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Theorem phiprmpw 12253
Description: Value of the Euler  phi function at a prime power. Theorem 2.5(a) in [ApostolNT] p. 28. (Contributed by Mario Carneiro, 24-Feb-2014.)
Assertion
Ref Expression
phiprmpw  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( phi `  ( P ^ K ) )  =  ( ( P ^
( K  -  1 ) )  x.  ( P  -  1 ) ) )

Proof of Theorem phiprmpw
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 prmnn 12141 . . . 4  |-  ( P  e.  Prime  ->  P  e.  NN )
2 nnnn0 9212 . . . 4  |-  ( K  e.  NN  ->  K  e.  NN0 )
3 nnexpcl 10563 . . . 4  |-  ( ( P  e.  NN  /\  K  e.  NN0 )  -> 
( P ^ K
)  e.  NN )
41, 2, 3syl2an 289 . . 3  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( P ^ K )  e.  NN )
5 phival 12244 . . 3  |-  ( ( P ^ K )  e.  NN  ->  ( phi `  ( P ^ K ) )  =  ( `  { x  e.  ( 1 ... ( P ^ K ) )  |  ( x  gcd  ( P ^ K ) )  =  1 } ) )
64, 5syl 14 . 2  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( phi `  ( P ^ K ) )  =  ( `  { x  e.  ( 1 ... ( P ^ K ) )  |  ( x  gcd  ( P ^ K ) )  =  1 } ) )
7 nnm1nn0 9246 . . . . . 6  |-  ( K  e.  NN  ->  ( K  -  1 )  e.  NN0 )
8 nnexpcl 10563 . . . . . 6  |-  ( ( P  e.  NN  /\  ( K  -  1
)  e.  NN0 )  ->  ( P ^ ( K  -  1 ) )  e.  NN )
91, 7, 8syl2an 289 . . . . 5  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( P ^ ( K  - 
1 ) )  e.  NN )
109nncnd 8962 . . . 4  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( P ^ ( K  - 
1 ) )  e.  CC )
111nncnd 8962 . . . . 5  |-  ( P  e.  Prime  ->  P  e.  CC )
1211adantr 276 . . . 4  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  P  e.  CC )
13 ax-1cn 7933 . . . . 5  |-  1  e.  CC
14 subdi 8371 . . . . 5  |-  ( ( ( P ^ ( K  -  1 ) )  e.  CC  /\  P  e.  CC  /\  1  e.  CC )  ->  (
( P ^ ( K  -  1 ) )  x.  ( P  -  1 ) )  =  ( ( ( P ^ ( K  -  1 ) )  x.  P )  -  ( ( P ^
( K  -  1 ) )  x.  1 ) ) )
1513, 14mp3an3 1337 . . . 4  |-  ( ( ( P ^ ( K  -  1 ) )  e.  CC  /\  P  e.  CC )  ->  ( ( P ^
( K  -  1 ) )  x.  ( P  -  1 ) )  =  ( ( ( P ^ ( K  -  1 ) )  x.  P )  -  ( ( P ^ ( K  - 
1 ) )  x.  1 ) ) )
1610, 12, 15syl2anc 411 . . 3  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  (
( P ^ ( K  -  1 ) )  x.  ( P  -  1 ) )  =  ( ( ( P ^ ( K  -  1 ) )  x.  P )  -  ( ( P ^
( K  -  1 ) )  x.  1 ) ) )
1710mulridd 8003 . . . 4  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  (
( P ^ ( K  -  1 ) )  x.  1 )  =  ( P ^
( K  -  1 ) ) )
1817oveq2d 5911 . . 3  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  (
( ( P ^
( K  -  1 ) )  x.  P
)  -  ( ( P ^ ( K  -  1 ) )  x.  1 ) )  =  ( ( ( P ^ ( K  -  1 ) )  x.  P )  -  ( P ^ ( K  -  1 ) ) ) )
19 phivalfi 12243 . . . . . . 7  |-  ( ( P ^ K )  e.  NN  ->  { x  e.  ( 1 ... ( P ^ K ) )  |  ( x  gcd  ( P ^ K ) )  =  1 }  e.  Fin )
204, 19syl 14 . . . . . 6  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  { x  e.  ( 1 ... ( P ^ K ) )  |  ( x  gcd  ( P ^ K ) )  =  1 }  e.  Fin )
21 1zzd 9309 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  1  e.  ZZ )
22 prmz 12142 . . . . . . . . 9  |-  ( P  e.  Prime  ->  P  e.  ZZ )
23 zexpcl 10565 . . . . . . . . 9  |-  ( ( P  e.  ZZ  /\  K  e.  NN0 )  -> 
( P ^ K
)  e.  ZZ )
2422, 2, 23syl2an 289 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( P ^ K )  e.  ZZ )
2521, 24fzfigd 10461 . . . . . . 7  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  (
1 ... ( P ^ K ) )  e. 
