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Theorem pcpremul 13172
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 13052 . . . . . . 7  |-  ( P  e.  Prime  ->  P  e.  ( ZZ>= `  2 )
)
213ad2ant1 978 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  P  e.  ( ZZ>= ` 
2 ) )
3 zmulcl 10280 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  x.  N
)  e.  ZZ )
43ad2ant2r 728 . . . . . . 7  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( M  x.  N
)  e.  ZZ )
543adant1 975 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( M  x.  N
)  e.  ZZ )
6 zcn 10243 . . . . . . . . 9  |-  ( M  e.  ZZ  ->  M  e.  CC )
76anim1i 552 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  M  =/=  0 )  -> 
( M  e.  CC  /\  M  =/=  0 ) )
8 zcn 10243 . . . . . . . . 9  |-  ( N  e.  ZZ  ->  N  e.  CC )
98anim1i 552 . . . . . . . 8  |-  ( ( N  e.  ZZ  /\  N  =/=  0 )  -> 
( N  e.  CC  /\  N  =/=  0 ) )
10 mulne0 9620 . . . . . . . 8  |-  ( ( ( M  e.  CC  /\  M  =/=  0 )  /\  ( N  e.  CC  /\  N  =/=  0 ) )  -> 
( M  x.  N
)  =/=  0 )
117, 9, 10syl2an 464 . . . . . . 7  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( M  x.  N
)  =/=  0 )
12113adant1 975 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( M  x.  N
)  =/=  0 )
13 eqid 2404 . . . . . . 7  |-  { n  e.  NN0  |  ( P ^ n )  ||  ( M  x.  N
) }  =  {
n  e.  NN0  | 
( P ^ n
)  ||  ( M  x.  N ) }
1413pclem 13167 . . . . . 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 1182 . . . . 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 969 . . . 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 971 . . . 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 983 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  M  e.  ZZ )
19 simp2r 984 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  M  =/=  0 )
20 eqid 2404 . . . . . . . . . 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 13168 . . . . . . . . 9  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  ( M  e.  ZZ  /\  M  =/=  0 ) )  -> 
( S  e.  NN0  /\  ( P ^ S
)  ||  M )
)
232, 18, 19, 22syl12anc 1182 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( S  e.  NN0  /\  ( P ^ S
)  ||  M )
)
2423simpld 446 . . . . . . 7  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  S  e.  NN0 )
25 simp3l 985 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  N  e.  ZZ )
26 simp3r 986 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  N  =/=  0 )
27 eqid 2404 . . . . . . . . . 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 13168 . . . . . . . . 9  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( T  e.  NN0  /\  ( P ^ T
)  ||  N )
)
302, 25, 26, 29syl12anc 1182 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( T  e.  NN0  /\  ( P ^ T
)  ||  N )
)
3130simpld 446 . . . . . . 7  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  T  e.  NN0 )
3224, 31nn0addcld 10234 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( S  +  T
)  e.  NN0 )
33 prmnn 13037 . . . . . . . . . . 11  |-  ( P  e.  Prime  ->  P  e.  NN )
34333ad2ant1 978 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  P  e.  NN )
3534nncnd 9972 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  P  e.  CC )
3635, 31, 24expaddd 11480 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ ( S  +  T )
)  =  ( ( P ^ S )  x.  ( P ^ T ) ) )
3723simprd 450 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ S
)  ||  M )
3834, 24nnexpcld 11499 . . . . . . . . . . 11  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ S
)  e.  NN )
3938nnzd 10330 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ S
)  e.  ZZ )
4034, 31nnexpcld 11499 . . . . . . . . . . 11  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ T
)  e.  NN )
4140nnzd 10330 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ T
)  e.  ZZ )
42 dvdsmulc 12832 . . . . . . . . . 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 1184 . . . . . . . . 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 4192 . . . . . . 7  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ ( S  +  T )
)  ||  ( M  x.  ( P ^ T
) ) )
4630simprd 450 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ T
)  ||  N )
47 dvdscmul 12831 . . . . . . . . 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 1184 . . . . . . . 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 11499 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ ( S  +  T )
)  e.  NN )
5150nnzd 10330 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ ( S  +  T )
