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Theorem pcpremul 13104
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 12984 . . . . . . 7  |-  ( P  e.  Prime  ->  P  e.  ( ZZ>= `  2 )
)
213ad2ant1 977 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  P  e.  ( ZZ>= ` 
2 ) )
3 zmulcl 10217 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  x.  N
)  e.  ZZ )
43ad2ant2r 727 . . . . . . 7  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( M  x.  N
)  e.  ZZ )
543adant1 974 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( M  x.  N
)  e.  ZZ )
6 zcn 10180 . . . . . . . . 9  |-  ( M  e.  ZZ  ->  M  e.  CC )
76anim1i 551 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  M  =/=  0 )  -> 
( M  e.  CC  /\  M  =/=  0 ) )
8 zcn 10180 . . . . . . . . 9  |-  ( N  e.  ZZ  ->  N  e.  CC )
98anim1i 551 . . . . . . . 8  |-  ( ( N  e.  ZZ  /\  N  =/=  0 )  -> 
( N  e.  CC  /\  N  =/=  0 ) )
10 mulne0 9557 . . . . . . . 8  |-  ( ( ( M  e.  CC  /\  M  =/=  0 )  /\  ( N  e.  CC  /\  N  =/=  0 ) )  -> 
( M  x.  N
)  =/=  0 )
117, 9, 10syl2an 463 . . . . . . 7  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( M  x.  N
)  =/=  0 )
12113adant1 974 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( M  x.  N
)  =/=  0 )
13 eqid 2366 . . . . . . 7  |-  { n  e.  NN0  |  ( P ^ n )  ||  ( M  x.  N
) }  =  {
n  e.  NN0  | 
( P ^ n
)  ||  ( M  x.  N ) }
1413pclem 13099 . . . . . 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 1181 . . . . 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 968 . . . 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 970 . . . 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 982 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  M  e.  ZZ )
19 simp2r 983 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  M  =/=  0 )
20 eqid 2366 . . . . . . . . . 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 13100 . . . . . . . . 9  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  ( M  e.  ZZ  /\  M  =/=  0 ) )  -> 
( S  e.  NN0  /\  ( P ^ S
)  ||  M )
)
232, 18, 19, 22syl12anc 1181 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( S  e.  NN0  /\  ( P ^ S
)  ||  M )
)
2423simpld 445 . . . . . . 7  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  S  e.  NN0 )
25 simp3l 984 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  N  e.  ZZ )
26 simp3r 985 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  N  =/=  0 )
27 eqid 2366 . . . . . . . . . 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 13100 . . . . . . . . 9  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( T  e.  NN0  /\  ( P ^ T
)  ||  N )
)
302, 25, 26, 29syl12anc 1181 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( T  e.  NN0  /\  ( P ^ T
)  ||  N )
)
3130simpld 445 . . . . . . 7  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  T  e.  NN0 )
3224, 31nn0addcld 10171 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( S  +  T
)  e.  NN0 )
33 prmnn 12969 . . . . . . . . . . 11  |-  ( P  e.  Prime  ->  P  e.  NN )
34333ad2ant1 977 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  P  e.  NN )
3534nncnd 9909 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  P  e.  CC )
3635, 31, 24expaddd 11412 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ ( S  +  T )
)  =  ( ( P ^ S )  x.  ( P ^ T ) ) )
3723simprd 449 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ S
)  ||  M )
3834, 24nnexpcld 11431 . . . . . . . . . . 11  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ S
)  e.  NN )
3938nnzd 10267 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ S
)  e.  ZZ )
4034, 31nnexpcld 11431 . . . . . . . . . . 11  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ T
)  e.  NN )
4140nnzd 10267 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ T
)  e.  ZZ )
42 dvdsmulc 12764 . . . . . . . . . 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 1183 . . . . . . . . 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 14 . . . . . . . 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 4145 . . . . . . 7  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ ( S  +  T )
)  ||  ( M  x.  ( P ^ T
) ) )
4630simprd 449 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ T
)  ||  N )
47 dvdscmul 12763 . . . . . . . . 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 1183 . . . . . . . 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 14 . . . . . . 7  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( M  x.  ( P ^ T ) ) 
||  ( M  x.  N ) )
