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Theorem pcprmpw2 13969
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 754 . . . . 5  |-  ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  ->  A  e.  NN )
21nnnn0d 10657 . . . 4  |-  ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  ->  A  e.  NN0 )
3 prmnn 13787 . . . . . . 7  |-  ( P  e.  Prime  ->  P  e.  NN )
43ad2antrr 725 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  ->  P  e.  NN )
5 pccl 13937 . . . . . . 7  |-  ( ( P  e.  Prime  /\  A  e.  NN )  ->  ( P  pCnt  A )  e. 
NN0 )
65adantr 465 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  ->  ( P  pCnt  A )  e.  NN0 )
74, 6nnexpcld 12050 . . . . 5  |-  ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  ->  ( P ^ ( P  pCnt  A ) )  e.  NN )
87nnnn0d 10657 . . . 4  |-  ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  ->  ( P ^ ( P  pCnt  A ) )  e.  NN0 )
96nn0red 10658 . . . . . . . . . . 11  |-  ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  ->  ( P  pCnt  A )  e.  RR )
109leidd 9927 . . . . . . . . . 10  |-  ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  ->  ( P  pCnt  A )  <_  ( P  pCnt  A ) )
11 simpll 753 . . . . . . . . . . 11  |-  ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  ->  P  e.  Prime )
126nn0zd 10766 . . . . . . . . . . 11  |-  ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  ->  ( P  pCnt  A )  e.  ZZ )
13 pcid 13960 . . . . . . . . . . 11  |-  ( ( P  e.  Prime  /\  ( P  pCnt  A )  e.  ZZ )  ->  ( P  pCnt  ( P ^
( P  pCnt  A
) ) )  =  ( P  pCnt  A
) )
1411, 12, 13syl2anc 661 . . . . . . . . . 10  |-  ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  ->  ( P  pCnt  ( P ^ ( P  pCnt  A ) ) )  =  ( P 
pCnt  A ) )
1510, 14breqtrrd 4339 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  ->  ( P  pCnt  A )  <_  ( P  pCnt  ( P ^
( P  pCnt  A
) ) ) )
1615ad2antrr 725 . . . . . . . 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 461 . . . . . . . . 9  |-  ( ( ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  (
n  e.  NN0  /\  A  ||  ( P ^
n ) ) )  /\  p  e.  Prime )  /\  p  =  P )  ->  p  =  P )
1817oveq1d 6127 . . . . . . . 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 6127 . . . . . . . 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 4343 . . . . . . 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 760 . . . . . . . . . . . . 13  |-  ( ( ( ( P  e. 
Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  /\  p  e. 
Prime )  ->  A  ||  ( P ^ n ) )
22 prmz 13788 . . . . . . . . . . . . . . 15  |-  ( p  e.  Prime  ->  p  e.  ZZ )
2322adantl 466 . . . . . . . . . . . . . 14  |-  ( ( ( ( P  e. 
Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  /\  p  e. 
Prime )  ->  p  e.  ZZ )
241adantr 465 . . . . . . . . . . . . . . 15  |-  ( ( ( ( P  e. 
Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  /\  p  e. 
Prime )  ->  A  e.  NN )
2524nnzd 10767 . . . . . . . . . . . . . 14  |-  ( ( ( ( P  e. 
Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  /\  p  e. 
Prime )  ->  A  e.  ZZ )
26 simprl 755 . . . . . . . . . . . . . . . . 17  |-  ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  ->  n  e.  NN0 )
274, 26nnexpcld 12050 . . . . . . . . . . . . . . . 16  |-  ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  ->  ( P ^ n )  e.  NN )
2827adantr 465 . . . . . . . . . . . . . . 15  |-  ( ( ( ( P  e. 
Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  /\  p  e. 
Prime )  ->  ( P ^ n )  e.  NN )
2928nnzd 10767 . . . . . . . . . . . . . 14  |-  ( ( ( ( P  e. 
Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  /\  p  e. 
Prime )  ->  ( P ^ n )  e.  ZZ )
30 dvdstr 13587 . . . . . . . . . . . . . 14  |-  ( ( p  e.  ZZ  /\  A  e.  ZZ  /\  ( P ^ n )  e.  ZZ )  ->  (
( p  ||  A  /\  A  ||  ( P ^ n ) )  ->  p  ||  ( P ^ n ) ) )
3123, 25, 29, 30syl3anc 1218 . . . . . . . . . . . . 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 674 . . . . . . . . . . . 12  |-  ( ( ( ( P  e. 
Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  /\  p  e. 
Prime )  ->  ( p 
||  A  ->  p  ||  ( P ^ n
) ) )
33 simpr 461 . . . . . . . . . . . . 13  |-  ( ( ( ( P  e. 
Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  /\  p  e. 
Prime )  ->  p  e. 
Prime )
3411adantr 465 . . . . . . . . . . . . 13  |-  ( ( ( ( P  e. 
Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  /\  p  e. 
Prime )  ->  P  e. 
Prime )
35 simplrl 759 . . . . . . . . . . . . 13  |-  ( ( ( ( P  e. 
Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  /\  p  e. 
Prime )  ->  n  e. 
NN0 )
36 prmdvdsexpr 13823 . . . . . . . . . . . . 13  |-  ( ( p  e.  Prime  /\  P  e.  Prime  /\  n  e.  NN0 )  ->  ( p  ||  ( P ^ n
)  ->  p  =  P ) )
3733, 34, 35, 36syl3anc 1218 . . . . . . . . . . . 12  |-  ( ( ( ( P  e. 
Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  /\  p  e. 
Prime )  ->  ( p 
||  ( P ^
n )  ->  p  =  P ) )
3832, 37syld 44 . . . . . . . . . . 11  |-  ( ( ( ( P  e. 
Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  /\  p  e. 
Prime )  ->  ( p 
||  A  ->  p  =  P ) )
3938necon3ad 2668 . . . . . . . . . 10  |-  ( ( ( ( P  e. 
Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  /\  p  e. 
Prime )  ->  ( p  =/=  P  ->  -.  p  ||  A ) )
4039imp 429 . . . . . . . . 9  |-  ( ( ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  (
n  e.  NN0  /\  A  ||  ( P ^
n ) ) )  /\  p  e.  Prime )  /\  p  =/=  P
)  ->  -.  p  ||  A )
41 simplr 754 . . . . . . . . . 10  |-  ( ( ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  (
n  e.  NN0  /\  A  ||  ( P ^
n ) ) )  /\  p  e.  Prime )  /\  p  =/=  P
)  ->  p  e.  Prime )
421ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  (
n  e.  NN0  /\  A  ||  ( P ^
n ) ) )  /\  p  e.  Prime )  /\  p  =/=  P
)  ->  A  e.  NN )
43 pceq0 13958 . . . . . . . . . 10  |-  ( ( p  e.  Prime  /\  A  e.  NN )  ->  (
( p  pCnt  A
)  =  0  <->  -.  p  ||  A ) )
4441, 42, 43syl2anc 661 . . . . . . . . 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 232 . . . . . . . 8  |-  ( ( ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  (
n  e.  NN0  /\  A  ||  ( P ^
n ) ) )  /\  p  e.  Prime )  /\  p  =/=  P
)  ->  ( p  pCnt  A )  =  0 )
467ad2antrr 725 . . . . . . . . . 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 13938 . . . . . . . . 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 10660 . . . . . . . 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 4333 . . . . . . 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 ) ) ) )
5020, 49pm2.61dane 2713 . . . . . 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 ) ) ) )
5150ralrimiva 2820 . . . . 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 ) ) ) )
521nnzd 10767 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  ->  A  e.  ZZ )
537nnzd 10767 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  ->  ( P ^ ( P  pCnt  A ) )  e.  ZZ )
54 pc2dvds 13966 . . . . . 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 ) ) ) ) )
5552, 53, 54syl2anc 661 . . . . 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 ) ) ) ) )
5651, 55mpbird 232 . . . 4  |-  ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  ->  A  ||  ( P ^ ( P  pCnt  A ) ) )
57 pcdvds 13951 . . . . 5  |-  ( ( P  e.  Prime  /\  A  e.  NN )  ->  ( P ^ ( P  pCnt  A ) )  ||  A
)
5857adantr 465 . . . 4  |-  ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  ->  ( P ^ ( P  pCnt  A ) )  ||  A
