MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isppw2 Structured version   Visualization version   Unicode version

Theorem isppw2 24042
Description: Two ways to say that  A is a prime power. (Contributed by Mario Carneiro, 7-Apr-2016.)
Assertion
Ref Expression
isppw2  |-  ( A  e.  NN  ->  (
(Λ `  A )  =/=  0  <->  E. p  e.  Prime  E. k  e.  NN  A  =  ( p ^
k ) ) )
Distinct variable group:    k, p, A

Proof of Theorem isppw2
Dummy variable  q is distinct from all other variables.
StepHypRef Expression
1 isppw 24041 . 2  |-  ( A  e.  NN  ->  (
(Λ `  A )  =/=  0  <->  E! q  e.  Prime  q 
||  A ) )
2 reu6 3227 . . 3  |-  ( E! q  e.  Prime  q  ||  A  <->  E. p  e.  Prime  A. q  e.  Prime  (
q  ||  A  <->  q  =  p ) )
3 equid 1855 . . . . . . . . 9  |-  p  =  p
4 breq1 4405 . . . . . . . . . . . 12  |-  ( q  =  p  ->  (
q  ||  A  <->  p  ||  A
) )
5 equequ1 1867 . . . . . . . . . . . 12  |-  ( q  =  p  ->  (
q  =  p  <->  p  =  p ) )
64, 5bibi12d 323 . . . . . . . . . . 11  |-  ( q  =  p  ->  (
( q  ||  A  <->  q  =  p )  <->  ( p  ||  A  <->  p  =  p
) ) )
76rspcva 3148 . . . . . . . . . 10  |-  ( ( p  e.  Prime  /\  A. q  e.  Prime  ( q 
||  A  <->  q  =  p ) )  -> 
( p  ||  A  <->  p  =  p ) )
87adantll 720 . . . . . . . . 9  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  A. q  e.  Prime  ( q  ||  A  <->  q  =  p ) )  -> 
( p  ||  A  <->  p  =  p ) )
93, 8mpbiri 237 . . . . . . . 8  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  A. q  e.  Prime  ( q  ||  A  <->  q  =  p ) )  ->  p  ||  A )
10 simplr 762 . . . . . . . . 9  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  A. q  e.  Prime  ( q  ||  A  <->  q  =  p ) )  ->  p  e.  Prime )
11 simpll 760 . . . . . . . . 9  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  A. q  e.  Prime  ( q  ||  A  <->  q  =  p ) )  ->  A  e.  NN )
12 pcelnn 14819 . . . . . . . . 9  |-  ( ( p  e.  Prime  /\  A  e.  NN )  ->  (
( p  pCnt  A
)  e.  NN  <->  p  ||  A
) )
1310, 11, 12syl2anc 667 . . . . . . . 8  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  A. q  e.  Prime  ( q  ||  A  <->  q  =  p ) )  -> 
( ( p  pCnt  A )  e.  NN  <->  p  ||  A
) )
149, 13mpbird 236 . . . . . . 7  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  A. q  e.  Prime  ( q  ||  A  <->  q  =  p ) )  -> 
( p  pCnt  A
)  e.  NN )
15 simpr 463 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  NN  /\  p  e. 
Prime )  /\  (
q  e.  Prime  /\  (
q  ||  A  <->  q  =  p ) ) )  /\  q  =  p )  ->  q  =  p )
1615oveq1d 6305 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  NN  /\  p  e. 
Prime )  /\  (
q  e.  Prime  /\  (
q  ||  A  <->  q  =  p ) ) )  /\  q  =  p )  ->  ( q  pCnt  A )  =  ( p  pCnt  A )
)
17 simpllr 769 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  NN  /\  p  e. 
Prime )  /\  (
q  e.  Prime  /\  (
q  ||  A  <->  q  =  p ) ) )  /\  q  =  p )  ->  p  e.  Prime )
18 pccl 14799 . . . . . . . . . . . . . . . . . 18  |-  ( ( p  e.  Prime  /\  A  e.  NN )  ->  (
p  pCnt  A )  e.  NN0 )
1918ancoms 455 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  NN  /\  p  e.  Prime )  -> 
( p  pCnt  A
)  e.  NN0 )
2019ad2antrr 732 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( A  e.  NN  /\  p  e. 
Prime )  /\  (
q  e.  Prime  /\  (
q  ||  A  <->  q  =  p ) ) )  /\  q  =  p )  ->  ( p  pCnt  A )  e.  NN0 )
2120nn0zd 11038 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  NN  /\  p  e. 