)  e.  ZZ )
5218, 41zmulcld 10337 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( M  x.  ( P ^ T ) )  e.  ZZ )
53 dvdstr 12838 . . . . . . . 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 1184 . . . . . . 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 661 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ ( S  +  T )
)  ||  ( M  x.  N ) )
56 oveq2 6048 . . . . . . . 8  |-  ( x  =  ( S  +  T )  ->  ( P ^ x )  =  ( P ^ ( S  +  T )
) )
5756breq1d 4182 . . . . . . 7  |-  ( x  =  ( S  +  T )  ->  (
( P ^ x
)  ||  ( M  x.  N )  <->  ( P ^ ( S  +  T ) )  ||  ( M  x.  N
) ) )
5857elrab 3052 . . . . . 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 646 . . . . 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 6048 . . . . . . 7  |-  ( x  =  n  ->  ( P ^ x )  =  ( P ^ n
) )
6160breq1d 4182 . . . . . 6  |-  ( x  =  n  ->  (
( P ^ x
)  ||  ( M  x.  N )  <->  ( P ^ n )  ||  ( M  x.  N
) ) )
6261cbvrabv 2915 . . . . 5  |-  { x  e.  NN0  |  ( P ^ x )  ||  ( M  x.  N
) }  =  {
n  e.  NN0  | 
( P ^ n
)  ||  ( M  x.  N ) }
6359, 62syl6eleq 2494 . . . 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 10523 . . . 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 1184 . . 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 4212 . 2  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( S  +  T
)  <_  U )
6820, 21pcprendvds2 13170 . . . . . 6  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  ( M  e.  ZZ  /\  M  =/=  0 ) )  ->  -.  P  ||  ( M  /  ( P ^ S ) ) )
692, 18, 19, 68syl12anc 1182 . . . . 5  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  -.  P  ||  ( M  /  ( P ^ S ) ) )
7027, 28pcprendvds2 13170 . . . . . 6  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  -.  P  ||  ( N  /  ( P ^ T ) ) )
712, 25, 26, 70syl12anc 1182 . . . . 5  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  -.  P  ||  ( N  /  ( P ^ T ) ) )
72 ioran 477 . . . . 5  |-  ( -.  ( P  ||  ( M  /  ( P ^ S ) )  \/  P  ||  ( N  /  ( P ^ T ) ) )  <-> 
( -.  P  ||  ( M  /  ( P ^ S ) )  /\  -.  P  ||  ( N  /  ( P ^ T ) ) ) )
7369, 71, 72sylanbrc 646 . . . 4  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  -.  ( P  ||  ( M  /  ( P ^ S ) )  \/  P  ||  ( N  /  ( P ^ T ) ) ) )
74 simp1 957 . . . . 5  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  P  e.  Prime )
7538nnne0d 10000 . . . . . . 7  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ S
)  =/=  0 )
76 dvdsval2 12810 . . . . . . 7  |-  ( ( ( P ^ S
)  e.  ZZ  /\  ( P ^ S )  =/=  0  /\  M  e.  ZZ )  ->  (
( P ^ S
)  ||  M  <->  ( M  /  ( P ^ S ) )  e.  ZZ ) )
7739, 75, 18, 76syl3anc 1184 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( ( P ^ S )  ||  M  <->  ( M  /  ( P ^ S ) )  e.  ZZ ) )
7837, 77mpbid 202 . . . . 5  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( M  /  ( P ^ S ) )  e.  ZZ )
7940nnne0d 10000 . . . . . . 7  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ T
)  =/=  0 )
80 dvdsval2 12810 . . . . . . 7  |-  ( ( ( P ^ T
)  e.  ZZ  /\  ( P ^ T )  =/=  0  /\  N  e.  ZZ )  ->  (
( P ^ T
)  ||  N  <->  ( N  /  ( P ^ T ) )  e.  ZZ ) )
8141, 79, 25, 80syl3anc 1184 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( ( P ^ T )  ||  N  <->  ( N  /  ( P ^ T ) )  e.  ZZ ) )
8246, 81mpbid 202 . . . . 5  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( N  /  ( P ^ T ) )  e.  ZZ )
83 euclemma 13063 . . . . 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 1184 . . . 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 293 . . 3  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  -.  P  ||  ( ( M  /  ( P ^ S ) )  x.  ( N  / 
( P ^ T
) ) ) )
8613, 66pcprecl 13168 . . . . . . 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 1182 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( U  e.  NN0  /\  ( P ^ U
)  ||  ( M  x.  N ) ) )
8887simpld 446 . . . . 5  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  U  e.  NN0 )
89 nn0ltp1le 10288 . . . . 5  |-  ( ( ( S  +  T
)  e.  NN0  /\  U  e.  NN0 )  -> 
( ( S  +  T )  <  U  <->  ( ( S  +  T
)  +  1 )  <_  U ) )
9032, 88, 89syl2anc 643 . . . 4  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( ( S  +  T )  <  U  <->  ( ( S  +  T
)  +  1 )  <_  U ) )
9134nnzd 10330 . . . . . . 7  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  P  e.  ZZ )
92 peano2nn0 10216 . . . . . . . 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 12860 . . . . . . . 8  |-  ( ( P  e.  ZZ  /\  ( ( S  +  T )  +  1 )  e.  NN0  /\  U  e.  ( ZZ>= `  ( ( S  +  T )  +  1 ) ) )  -> 
