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Theorem isppw2 23514
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 23513 . 2  |-  ( A  e.  NN  ->  (
(Λ `  A )  =/=  0  <->  E! q  e.  Prime  q 
||  A ) )
2 reu6 3288 . . 3  |-  ( E! q  e.  Prime  q  ||  A  <->  E. p  e.  Prime  A. q  e.  Prime  (
q  ||  A  <->  q  =  p ) )
3 equid 1792 . . . . . . . . 9  |-  p  =  p
4 breq1 4459 . . . . . . . . . . . 12  |-  ( q  =  p  ->  (
q  ||  A  <->  p  ||  A
) )
5 equequ1 1799 . . . . . . . . . . . 12  |-  ( q  =  p  ->  (
q  =  p  <->  p  =  p ) )
64, 5bibi12d 321 . . . . . . . . . . 11  |-  ( q  =  p  ->  (
( q  ||  A  <->  q  =  p )  <->  ( p  ||  A  <->  p  =  p
) ) )
76rspcva 3208 . . . . . . . . . 10  |-  ( ( p  e.  Prime  /\  A. q  e.  Prime  ( q 
||  A  <->  q  =  p ) )  -> 
( p  ||  A  <->  p  =  p ) )
87adantll 713 . . . . . . . . 9  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  A. q  e.  Prime  ( q  ||  A  <->  q  =  p ) )  -> 
( p  ||  A  <->  p  =  p ) )
93, 8mpbiri 233 . . . . . . . 8  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  A. q  e.  Prime  ( q  ||  A  <->  q  =  p ) )  ->  p  ||  A )
10 simplr 755 . . . . . . . . 9  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  A. q  e.  Prime  ( q  ||  A  <->  q  =  p ) )  ->  p  e.  Prime )
11 simpll 753 . . . . . . . . 9  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  A. q  e.  Prime  ( q  ||  A  <->  q  =  p ) )  ->  A  e.  NN )
12 pcelnn 14404 . . . . . . . . 9  |-  ( ( p  e.  Prime  /\  A  e.  NN )  ->  (
( p  pCnt  A
)  e.  NN  <->  p  ||  A
) )
1310, 11, 12syl2anc 661 . . . . . . . 8  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  A. q  e.  Prime  ( q  ||  A  <->  q  =  p ) )  -> 
( ( p  pCnt  A )  e.  NN  <->  p  ||  A
) )
149, 13mpbird 232 . . . . . . 7  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  A. q  e.  Prime  ( q  ||  A  <->  q  =  p ) )  -> 
( p  pCnt  A
)  e.  NN )
15 simpr 461 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  NN  /\  p  e. 
Prime )  /\  (
q  e.  Prime  /\  (
q  ||  A  <->  q  =  p ) ) )  /\  q  =  p )  ->  q  =  p )
1615oveq1d 6311 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  NN  /\  p  e. 
Prime )  /\  (
q  e.  Prime  /\  (
q  ||  A  <->  q  =  p ) ) )  /\  q  =  p )  ->  ( q  pCnt  A )  =  ( p  pCnt  A )
)
17 simpllr 760 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  NN  /\  p  e. 
Prime )  /\  (
q  e.  Prime  /\  (
q  ||  A  <->  q  =  p ) ) )  /\  q  =  p )  ->  p  e.  Prime )
18 pccl 14384 . . . . . . . . . . . . . . . . . 18  |-  ( ( p  e.  Prime  /\  A  e.  NN )  ->  (
p  pCnt  A )  e.  NN0 )
1918ancoms 453 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  NN  /\  p  e.  Prime )  -> 
( p  pCnt  A
)  e.  NN0 )
2019ad2antrr 725 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( A  e.  NN  /\  p  e. 
Prime )  /\  (
q  e.  Prime  /\  (
q  ||  A  <->  q  =  p ) ) )  /\  q  =  p )  ->  ( p  pCnt  A )  e.  NN0 )
2120nn0zd 10988 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  NN  /\  p  e. 
