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

Theorem pcpremul 13929
Description: Multiplicative property of the prime count pre-function. Note that the primality of  P is essential for this property;  ( 4  pCnt  2
)  =  0 but  ( 4  pCnt 
( 2  x.  2 ) )  =  1  =/=  2  x.  (
4  pCnt  2 )  =  0. Since this is needed to show uniqueness for the real prime count function (over  QQ), we don't bother to define it off the primes. (Contributed by Mario Carneiro, 23-Feb-2014.)
Hypotheses
Ref Expression
pcpremul.1  |-  S  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  M } ,  RR ,  <  )
pcpremul.2  |-  T  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  N } ,  RR ,  <  )
pcpremul.3  |-  U  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  ( M  x.  N
) } ,  RR ,  <  )
Assertion
Ref Expression
pcpremul  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( S  +  T
)  =  U )
Distinct variable groups:    n, M    n, N    P, n
Allowed substitution hints:    S( n)    T( n)    U( n)

Proof of Theorem pcpremul
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prmuz2 13800 . . . . . . 7  |-  ( P  e.  Prime  ->  P  e.  ( ZZ>= `  2 )
)
213ad2ant1 1009 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  P  e.  ( ZZ>= ` 
2 ) )
3 zmulcl 10712 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  x.  N
)  e.  ZZ )
43ad2ant2r 746 . . . . . . 7  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( M  x.  N
)  e.  ZZ )
543adant1 1006 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( M  x.  N
)  e.  ZZ )
6 zcn 10670 . . . . . . . . 9  |-  ( M  e.  ZZ  ->  M  e.  CC )
76anim1i 568 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  M  =/=  0 )  -> 
( M  e.  CC  /\  M  =/=  0 ) )
8 zcn 10670 . . . . . . . . 9  |-  ( N  e.  ZZ  ->  N  e.  CC )
98anim1i 568 . . . . . . . 8  |-  ( ( N  e.  ZZ  /\  N  =/=  0 )  -> 
( N  e.  CC  /\  N  =/=  0 ) )
10 mulne0 9997 . . . . . . . 8  |-  ( ( ( M  e.  CC  /\  M  =/=  0 )  /\  ( N  e.  CC  /\  N  =/=  0 ) )  -> 
( M  x.  N
)  =/=  0 )
117, 9, 10syl2an 477 . . . . . . 7  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( M  x.  N
)  =/=  0 )
12113adant1 1006 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( M  x.  N
)  =/=  0 )
13 eqid 2443 . . . . . . 7  |-  { n  e.  NN0  |  ( P ^ n )  ||  ( M  x.  N
) }  =  {
n  e.  NN0  | 
( P ^ n
)  ||  ( M  x.  N ) }
1413pclem 13924 . . . . . 6  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  (
( M  x.  N
)  e.  ZZ  /\  ( M  x.  N
)  =/=  0 ) )  ->  ( {
n  e.  NN0  | 
( P ^ n
)  ||  ( M  x.  N ) }  C_  ZZ  /\  { n  e. 
NN0  |  ( P ^ n )  ||  ( M  x.  N
) }  =/=  (/)  /\  E. x  e.  ZZ  A. y  e.  { n  e.  NN0  |  ( P ^ n
)  ||  ( M  x.  N ) } y  <_  x ) )
152, 5, 12, 14syl12anc 1216 . . . . 5  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( { n  e. 