)
59 dvdseq 13601 . . . 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
) ) )
602, 8, 56, 58, 59syl22anc 1219 . . 3  |-  ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  ->  A  =  ( P ^ ( P 
pCnt  A ) ) )
6160rexlimdvaa 2863 . 2  |-  ( ( P  e.  Prime  /\  A  e.  NN )  ->  ( E. n  e.  NN0  A 
||  ( P ^
n )  ->  A  =  ( P ^
( P  pCnt  A
) ) ) )
623adantr 465 . . . . . . 7  |-  ( ( P  e.  Prime  /\  A  e.  NN )  ->  P  e.  NN )
6362, 5nnexpcld 12050 . . . . . 6  |-  ( ( P  e.  Prime  /\  A  e.  NN )  ->  ( P ^ ( P  pCnt  A ) )  e.  NN )
6463nnzd 10767 . . . . 5  |-  ( ( P  e.  Prime  /\  A  e.  NN )  ->  ( P ^ ( P  pCnt  A ) )  e.  ZZ )
65 iddvds 13567 . . . . 5  |-  ( ( P ^ ( P 
pCnt  A ) )  e.  ZZ  ->  ( P ^ ( P  pCnt  A ) )  ||  ( P ^ ( P  pCnt  A ) ) )
6664, 65syl 16 . . . 4  |-  ( ( P  e.  Prime  /\  A  e.  NN )  ->  ( P ^ ( P  pCnt  A ) )  ||  ( P ^ ( P  pCnt  A ) ) )
67 oveq2 6120 . . . . . 6  |-  ( n  =  ( P  pCnt  A )  ->  ( P ^ n )  =  ( P ^ ( P  pCnt  A ) ) )
6867breq2d 4325 . . . . 5  |-  ( n  =  ( P  pCnt  A )  ->  ( ( P ^ ( P  pCnt  A ) )  ||  ( P ^ n )  <->  ( P ^ ( P  pCnt  A ) )  ||  ( P ^ ( P  pCnt  A ) ) ) )
6968rspcev 3094 . . . 4  |-  ( ( ( P  pCnt  A
)  e.  NN0  /\  ( P ^ ( P 
pCnt  A ) )  ||  ( P ^ ( P 
pCnt  A ) ) )  ->  E. n  e.  NN0  ( P ^ ( P 
pCnt  A ) )  ||  ( P ^ n ) )
705, 66, 69syl2anc 661 . . 3  |-  ( ( P  e.  Prime  /\  A  e.  NN )  ->  E. n  e.  NN0  ( P ^
( P  pCnt  A
) )  ||  ( P ^ n ) )
71 breq1 4316 . . . 4  |-  ( A  =  ( P ^
( P  pCnt  A
) )  ->  ( A  ||  ( P ^
n )  <->  ( P ^ ( P  pCnt  A ) )  ||  ( P ^ n ) ) )
7271rexbidv 2757 . . 3  |-  ( A  =  ( P ^
( P  pCnt  A
) )  ->  ( E. n  e.  NN0  A 
||  ( P ^
n )  <->  E. n  e.  NN0  ( P ^
( P  pCnt  A
) )  ||  ( P ^ n ) ) )
7370, 72syl5ibrcom 222 . 2  |-  ( ( P  e.  Prime  /\  A  e.  NN )  ->  ( A  =  ( P ^ ( P  pCnt  A ) )  ->  E. n  e.  NN0  A  ||  ( P ^ n ) ) )
7461, 73impbid 191 1  |-  ( ( P  e.  Prime  /\  A  e.  NN )  ->  ( E. n  e.  NN0  A 
||  ( P ^
n )  <->  A  =  ( P ^ ( P 
pCnt  A ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2620   A.wral 2736   E.wrex 2737   class class class wbr 4313  (class class class)co 6112   0cc0 9303    <_ cle 9440   NNcn 10343   NN0cn0 10600   ZZcz 10667   ^cexp 11886    || cdivides 13556   Primecprime 13784    pCnt cpc 13924
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 2423  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380  ax-pre-sup 9381
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 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-int 4150  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-1st 6598  df-2nd 6599  df-recs 6853  df-rdg 6887  df-1o 6941  df-2o 6942  df-oadd 6945  df-er 7122  df-en 7332  df-dom 7333  df-sdom 7334  df-fin 7335  df-sup 7712  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-div 10015  df-nn 10344  df-2 10401  df-3 10402  df-n0 10601  df-z 10668  df-uz 10883  df-q 10975  df-rp 11013  df-fz 11459  df-fl 11663  df-mod 11730  df-seq 11828  df-exp 11887  df-cj 12609  df-re 12610  df-im 12611  df-sqr 12745  df-abs 12746  df-dvds 13557  df-gcd 13712  df-prm 13785  df-pc 13925
This theorem is referenced by:  pcprmpw  13970  pgpfi1  16115  pgpfi  16125  sylow2alem2  16138  lt6abl  16392  pgpfac1lem3a  16599  dvdsppwf1o  22548
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