Prime )  /\  (
q  e.  Prime  /\  (
q  ||  A  <->  q  =  p ) ) )  /\  q  =  p )  ->  ( p  pCnt  A )  e.  ZZ )
22 pcid 14407 . . . . . . . . . . . . . . 15  |-  ( ( p  e.  Prime  /\  (
p  pCnt  A )  e.  ZZ )  ->  (
p  pCnt  ( p ^ ( p  pCnt  A ) ) )  =  ( p  pCnt  A
) )
2317, 21, 22syl2anc 661 . . . . . . . . . . . . . 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 2501 . . . . . . . . . . . . 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 6311 . . . . . . . . . . . . 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 2501 . . . . . . . . . . . 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 757 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  ( q  e.  Prime  /\  ( q  ||  A  <->  q  =  p ) ) )  ->  ( q  ||  A  <->  q  =  p ) )
2827notbid 294 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  ( q  e.  Prime  /\  ( q  ||  A  <->  q  =  p ) ) )  ->  ( -.  q  ||  A  <->  -.  q  =  p ) )
2928biimpar 485 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  NN  /\  p  e. 
Prime )  /\  (
q  e.  Prime  /\  (
q  ||  A  <->  q  =  p ) ) )  /\  -.  q  =  p )  ->  -.  q  ||  A )
30 simplrl 761 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  NN  /\  p  e. 
Prime )  /\  (
q  e.  Prime  /\  (
q  ||  A  <->  q  =  p ) ) )  /\  -.  q  =  p )  ->  q  e.  Prime )
31 simplll 759 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  NN  /\  p  e. 
Prime )  /\  (
q  e.  Prime  /\  (
q  ||  A  <->  q  =  p ) ) )  /\  -.  q  =  p )  ->  A  e.  NN )
32 pceq0 14405 . . . . . . . . . . . . . . 15  |-  ( ( q  e.  Prime  /\  A  e.  NN )  ->  (
( q  pCnt  A
)  =  0  <->  -.  q  ||  A ) )
3330, 31, 32syl2anc 661 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  NN  /\  p  e. 
Prime )  /\  (
q  e.  Prime  /\  (
q  ||  A  <->  q  =  p ) ) )  /\  -.  q  =  p )  ->  (
( q  pCnt  A
)  =  0  <->  -.  q  ||  A ) )
3429, 33mpbird 232 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  NN  /\  p  e. 
Prime )  /\  (
q  e.  Prime  /\  (
q  ||  A  <->  q  =  p ) ) )  /\  -.  q  =  p )  ->  (
q  pCnt  A )  =  0 )
35 simprl 756 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  ( q  e.  Prime  /\  ( q  ||  A  <->  q  =  p ) ) )  ->  q  e.  Prime )
36 simplr 755 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  ( q  e.  Prime  /\  ( q  ||  A  <->  q  =  p ) ) )  ->  p  e.  Prime )
3719adantr 465 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  ( q  e.  Prime  /\  ( q  ||  A  <->  q  =  p ) ) )  ->  ( p  pCnt  A )  e.  NN0 )
38 prmdvdsexpr 14268 . . . . . . . . . . . . . . . 16  |-  ( ( q  e.  Prime  /\  p  e.  Prime  /\  ( p  pCnt  A )  e.  NN0 )  ->  ( q  ||  ( p ^ (
p  pCnt  A )
)  ->  q  =  p ) )
3935, 36, 37, 38syl3anc 1228 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  ( q  e.  Prime  /\  ( q  ||  A  <->  q  =  p ) ) )  ->  ( q  ||  ( p ^ (
p  pCnt  A )
)  ->  q  =  p ) )
4039con3dimp 441 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  NN  /\  p  e. 
Prime )  /\  (
q  e.  Prime  /\  (
q  ||  A  <->  q  =  p ) ) )  /\  -.  q  =  p )  ->  -.  q  ||  ( p ^
( p  pCnt  A
) ) )
41 prmnn 14231 . . . . . . . . . . . . . . . . . 18  |-  ( p  e.  Prime  ->  p  e.  NN )
4241adantl 466 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  NN  /\  p  e.  Prime )  ->  p  e.  NN )
4342, 19nnexpcld 12333 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  NN  /\  p  e.  Prime )  -> 
( p ^ (
p  pCnt  A )
)  e.  NN )
4443ad2antrr 725 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  NN  /\  p  e. 