NN0  |  ( P ^ n )  ||  ( M  x.  N
) }  C_  ZZ  /\ 
{ n  e.  NN0  |  ( P ^ n
)  ||  ( M  x.  N ) }  =/=  (/) 
/\  E. x  e.  ZZ  A. y  e.  { n  e.  NN0  |  ( P ^ n )  ||  ( M  x.  N
) } y  <_  x ) )
1615simp1d 1000 . . . 4  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  { n  e.  NN0  |  ( P ^ n
)  ||  ( M  x.  N ) }  C_  ZZ )
1715simp3d 1002 . . . 4  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  E. x  e.  ZZ  A. y  e.  { n  e.  NN0  |  ( P ^ n )  ||  ( M  x.  N
) } y  <_  x )
18 simp2l 1014 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  M  e.  ZZ )
19 simp2r 1015 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  M  =/=  0 )
20 eqid 2443 . . . . . . . . . 10  |-  { n  e.  NN0  |  ( P ^ n )  ||  M }  =  {
n  e.  NN0  | 
( P ^ n
)  ||  M }
21 pcpremul.1 . . . . . . . . . 10  |-  S  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  M } ,  RR ,  <  )
2220, 21pcprecl 13925 . . . . . . . . 9  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  ( M  e.  ZZ  /\  M  =/=  0 ) )  -> 
( S  e.  NN0  /\  ( P ^ S
)  ||  M )
)
232, 18, 19, 22syl12anc 1216 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( S  e.  NN0  /\  ( P ^ S
)  ||  M )
)
2423simpld 459 . . . . . . 7  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  S  e.  NN0 )
25 simp3l 1016 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  N  e.  ZZ )
26 simp3r 1017 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  N  =/=  0 )
27 eqid 2443 . . . . . . . . . 10  |-  { n  e.  NN0  |  ( P ^ n )  ||  N }  =  {
n  e.  NN0  | 
( P ^ n
)  ||  N }
28 pcpremul.2 . . . . . . . . . 10  |-  T  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  N } ,  RR ,  <  )
2927, 28pcprecl 13925 . . . . . . . . 9  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( T  e.  NN0  /\  ( P ^ T
)  ||  N )
)
302, 25, 26, 29syl12anc 1216 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( T  e.  NN0  /\  ( P ^ T
)  ||  N )
)
3130simpld 459 . . . . . . 7  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  T  e.  NN0 )
3224, 31nn0addcld 10659 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( S  +  T
)  e.  NN0 )
33 prmnn 13785 . . . . . . . . . . 11  |-  ( P  e.  Prime  ->  P  e.  NN )
34333ad2ant1 1009 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  P  e.  NN )
3534nncnd 10357 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  P  e.  CC )
3635, 31, 24expaddd 12029 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ ( S  +  T )
)  =  ( ( P ^ S )  x.  ( P ^ T ) ) )
3723simprd 463 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ S
)  ||  M )
3834, 24nnexpcld 12048 . . . . . . . . . . 11  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ S
)  e.  NN )
3938nnzd 10765 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ S
)  e.  ZZ )
4034, 31nnexpcld 12048 . . . . . . . . . . 11  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ T
)  e.  NN )
4140nnzd 10765 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ T
)  e.  ZZ )
42 dvdsmulc 13579 . . . . . . . . . 10  |-  ( ( ( P ^ S
)  e.  ZZ  /\  M  e.  ZZ  /\  ( P ^ T )  e.  ZZ )  ->  (
( P ^ S
