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Theorem pcprmpw2 12279
Description: Self-referential expression for a prime power. (Contributed by Mario Carneiro, 16-Jan-2015.)
Assertion
Ref Expression
pcprmpw2  |-  ( ( P  e.  Prime  /\  A  e.  NN )  ->  ( E. n  e.  NN0  A 
||  ( P ^
n )  <->  A  =  ( P ^ ( P 
pCnt  A ) ) ) )
Distinct variable groups:    A, n    P, n

Proof of Theorem pcprmpw2
Dummy variable  p is distinct from all other variables.
StepHypRef Expression
1 simplr 525 . . . . 5  |-  ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  ->  A  e.  NN )
21nnnn0d 9181 . . . 4  |-  ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  ->  A  e.  NN0 )
3 prmnn 12057 . . . . . . 7  |-  ( P  e.  Prime  ->  P  e.  NN )
43ad2antrr 485 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  ->  P  e.  NN )
5 pccl 12246 . . . . . . 7  |-  ( ( P  e.  Prime  /\  A  e.  NN )  ->  ( P  pCnt  A )  e. 
NN0 )
65adantr 274 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  ->  ( P  pCnt  A )  e.  NN0 )
74, 6nnexpcld 10624 . . . . 5  |-  ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  ->  ( P ^ ( P  pCnt  A ) )  e.  NN )
87nnnn0d 9181 . . . 4  |-  ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  ->  ( P ^ ( P  pCnt  A ) )  e.  NN0 )
96nn0red 9182 . . . . . . . . . . 11  |-  ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  ->  ( P  pCnt  A )  e.  RR )
109leidd 8426 . . . . . . . . . 10  |-  ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  ->  ( P  pCnt  A )  <_  ( P  pCnt  A ) )
11 simpll 524 . . . . . . . . . . 11  |-  ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  ->  P  e.  Prime )
126nn0zd 9325 . . . . . . . . . . 11  |-  ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  ->  ( P  pCnt  A )  e.  ZZ )
13 pcid 12270 . . . . . . . . . . 11  |-  ( ( P  e.  Prime  /\  ( P  pCnt  A )  e.  ZZ )  ->  ( P  pCnt  ( P ^
( P  pCnt  A
) ) )  =  ( P  pCnt  A
) )
1411, 12, 13syl2anc 409 . . . . . . . . . 10  |-  ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  ->  ( P  pCnt  ( P ^ ( P  pCnt  A ) ) )  =  ( P 
pCnt  A ) )
1510, 14breqtrrd 4015 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  ->  ( P  pCnt  A )  <_  ( P  pCnt  ( P ^
( P  pCnt  A
) ) ) )
1615ad2antrr 485 . . . . . . . 8  |-  ( ( ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  (
n  e.  NN0  /\  A  ||  ( P ^
n ) ) )  /\  p  e.  Prime )  /\  p  =  P )  ->  ( P  pCnt  A )  <_  ( P  pCnt  ( P ^
( P  pCnt  A
) ) ) )
17 simpr 109 . . . . . . . . 9  |-  ( ( ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  (
n  e.  NN0  /\  A  ||  ( P ^
n ) ) )  /\  p  e.  Prime )  /\  p  =  P )  ->  p  =  P )
1817oveq1d 5866 . . . . . . . 8  |-  ( ( ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  (
n  e.  NN0  /\  A  ||  ( P ^
n ) ) )  /\  p  e.  Prime )  /\  p  =  P )  ->  ( p  pCnt  A )  =  ( P  pCnt  A )
)
1917oveq1d 5866 . . . . . . . 8  |-  ( ( ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  (
n  e.  NN0  /\  A  ||  ( P ^
n ) ) )  /\  p  e.  Prime )  /\  p  =  P )  ->  ( p  pCnt  ( P ^ ( P  pCnt  A ) ) )  =  ( P 
pCnt  ( P ^
( P  pCnt  A
) ) ) )
2016, 18, 193brtr4d 4019 . . . . . . 7  |-  ( ( ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  (
n  e.  NN0  /\  A  ||  ( P ^
n ) ) )  /\  p  e.  Prime )  /\  p  =  P )  ->  ( p  pCnt  A )  <_  (
p  pCnt  ( P ^ ( P  pCnt  A ) ) ) )
21 simplrr 531 . . . . . . . . . . . . 13  |-  ( ( ( ( P  e. 
Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  /\  p  e. 
Prime )  ->  A  ||  ( P ^ n ) )
22 prmz 12058 . . . . . . . . . . . . . . 15  |-  ( p  e.  Prime  ->  p  e.  ZZ )
2322adantl 275 . . . . . . . . . . . . . 14  |-  ( ( ( ( P  e. 
Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  /\  p  e. 
Prime )  ->  p  e.  ZZ )
241adantr 274 . . . . . . . . . . . . . . 15  |-  ( ( ( ( P  e. 
Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  /\  p  e. 
Prime )  ->  A  e.  NN )
2524nnzd 9326 . . . . . . . . . . . . . 14  |-  ( ( ( ( P  e. 
Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  /\  p  e. 
Prime )  ->  A  e.  ZZ )
26 simprl 526 . . . . . . . . . . . . . . . . 17  |-  ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  ->  n  e.  NN0 )
274, 26nnexpcld 10624 . . . . . . . . . . . . . . . 16  |-  ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  ->  ( P ^ n )  e.  NN )
2827adantr 274 . . . . . . . . . . . . . . 15  |-  ( ( ( ( P  e. 
Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  /\  p  e. 
Prime )  ->  ( P ^ n )  e.  NN )
2928nnzd 9326 . . . . . . . . . . . . . 14  |-  ( ( ( ( P  e. 
Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  /\  p  e. 
Prime )  ->  ( P ^ n )  e.  ZZ )
30 dvdstr 11783 . . . . . . . . . . . . . 14  |-  ( ( p  e.  ZZ  /\  A  e.  ZZ  /\  ( P ^ n )  e.  ZZ )  ->  (
( p  ||  A  /\  A  ||  ( P ^ n ) )  ->  p  ||  ( P ^ n ) ) )
3123, 25, 29, 30syl3anc 1233 . . . . . . . . . . . . 13  |-  ( ( ( ( P  e. 
Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  /\  p  e. 
Prime )  ->  ( ( p  ||  A  /\  A  ||  ( P ^
n ) )  ->  p  ||  ( P ^
n ) ) )
3221, 31mpan2d 426 . . . . . . . . . . . 12  |-  ( ( ( ( P  e. 
Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  /\  p  e. 
Prime )  ->  ( p 
||  A  ->  p  ||  ( P ^ n
) ) )
33 simpr 109 . . . . . . . . . . . . 13  |-  ( ( ( ( P  e. 
Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  /\  p  e. 
Prime )  ->  p  e. 
Prime )
3411adantr 274 . . . . . . . . . . . . 13  |-  ( ( ( ( P  e. 
Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  /\  p  e. 
Prime )  ->  P  e. 
Prime )
35 simplrl 530 . . . . . . . . . . . . 13  |-  ( ( ( ( P  e. 
Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  /\  p  e. 
Prime )  ->  n  e. 
NN0 )
36 prmdvdsexpr 12097 . . . . . . . . . . . . 13  |-  ( ( p  e.  Prime  /\  P  e.  Prime  /\  n  e.  NN0 )  ->  ( p  ||  ( P ^ n
)  ->  p  =  P ) )
3733, 34, 35, 36syl3anc 1233 . . . . . . . . . . . 12  |-  ( ( ( ( P  e. 
Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  /\  p  e. 
Prime )  ->  ( p 
||  ( P ^
n )  ->  p  =  P ) )
3832, 37syld 45 . . . . . . . . . . 11  |-  ( ( ( ( P  e. 
Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  /\  p  e. 
Prime )  ->  ( p 
||  A  ->  p  =  P ) )
3938necon3ad 2382 . . . . . . . . . 10  |-  ( ( ( ( P  e. 
Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  /\  p  e. 
