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Theorem isppw2 24121
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 24120 . 2  |-  ( A  e.  NN  ->  (
(Λ `  A )  =/=  0  <->  E! q  e.  Prime  q 
||  A ) )
2 reu6 3215 . . 3  |-  ( E! q  e.  Prime  q  ||  A  <->  E. p  e.  Prime  A. q  e.  Prime  (
q  ||  A  <->  q  =  p ) )
3 equid 1863 . . . . . . . . 9  |-  p  =  p
4 breq1 4398 . . . . . . . . . . . 12  |-  ( q  =  p  ->  (
q  ||  A  <->  p  ||  A
) )
5 equequ1 1875 . . . . . . . . . . . 12  |-  ( q  =  p  ->  (
q  =  p  <->  p  =  p ) )
64, 5bibi12d 328 . . . . . . . . . . 11  |-  ( q  =  p  ->  (
( q  ||  A  <->  q  =  p )  <->  ( p  ||  A  <->  p  =  p
) ) )
76rspcva 3134 . . . . . . . . . 10  |-  ( ( p  e.  Prime  /\  A. q  e.  Prime  ( q 
||  A  <->  q  =  p ) )  -> 
( p  ||  A  <->  p  =  p ) )
87adantll 728 . . . . . . . . 9  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  A. q  e.  Prime  ( q  ||  A  <->  q  =  p ) )  -> 
( p  ||  A  <->  p  =  p ) )
93, 8mpbiri 241 . . . . . . . 8  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  A. q  e.  Prime  ( q  ||  A  <->  q  =  p ) )  ->  p  ||  A )
10 simplr 770 . . . . . . . . 9  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  A. q  e.  Prime  ( q  ||  A  <->  q  =  p ) )  ->  p  e.  Prime )
11 simpll 768 . . . . . . . . 9  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  A. q  e.  Prime  ( q  ||  A  <->  q  =  p ) )  ->  A  e.  NN )
12 pcelnn 14898 . . . . . . . . 9  |-  ( ( p  e.  Prime  /\  A  e.  NN )  ->  (
( p  pCnt  A
)  e.  NN  <->  p  ||  A
) )
1310, 11, 12syl2anc 673 . . . . . . . 8  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  A. q  e.  Prime  ( q  ||  A  <->  q  =  p ) )  -> 
( ( p  pCnt  A )  e.  NN  <->  p  ||  A
) )
149, 13mpbird 240 . . . . . . 7  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  A. q  e.  Prime  ( q  ||  A  <->  q  =  p ) )  -> 
( p  pCnt  A
)  e.  NN )
15 simpr 468 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  NN  /\  p  e. 
Prime )  /\  (
q  e.  Prime  /\  (
q  ||  A  <->  q  =  p ) ) )  /\  q  =  p )  ->  q  =  p )
1615oveq1d 6323 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  NN  /\  p  e. 
Prime )  /\  (
q  e.  Prime  /\  (
q  ||  A  <->  q  =  p ) ) )  /\  q  =  p )  ->  ( q  pCnt  A )  =  ( p  pCnt  A )
)
17 simpllr 777 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  NN  /\  p  e. 
Prime )  /\  (
q  e.  Prime  /\  (
q  ||  A  <->  q  =  p ) ) )  /\  q  =  p )  ->  p  e.  Prime )
18 pccl 14878 . . . . . . . . . . . . . . . . . 18  |-  ( ( p  e.  Prime  /\  A  e.  NN )  ->  (
p  pCnt  A )  e.  NN0 )
1918ancoms 460 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  NN  /\  p  e.  Prime )  -> 
( p  pCnt  A
)  e.  NN0 )
2019ad2antrr 740 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( A  e.  NN  /\  p  e. 
Prime )  /\  (
q  e.  Prime  /\  (
q  ||  A  <->  q  =  p ) ) )  /\  q  =  p )  ->  ( p  pCnt  A )  e.  NN0 )
2120nn0zd 11061 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  NN  /\  p  e. 