)  ||  M  ->  ( ( P ^ S
)  x.  ( P ^ T ) ) 
||  ( M  x.  ( P ^ T ) ) ) )
4339, 18, 41, 42syl3anc 1218 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( ( P ^ S )  ||  M  ->  ( ( P ^ S )  x.  ( P ^ T ) ) 
||  ( M  x.  ( P ^ T ) ) ) )
4437, 43mpd 15 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( ( P ^ S )  x.  ( P ^ T ) ) 
||  ( M  x.  ( P ^ T ) ) )
4536, 44eqbrtrd 4331 . . . . . . 7  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ ( S  +  T )
)  ||  ( M  x.  ( P ^ T
) ) )
4630simprd 463 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ T
)  ||  N )
47 dvdscmul 13578 . . . . . . . . 9  |-  ( ( ( P ^ T
)  e.  ZZ  /\  N  e.  ZZ  /\  M  e.  ZZ )  ->  (
( P ^ T
)  ||  N  ->  ( M  x.  ( P ^ T ) ) 
||  ( M  x.  N ) ) )
4841, 25, 18, 47syl3anc 1218 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( ( P ^ T )  ||  N  ->  ( M  x.  ( P ^ T ) ) 
||  ( M  x.  N ) ) )
4946, 48mpd 15 . . . . . . 7  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( M  x.  ( P ^ T ) ) 
||  ( M  x.  N ) )
5034, 32nnexpcld 12048 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ ( S  +  T )
)  e.  NN )
5150nnzd 10765 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ ( S  +  T )
)  e.  ZZ )
5218, 41zmulcld 10772 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( M  x.  ( P ^ T ) )  e.  ZZ )
53 dvdstr 13585 . . . . . . . 8  |-  ( ( ( P ^ ( S  +  T )
)  e.  ZZ  /\  ( M  x.  ( P ^ T ) )  e.  ZZ  /\  ( M  x.  N )  e.  ZZ )  ->  (
( ( P ^
( S  +  T
) )  ||  ( M  x.  ( P ^ T ) )  /\  ( M  x.  ( P ^ T ) ) 
||  ( M  x.  N ) )  -> 
( P ^ ( S  +  T )
)  ||  ( M  x.  N ) ) )
5451, 52, 5, 53syl3anc 1218 . . . . . . 7  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( ( ( P ^ ( S  +  T ) )  ||  ( M  x.  ( P ^ T ) )  /\  ( M  x.  ( P ^ T ) )  ||  ( M  x.  N ) )  ->  ( P ^
( S  +  T
) )  ||  ( M  x.  N )
) )
5545, 49, 54mp2and 679 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ ( S  +  T )
)  ||  ( M  x.  N ) )
56 oveq2 6118 . . . . . . . 8  |-  ( x  =  ( S  +  T )  ->  ( P ^ x )  =  ( P ^ ( S  +  T )
) )
5756breq1d 4321 . . . . . . 7  |-  ( x  =  ( S  +  T )  ->  (
( P ^ x
)  ||  ( M  x.  N )  <->  ( P ^ ( S  +  T ) )  ||  ( M  x.  N
) ) )
5857elrab 3136 . . . . . 6  |-  ( ( S  +  T )  e.  { x  e. 
NN0  |  ( P ^ x )  ||  ( M  x.  N
) }  <->  ( ( S  +  T )  e.  NN0  /\  ( P ^ ( S  +  T ) )  ||  ( M  x.  N
) ) )
5932, 55, 58sylanbrc 664 . . . . 5  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( S  +  T
)  e.  { x  e.  NN0  |  ( P ^ x )  ||  ( M  x.  N
) } )
60 oveq2 6118 . . . . . . 7  |-  ( x  =  n  ->  ( P ^ x )  =  ( P ^ n
) )
6160breq1d 4321 . . . . . 6  |-  ( x  =  n  ->  (
( P ^ x
)  ||  ( M  x.  N )  <->  ( P ^ n )  ||  ( M  x.  N
) ) )
6261cbvrabv 2990 . . . . 5  |-  { x  e.  NN0  |  ( P ^ x )  ||  ( M  x.  N
) }  =  {
n  e.  NN0  | 
( P ^ n
)  ||  ( M  x.  N ) }
6359, 62syl6eleq 2533 . . . 4  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( S  +  T
)  e.  { n  e.  NN0  |  ( P ^ n )  ||  ( M  x.  N
) } )
64 suprzub 10965 . . . 4  |-  ( ( { n  e.  NN0  |  ( P ^ n
)  ||  ( M  x.  N ) }  C_  ZZ  /\  E. x  e.  ZZ  A. y  e. 