Fin )
2622ad2antrr 488 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  K  e.  NN )  /\  x  e.  ( 1 ... ( P ^ K ) ) )  ->  P  e.  ZZ )
27 elfzelz 10054 . . . . . . . . . . 11  |-  ( x  e.  ( 1 ... ( P ^ K
) )  ->  x  e.  ZZ )
2827adantl 277 . . . . . . . . . 10  |-  ( ( ( P  e.  Prime  /\  K  e.  NN )  /\  x  e.  ( 1 ... ( P ^ K ) ) )  ->  x  e.  ZZ )
29 0zd 9294 . . . . . . . . . 10  |-  ( ( ( P  e.  Prime  /\  K  e.  NN )  /\  x  e.  ( 1 ... ( P ^ K ) ) )  ->  0  e.  ZZ )
3028, 29zsubcld 9409 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  K  e.  NN )  /\  x  e.  ( 1 ... ( P ^ K ) ) )  ->  ( x  -  0 )  e.  ZZ )
31 zdvdsdc 11850 . . . . . . . . 9  |-  ( ( P  e.  ZZ  /\  ( x  -  0
)  e.  ZZ )  -> DECID 
P  ||  ( x  -  0 ) )
3226, 30, 31syl2anc 411 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  K  e.  NN )  /\  x  e.  ( 1 ... ( P ^ K ) ) )  -> DECID  P  ||  ( x  -  0 ) )
3332ralrimiva 2563 . . . . . . 7  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  A. x  e.  ( 1 ... ( P ^ K ) )DECID  P 
||  ( x  - 
0 ) )
3425, 33ssfirab 6961 . . . . . 6  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  { x  e.  ( 1 ... ( P ^ K ) )  |  P  ||  (
x  -  0 ) }  e.  Fin )
35 inrab 3422 . . . . . . 7  |-  ( { x  e.  ( 1 ... ( P ^ K ) )  |  ( x  gcd  ( P ^ K ) )  =  1 }  i^i  { x  e.  ( 1 ... ( P ^ K ) )  |  P  ||  ( x  -  0 ) } )  =  { x  e.  ( 1 ... ( P ^ K ) )  |  ( ( x  gcd  ( P ^ K ) )  =  1  /\  P  ||  ( x  -  0
) ) }
36 rpexp 12184 . . . . . . . . . . . . . . . . 17  |-  ( ( P  e.  ZZ  /\  x  e.  ZZ  /\  K  e.  NN )  ->  (
( ( P ^ K )  gcd  x
)  =  1  <->  ( P  gcd  x )  =  1 ) )
3722, 36syl3an1 1282 . . . . . . . . . . . . . . . 16  |-  ( ( P  e.  Prime  /\  x  e.  ZZ  /\  K  e.  NN )  ->  (
( ( P ^ K )  gcd  x
)  =  1  <->  ( P  gcd  x )  =  1 ) )
38373expa 1205 . . . . . . . . . . . . . . 15  |-  ( ( ( P  e.  Prime  /\  x  e.  ZZ )  /\  K  e.  NN )  ->  ( ( ( P ^ K )  gcd  x )  =  1  <->  ( P  gcd  x )  =  1 ) )
3938an32s 568 . . . . . . . . . . . . . 14  |-  ( ( ( P  e.  Prime  /\  K  e.  NN )  /\  x  e.  ZZ )  ->  ( ( ( P ^ K )  gcd  x )  =  1  <->  ( P  gcd  x )  =  1 ) )
40 simpr 110 . . . . . . . . . . . . . . . 16  |-  ( ( ( P  e.  Prime  /\  K  e.  NN )  /\  x  e.  ZZ )  ->  x  e.  ZZ )
