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Theorem pcpremul 14576
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 14444 . . . . . . 7  |-  ( P  e.  Prime  ->  P  e.  ( ZZ>= `  2 )
)
213ad2ant1 1018 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  P  e.  ( ZZ>= ` 
2 ) )
3 zmulcl 10953 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  x.  N
)  e.  ZZ )
43ad2ant2r 745 . . . . . . 7  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( M  x.  N
)  e.  ZZ )
543adant1 1015 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( M  x.  N
)  e.  ZZ )
6 zcn 10910 . . . . . . . . 9  |-  ( M  e.  ZZ  ->  M  e.  CC )
76anim1i 566 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  M  =/=  0 )  -> 
( M  e.  CC  /\  M  =/=  0 ) )
8 zcn 10910 . . . . . . . . 9  |-  ( N  e.  ZZ  ->  N  e.  CC )
98anim1i 566 . . . . . . . 8  |-  ( ( N  e.  ZZ  /\  N  =/=  0 )  -> 
( N  e.  CC  /\  N  =/=  0 ) )
10 mulne0 10232 . . . . . . . 8  |-  ( ( ( M  e.  CC  /\  M  =/=  0 )  /\  ( N  e.  CC  /\  N  =/=  0 ) )  -> 
( M  x.  N
)  =/=  0 )
117, 9, 10syl2an 475 . . . . . . 7  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( M  x.  N
)  =/=  0 )
12113adant1 1015 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( M  x.  N
)  =/=  0 )
13 eqid 2402 . . . . . . 7  |-  { n  e.  NN0  |  ( P ^ n )  ||  ( M  x.  N
) }  =  {
n  e.  NN0  | 
( P ^ n
)  ||  ( M  x.  N ) }
1413pclem 14571 . . . . . 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 1228 . . . . 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 1009 . . . 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 1011 . . . 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 1023 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  M  e.  ZZ )
19 simp2r 1024 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  M  =/=  0 )
20 eqid 2402 . . . . . . . . . 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 14572 . . . . . . . . 9  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  ( M  e.  ZZ  /\  M  =/=  0 ) )  -> 
( S  e.  NN0  /\  ( P ^ S
)  ||  M )
)
232, 18, 19, 22syl12anc 1228 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( S  e.  NN0  /\  ( P ^ S
)  ||  M )
)
2423simpld 457 . . . . . . 7  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  S  e.  NN0 )
25 simp3l 1025 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  N  e.  ZZ )
26 simp3r 1026 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  N  =/=  0 )
27 eqid 2402 . . . . . . . . . 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 14572 . . . . . . . . 9  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( T  e.  NN0  /\  ( P ^ T
)  ||  N )
)
302, 25, 26, 29syl12anc 1228 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( T  e.  NN0  /\  ( P ^ T
)  ||  N )
)
3130simpld 457 . . . . . . 7  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  T  e.  NN0 )
3224, 31nn0addcld 10897 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( S  +  T
)  e.  NN0 )
33 prmnn 14429 . . . . . . . . . . 11  |-  ( P  e.  Prime  ->  P  e.  NN )
34333ad2ant1 1018 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  P  e.  NN )
3534nncnd 10592 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  P  e.  CC )
3635, 31, 24expaddd 12356 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ ( S  +  T )
)  =  ( ( P ^ S )  x.  ( P ^ T ) ) )
3723simprd 461 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ S
)  ||  M )
3834, 24nnexpcld 12375 . . . . . . . . . . 11  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ S
)  e.  NN )
3938nnzd 11007 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ S
)  e.  ZZ )
4034, 31nnexpcld 12375 . . . . . . . . . . 11  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ T
)  e.  NN )
4140nnzd 11007 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ T
)  e.  ZZ )
42 dvdsmulc 14220 . . . . . . . . . 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 1230 . . . . . . . . 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 4415 . . . . . . 7  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ ( S  +  T )
)  ||  ( M  x.  ( P ^ T
) ) )
4630simprd 461 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ T
)  ||  N )