{ n  e.  NN0  |  ( P ^ n
)  ||  ( M  x.  N ) } y  <_  x  /\  ( S  +  T )  e.  { n  e.  NN0  |  ( P ^ n
)  ||  ( M  x.  N ) } )  ->  ( S  +  T )  <_  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  ( M  x.  N ) } ,  RR ,  <  ) )
6516, 17, 63, 64syl3anc 1218 . . 3  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( S  +  T
)  <_  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  ( M  x.  N ) } ,  RR ,  <  ) )
66 pcpremul.3 . . 3  |-  U  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  ( M  x.  N
) } ,  RR ,  <  )
6765, 66syl6breqr 4351 . 2  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( S  +  T
)  <_  U )
6820, 21pcprendvds2 13927 . . . . . 6  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  ( M  e.  ZZ  /\  M  =/=  0 ) )  ->  -.  P  ||  ( M  /  ( P ^ S ) ) )
692, 18, 19, 68syl12anc 1216 . . . . 5  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  -.  P  ||  ( M  /  ( P ^ S ) ) )
7027, 28pcprendvds2 13927 . . . . . 6  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  -.  P  ||  ( N  /  ( P ^ T ) ) )
712, 25, 26, 70syl12anc 1216 . . . . 5  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  -.  P  ||  ( N  /  ( P ^ T ) ) )
72 ioran 490 . . . . 5  |-  ( -.  ( P  ||  ( M  /  ( P ^ S ) )  \/  P  ||  ( N  /  ( P ^ T ) ) )  <-> 
( -.  P  ||  ( M  /  ( P ^ S ) )  /\  -.  P  ||  ( N  /  ( P ^ T ) ) ) )
7369, 71, 72sylanbrc 664 . . . 4  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  -.  ( P  ||  ( M  /  ( P ^ S ) )  \/  P  ||  ( N  /  ( P ^ T ) ) ) )
74 simp1 988 . . . . 5  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  P  e.  Prime )
7538nnne0d 10385 . . . . . . 7  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ S
)  =/=  0 )
76 dvdsval2 13557 . . . . . . 7  |-  ( ( ( P ^ S
)  e.  ZZ  /\  ( P ^ S )  =/=  0  /\  M  e.  ZZ )  ->  (
( P ^ S
)  ||  M  <->  ( M  /  ( P ^ S ) )  e.  ZZ ) )
7739, 75, 18, 76syl3anc 1218 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( ( P ^ S )  ||  M  <->  ( M  /  ( P ^ S ) )  e.  ZZ ) )
7837, 77mpbid 210 . . . . 5  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( M  /  ( P ^ S ) )  e.  ZZ )
7940nnne0d 10385 . . . . . . 7  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ T
)  =/=  0 )
80 dvdsval2 13557 . . . . . . 7  |-  ( ( ( P ^ T
)  e.  ZZ  /\  ( P ^ T )  =/=  0  /\  N  e.  ZZ )  ->  (
( P ^ T
)  ||  N  <->  ( N  /  ( P ^ T ) )  e.  ZZ ) )
8141, 79, 25, 80syl3anc 1218 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( ( P ^ T )  ||  N  <->  ( N  /  ( P ^ T ) )  e.  ZZ ) )
8246, 81mpbid 210 . . . . 5  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( N  /  ( P ^ T ) )  e.  ZZ )
83 euclemma 13813 . . . . 5  |-  ( ( P  e.  Prime  /\  ( M  /  ( P ^ S ) )  e.  ZZ  /\  ( N  /  ( P ^ T ) )  e.  ZZ )  ->  ( P  ||  ( ( M  /  ( P ^ S ) )  x.  ( N  /  ( P ^ T ) ) )  <->  ( P  ||  ( M  /  ( P ^ S ) )  \/  P  ||  ( N  /  ( P ^ T ) ) ) ) )
8474, 78, 82, 83syl3anc 1218 . . . 4  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P  ||  (
( M  /  ( P ^ S ) )  x.  ( N  / 
( P ^ T
) ) )  <->  ( P  ||  ( M  /  ( P ^ S ) )  \/  P  ||  ( N  /  ( P ^ T ) ) ) ) )
8573, 84mtbird 301 . . 3  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  -.  P  ||  ( ( M  /  ( P ^ S ) )  x.  ( N  / 
( P ^ T
) ) ) )
8613, 66pcprecl 13925 . . . . . . 7  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  (
( M  x.  N
)  e.  ZZ  /\  ( M  x.  N
)  =/=  0 ) )  ->  ( U  e.  NN0  /\  ( P ^ U )  ||  ( M  x.  N
) ) )
872, 5, 12, 86syl12anc 1216 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( U  e.  NN0  /\  ( P ^ U
)  ||  ( M  x.  N ) ) )
8887simpld 459 . . . . 5  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  U  e.  NN0 )
89 nn0ltp1le 10721 . . . . 5  |-  ( ( ( S  +  T
)  e.  NN0  /\  U  e.  NN0 )  -> 
( ( S  +  T )  <  U  <->  ( ( S  +  T
)  +  1 )  <_  U ) )
9032, 88, 89syl2anc 661 . . . 4  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( ( S  +  T )  <  U  <->  ( ( S  +  T
)  +  1 )  <_  U ) )
9134nnzd 10765 . . . . . . 7  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  P  e.  ZZ )
92 peano2nn0 10639 . . . . . . . 8  |-  ( ( S  +  T )  e.  NN0  ->  ( ( S  +  T )  +  1 )  e. 