Prime )  /\  (
q  e.  Prime  /\  (
q  ||  A  <->  q  =  p ) ) )  /\  q  =  p )  ->  ( p  pCnt  A )  e.  ZZ )
22 pcid 14901 . . . . . . . . . . . . . . 15  |-  ( ( p  e.  Prime  /\  (
p  pCnt  A )  e.  ZZ )  ->  (
p  pCnt  ( p ^ ( p  pCnt  A ) ) )  =  ( p  pCnt  A
) )
2317, 21, 22syl2anc 673 . . . . . . . . . . . . . 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 2508 . . . . . . . . . . . . 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 6323 . . . . . . . . . . . . 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 2508 . . . . . . . . . . . 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 774 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  ( q  e.  Prime  /\  ( q  ||  A  <->  q  =  p ) ) )  ->  ( q  ||  A  <->  q  =  p ) )
2827notbid 301 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  ( q  e.  Prime  /\  ( q  ||  A  <->  q  =  p ) ) )  ->  ( -.  q  ||  A  <->  -.  q  =  p ) )
2928biimpar 493 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  NN  /\  p  e. 
Prime )  /\  (
q  e.  Prime  /\  (
q  ||  A  <->  q  =  p ) ) )  /\  -.  q  =  p )  ->  -.  q  ||  A )
30 simplrl 778 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  NN  /\  p  e. 
Prime )  /\  (
q  e.  Prime  /\  (
q  ||  A  <->  q  =  p ) ) )  /\  -.  q  =  p )  ->  q  e.  Prime )
31 simplll 776 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  NN  /\  p  e. 
Prime )  /\  (
q  e.  Prime  /\  (
q  ||  A  <->  q  =  p ) ) )  /\  -.  q  =  p )  ->  A  e.  NN )
32 pceq0 14899 . . . . . . . . . . . . . . 15  |-  ( ( q  e.  Prime  /\  A  e.  NN )  ->  (
( q  pCnt  A
)  =  0  <->  -.  q  ||  A ) )
3330, 31, 32syl2anc 673 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  NN  /\  p  e. 
Prime )  /\  (
q  e.  Prime  /\  (
q  ||  A  <->  q  =  p ) ) )  /\  -.  q  =  p )  ->  (
( q  pCnt  A
)  =  0  <->  -.  q  ||  A ) )
3429, 33mpbird 240 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  NN  /\  p  e. 
Prime )  /\  (
q  e.  Prime  /\  (
q  ||  A  <->  q  =  p ) ) )  /\  -.  q  =  p )  ->  (
q  pCnt  A )  =  0 )
35 simprl 772 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  ( q  e.  Prime  /\  ( q  ||  A  <->  q  =  p ) ) )  ->  q  e.  Prime )
36 simplr 770 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  ( q  e.  Prime  /\  ( q  ||  A  <->  q  =  p ) ) )  ->  p  e.  Prime )
3719adantr 472 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  ( q  e.  Prime  /\  ( q  ||  A  <->  q  =  p ) ) )  ->  ( p  pCnt  A )  e.  NN0 )
38 prmdvdsexpr 14748 . . . . . . . . . . . . . . . 16  |-  ( ( q  e.  Prime  /\  p  e.  Prime  /\  ( p  pCnt  A )  e.  NN0 )  ->  ( q  ||  ( p ^ (
p  pCnt  A )
)  ->  q  =  p ) )
3935, 36, 37, 38syl3anc 1292 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  ( q  e.  Prime  /\  ( q  ||  A  <->  q  =  p ) ) )  ->  ( q  ||  ( p ^ (
p  pCnt  A )
)  ->  q  =  p ) )
4039con3dimp 448 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  NN  /\  p  e. 
Prime )  /\  (
q  e.  Prime  /\  (
q  ||  A  <->  q  =  p ) ) )  /\  -.  q  =  p )  ->  -.  q  ||  ( p ^
( p  pCnt  A
) ) )
41 prmnn 14704 . . . . . . . . . . . . . . . . . 18  |-  ( p  e.  Prime  ->  p  e.  NN )
4241adantl 473 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  NN  /\  p  e.  Prime )  ->  p  e.  NN )
4342, 19nnexpcld 12475 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  NN  /\  p  e.  Prime )  -> 
( p ^ (
p  pCnt  A )
)  e.  NN )
4443ad2antrr 740 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  NN  /\  p  e. 