Prime )  /\  (
q  e.  Prime  /\  (
q  ||  A  <->  q  =  p ) ) )  /\  -.  q  =  p )  ->  (
p ^ ( p 
pCnt  A ) )  e.  NN )
45 pceq0 14405 . . . . . . . . . . . . . . 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 661 . . . . . . . . . . . . . 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 232 . . . . . . . . . . . . 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 2501 . . . . . . . . . . . 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 791 . . . . . . . . . . 11  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  ( q  e.  Prime  /\  ( q  ||  A  <->  q  =  p ) ) )  ->  ( q  pCnt  A )  =  ( q  pCnt  ( p ^ ( p  pCnt  A ) ) ) )
5049expr 615 . . . . . . . . . 10  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  q  e.  Prime )  ->  ( ( q 
||  A  <->  q  =  p )  ->  (
q  pCnt  A )  =  ( q  pCnt  ( p ^ ( p 
pCnt  A ) ) ) ) )
5150ralimdva 2865 . . . . . . . . 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 429 . . . . . . . 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 10823 . . . . . . . . . 10  |-  ( A  e.  NN  ->  A  e.  NN0 )
5453ad2antrr 725 . . . . . . . . 9  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  A. q  e.  Prime  ( q  ||  A  <->  q  =  p ) )  ->  A  e.  NN0 )
5543adantr 465 . . . . . . . . . 10  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  A. q  e.  Prime  ( q  ||  A  <->  q  =  p ) )  -> 
( p ^ (
p  pCnt  A )
)  e.  NN )
5655nnnn0d 10873 . . . . . . . . 9  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  A. q  e.  Prime  ( q  ||  A  <->  q  =  p ) )  -> 
( p ^ (
p  pCnt  A )
)  e.  NN0 )
57 pc11 14414 . . . . . . . . 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 661 . . . . . . . 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 232 . . . . . . 7  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  A. q  e.  Prime  ( q  ||  A  <->  q  =  p ) )  ->  A  =  ( p ^ ( p  pCnt  A ) ) )
60 oveq2 6304 . . . . . . . . 9  |-  ( k  =  ( p  pCnt  A )  ->  ( p ^ k )  =  ( p ^ (
p  pCnt  A )
) )
6160eqeq2d 2471 . . . . . . . 8  |-  ( k  =  ( p  pCnt  A )  ->  ( A  =  ( p ^
k )  <->  A  =  ( p ^ (
p  pCnt  A )
) ) )
6261rspcev 3210 . . . . . . 7  |-  ( ( ( p  pCnt  A
)  e.  NN  /\  A  =  ( p ^ ( p  pCnt  A ) ) )  ->  E. k  e.  NN  A  =  ( p ^ k ) )
6314, 59, 62syl2anc 661 . . . . . 6  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  A. q  e.  Prime  ( q  ||  A  <->  q  =  p ) )  ->  E. k  e.  NN  A  =  ( p ^ k ) )
6463ex 434 . . . . 5  |-  ( ( A  e.  NN  /\  p  e.  Prime )  -> 
( A. q  e. 
Prime  ( q  ||  A  <->  q  =  p )  ->  E. k  e.  NN  A  =  ( p ^ k ) ) )
65 prmdvdsexpb 14267 . . . . . . . . . . 11  |-  ( ( q  e.  Prime  /\  p  e.  Prime  /\  k  e.  NN )  ->  ( q 
||  ( p ^
k )  <->  q  =  p ) )
66653coml 1203 . . . . . . . . . 10  |-  ( ( p  e.  Prime  /\  k  e.  NN  /\  q  e. 
Prime )  ->  ( q 
||  ( p ^
k )  <->  q  =  p ) )
67663expa 1196 . . . . . . . . 9  |-  ( ( ( p  e.  Prime  /\  k  e.  NN )  /\  q  e.  Prime )  ->  ( q  ||  ( p ^ k
)  <->  q  =  p ) )
6867ralrimiva 2871 . . . . . . . 8  |-  ( ( p  e.  Prime  /\  k  e.  NN )  ->  A. q  e.  Prime  ( q  ||  ( p ^ k
)  <->  q  =  p ) )
6968adantll 713 . . . . . . 7  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  k  e.  NN )  ->  A. q  e.  Prime  ( q  ||  ( p ^ k )  <->  q  =  p ) )
70 breq2 4460 . . . . . . . . 9  |-  ( A  =  ( p ^
k )  ->  (
q  ||  A  <->  q  ||  ( p ^ k
) ) )
7170bibi1d 319 . . . . . . . 8  |-  ( A  =  ( p ^
k )  ->  (
( q  ||  A  <->  q  =  p )  <->  ( q  ||  ( p ^ k
)  <->  q  =  p ) ) )
7271ralbidv 2896 . . . . . . 7  |-  ( A  =  ( p ^
k )  ->  ( A. q  e.  Prime  ( q  ||  A  <->  q  =  p )  <->  A. q  e.  Prime  ( q  ||  ( p ^ k
)  <->  q  =  p ) ) )
7369, 72syl5ibrcom 222 . . . . . 6  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  k  e.  NN )  ->  ( A  =  ( p ^ k
)  ->  A. q  e.  Prime  ( q  ||  A 
<->  q  =  p ) ) )
7473rexlimdva 2949 . . . . 5  |-  ( ( A  e.  NN  /\  p  e.  Prime )  -> 
( E. k  e.  NN  A  =  ( p ^ k )  ->  A. q  e.  Prime  ( q  ||  A  <->  q  =  p ) ) )
7564, 74impbid 191 . . . 4  |-  ( ( A  e.  NN  /\  p  e.  Prime )  -> 
( A. q  e. 