NN0 )
9332, 92syl 16 . . . . . . 7  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( ( S  +  T )  +  1 )  e.  NN0 )
94 dvdsexp 13608 . . . . . . . 8  |-  ( ( P  e.  ZZ  /\  ( ( S  +  T )  +  1 )  e.  NN0  /\  U  e.  ( ZZ>= `  ( ( S  +  T )  +  1 ) ) )  -> 
( P ^ (
( S  +  T
)  +  1 ) )  ||  ( P ^ U ) )
95943expia 1189 . . . . . . 7  |-  ( ( P  e.  ZZ  /\  ( ( S  +  T )  +  1 )  e.  NN0 )  ->  ( U  e.  (
ZZ>= `  ( ( S  +  T )  +  1 ) )  -> 
( P ^ (
( S  +  T
)  +  1 ) )  ||  ( P ^ U ) ) )
9691, 93, 95syl2anc 661 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( U  e.  (
ZZ>= `  ( ( S  +  T )  +  1 ) )  -> 
( P ^ (
( S  +  T
)  +  1 ) )  ||  ( P ^ U ) ) )
9787simprd 463 . . . . . . 7  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ U
)  ||  ( M  x.  N ) )
9834, 93nnexpcld 12048 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ (
( S  +  T
)  +  1 ) )  e.  NN )
9998nnzd 10765 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ (
( S  +  T
)  +  1 ) )  e.  ZZ )
10034, 88nnexpcld 12048 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ U
)  e.  NN )
101100nnzd 10765 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ U
)  e.  ZZ )
102 dvdstr 13585 . . . . . . . 8  |-  ( ( ( P ^ (
( S  +  T
)  +  1 ) )  e.  ZZ  /\  ( P ^ U )  e.  ZZ  /\  ( M  x.  N )  e.  ZZ )  ->  (
( ( P ^
( ( S  +  T )  +  1 ) )  ||  ( P ^ U )  /\  ( P ^ U ) 
||  ( M  x.  N ) )  -> 
( P ^ (
( S  +  T
)  +  1 ) )  ||  ( M  x.  N ) ) )
10399, 101, 5, 102syl3anc 1218 . . . . . . 7  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( ( ( P ^ ( ( S  +  T )  +  1 ) )  ||  ( P ^ U )  /\  ( P ^ U )  ||  ( M  x.  N )
)  ->  ( P ^ ( ( S  +  T )  +  1 ) )  ||  ( M  x.  N
) ) )
10497, 103mpan2d 674 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( ( P ^
( ( S  +  T )  +  1 ) )  ||  ( P ^ U )  -> 
( P ^ (
( S  +  T
)  +  1 ) )  ||  ( M  x.  N ) ) )
10596, 104syld 44 . . . . 5  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( U  e.  (
ZZ>= `  ( ( S  +  T )  +  1 ) )  -> 
( P ^ (
( S  +  T
)  +  1 ) )  ||  ( M  x.  N ) ) )
10693nn0zd 10764 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( ( S  +  T )  +  1 )  e.  ZZ )
10788nn0zd 10764 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  U  e.  ZZ )
108 eluz 10893 . . . . . 6  |-  ( ( ( ( S  +  T )  +  1 )  e.  ZZ  /\  U  e.  ZZ )  ->  ( U  e.  (
ZZ>= `  ( ( S  +  T )  +  1 ) )  <->  ( ( S  +  T )  +  1 )  <_  U ) )
109106, 107, 108syl2anc 661 . . . . 5  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( U  e.  (
ZZ>= `  ( ( S  +  T )  +  1 ) )  <->  ( ( S  +  T )  +  1 )  <_  U ) )
11035, 32expp1d 12028 . . . . . . 7  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ (
( S  +  T
)  +  1 ) )  =  ( ( P ^ ( S  +  T ) )  x.  P ) )
11118zcnd 10767 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  M  e.  CC )
11225zcnd 10767 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  N  e.  CC )
113111, 112mulcld 9425 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( M  x.  N
)  e.  CC )