Prime )  /\  (
q  e.  Prime  /\  (
q  ||  A  <->  q  =  p ) ) )  /\  q  =  p )  ->  ( p  pCnt  A )  e.  ZZ )
22 pcid 14822 . . . . . . . . . . . . . . 15  |-  ( ( p  e.  Prime  /\  (
p  pCnt  A )  e.  ZZ )  ->  (
p  pCnt  ( p ^ ( p  pCnt  A ) ) )  =  ( p  pCnt  A
) )
2317, 21, 22syl2anc 667 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  NN  /\  p  e. 
Prime )  /\  (
q  e.  Prime  /\  (
q  ||  A  <->  q  =  p ) ) )  /\  q  =  p )  ->  ( p  pCnt  ( p ^ (
p  pCnt  A )
) )  =  ( p  pCnt  A )
)
2416, 23eqtr4d 2488 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  NN  /\  p  e. 
Prime )  /\  (
q  e.  Prime  /\  (
q  ||  A  <->  q  =  p ) ) )  /\  q  =  p )  ->  ( q  pCnt  A )  =  ( p  pCnt  ( p ^ ( p  pCnt  A ) ) ) )
2515oveq1d 6305 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  NN  /\  p  e. 
Prime )  /\  (
q  e.  Prime  /\  (
q  ||  A  <->  q  =  p ) ) )  /\  q  =  p )  ->  ( q  pCnt  ( p ^ (
p  pCnt  A )
) )  =  ( p  pCnt  ( p ^ ( p  pCnt  A ) ) ) )
2624, 25eqtr4d 2488 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  NN  /\  p  e. 
Prime )  /\  (
q  e.  Prime  /\  (
q  ||  A  <->  q  =  p ) ) )  /\  q  =  p )  ->  ( q  pCnt  A )  =  ( q  pCnt  ( p ^ ( p  pCnt  A ) ) ) )
27 simprr 766 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  ( q  e.  Prime  /\  ( q  ||  A  <->  q  =  p ) ) )  ->  ( q  ||  A  <->  q  =  p ) )
2827notbid 296 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  ( q  e.  Prime  /\  ( q  ||  A  <->  q  =  p ) ) )  ->  ( -.  q  ||  A  <->  -.  q  =  p ) )
2928biimpar 488 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  NN  /\  p  e. 
Prime )  /\  (
q  e.  Prime  /\  (
q  ||  A  <->  q  =  p ) ) )  /\  -.  q  =  p )  ->  -.  q  ||  A )
30 simplrl 770 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  NN  /\  p  e. 
Prime )  /\  (
q  e.  Prime  /\  (
q  ||  A  <->  q  =  p ) ) )  /\  -.  q  =  p )  ->  q  e.  Prime )
31 simplll 768 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  NN  /\  p  e. 
Prime )  /\  (
q  e.  Prime  /\  (
q  ||  A  <->  q  =  p ) ) )  /\  -.  q  =  p )  ->  A  e.  NN )
32 pceq0 14820 . . . . . . . . . . . . . . 15  |-  ( ( q  e.  Prime  /\  A  e.  NN )  ->  (
( q  pCnt  A
)  =  0  <->  -.  q  ||  A ) )
3330, 31, 32syl2anc 667 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  NN  /\  p  e. 
Prime )  /\  (
q  e.  Prime  /\  (
q  ||  A  <->  q  =  p ) ) )  /\  -.  q  =  p )  ->  (
( q  pCnt  A
)  =  0  <->  -.  q  ||  A ) )
3429, 33mpbird 236 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  NN  /\  p  e. 
Prime )  /\  (
q  e.  Prime  /\  (
q  ||  A  <->  q  =  p ) ) )  /\  -.  q  =  p )  ->  (
q  pCnt  A )  =  0 )
35 simprl 764 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  ( q  e.  Prime  /\  ( q  ||  A  <->  q  =  p ) ) )  ->  q  e.  Prime )
36 simplr 762 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  ( q  e.  Prime  /\  ( q  ||  A  <->  q  =  p ) ) )  ->  p  e.  Prime )
3719adantr 467 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  ( q  e.  Prime  /\  ( q  ||  A  <->  q  =  p ) ) )  ->  ( p  pCnt  A )  e.  NN0 )
38 prmdvdsexpr 14669 . . . . . . . . . . . . . . . 16  |-  ( ( q  e.  Prime  /\  p  e.  Prime  /\  ( p  pCnt  A )  e.  NN0 )  ->  ( q  ||  ( p ^ (
p  pCnt  A )
)  ->  q  =  p ) )
3935, 36, 37, 38syl3anc 1268 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  ( q  e.  Prime  /\  ( q  ||  A  <->  q  =  p ) ) )  ->  ( q  ||  ( p ^ (
p  pCnt  A )
)  ->  q  =  p ) )
4039con3dimp 443 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  NN  /\  p  e. 