Prime )  /\  (
q  e.  Prime  /\  (
q  ||  A  <->  q  =  p ) ) )  /\  -.  q  =  p )  ->  (
p ^ ( p 
pCnt  A ) )  e.  NN )
45 pceq0 14899 . . . . . . . . . . . . . . 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 673 . . . . . . . . . . . . . 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 240 . . . . . . . . . . . . 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 2508 . . . . . . . . . . . 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 808 . . . . . . . . . . 11  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  ( q  e.  Prime  /\  ( q  ||  A  <->  q  =  p ) ) )  ->  ( q  pCnt  A )  =  ( q  pCnt  ( p ^ ( p  pCnt  A ) ) ) )
5049expr 626 . . . . . . . . . 10  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  q  e.  Prime )  ->  ( ( q 
||  A  <->  q  =  p )  ->  (
q  pCnt  A )  =  ( q  pCnt  ( p ^ ( p 
pCnt  A ) ) ) ) )
5150ralimdva 2805 . . . . . . . . 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 436 . . . . . . . 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 10900 . . . . . . . . . 10  |-  ( A  e.  NN  ->  A  e.  NN0 )
5453ad2antrr 740 . . . . . . . . 9  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  A. q  e.  Prime  ( q  ||  A  <->  q  =  p ) )  ->  A  e.  NN0 )
5543adantr 472 . . . . . . . . . 10  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  A. q  e.  Prime  ( q  ||  A  <->  q  =  p ) )  -> 
( p ^ (
p  pCnt  A )
)  e.  NN )
5655nnnn0d 10949 . . . . . . . . 9  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  A. q  e.  Prime  ( q  ||  A  <->  q  =  p ) )  -> 
( p ^ (
p  pCnt  A )
)  e.  NN0 )
57 pc11 14908 . . . . . . . . 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 673 . . . . . . . 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 240 . . . . . . 7  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  A. q  e.  Prime  ( q  ||  A  <->  q  =  p ) )  ->  A  =  ( p ^ ( p  pCnt  A ) ) )
60 oveq2 6316 . . . . . . . . 9  |-  ( k  =  ( p  pCnt  A )  ->  ( p ^ k )  =  ( p ^ (
p  pCnt  A )
) )
6160eqeq2d 2481 . . . . . . . 8  |-  ( k  =  ( p  pCnt  A )  ->  ( A  =  ( p ^
k )  <->  A  =  ( p ^ (
p  pCnt  A )
) ) )
6261rspcev 3136 . . . . . . 7  |-  ( ( ( p  pCnt  A
)  e.  NN  /\  A  =  ( p ^ ( p  pCnt  A ) ) )  ->  E. k  e.  NN  A  =  ( p ^ k ) )
6314, 59, 62syl2anc 673 . . . . . 6  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  A. q  e.  Prime  ( q  ||  A  <->  q  =  p ) )  ->  E. k  e.  NN  A  =  ( p ^ k ) )
6463ex 441 . . . . 5  |-  ( ( A  e.  NN  /\  p  e.  Prime )  -> 
( A. q  e. 
Prime  ( q  ||  A  <->  q  =  p )  ->  E. k  e.  NN  A  =  ( p ^ k ) ) )
65 prmdvdsexpb 14747 . . . . . . . . . . 11  |-  ( ( q  e.  Prime  /\  p  e.  Prime  /\  k  e.  NN )  ->  ( q 
||  ( p ^
k )  <->  q  =  p ) )
66653coml 1238 . . . . . . . . . 10  |-  ( ( p  e.  Prime  /\  k  e.  NN  /\  q  e. 