5034, 32nnexpcld 11431 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ ( S  +  T )
)  e.  NN )
5150nnzd 10267 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ ( S  +  T )
)  e.  ZZ )
5218, 41zmulcld 10274 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( M  x.  ( P ^ T ) )  e.  ZZ )
53 dvdstr 12770 . . . . . . . 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 1183 . . . . . . 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 660 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ ( S  +  T )
)  ||  ( M  x.  N ) )
56 oveq2 5989 . . . . . . . 8  |-  ( x  =  ( S  +  T )  ->  ( P ^ x )  =  ( P ^ ( S  +  T )
) )
5756breq1d 4135 . . . . . . 7  |-  ( x  =  ( S  +  T )  ->  (
( P ^ x
)  ||  ( M  x.  N )  <->  ( P ^ ( S  +  T ) )  ||  ( M  x.  N
) ) )
5857elrab 3009 . . . . . 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 645 . . . . 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 5989 . . . . . . 7  |-  ( x  =  n  ->  ( P ^ x )  =  ( P ^ n
) )
6160breq1d 4135 . . . . . 6  |-  ( x  =  n  ->  (
( P ^ x
)  ||  ( M  x.  N )  <->  ( P ^ n )  ||  ( M  x.  N
) ) )
6261cbvrabv 2872 . . . . 5  |-  { x  e.  NN0  |  ( P ^ x )  ||  ( M  x.  N
) }  =  {
n  e.  NN0  | 
( P ^ n
)  ||  ( M  x.  N ) }
6359, 62syl6eleq 2456 . . . 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 10460 . . . 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 1183 . . 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 4165 . 2  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( S  +  T
)  <_  U )
6820, 21pcprendvds2 13102 . . . . . 6  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  ( M  e.  ZZ  /\  M  =/=  0 ) )  ->  -.  P  ||  ( M  /  ( P ^ S ) ) )
692, 18, 19, 68syl12anc 1181 . . . . 5  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  -.  P  ||  ( M  /  ( P ^ S ) ) )
7027, 28pcprendvds2 13102 . . . . . 6  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  -.  P  ||  ( N  /  ( P ^ T ) ) )
712, 25, 26, 70syl12anc 1181 . . . . 5  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  -.  P  ||  ( N  /  ( P ^ T ) ) )
72 ioran 476 . . . . 5  |-  ( -.  ( P  ||  ( M  /  ( P ^ S ) )  \/  P  ||  ( N  /  ( P ^ T ) ) )  <-> 
( -.  P  ||  ( M  /  ( P ^ S ) )  /\  -.  P  ||  ( N  /  ( P ^ T ) ) ) )
7369, 71, 72sylanbrc 645 . . . 4  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  -.  ( P  ||  ( M  /  ( P ^ S ) )  \/  P  ||  ( N  /  ( P ^ T ) ) ) )
74 simp1 956 . . . . 5  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  P  e.  Prime )
7538nnne0d 9937 . . . . . . 7  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ S
)  =/=  0 )
76 dvdsval2 12742 . . . . . . 7  |-  ( ( ( P ^ S
)  e.  ZZ  /\  ( P ^ S )  =/=  0  /\  M  e.  ZZ )  ->  (
( P ^ S
)  ||  M  <->  ( M  /  ( P ^ S ) )  e.  ZZ ) )
7739, 75, 18, 76syl3anc 1183 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( ( P ^ S )  ||  M  <->  ( M  /  ( P ^ S ) )  e.  ZZ ) )
7837, 77mpbid 201 . . . . 5  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( M  /  ( P ^ S ) )  e.  ZZ )
7940nnne0d 9937 . . . . . . 7  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ T
)  =/=  0 )
80 dvdsval2 12742 . . . . . . 7  |-  ( ( ( P ^ T
)  e.  ZZ  /\  ( P ^ T )  =/=  0  /\  N  e.  ZZ )  ->  (
( P ^ T
)  ||  N  <->  ( N  /  ( P ^ T ) )  e.  ZZ ) )
8141, 79, 25, 80syl3anc 1183 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( ( P ^ T )  ||  N  <->  ( N  /  ( P ^ T ) )  e.  ZZ ) )
8246, 81mpbid 201 . . . . 5  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( N  /  ( P ^ T ) )  e.  ZZ )
83 euclemma 12995 . . . . 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 1183 . . . 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 292 . . 3  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  -.  P  ||  ( ( M  /  ( P ^ S ) )  x.  ( N  / 
( P ^ T
) ) ) )
8613, 66pcprecl 13100 . . . . . . 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 1181 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( U  e.  NN0  /\  ( P ^ U
)  ||  ( M  x.  N ) ) )
8887simpld 445 . . . . 5  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  U  e.  NN0 )
89 nn0ltp1le 10225 . . . . 5  |-  ( ( ( S  +  T
)  e.  NN0  /\  U  e.  NN0 )  -> 
( ( S  +  T )  <  U  <->  ( ( S  +  T
)  +  1 )  <_  U ) )
9032, 88, 89syl2anc 642 . . . 4  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( ( S  +  T )  <  U  <->  ( ( S  +  T
)  +  1 )  <_  U ) )
9134nnzd 10267 . . . . . . 7  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  P  e.  ZZ )
92 peano2nn0 10153 . . . . . . . 8  |-  ( ( S  +  T )  e.  NN0  ->  ( ( S  +  T )  +  1 )  e. 