47 dvdscmul 14219 . . . . . . . . 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 1230 . . . . . . . 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 12375 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ ( S  +  T )
)  e.  NN )
5150nnzd 11007 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ ( S  +  T )
)  e.  ZZ )
5218, 41zmulcld 11014 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( M  x.  ( P ^ T ) )  e.  ZZ )
53 dvdstr 14227 . . . . . . . 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 1230 . . . . . . 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 677 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ ( S  +  T )
)  ||  ( M  x.  N ) )
56 oveq2 6286 . . . . . . . 8  |-  ( x  =  ( S  +  T )  ->  ( P ^ x )  =  ( P ^ ( S  +  T )
) )
5756breq1d 4405 . . . . . . 7  |-  ( x  =  ( S  +  T )  ->  (
( P ^ x
)  ||  ( M  x.  N )  <->  ( P ^ ( S  +  T ) )  ||  ( M  x.  N
) ) )
5857elrab 3207 . . . . . 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 662 . . . . 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 6286 . . . . . . 7  |-  ( x  =  n  ->  ( P ^ x )  =  ( P ^ n
) )
6160breq1d 4405 . . . . . 6  |-  ( x  =  n  ->  (
( P ^ x
)  ||  ( M  x.  N )  <->  ( P ^ n )  ||  ( M  x.  N
) ) )
6261cbvrabv 3058 . . . . 5  |-  { x  e.  NN0  |  ( P ^ x )  ||  ( M  x.  N
) }  =  {
n  e.  NN0  | 
( P ^ n
)  ||  ( M  x.  N ) }
6359, 62syl6eleq 2500 . . . 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 11218 . . . 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 1230 . . 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 4435 . 2  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( S  +  T
)  <_  U )
6820, 21pcprendvds2 14574 . . . . . 6  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  ( M  e.  ZZ  /\  M  =/=  0 ) )  ->  -.  P  ||  ( M  /  ( P ^ S ) ) )
692, 18, 19, 68syl12anc 1228 . . . . 5  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  -.  P  ||  ( M  /  ( P ^ S ) ) )
7027, 28pcprendvds2 14574 . . . . . 6  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  -.  P  ||  ( N  /  ( P ^ T ) ) )
712, 25, 26, 70syl12anc 1228 . . . . 5  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  -.  P  ||  ( N  /  ( P ^ T ) ) )
72 ioran 488 . . . . 5  |-  ( -.  ( P  ||  ( M  /  ( P ^ S ) )  \/  P  ||  ( N  /  ( P ^ T ) ) )  <-> 
( -.  P  ||  ( M  /  ( P ^ S ) )  /\  -.  P  ||  ( N  /  ( P ^ T ) ) ) )
7369, 71, 72sylanbrc 662 . . . 4  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  -.  ( P  ||  ( M  /  ( P ^ S ) )  \/  P  ||  ( N  /  ( P ^ T ) ) ) )
74 simp1 997 . . . . 5  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  P  e.  Prime )
7538nnne0d 10621 . . . . . . 7  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ S
)  =/=  0 )
76 dvdsval2 14198 . . . . . . 7  |-  ( ( ( P ^ S
)  e.  ZZ  /\  ( P ^ S )  =/=  0  /\  M  e.  ZZ )  ->  (
( P ^ S
)  ||  M  <->  ( M  /  ( P ^ S ) )  e.  ZZ ) )
7739, 75, 18, 76syl3anc 1230 . . . . . 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 10621 . . . . . . 7  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ T
)  =/=  0 )
80 dvdsval2 14198 . . . . . . 7  |-  ( ( ( P ^ T
)  e.  ZZ  /\  ( P ^ T )  =/=  0  /\  N  e.  ZZ )  ->  (
( P ^ T
)  ||  N  <->  ( N  /  ( P ^ T ) )  e.  ZZ ) )
8141, 79, 25, 80syl3anc 1230 . . . . . 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 14458 . . . . 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 1230 . . . 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 299 . . 3  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  -.  P  ||  ( ( M  /  ( P ^ S ) )  x.  ( N  / 
( P ^ T
) ) ) )
8613, 66pcprecl 14572 . . . . . . 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 1228 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( U  e.  NN0  /\  ( P ^ U
)  ||  ( M  x.  N ) ) )
8887simpld 457 . . . . 5  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  U  e.  NN0 )
89 nn0ltp1le 10962 . . . . 5  |-  ( ( ( S  +  T
)  e.  NN0  /\  U  e.  NN0 )  -> 
( ( S  +  T )  <  U  <->  ( ( S  +  T
)  +  1 )  <_  U ) )
9032, 88, 89syl2anc 659 . . . 4  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( ( S  +  T )  <  U  <->  ( ( S  +  T
)  +  1 )  <_  U ) )
9134nnzd 11007 . . . . . . 7  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  P  e.  ZZ )
92 peano2nn0 10877 . . . . . . . 8  |-  ( ( S  +  T )  e.  NN0  ->  ( ( S  +  T )  +  1 )  e. 
