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Theorem isppw2 23114
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 23113 . 2  |-  ( A  e.  NN  ->  (
(Λ `  A )  =/=  0  <->  E! q  e.  Prime  q 
||  A ) )
2 reu6 3292 . . 3  |-  ( E! q  e.  Prime  q  ||  A  <->  E. p  e.  Prime  A. q  e.  Prime  (
q  ||  A  <->  q  =  p ) )
3 equid 1740 . . . . . . . . 9  |-  p  =  p
4 breq1 4450 . . . . . . . . . . . 12  |-  ( q  =  p  ->  (
q  ||  A  <->  p  ||  A
) )
5 equequ1 1747 . . . . . . . . . . . 12  |-  ( q  =  p  ->  (
q  =  p  <->  p  =  p ) )
64, 5bibi12d 321 . . . . . . . . . . 11  |-  ( q  =  p  ->  (
( q  ||  A  <->  q  =  p )  <->  ( p  ||  A  <->  p  =  p
) ) )
76rspcva 3212 . . . . . . . . . 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 754 . . . . . . . . 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 14245 . . . . . . . . 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 6297 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  NN  /\  p  e. 
Prime )  /\  (
q  e.  Prime  /\  (
q  ||  A  <->  q  =  p ) ) )  /\  q  =  p )  ->  ( q  pCnt  A )  =  ( p  pCnt  A )
)
17 simpllr 758 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  NN  /\  p  e. 
Prime )  /\  (
q  e.  Prime  /\  (
q  ||  A  <->  q  =  p ) ) )  /\  q  =  p )  ->  p  e.  Prime )
18 pccl 14225 . . . . . . . . . . . . . . . . . 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 10960 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  NN  /\  p  e. 
Prime )  /\  (
q  e.  Prime  /\  (
q  ||  A  <->  q  =  p ) ) )  /\  q  =  p )  ->  ( p  pCnt  A )  e.  ZZ )
22 pcid 14248 . . . . . . . . . . . . . . 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 2511 . . . . . . . . . . . . 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 6297 . . . . . . . . . . . . 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 2511 . . . . . . . . . . . 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 756 . . . . . . . . . . . . . . . 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 759 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  NN  /\  p  e. 
Prime )  /\  (
q  e.  Prime  /\  (
q  ||  A  <->  q  =  p ) ) )  /\  -.  q  =  p )  ->  q  e.  Prime )
31 simplll 757 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  NN  /\  p  e. 
Prime )  /\  (
q  e.  Prime  /\  (
q  ||  A  <->  q  =  p ) ) )  /\  -.  q  =  p )  ->  A  e.  NN )
32 pceq0 14246 . . . . . . . . . . . . . . 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 755 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  ( q  e.  Prime  /\  ( q  ||  A  <->  q  =  p ) ) )  ->  q  e.  Prime )
36 simplr 754 . . . . . . . . . . . . . . . 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 14109 . . . . . . . . . . . . . . . 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 14072 . . . . . . . . . . . . . . . . . 18  |-  ( p  e.  Prime  ->  p  e.  NN )
4241adantl 466 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  NN  /\  p  e.  Prime )  ->  p  e.  NN )
4342, 19nnexpcld 12293 . . . . . . . . . . . . . . . 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 14246 . . . . . . . . . . . . . . 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 2511 . . . . . . . . . . . 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 789 . . . . . . . . . . 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 2872 . . . . . . . . 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 10798 . . . . . . . . . 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 10848 . . . . . . . . 9  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  A. q  e.  Prime  ( q  ||  A  <->  q  =  p ) )  -> 
( p ^ (
p  pCnt  A )
)  e.  NN0 )
57 pc11 14255 . . . . . . . . 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 6290 . . . . . . . . 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 3214 . . . . . . 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 14108 . . . . . . . . . . 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 2878 . . . . . . . 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 4451 . . . . . . . . 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 2903 . . . . . . 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 2955 . . . . 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 2970 . . 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 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   E.wrex 2815   E!wreu 2816   class class class wbr 4447   ` cfv 5586  (class class class)co 6282   0cc0 9488   NNcn 10532   NN0cn0 10791   ZZcz 10860   ^cexp 12129    || cdivides 13840   Primecprime 14069    pCnt cpc 14212  Λcvma 23090
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566  ax-addf 9567  ax-mulf 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-of 6522  df-om 6679  df-1st 6781  df-2nd 6782  df-supp 6899  df-recs 7039  df-rdg 7073  df-1o 7127  df-2o 7128  df-oadd 7131  df-er 7308  df-map 7419  df-pm 7420  df-ixp 7467  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-fsupp 7826  df-fi 7867  df-sup 7897  df-oi 7931  df-card 8316  df-cda 8544  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10973  df-uz 11079  df-q 11179  df-rp 11217  df-xneg 11314  df-xadd 11315  df-xmul 11316  df-ioo 11529  df-ioc 11530  df-ico 11531  df-icc 11532  df-fz 11669  df-fzo 11789  df-fl 11893  df-mod 11960  df-seq 12071  df-exp 12130  df-fac 12316  df-bc 12343  df-hash 12368  df-shft 12857  df-cj 12889  df-re 12890  df-im 12891  df-sqrt 13025  df-abs 13026  df-limsup 13250  df-clim 13267  df-rlim 13268  df-sum 13465  df-ef 13658  df-sin 13660  df-cos 13661  df-pi 13663  df-dvds 13841  df-gcd 13997  df-prm 14070  df-pc 14213  df-struct 14485  df-ndx 14486  df-slot 14487  df-base 14488  df-sets 14489  df-ress 14490  df-plusg 14561  df-mulr 14562  df-starv 14563  df-sca 14564  df-vsca 14565  df-ip 14566  df-tset 14567  df-ple 14568  df-ds 14570  df-unif 14571  df-hom 14572  df-cco 14573  df-rest 14671  df-topn 14672  df-0g 14690  df-gsum 14691  df-topgen 14692  df-pt 14693  df-prds 14696  df-xrs 14750  df-qtop 14755  df-imas 14756  df-xps 14758  df-mre 14834  df-mrc 14835  df-acs 14837  df-mnd 15725  df-submnd 15775  df-mulg 15858  df-cntz 16147  df-cmn 16593  df-psmet 18179  df-xmet 18180  df-met 18181  df-bl 18182  df-mopn 18183  df-fbas 18184  df-fg 18185  df-cnfld 18189  df-top 19163  df-bases 19165  df-topon 19166  df-topsp 19167  df-cld 19283  df-ntr 19284  df-cls 19285  df-nei 19362  df-lp 19400  df-perf 19401  df-cn 19491  df-cnp 19492  df-haus 19579  df-tx 19795  df-hmeo 19988  df-fil 20079  df-fm 20171  df-flim 20172  df-flf 20173  df-xms 20555  df-ms 20556  df-tms 20557  df-cncf 21114  df-limc 22002  df-dv 22003  df-log 22669  df-vma 23096
This theorem is referenced by:  vmacl  23117  efvmacl  23119  vma1  23165  vmalelog  23205  fsumvma  23213
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