11450nncnd 10357 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ ( S  +  T )
)  e.  CC )
11550nnne0d 10385 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ ( S  +  T )
)  =/=  0 )
116113, 114, 115divcan2d 10128 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( ( P ^
( S  +  T
) )  x.  (
( M  x.  N
)  /  ( P ^ ( S  +  T ) ) ) )  =  ( M  x.  N ) )
11736oveq2d 6126 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( ( M  x.  N )  /  ( P ^ ( S  +  T ) ) )  =  ( ( M  x.  N )  / 
( ( P ^ S )  x.  ( P ^ T ) ) ) )
11838nncnd 10357 . . . . . . . . . . 11  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ S
)  e.  CC )
11940nncnd 10357 . . . . . . . . . . 11  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ T
)  e.  CC )
120111, 118, 112, 119, 75, 79divmuldivd 10167 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( ( M  / 
( P ^ S
) )  x.  ( N  /  ( P ^ T ) ) )  =  ( ( M  x.  N )  / 
( ( P ^ S )  x.  ( P ^ T ) ) ) )
121117, 120eqtr4d 2478 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( ( M  x.  N )  /  ( P ^ ( S  +  T ) ) )  =  ( ( M  /  ( P ^ S ) )  x.  ( N  /  ( P ^ T ) ) ) )
122121oveq2d 6126 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( ( P ^
( S  +  T
) )  x.  (
( M  x.  N
)  /  ( P ^ ( S  +  T ) ) ) )  =  ( ( P ^ ( S  +  T ) )  x.  ( ( M  /  ( P ^ S ) )  x.  ( N  /  ( P ^ T ) ) ) ) )
123116, 122eqtr3d 2477 . . . . . . 7  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( M  x.  N
)  =  ( ( P ^ ( S  +  T ) )  x.  ( ( M  /  ( P ^ S ) )  x.  ( N  /  ( P ^ T ) ) ) ) )
124110, 123breq12d 4324 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( ( P ^
( ( S  +  T )  +  1 ) )  ||  ( M  x.  N )  <->  ( ( P ^ ( S  +  T )
)  x.  P ) 
||  ( ( P ^ ( S  +  T ) )  x.  ( ( M  / 
( P ^ S
) )  x.  ( N  /  ( P ^ T ) ) ) ) ) )
12578, 82zmulcld 10772 . . . . . . 7  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( ( M  / 
( P ^ S
) )  x.  ( N  /  ( P ^ T ) ) )  e.  ZZ )
126 dvdscmulr 13580 . . . . . . 7  |-  ( ( P  e.  ZZ  /\  ( ( M  / 
( P ^ S
) )  x.  ( N  /  ( P ^ T ) ) )  e.  ZZ  /\  (
( P ^ ( S  +  T )
)  e.  ZZ  /\  ( P ^ ( S  +  T ) )  =/=  0 ) )  ->  ( ( ( P ^ ( S  +  T ) )  x.  P )  ||  ( ( P ^
( S  +  T
) )  x.  (
( M  /  ( P ^ S ) )  x.  ( N  / 
( P ^ T
) ) ) )  <-> 
P  ||  ( ( M  /  ( P ^ S ) )  x.  ( N  /  ( P ^ T ) ) ) ) )
12791, 125, 51, 115, 126syl112anc 1222 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( ( ( P ^ ( S  +  T ) )  x.  P )  ||  (
( P ^ ( S  +  T )
)  x.  ( ( M  /  ( P ^ S ) )  x.  ( N  / 
( P ^ T
) ) ) )  <-> 
P  ||  ( ( M  /  ( P ^ S ) )  x.  ( N  /  ( P ^ T ) ) ) ) )
128124, 127bitrd 253 . . . . 5  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( ( P ^
( ( S  +  T )  +  1 ) )  ||  ( M  x.  N )  <->  P 
||  ( ( M  /  ( P ^ S ) )  x.  ( N  /  ( P ^ T ) ) ) ) )
129105, 109, 1283imtr3d 267 . . . 4  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( ( ( S  +  T )  +  1 )  <_  U  ->  P  ||  ( ( M  /  ( P ^ S ) )  x.  ( N  / 
( P ^ T
) ) ) ) )