NN0 )
9332, 92syl 15 . . . . . . 7  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( ( S  +  T )  +  1 )  e.  NN0 )
94 dvdsexp 12792 . . . . . . . 8  |-  ( ( P  e.  ZZ  /\  ( ( S  +  T )  +  1 )  e.  NN0  /\  U  e.  ( ZZ>= `  ( ( S  +  T )  +  1 ) ) )  -> 
( P ^ (
( S  +  T
)  +  1 ) )  ||  ( P ^ U ) )
95943expia 1154 . . . . . . 7  |-  ( ( P  e.  ZZ  /\  ( ( S  +  T )  +  1 )  e.  NN0 )  ->  ( U  e.  (
ZZ>= `  ( ( S  +  T )  +  1 ) )  -> 
( P ^ (
( S  +  T
)  +  1 ) )  ||  ( P ^ U ) ) )
9691, 93, 95syl2anc 642 . . . . . 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 449 . . . . . . 7  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ U
)  ||  ( M  x.  N ) )
9834, 93nnexpcld 11431 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ (
( S  +  T
)  +  1 ) )  e.  NN )
9998nnzd 10267 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ (
( S  +  T
)  +  1 ) )  e.  ZZ )
10034, 88nnexpcld 11431 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ U
)  e.  NN )
101100nnzd 10267 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ U
)  e.  ZZ )
102 dvdstr 12770 . . . . . . . 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 1183 . . . . . . 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 655 . . . . . 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 40 . . . . 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 10266 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( ( S  +  T )  +  1 )  e.  ZZ )
10788nn0zd 10266 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  U  e.  ZZ )
108 eluz 10392 . . . . . 6  |-  ( ( ( ( S  +  T )  +  1 )  e.  ZZ  /\  U  e.  ZZ )  ->  ( U  e.  (
ZZ>= `  ( ( S  +  T )  +  1 ) )  <->  ( ( S  +  T )  +  1 )  <_  U ) )
109106, 107, 108syl2anc 642 . . . . 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 11411 . . . . . . 7  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ (
( S  +  T
)  +  1 ) )  =  ( ( P ^ ( S  +  T ) )  x.  P ) )
11118zcnd 10269 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  M  e.  CC )
11225zcnd 10269 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  N  e.  CC )
113111, 112mulcld 9002 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( M  x.  N
)  e.  CC )
11450nncnd 9909 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ ( S  +  T )
)  e.  CC )
11550nnne0d 9937 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ ( S  +  T )
)  =/=  0 )
116113, 114, 115divcan2d 9685 . . . . . . . 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 5997 . . . . . . . . . 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 9909 . . . . . . . . . . 11  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ S
)  e.  CC )
11940nncnd 9909 . . . . . . . . . . 11  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ T
)  e.  CC )
120111, 118, 112, 119, 75, 79divmuldivd 9724 . . . . . . . . . 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 2401 . . . . . . . . 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 5997 . . . . . . . 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 2400 . . . . . . 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 4138 . . . . . 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 10274 . . . . . . 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 12765 . . . . . . 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 1187 . . . . . 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 244 . . . . 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 258 . . . 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 206 . . 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 168 . 2  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  -.  ( S  +  T
)  <  U )
13232nn0red 10168 . . 3  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( S  +  T
)  e.  RR )
13388nn0red 10168 . . 3  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  U  e.  RR )
134132, 133eqleltd 9110 . 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 888 1  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( S  +  T
)  =  U )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    /\ w3a 935    = wceq 1647    e. wcel 1715    =/= wne 2529   A.wral 2628   E.wrex 2629   {crab 2632    C_ wss 3238   (/)c0 3543   class class class wbr 4125   ` cfv 5358  (class class class)co 5981   supcsup 7340   CCcc 8882   RRcr 8883   0cc0 8884   1c1 8885    + caddc 8887    x. cmul 8889    < clt 9014    <_ cle 9015    / cdiv 9570   NNcn 9893   2c2 9942   NN0cn0 10114   ZZcz 10175   ZZ>=cuz 10381   ^cexp 11269    || cdivides 12739   Primecprime 12966
This theorem is referenced by:  pceulem  13106  pcmul  13112
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-cnex 8940  ax-resscn 8941  ax-1cn 8942  ax-icn 8943  ax-addcl 8944  ax-addrcl 8945  ax-mulcl 8946  ax-mulrcl 8947  ax-mulcom 8948  ax-addass 8949  ax-mulass 8950  ax-distr 8951  ax-i2m1 8952  ax-1ne0 8953  ax-1rid 8954  ax-rnegex 8955  ax-rrecex 8956  ax-cnre 8957  ax-pre-lttri 8958  ax-pre-lttrn 8959  ax-pre-ltadd 8960  ax-pre-mulgt0 8961  ax-pre-sup 8962
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-int 3965  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-2nd 6250  df-riota 6446  df-recs 6530  df-rdg 6565  df-1o 6621  df-2o 6622  df-oadd 6625  df-er 6802  df-en 7007  df-dom 7008  df-sdom 7009  df-fin 7010  df-sup 7341  df-pnf 9016  df-mnf 9017  df-xr 9018  df-ltxr 9019  df-le 9020  df-sub 9186  df-neg 9187  df-div 9571  df-nn 9894  df-2 9951  df-3 9952  df-n0 10115  df-z 10176  df-uz 10382  df-rp 10506  df-fl 11089  df-mod 11138  df-seq 11211  df-exp 11270  df-cj 11791  df-re 11792  df-im 11793  df-sqr 11927  df-abs 11928  df-dvds 12740  df-gcd 12894  df-prm 12967
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