Prime )  /\  (
q  e.  Prime  /\  (
q  ||  A  <->  q  =  p ) ) )  /\  -.  q  =  p )  ->  -.  q  ||  ( p ^
( p  pCnt  A
) ) )
41 prmnn 14625 . . . . . . . . . . . . . . . . . 18  |-  ( p  e.  Prime  ->  p  e.  NN )
4241adantl 468 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  NN  /\  p  e.  Prime )  ->  p  e.  NN )
4342, 19nnexpcld 12437 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  NN  /\  p  e.  Prime )  -> 
( p ^ (
p  pCnt  A )
)  e.  NN )
4443ad2antrr 732 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  NN  /\  p  e. 
Prime )  /\  (
q  e.  Prime  /\  (
q  ||  A  <->  q  =  p ) ) )  /\  -.  q  =  p )  ->  (
p ^ ( p 
pCnt  A ) )  e.  NN )
45 pceq0 14820 . . . . . . . . . . . . . . 15  |-  ( ( q  e.  Prime  /\  (
p ^ ( p 
pCnt  A ) )  e.  NN )  ->  (
( q  pCnt  (
p ^ ( p 
pCnt  A ) ) )  =  0  <->  -.  q  ||  ( p ^ (
p  pCnt  A )
) ) )
4630, 44, 45syl2anc 667 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  NN  /\  p  e. 
Prime )  /\  (
q  e.  Prime  /\  (
q  ||  A  <->  q  =  p ) ) )  /\  -.  q  =  p )  ->  (
( q  pCnt  (
p ^ ( p 
pCnt  A ) ) )  =  0  <->  -.  q  ||  ( p ^ (
p  pCnt  A )
) ) )
4740, 46mpbird 236 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  NN  /\  p  e. 
Prime )  /\  (
q  e.  Prime  /\  (
q  ||  A  <->  q  =  p ) ) )  /\  -.  q  =  p )  ->  (
q  pCnt  ( p ^ ( p  pCnt  A ) ) )  =  0 )
4834, 47eqtr4d 2488 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  NN  /\  p  e. 
Prime )  /\  (
q  e.  Prime  /\  (
q  ||  A  <->  q  =  p ) ) )  /\  -.  q  =  p )  ->  (
q  pCnt  A )  =  ( q  pCnt  ( p ^ ( p 
pCnt  A ) ) ) )
4926, 48pm2.61dan 800 . . . . . . . . . . 11  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  ( q  e.  Prime  /\  ( q  ||  A  <->  q  =  p ) ) )  ->  ( q  pCnt  A )  =  ( q  pCnt  ( p ^ ( p  pCnt  A ) ) ) )
5049expr 620 . . . . . . . . . 10  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  q  e.  Prime )  ->  ( ( q 
||  A  <->  q  =  p )  ->  (
q  pCnt  A )  =  ( q  pCnt  ( p ^ ( p 
pCnt  A ) ) ) ) )
5150ralimdva 2796 . . . . . . . . 9  |-  ( ( A  e.  NN  /\  p  e.  Prime )  -> 
( A. q  e. 