13090, 129sylbid 215 . . 3  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( ( S  +  T )  <  U  ->  P  ||  ( ( M  /  ( P ^ S ) )  x.  ( N  / 
( P ^ T
) ) ) ) )
13185, 130mtod 177 . 2  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  -.  ( S  +  T
)  <  U )
13232nn0red 10656 . . 3  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( S  +  T
)  e.  RR )
13388nn0red 10656 . . 3  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  U  e.  RR )
134132, 133eqleltd 9537 . 2  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( ( S  +  T )  =  U  <-> 
( ( S  +  T )  <_  U  /\  -.  ( S  +  T )  <  U
) ) )
13567, 131, 134mpbir2and 913 1  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( S  +  T
)  =  U )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2620   A.wral 2734   E.wrex 2735   {crab 2738    C_ wss 3347   (/)c0 3656   class class class wbr 4311   ` cfv 5437  (class class class)co 6110   supcsup 7709   CCcc 9299   RRcr 9300   0cc0 9301   1c1 9302    + caddc 9304    x. cmul 9306    < clt 9437    <_ cle 9438    / cdiv 10012   NNcn 10341   2c2 10390   NN0cn0 10598   ZZcz 10665   ZZ>=cuz 10880   ^cexp 11884    || cdivides 13554   Primecprime 13782
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 4432  ax-nul 4440  ax-pow 4489  ax-pr 4550  ax-un 6391  ax-cnex 9357  ax-resscn 9358  ax-1cn 9359  ax-icn 9360  ax-addcl 9361  ax-addrcl 9362  ax-mulcl 9363  ax-mulrcl 9364  ax-mulcom 9365  ax-addass 9366  ax-mulass 9367  ax-distr 9368  ax-i2m1 9369  ax-1ne0 9370  ax-1rid 9371  ax-rnegex 9372  ax-rrecex 9373  ax-cnre 9374  ax-pre-lttri 9375  ax-pre-lttrn 9376  ax-pre-ltadd 9377  ax-pre-mulgt0 9378  ax-pre-sup 9379
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 2739  df-rex 2740  df-reu 2741  df-rmo 2742  df-rab 2743  df-v 2993  df-sbc 3206  df-csb 3308  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-pss 3363  df-nul 3657  df-if 3811  df-pw 3881  df-sn 3897  df-pr 3899  df-tp 3901  df-op 3903  df-uni 4111  df-int 4148  df-iun 4192  df-br 4312  df-opab 4370  df-mpt 4371  df-tr 4405  df-eprel 4651  df-id 4655  df-po 4660  df-so 4661  df-fr 4698  df-we 4700  df-ord 4741  df-on 4742  df-lim 4743  df-suc 4744  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-res 4871  df-ima 4872  df-iota 5400  df-fun 5439  df-fn 5440  df-f 5441  df-f1 5442  df-fo 5443  df-f1o 5444  df-fv 5445  df-riota 6071  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-om 6496  df-2nd 6597  df-recs 6851  df-rdg 6885  df-1o 6939  df-2o 6940  df-oadd 6943  df-er 7120  df-en 7330  df-dom 7331  df-sdom 7332  df-fin 7333  df-sup 7710  df-pnf 9439  df-mnf 9440  df-xr 9441  df-ltxr 9442  df-le 9443  df-sub 9616  df-neg 9617  df-div 10013  df-nn 10342  df-2 10399  df-3 10400  df-n0 10599  df-z 10666  df-uz 10881  df-rp 11011  df-fl 11661  df-mod 11728  df-seq 11826  df-exp 11885  df-cj 12607  df-re 12608  df-im 12609  df-sqr 12743  df-abs 12744  df-dvds 13555  df-gcd 13710  df-prm 13783
This theorem is referenced by:  pceulem  13931  pcmul  13937
  Copyright terms: Public domain W3C validator