Prime )  ->  ( p  =/=  P  ->  -.  p  ||  A ) )
4039imp 123 . . . . . . . . 9  |-  ( ( ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  (
n  e.  NN0  /\  A  ||  ( P ^
n ) ) )  /\  p  e.  Prime )  /\  p  =/=  P
)  ->  -.  p  ||  A )
41 simplr 525 . . . . . . . . . 10  |-  ( ( ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  (
n  e.  NN0  /\  A  ||  ( P ^
n ) ) )  /\  p  e.  Prime )  /\  p  =/=  P
)  ->  p  e.  Prime )
421ad2antrr 485 . . . . . . . . . 10  |-  ( ( ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  (
n  e.  NN0  /\  A  ||  ( P ^
n ) ) )  /\  p  e.  Prime )  /\  p  =/=  P
)  ->  A  e.  NN )
43 pceq0 12268 . . . . . . . . . 10  |-  ( ( p  e.  Prime  /\  A  e.  NN )  ->  (
( p  pCnt  A
)  =  0  <->  -.  p  ||  A ) )
4441, 42, 43syl2anc 409 . . . . . . . . 9  |-  ( ( ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  (
n  e.  NN0  /\  A  ||  ( P ^
n ) ) )  /\  p  e.  Prime )  /\  p  =/=  P
)  ->  ( (
p  pCnt  A )  =  0  <->  -.  p  ||  A ) )
4540, 44mpbird 166 . . . . . . . 8  |-  ( ( ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  (
n  e.  NN0  /\  A  ||  ( P ^
n ) ) )  /\  p  e.  Prime )  /\  p  =/=  P
)  ->  ( p  pCnt  A )  =  0 )
467ad2antrr 485 . . . . . . . . . 10  |-  ( ( ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  (
n  e.  NN0  /\  A  ||  ( P ^
n ) ) )  /\  p  e.  Prime )  /\  p  =/=  P
)  ->  ( P ^ ( P  pCnt  A ) )  e.  NN )
4741, 46pccld 12247 . . . . . . . . 9  |-  ( ( ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  (
n  e.  NN0  /\  A  ||  ( P ^
n ) ) )  /\  p  e.  Prime )  /\  p  =/=  P
)  ->  ( p  pCnt  ( P ^ ( P  pCnt  A ) ) )  e.  NN0 )
4847nn0ge0d 9184 . . . . . . . 8  |-  ( ( ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  (
n  e.  NN0  /\  A  ||  ( P ^
n ) ) )  /\  p  e.  Prime )  /\  p  =/=  P
)  ->  0  <_  ( p  pCnt  ( P ^ ( P  pCnt  A ) ) ) )
4945, 48eqbrtrd 4009 . . . . . . 7  |-  ( ( ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  (
n  e.  NN0  /\  A  ||  ( P ^
n ) ) )  /\  p  e.  Prime )  /\  p  =/=  P
)  ->  ( p  pCnt  A )  <_  (
p  pCnt  ( P ^ ( P  pCnt  A ) ) ) )
50 prmz 12058 . . . . . . . . . . 11  |-  ( P  e.  Prime  ->  P  e.  ZZ )
5150adantr 274 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  A  e.  NN )  ->  P  e.  ZZ )
5251ad2antrr 485 . . . . . . . . 9  |-  ( ( ( ( P  e. 
Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  /\  p  e. 
Prime )  ->  P  e.  ZZ )
53 zdceq 9280 . . . . . . . . 9  |-  ( ( p  e.  ZZ  /\  P  e.  ZZ )  -> DECID  p  =  P )
5423, 52, 53syl2anc 409 . . . . . . . 8  |-  ( ( ( ( P  e. 
Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  /\  p  e. 
Prime )  -> DECID  p  =  P
)
55 dcne 2351 . . . . . . . 8  |-  (DECID  p  =  P  <->  ( p  =  P  \/  p  =/= 
P ) )
5654, 55sylib 121 . . . . . . 7  |-  ( ( ( ( P  e. 
Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  /\  p  e. 
Prime )  ->  ( p  =  P  \/  p  =/=  P ) )
5720, 49, 56mpjaodan 793 . . . . . 6  |-  ( ( ( ( P  e. 
Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  /\  p  e. 
