  /\  ( 0  <_  N  /\  N  <  P ) )  ->  ( N  mod  P )  =  N )
5645, 46, 48, 54, 55syl22anc 1270 . . . . 5  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( N  mod  P )  =  N )
5739nnred 10631 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  P  e.  RR )
58 prmuz2 14654 . . . . . . . 8  |-  ( P  e.  Prime  ->  P  e.  ( ZZ>= `  2 )
)
5929, 58syl 17 . . . . . . 7  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  P  e.  ( ZZ>= `  2 )
)
60 eluz2b2 11238 . . . . . . . 8  |-  ( P  e.  ( ZZ>= `  2
)  <->  ( P  e.  NN  /\  1  < 
P ) )
6160simprbi 466 . . . . . . 7  |-  ( P  e.  ( ZZ>= `  2
)  ->  1  <  P )
6259, 61syl 17 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  1  <  P )
63 1mod 12136 . . . . . 6  |-  ( ( P  e.  RR  /\  1  <  P )  -> 
( 1  mod  P
)  =  1 )
6457, 62, 63syl2anc 667 . . . . 5  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  (
1  mod  P )  =  1 )
6556, 64eqeq12d 2468 . . . 4  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  (
( N  mod  P
)  =  ( 1  mod  P )  <->  N  = 
1 ) )
6642, 65bitr3d 259 . . 3  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( P  ||  ( N  - 
1 )  <->  N  = 
1 ) )
6740znegcld 11049 . . . . 5  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  -u 1  e.  ZZ )
68 moddvds 14324 . . . . 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 1269 . . . 4  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  (
( N  mod  P
)  =  ( -u
1  mod  P )  <->  P 
||  ( N  -  -u 1 ) ) )
7039nncnd 10632 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  P  e.  CC )
7170mulid2d 9666 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  (
1  x.  P )  =  P )
7271oveq2d 6311 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( -u 1  +  ( 1  x.  P ) )  =  ( -u 1  +  P ) )
73 neg1cn 10720 . . . . . . . . 9  |-  -u 1  e.  CC
74 addcom 9824 . . . . . . . . 9  |-  ( (
-u 1  e.  CC  /\  P  e.  CC )  ->  ( -u 1  +  P )  =  ( P  +  -u 1
) )
7573, 70, 74sylancr 670 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( -u 1  +  P )  =  ( P  +  -u 1 ) )
76 negsub 9927 . . . . . . . . 9  |-  ( ( P  e.  CC  /\  1  e.  CC )  ->  ( P  +  -u
1 )  =  ( P  -  1 ) )
7770, 10, 76sylancl 669 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( P  +  -u 1 )  =  ( P  - 
1 ) )
7872, 75, 773eqtrd 2491 . . . . . . 7  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( -u 1  +  ( 1  x.  P ) )  =  ( P  - 
1 ) )
7978oveq1d 6310 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  (
( -u 1  +  ( 1  x.  P ) )  mod  P )  =  ( ( P  -  1 )  mod 
P ) )
80 neg1rr 10721 . . . . . . . 8  |-  -u 1  e.  RR
8180a1i 11 . . . . . . 7  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  -u 1  e.  RR )
82 modcyc 12139 . . . . . . 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 1269 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  (
( -u 1  +  ( 1  x.  P ) )  mod  P )  =  ( -u 1  mod  P ) )
84 peano2rem 9946 . . . . . . . 8  |-  ( P  e.  RR  ->  ( P  -  1 )  e.  RR )
8557, 84syl 17 . . . . . . 7  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( P  -  1 )  e.  RR )
86 nnm1nn0 10918 . . . . . . . . 9  |-  ( P  e.  NN  ->  ( P  -  1 )  e.  NN0 )
8739, 86syl 17 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( P  -  1 )  e.  NN0 )
8887nn0ge0d 10935 . . . . . . 7  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  0  <_  ( P  -  1 ) )
8957ltm1d 10546 . . . . . . 7  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( P  -  1 )  <  P )
90 modid 12128 . . . . . . 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 1270 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  (
( P  -  1 )  mod  P )  =  ( P  - 
1 ) )
9279, 83, 913eqtr3d 2495 . . . . 5  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( -u 1  mod  P )  =  ( P  - 
1 ) )
9356, 92eqeq12d 2468 . . . 4  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  (
( N  mod  P
)  =  ( -u
1  mod  P )  <->  N  =  ( P  - 
1 ) ) )
94 subneg 9928 . . . . . 6  |-  ( ( N  e.  CC  /\  1  e.  CC )  ->  ( N  -  -u 1
)  =  ( N  +  1 ) )
959, 10, 94sylancl 669 . . . . 5  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( N  -  -u 1 )  =  ( N  + 
1 ) )
9695breq2d 4417 . . . 4  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( P  ||  ( N  -  -u 1 )  <->  P  ||  ( N  +  1 ) ) )
9769, 93, 963bitr3rd 288 . . 3  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( P  ||  ( N  + 
1 )  <->  N  =  ( P  -  1
) ) )
9866, 97orbi12d 717 . 2  |-  ( ( P  e.  Prime  /\  N  e.  ( 1 ... ( P  -  1 ) ) )  ->  (
( P  ||  ( N  -  1 )  \/  P  ||  ( N  +  1 ) )  <->  ( N  =  1  \/  N  =  ( P  -  1 ) ) ) )
9938, 98bitrd 257 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 188    \/ wo 370    /\ wa 371    = wceq 1446    e. wcel 1889    C_ wss 3406   class class class wbr 4405   ` cfv 5585  (class class class)co 6295   CCcc 9542   RRcr 9543   0cc0 9544   1c1 9545    + caddc 9547    x. cmul 9549    < clt 9680    <_ cle 9681    - cmin 9865   -ucneg 9866   NNcn 10616   2c2 10666   NN0cn0 10876   ZZcz 10944   ZZ>=cuz 11166   RR+crp 11309   ...cfz 11791    mod cmo 12103   ^cexp 12279    || cdvds 14317   Primecprime 14634
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621  ax-pre-sup 9622
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-int 4238  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-2o 7188  df-oadd 7191  df-er 7368  df-map 7479  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-sup 7961  df-inf 7962  df-card 8378  df-cda 8603  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868  df-div 10277  df-nn 10617  df-2 10675  df-3 10676  df-n0 10877  df-z 10945  df-uz 11167  df-rp 11310  df-fz 11792  df-fzo 11923  df-fl 12035  df-mod 12104  df-seq 12221  df-exp 12280  df-hash 12523  df-cj 13174  df-re 13175  df-im 13176  df-sqrt 13310  df-abs 13311  df-dvds 14318  df-gcd 14481  df-prm 14635  df-phi 14726
This theorem is referenced by:  wilthlem2  24006
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