Prime  ( q  ||  A  <->  q  =  p )  ->  A. q  e.  Prime  ( q  pCnt  A )  =  ( q  pCnt  ( p ^ ( p 
pCnt  A ) ) ) ) )
5251imp 431 . . . . . . . 8  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  A. q  e.  Prime  ( q  ||  A  <->  q  =  p ) )  ->  A. q  e.  Prime  ( q  pCnt  A )  =  ( q  pCnt  ( p ^ ( p 
pCnt  A ) ) ) )
53 nnnn0 10876 . . . . . . . . . 10  |-  ( A  e.  NN  ->  A  e.  NN0 )
5453ad2antrr 732 . . . . . . . . 9  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  A. q  e.  Prime  ( q  ||  A  <->  q  =  p ) )  ->  A  e.  NN0 )
5543adantr 467 . . . . . . . . . 10  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  A. q  e.  Prime  ( q  ||  A  <->  q  =  p ) )  -> 
( p ^ (
p  pCnt  A )
)  e.  NN )
5655nnnn0d 10925 . . . . . . . . 9  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  A. q  e.  Prime  ( q  ||  A  <->  q  =  p ) )  -> 
( p ^ (
p  pCnt  A )
)  e.  NN0 )
57 pc11 14829 . . . . . . . . 9  |-  ( ( A  e.  NN0  /\  ( p ^ (
p  pCnt  A )
)  e.  NN0 )  ->  ( A  =  ( p ^ ( p 
pCnt  A ) )  <->  A. q  e.  Prime  ( q  pCnt  A )  =  ( q 
pCnt  ( p ^
( p  pCnt  A
) ) ) ) )
5854, 56, 57syl2anc 667 . . . . . . . 8  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  A. q  e.  Prime  ( q  ||  A  <->  q  =  p ) )  -> 
( A  =  ( p ^ ( p 
pCnt  A ) )  <->  A. q  e.  Prime  ( q  pCnt  A )  =  ( q 
pCnt  ( p ^
( p  pCnt  A
) ) ) ) )
5952, 58mpbird 236 . . . . . . 7  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  A. q  e.  Prime  ( q  ||  A  <->  q  =  p ) )  ->  A  =  ( p ^ ( p  pCnt  A ) ) )
60 oveq2 6298 . . . . . . . . 9  |-  ( k  =  ( p  pCnt  A )  ->  ( p ^ k )  =  ( p ^ (
p  pCnt  A )
) )
6160eqeq2d 2461 . . . . . . . 8  |-  ( k  =  ( p  pCnt  A )  ->  ( A  =  ( p ^
k )  <->  A  =  ( p ^ (
p  pCnt  A )
) ) )
6261rspcev 3150 . . . . . . 7  |-  ( ( ( p  pCnt  A
)  e.  NN  /\  A  =  ( p ^ ( p  pCnt  A ) ) )  ->  E. k  e.  NN  A  =  ( p ^ k ) )
6314, 59, 62syl2anc 667 . . . . . 6  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  A. q  e.  Prime  ( q  ||  A  <->  q  =  p ) )  ->  E. k  e.  NN  A  =  ( p ^ k ) )
6463ex 436 . . . . 5  |-  ( ( A  e.  NN  /\  p  e.  Prime )  -> 
( A. q  e. 
Prime  ( q  ||  A  <->  q  =  p )  ->  E. k  e.  NN  A  =  ( p ^ k ) ) )
65 prmdvdsexpb 14668 . . . . . . . . . . 11  |-  ( ( q  e.  Prime  /\  p  e.  Prime  /\  k  e.  NN )  ->  ( q 
||  ( p ^
k )  <->  q  =  p ) )
66653coml 1215 . . . . . . . . . 10  |-  ( ( p  e.  Prime  /\  k  e.  NN  /\  q  e. 
Prime )  ->  ( q 
||  ( p ^
k )  <->  q  =  p ) )
67663expa 1208 . . . . . . . . 9  |-  ( ( ( p  e.  Prime  /\  k  e.  NN )  /\  q  e.  Prime )  ->  ( q  ||  ( p ^ k
)  <->  q  =  p ) )
6867ralrimiva 2802 . . . . . . . 8  |-  ( ( p  e.  Prime  /\  k  e.  NN )  ->  A. q  e.  Prime  ( q  ||  ( p ^ k
)  <->  q  =  p ) )
6968adantll 720 . . . . . . 7  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  k  e.  NN )  ->  A. q  e.  Prime  ( q  ||  ( p ^ k )  <->  q  =  p ) )
70 breq2 4406 . . . . . . . . 9  |-  ( A  =  ( p ^
k )  ->  (
q  ||  A  <->  q  ||  ( p ^ k
) ) )
7170bibi1d 321 . . . . . . . 8  |-  ( A  =  ( p ^
k )  ->  (
( q  ||  A  <->  q  =  p )  <->  ( q  ||  ( p ^ k
)  <->  q  =  p ) ) )
7271ralbidv 2827 . . . . . . 7  |-  ( A  =  ( p ^
k )  ->  ( A. q  e.  Prime  ( q  ||  A  <->  q  =  p )  <->  A. q  e.  Prime  ( q  ||  ( p ^ k
)  <->  q  =  p ) ) )
7369, 72syl5ibrcom 226 . . . . . 6  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  k  e.  NN )  ->  ( A  =  ( p ^ k
)  ->  A. q  e.  Prime  ( q  ||  A 
<->  q  =  p ) ) )
7473rexlimdva 2879 . . . . 5  |-  ( ( A  e.  NN  /\  p  e.  Prime )  -> 
( E. k  e.  NN  A  =  ( p ^ k )  ->  A. q  e.  Prime  ( q  ||  A  <->  q  =  p ) ) )
7564, 74impbid 194 . . . 4  |-  ( ( A  e.  NN  /\  p  e.  Prime )  -> 
( A. q  e. 