( P ^ (
( S  +  T
)  +  1 ) )  ||  ( P ^ U ) )
95943expia 1155 . . . . . . 7  |-  ( ( P  e.  ZZ  /\  ( ( S  +  T )  +  1 )  e.  NN0 )  ->  ( U  e.  (
ZZ>= `  ( ( S  +  T )  +  1 ) )  -> 
( P ^ (
( S  +  T
)  +  1 ) )  ||  ( P ^ U ) ) )
9691, 93, 95syl2anc 643 . . . . . 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 450 . . . . . . 7  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ U
)  ||  ( M  x.  N ) )
9834, 93nnexpcld 11499 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ (
( S  +  T
)  +  1 ) )  e.  NN )
9998nnzd 10330 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ (
( S  +  T
)  +  1 ) )  e.  ZZ )
10034, 88nnexpcld 11499 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ U
)  e.  NN )
101100nnzd 10330 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ U
)  e.  ZZ )
102 dvdstr 12838 . . . . . . . 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 1184 . . . . . . 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 656 . . . . . 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 10329 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( ( S  +  T )  +  1 )  e.  ZZ )
10788nn0zd 10329 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  U  e.  ZZ )
108 eluz 10455 . . . . . 6  |-  ( ( ( ( S  +  T )  +  1 )  e.  ZZ  /\  U  e.  ZZ )  ->  ( U  e.  (
ZZ>= `  ( ( S  +  T )  +  1 ) )  <->  ( ( S  +  T )  +  1 )  <_  U ) )
109106, 107, 108syl2anc 643 . . . . 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 11479 . . . . . . 7  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ (
( S  +  T
)  +  1 ) )  =  ( ( P ^ ( S  +  T ) )  x.  P ) )
11118zcnd 10332 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  M  e.  CC )
11225zcnd 10332 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  N  e.  CC )
113111, 112mulcld 9064 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( M  x.  N
)  e.  CC )
11450nncnd 9972 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ ( S  +  T )
)  e.  CC )
11550nnne0d 10000 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ ( S  +  T )
)  =/=  0 )
116113, 114, 115divcan2d 9748 . . . . . . . 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 6056 . . . . . . . . . 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 9972 . . . . . . . . . . 11  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ S
)  e.  CC )
11940nncnd 9972 . . . . . . . . . . 11  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ T
)  e.  CC )
120111, 118, 112, 119, 75, 79divmuldivd 9787 . . . . . . . . . 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 2439 . . . . . . . . 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 6056 . . . . . . . 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 2438 . . . . . . 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 4185 . . . . . 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 10337 . . . . . . 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 12833 . . . . . . 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 1188 . . . . . 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 245 . . . . 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 259 . . . 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 207 . . 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 170 . 2  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  -.  ( S  +  T
)  <  U )
13232nn0red 10231 . . 3  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( S  +  T
)  e.  RR )
13388nn0red 10231 . . 3  |-  ( ( P  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  U  e.  RR )
134132, 133eqleltd 9173 . 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 889 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 177    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567   A.wral 2666   E.wrex 2667   {crab 2670    C_ wss 3280   (/)c0 3588   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   supcsup 7403   CCcc 8944   RRcr 8945   0cc0 8946   1c1 8947    + caddc 8949    x. cmul 8951    < clt 9076    <_ cle 9077    / cdiv 9633   NNcn 9956   2c2 10005   NN0cn0 10177   ZZcz 10238   ZZ>=cuz 10444   ^cexp 11337    || cdivides 12807   Primecprime 13034
This theorem is referenced by:  pceulem  13174  pcmul  13180
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-fl 11157  df-mod 11206  df-seq 11279  df-exp 11338  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-dvds 12808  df-gcd 12962  df-prm 13035
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