Prime )  ->  ( q 
||  ( p ^
k )  <->  q  =  p ) )
67663expa 1231 . . . . . . . . 9  |-  ( ( ( p  e.  Prime  /\  k  e.  NN )  /\  q  e.  Prime )  ->  ( q  ||  ( p ^ k
)  <->  q  =  p ) )
6867ralrimiva 2809 . . . . . . . 8  |-  ( ( p  e.  Prime  /\  k  e.  NN )  ->  A. q  e.  Prime  ( q  ||  ( p ^ k
)  <->  q  =  p ) )
6968adantll 728 . . . . . . 7  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  k  e.  NN )  ->  A. q  e.  Prime  ( q  ||  ( p ^ k )  <->  q  =  p ) )
70 breq2 4399 . . . . . . . . 9  |-  ( A  =  ( p ^
k )  ->  (
q  ||  A  <->  q  ||  ( p ^ k
) ) )
7170bibi1d 326 . . . . . . . 8  |-  ( A  =  ( p ^
k )  ->  (
( q  ||  A  <->  q  =  p )  <->  ( q  ||  ( p ^ k
)  <->  q  =  p ) ) )
7271ralbidv 2829 . . . . . . 7  |-  ( A  =  ( p ^
k )  ->  ( A. q  e.  Prime  ( q  ||  A  <->  q  =  p )  <->  A. q  e.  Prime  ( q  ||  ( p ^ k
)  <->  q  =  p ) ) )
7369, 72syl5ibrcom 230 . . . . . 6  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  k  e.  NN )  ->  ( A  =  ( p ^ k
)  ->  A. q  e.  Prime  ( q  ||  A 
<->  q  =  p ) ) )
7473rexlimdva 2871 . . . . 5  |-  ( ( A  e.  NN  /\  p  e.  Prime )  -> 
( E. k  e.  NN  A  =  ( p ^ k )  ->  A. q  e.  Prime  ( q  ||  A  <->  q  =  p ) ) )
7564, 74impbid 195 . . . 4  |-  ( ( A  e.  NN  /\  p  e.  Prime )  -> 
( A. q  e. 
Prime  ( q  ||  A  <->  q  =  p )  <->  E. k  e.  NN  A  =  ( p ^ k ) ) )
7675rexbidva 2889 . . 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 265 . 2  |-  ( A  e.  NN  ->  ( E! q  e.  Prime  q 
||  A  <->  E. p  e.  Prime  E. k  e.  NN  A  =  ( p ^ k ) ) )
781, 77bitrd 261 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 189    /\ wa 376    = wceq 1452    e. wcel 1904    =/= wne 2641   A.wral 2756   E.wrex 2757   E!wreu 2758   class class class wbr 4395   ` cfv 5589  (class class class)co 6308   0cc0 9557   NNcn 10631   NN0cn0 10893   ZZcz 10961   ^cexp 12310    || cdvds 14382   Primecprime 14701    pCnt cpc 14865  Λcvma 24097
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635  ax-addf 9636  ax-mulf 9637
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-ixp 7541  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-fi 7943  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-q 11288  df-rp 11326  df-xneg 11432  df-xadd 11433  df-xmul 11434  df-ioo 11664  df-ioc 11665  df-ico 11666  df-icc 11667  df-fz 11811  df-fzo 11943  df-fl 12061  df-mod 12130  df-seq 12252  df-exp 12311  df-fac 12498  df-bc 12526  df-hash 12554  df-shft 13207  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-limsup 13603  df-clim 13629  df-rlim 13630  df-sum 13830  df-ef 14198  df-sin 14200  df-cos 14201  df-pi 14203  df-dvds 14383  df-gcd 14548  df-prm 14702  df-pc 14866  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-starv 15283  df-sca 15284  df-vsca 15285  df-ip 15286  df-tset 15287  df-ple 15288  df-ds 15290  df-unif 15291  df-hom 15292  df-cco 15293  df-rest 15399  df-topn 15400  df-0g 15418  df-gsum 15419  df-topgen 15420  df-pt 15421  df-prds 15424  df-xrs 15478  df-qtop 15484  df-imas 15485  df-xps 15488  df-mre 15570  df-mrc 15571  df-acs 15573  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-submnd 16661  df-mulg 16754  df-cntz 17049  df-cmn 17510  df-psmet 19039  df-xmet 19040  df-met 19041  df-bl 19042  df-mopn 19043  df-fbas 19044  df-fg 19045  df-cnfld 19048  df-top 19998  df-bases 19999  df-topon 20000  df-topsp 20001  df-cld 20111  df-ntr 20112  df-cls 20113  df-nei 20191  df-lp 20229  df-perf 20230  df-cn 20320  df-cnp 20321  df-haus 20408  df-tx 20654  df-hmeo 20847  df-fil 20939  df-fm 21031  df-flim 21032  df-flf 21033  df-xms 21413  df-ms 21414  df-tms 21415  df-cncf 21988  df-limc 22900  df-dv 22901  df-log 23585  df-vma 24103
This theorem is referenced by:  vmacl  24124  efvmacl  24126  vma1  24172  vmalelog  24212  fsumvma  24220
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