NN0 )
9332, 92syl 17 . . . . . . 7  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( ( S  +  T )  +  1 )  e.  NN0 )
94 dvdsexp 14251 . . . . . . . 8  |-  ( ( P  e.  ZZ  /\  ( ( S  +  T )  +  1 )  e.  NN0  /\  U  e.  ( ZZ>= `  ( ( S  +  T )  +  1 ) ) )  -> 
( P ^ (
( S  +  T
)  +  1 ) )  ||  ( P ^ U ) )
95943expia 1199 . . . . . . 7  |-  ( ( P  e.  ZZ  /\  ( ( S  +  T )  +  1 )  e.  NN0 )  ->  ( U  e.  (
ZZ>= `  ( ( S  +  T )  +  1 ) )  -> 
( P ^ (
( S  +  T
)  +  1 ) )  ||  ( P ^ U ) ) )
9691, 93, 95syl2anc 659 . . . . . 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 461 . . . . . . 7  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ U
)  ||  ( M  x.  N ) )
9834, 93nnexpcld 12375 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ (
( S  +  T
)  +  1 ) )  e.  NN )
9998nnzd 11007 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ (
( S  +  T
)  +  1 ) )  e.  ZZ )
10034, 88nnexpcld 12375 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ U
)  e.  NN )
101100nnzd 11007 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ U
)  e.  ZZ )
102 dvdstr 14227 . . . . . . . 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 1230 . . . . . . 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 672 . . . . . 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 42 . . . . 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 11006 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( ( S  +  T )  +  1 )  e.  ZZ )
10788nn0zd 11006 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  U  e.  ZZ )
108 eluz 11140 . . . . . 6  |-  ( ( ( ( S  +  T )  +  1 )  e.  ZZ  /\  U  e.  ZZ )  ->  ( U  e.  (
ZZ>= `  ( ( S  +  T )  +  1 ) )  <->  ( ( S  +  T )  +  1 )  <_  U ) )
109106, 107, 108syl2anc 659 . . . . 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 12355 . . . . . . 7  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ (
( S  +  T
)  +  1 ) )  =  ( ( P ^ ( S  +  T ) )  x.  P ) )
11118zcnd 11009 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  M  e.  CC )
11225zcnd 11009 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  N  e.  CC )
113111, 112mulcld 9646 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( M  x.  N
)  e.  CC )
11450nncnd 10592 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ ( S  +  T )
)  e.  CC )
11550nnne0d 10621 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ ( S  +  T )
)  =/=  0 )
116113, 114, 115divcan2d 10363 . . . . . . . 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 6294 . . . . . . . . . 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 10592 . . . . . . . . . . 11  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ S
)  e.  CC )
11940nncnd 10592 . . . . . . . . . . 11  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ T
)  e.  CC )
120111, 118, 112, 119, 75, 79divmuldivd 10402 . . . . . . . . . 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 2446 . . . . . . . . 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 6294 . . . . . . . 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 2445 . . . . . . 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 4408 . . . . . 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 11014 . . . . . . 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 14221 . . . . . . 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 1234 . . . . . 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 10894 . . 3  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( S  +  T
)  e.  RR )
13388nn0red 10894 . . 3  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  U  e.  RR )
134132, 133eqleltd 9761 . 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 923 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 366    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842    =/= wne 2598   A.wral 2754   E.wrex 2755   {crab 2758    C_ wss 3414   (/)c0 3738   class class class wbr 4395   ` cfv 5569  (class class class)co 6278   supcsup 7934   CCcc 9520   RRcr 9521   0cc0 9522   1c1 9523    + caddc 9525    x. cmul 9527    < clt 9658    <_ cle 9659    / cdiv 10247   NNcn 10576   2c2 10626   NN0cn0 10836   ZZcz 10905   ZZ>=cuz 11127   ^cexp 12210    || cdvds 14195   Primecprime 14426
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599  ax-pre-sup 9600
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-2nd 6785  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-2o 7168  df-oadd 7171  df-er 7348  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-sup 7935  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-div 10248  df-nn 10577  df-2 10635  df-3 10636  df-n0 10837  df-z 10906  df-uz 11128  df-rp 11266  df-fl 11966  df-mod 12035  df-seq 12152  df-exp 12211  df-cj 13081  df-re 13082  df-im 13083  df-sqrt 13217  df-abs 13218  df-dvds 14196  df-gcd 14354  df-prm 14427
This theorem is referenced by:  pceulem  14578  pcmul  14584
  Copyright terms: Public domain W3C validator