Prime  ( q  ||  A  <->  q  =  p )  <->  E. k  e.  NN  A  =  ( p ^ k ) ) )
7675rexbidva 2965 . . 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 257 . 2  |-  ( A  e.  NN  ->  ( E! q  e.  Prime  q 
||  A  <->  E. p  e.  Prime  E. k  e.  NN  A  =  ( p ^ k ) ) )
781, 77bitrd 253 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 184    /\ wa 369    = wceq 1395    e. wcel 1819    =/= wne 2652   A.wral 2807   E.wrex 2808   E!wreu 2809   class class class wbr 4456   ` cfv 5594  (class class class)co 6296   0cc0 9509   NNcn 10556   NN0cn0 10816   ZZcz 10885   ^cexp 12168    || cdvds 13997   Primecprime 14228    pCnt cpc 14371  Λcvma 23490
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587  ax-addf 9588  ax-mulf 9589
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-om 6700  df-1st 6799  df-2nd 6800  df-supp 6918  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-er 7329  df-map 7440  df-pm 7441  df-ixp 7489  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fsupp 7848  df-fi 7889  df-sup 7919  df-oi 7953  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-dec 11001  df-uz 11107  df-q 11208  df-rp 11246  df-xneg 11343  df-xadd 11344  df-xmul 11345  df-ioo 11558  df-ioc 11559  df-ico 11560  df-icc 11561  df-fz 11698  df-fzo 11821  df-fl 11931  df-mod 11999  df-seq 12110  df-exp 12169  df-fac 12356  df-bc 12383  df-hash 12408  df-shft 12911  df-cj 12943  df-re 12944  df-im 12945  df-sqrt 13079  df-abs 13080  df-limsup 13305  df-clim 13322  df-rlim 13323  df-sum 13520  df-ef 13814  df-sin 13816  df-cos 13817  df-pi 13819  df-dvds 13998  df-gcd 14156  df-prm 14229  df-pc 14372  df-struct 14645  df-ndx 14646  df-slot 14647  df-base 14648  df-sets 14649  df-ress 14650  df-plusg 14724  df-mulr 14725  df-starv 14726  df-sca 14727  df-vsca 14728  df-ip 14729  df-tset 14730  df-ple 14731  df-ds 14733  df-unif 14734  df-hom 14735  df-cco 14736  df-rest 14839  df-topn 14840  df-0g 14858  df-gsum 14859  df-topgen 14860  df-pt 14861  df-prds 14864  df-xrs 14918  df-qtop 14923  df-imas 14924  df-xps 14926  df-mre 15002  df-mrc 15003  df-acs 15005  df-mgm 15998  df-sgrp 16037  df-mnd 16047  df-submnd 16093  df-mulg 16186  df-cntz 16481  df-cmn 16926  df-psmet 18537  df-xmet 18538  df-met 18539  df-bl 18540  df-mopn 18541  df-fbas 18542  df-fg 18543  df-cnfld 18547  df-top 19525  df-bases 19527  df-topon 19528  df-topsp 19529  df-cld 19646  df-ntr 19647  df-cls 19648  df-nei 19725  df-lp 19763  df-perf 19764  df-cn 19854  df-cnp 19855  df-haus 19942  df-tx 20188  df-hmeo 20381  df-fil 20472  df-fm 20564  df-flim 20565  df-flf 20566  df-xms 20948  df-ms 20949  df-tms 20950  df-cncf 21507  df-limc 22395  df-dv 22396  df-log 23069  df-vma 23496
This theorem is referenced by:  vmacl  23517  efvmacl  23519  vma1  23565  vmalelog  23605  fsumvma  23613
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