Prime  ( q  ||  A  <->  q  =  p )  <->  E. k  e.  NN  A  =  ( p ^ k ) ) )
7675rexbidva 2898 . . 3  |-  ( A  e.  NN  ->  ( E. p  e.  Prime  A. q  e.  Prime  (
q  ||  A  <->  q  =  p )  <->  E. p  e.  Prime  E. k  e.  NN  A  =  ( p ^ k ) ) )
772, 76syl5bb 261 . 2  |-  ( A  e.  NN  ->  ( E! q  e.  Prime  q 
||  A  <->  E. p  e.  Prime  E. k  e.  NN  A  =  ( p ^ k ) ) )
781, 77bitrd 257 1  |-  ( A  e.  NN  ->  (
(Λ `  A )  =/=  0  <->  E. p  e.  Prime  E. k  e.  NN  A  =  ( p ^
k ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1444    e. wcel 1887    =/= wne 2622   A.wral 2737   E.wrex 2738   E!wreu 2739   class class class wbr 4402   ` cfv 5582  (class class class)co 6290   0cc0 9539   NNcn 10609   NN0cn0 10869   ZZcz 10937   ^cexp 12272    || cdvds 14305   Primecprime 14622    pCnt cpc 14786  Λcvma 24018
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617  ax-addf 9618  ax-mulf 9619
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-iin 4281  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-of 6531  df-om 6693  df-1st 6793  df-2nd 6794  df-supp 6915  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-2o 7183  df-oadd 7186  df-er 7363  df-map 7474  df-pm 7475  df-ixp 7523  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-fsupp 7884  df-fi 7925  df-sup 7956  df-inf 7957  df-oi 8025  df-card 8373  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-ioo 11639  df-ioc 11640  df-ico 11641  df-icc 11642  df-fz 11785  df-fzo 11916  df-fl 12028  df-mod 12097  df-seq 12214  df-exp 12273  df-fac 12460  df-bc 12488  df-hash 12516  df-shft 13130  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-limsup 13526  df-clim 13552  df-rlim 13553  df-sum 13753  df-ef 14121  df-sin 14123  df-cos 14124  df-pi 14126  df-dvds 14306  df-gcd 14469  df-prm 14623  df-pc 14787  df-struct 15123  df-ndx 15124  df-slot 15125  df-base 15126  df-sets 15127  df-ress 15128  df-plusg 15203  df-mulr 15204  df-starv 15205  df-sca 15206  df-vsca 15207  df-ip 15208  df-tset 15209  df-ple 15210  df-ds 15212  df-unif 15213  df-hom 15214  df-cco 15215  df-rest 15321  df-topn 15322  df-0g 15340  df-gsum 15341  df-topgen 15342  df-pt 15343  df-prds 15346  df-xrs 15400  df-qtop 15406  df-imas 15407  df-xps 15410  df-mre 15492  df-mrc 15493  df-acs 15495  df-mgm 16488  df-sgrp 16527  df-mnd 16537  df-submnd 16583  df-mulg 16676  df-cntz 16971  df-cmn 17432  df-psmet 18962  df-xmet 18963  df-met 18964  df-bl 18965  df-mopn 18966  df-fbas 18967  df-fg 18968  df-cnfld 18971  df-top 19921  df-bases 19922  df-topon 19923  df-topsp 19924  df-cld 20034  df-ntr 20035  df-cls 20036  df-nei 20114  df-lp 20152  df-perf 20153  df-cn 20243  df-cnp 20244  df-haus 20331  df-tx 20577  df-hmeo 20770  df-fil 20861  df-fm 20953  df-flim 20954  df-flf 20955  df-xms 21335  df-ms 21336  df-tms 21337  df-cncf 21910  df-limc 22821  df-dv 22822  df-log 23506  df-vma 24024
This theorem is referenced by:  vmacl  24045  efvmacl  24047  vma1  24093  vmalelog  24133  fsumvma  24141
  Copyright terms: Public domain W3C validator