Prime )  ->  ( p 
pCnt  A )  <_  (
p  pCnt  ( P ^ ( P  pCnt  A ) ) ) )
5857ralrimiva 2543 . . . . 5  |-  ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  ->  A. p  e.  Prime  ( p  pCnt  A )  <_  ( p  pCnt  ( P ^ ( P  pCnt  A ) ) ) )
591nnzd 9326 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  ->  A  e.  ZZ )
607nnzd 9326 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  ->  ( P ^ ( P  pCnt  A ) )  e.  ZZ )
61 pc2dvds 12276 . . . . . 6  |-  ( ( A  e.  ZZ  /\  ( P ^ ( P 
pCnt  A ) )  e.  ZZ )  ->  ( A  ||  ( P ^
( P  pCnt  A
) )  <->  A. p  e.  Prime  ( p  pCnt  A )  <_  ( p  pCnt  ( P ^ ( P  pCnt  A ) ) ) ) )
6259, 60, 61syl2anc 409 . . . . 5  |-  ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  ->  ( A  ||  ( P ^ ( P  pCnt  A ) )  <->  A. p  e.  Prime  ( p  pCnt  A )  <_  ( p  pCnt  ( P ^ ( P  pCnt  A ) ) ) ) )
6358, 62mpbird 166 . . . 4  |-  ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  ->  A  ||  ( P ^ ( P  pCnt  A ) ) )
64 pcdvds 12261 . . . . 5  |-  ( ( P  e.  Prime  /\  A  e.  NN )  ->  ( P ^ ( P  pCnt  A ) )  ||  A
)
6564adantr 274 . . . 4  |-  ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  ->  ( P ^ ( P  pCnt  A ) )  ||  A
)
66 dvdseq 11801 . . . 4  |-  ( ( ( A  e.  NN0  /\  ( P ^ ( P  pCnt  A ) )  e.  NN0 )  /\  ( A  ||  ( P ^ ( P  pCnt  A ) )  /\  ( P ^ ( P  pCnt  A ) )  ||  A
) )  ->  A  =  ( P ^
( P  pCnt  A
) ) )
672, 8, 63, 65, 66syl22anc 1234 . . 3  |-  ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  ->  A  =  ( P ^ ( P 
pCnt  A ) ) )
6867rexlimdvaa 2588 . 2  |-  ( ( P  e.  Prime  /\  A  e.  NN )  ->  ( E. n  e.  NN0  A 
||  ( P ^
n )  ->  A  =  ( P ^
( P  pCnt  A
) ) ) )
693adantr 274 . . . . . . 7  |-  ( ( P  e.  Prime  /\  A  e.  NN )  ->  P  e.  NN )
7069, 5nnexpcld 10624 . . . . . 6  |-  ( ( P  e.  Prime  /\  A  e.  NN )  ->  ( P ^ ( P  pCnt  A ) )  e.  NN )
7170nnzd 9326 . . . . 5  |-  ( ( P  e.  Prime  /\  A  e.  NN )  ->  ( P ^ ( P  pCnt  A ) )  e.  ZZ )
72 iddvds 11759 . . . . 5  |-  ( ( P ^ ( P 
pCnt  A ) )  e.  ZZ  ->  ( P ^ ( P  pCnt  A ) )  ||  ( P ^ ( P  pCnt  A ) ) )
7371, 72syl 14 . . . 4  |-  ( ( P  e.  Prime  /\  A  e.  NN )  ->  ( P ^ ( P  pCnt  A ) )  ||  ( P ^ ( P  pCnt  A ) ) )
74 oveq2 5859 . . . . . 6  |-  ( n  =  ( P  pCnt  A )  ->  ( P ^ n )  =  ( P ^ ( P  pCnt  A ) ) )
7574breq2d 3999 . . . . 5  |-  ( n  =  ( P  pCnt  A )  ->  ( ( P ^ ( P  pCnt  A ) )  ||  ( P ^ n )  <->  ( P ^ ( P  pCnt  A ) )  ||  ( P ^ ( P  pCnt  A ) ) ) )
7675rspcev 2834 . . . 4  |-  ( ( ( P  pCnt  A
)  e.  NN0  /\  ( P ^ ( P 
pCnt  A ) )  ||  ( P ^ ( P 
pCnt  A ) ) )  ->  E. n  e.  NN0  ( P ^ ( P 
pCnt  A ) )  ||  ( P ^ n ) )
775, 73, 76syl2anc 409 . . 3  |-  ( ( P  e.  Prime  /\  A  e.  NN )  ->  E. n  e.  NN0  ( P ^
( P  pCnt  A
) )  ||  ( P ^ n ) )
78 breq1 3990 . . . 4  |-  ( A  =  ( P ^
( P  pCnt  A
) )  ->  ( A  ||  ( P ^
n )  <->  ( P ^ ( P  pCnt  A ) )  ||  ( P ^ n ) ) )
7978rexbidv 2471 . . 3  |-  ( A  =  ( P ^
( P  pCnt  A
) )  ->  ( E. n  e.  NN0  A 
||  ( P ^
n )  <->  E. n  e.  NN0  ( P ^
( P  pCnt  A
) )  ||  ( P ^ n ) ) )
8077, 79syl5ibrcom 156 . 2  |-  ( ( P  e.  Prime  /\  A  e.  NN )  ->  ( A  =  ( P ^ ( P  pCnt  A ) )  ->  E. n  e.  NN0  A  ||  ( P ^ n ) ) )
8168, 80impbid 128 1  |-  ( ( P  e.  Prime  /\  A  e.  NN )  ->  ( E. n  e.  NN0  A 
||  ( P ^
n )  <->  A  =  ( P ^ ( P 
pCnt  A ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 703  DECID wdc 829    = wceq 1348    e. wcel 2141    =/= wne 2340   A.wral 2448   E.wrex 2449   class class class wbr 3987  (class class class)co 5851   0cc0 7767    <_ cle 7948   NNcn 8871   NN0cn0 9128   ZZcz 9205   ^cexp 10468    || cdvds 11742   Primecprime 12054    pCnt cpc 12231
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-iinf 4570  ax-cnex 7858  ax-resscn 7859  ax-1cn 7860  ax-1re 7861  ax-icn 7862  ax-addcl 7863  ax-addrcl 7864  ax-mulcl 7865  ax-mulrcl 7866  ax-addcom 7867  ax-mulcom 7868  ax-addass 7869  ax-mulass 7870  ax-distr 7871  ax-i2m1 7872  ax-0lt1 7873  ax-1rid 7874  ax-0id 7875  ax-rnegex 7876  ax-precex 7877  ax-cnre 7878  ax-pre-ltirr 7879  ax-pre-ltwlin 7880  ax-pre-lttrn 7881  ax-pre-apti 7882  ax-pre-ltadd 7883  ax-pre-mulgt0 7884  ax-pre-mulext 7885  ax-arch 7886  ax-caucvg 7887
This theorem depends on definitions:  df-bi 116  df-stab 826  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3526  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-tr 4086  df-id 4276  df-po 4279  df-iso 4280  df-iord 4349  df-on 4351  df-ilim 4352  df-suc 4354  df-iom 4573  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-isom 5205  df-riota 5807  df-ov 5854  df-oprab 5855  df-mpo 5856  df-1st 6117  df-2nd 6118  df-recs 6282  df-frec 6368  df-1o 6393  df-2o 6394  df-er 6511  df-en 6717  df-sup 6959  df-inf 6960  df-pnf 7949  df-mnf 7950  df-xr 7951  df-ltxr 7952  df-le 7953  df-sub 8085  df-neg 8086  df-reap 8487  df-ap 8494  df-div 8583  df-inn 8872  df-2 8930  df-3 8931  df-4 8932  df-n0 9129  df-xnn0 9192  df-z 9206  df-uz 9481  df-q 9572  df-rp 9604  df-fz 9959  df-fzo 10092  df-fl 10219  df-mod 10272  df-seqfrec 10395  df-exp 10469  df-cj 10799  df-re 10800  df-im 10801  df-rsqrt 10955  df-abs 10956  df-dvds 11743  df-gcd 11891  df-prm 12055  df-pc 12232
This theorem is referenced by:  pcprmpw  12280  dvdsprmpweq  12281
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