4124adantr 276 . . . . . . . . . . . . . . . 16  |-  ( ( ( P  e.  Prime  /\  K  e.  NN )  /\  x  e.  ZZ )  ->  ( P ^ K )  e.  ZZ )
42 gcdcom 12005 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  ZZ  /\  ( P ^ K )  e.  ZZ )  -> 
( x  gcd  ( P ^ K ) )  =  ( ( P ^ K )  gcd  x ) )
4340, 41, 42syl2anc 411 . . . . . . . . . . . . . . 15  |-  ( ( ( P  e.  Prime  /\  K  e.  NN )  /\  x  e.  ZZ )  ->  ( x  gcd  ( P ^ K ) )  =  ( ( P ^ K )  gcd  x ) )
4443eqeq1d 2198 . . . . . . . . . . . . . 14  |-  ( ( ( P  e.  Prime  /\  K  e.  NN )  /\  x  e.  ZZ )  ->  ( ( x  gcd  ( P ^ K ) )  =  1  <->  ( ( P ^ K )  gcd  x )  =  1 ) )
45 coprm 12175 . . . . . . . . . . . . . . 15  |-  ( ( P  e.  Prime  /\  x  e.  ZZ )  ->  ( -.  P  ||  x  <->  ( P  gcd  x )  =  1 ) )
4645adantlr 477 . . . . . . . . . . . . . 14  |-  ( ( ( P  e.  Prime  /\  K  e.  NN )  /\  x  e.  ZZ )  ->  ( -.  P  ||  x  <->  ( P  gcd  x )  =  1 ) )
4739, 44, 463bitr4d 220 . . . . . . . . . . . . 13  |-  ( ( ( P  e.  Prime  /\  K  e.  NN )  /\  x  e.  ZZ )  ->  ( ( x  gcd  ( P ^ K ) )  =  1  <->  -.  P  ||  x
) )
48 zcn 9287 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  ZZ  ->  x  e.  CC )
4948adantl 277 . . . . . . . . . . . . . . . 16  |-  ( ( ( P  e.  Prime  /\  K  e.  NN )  /\  x  e.  ZZ )  ->  x  e.  CC )
5049subid1d 8286 . . . . . . . . . . . . . . 15  |-  ( ( ( P  e.  Prime  /\  K  e.  NN )  /\  x  e.  ZZ )  ->  ( x  - 
0 )  =  x )
5150breq2d 4030 . . . . . . . . . . . . . 14  |-  ( ( ( P  e.  Prime  /\  K  e.  NN )  /\  x  e.  ZZ )  ->  ( P  ||  ( x  -  0
)  <->  P  ||  x ) )
5251notbid 668 . . . . . . . . . . . . 13  |-  ( ( ( P  e.  Prime  /\  K  e.  NN )  /\  x  e.  ZZ )  ->  ( -.  P  ||  ( x  -  0 )  <->  -.  P  ||  x
) )
5347, 52bitr4d 191 . . . . . . . . . . . 12  |-  ( ( ( P  e.  Prime  /\  K  e.  NN )  /\  x  e.  ZZ )  ->  ( ( x  gcd  ( P ^ K ) )  =  1  <->  -.  P  ||  (
x  -  0 ) ) )
5427, 53sylan2 286 . . . . . . . . . . 11  |-  ( ( ( P  e.  Prime  /\  K  e.  NN )  /\  x  e.  ( 1 ... ( P ^ K ) ) )  ->  ( (
x  gcd  ( P ^ K ) )  =  1  <->  -.  P  ||  (
x  -  0 ) ) )
5554biimpd 144 . . . . . . . . . 10  |-  ( ( ( P  e.  Prime  /\  K  e.  NN )  /\  x  e.  ( 1 ... ( P ^ K ) ) )  ->  ( (
x  gcd  ( P ^ K ) )  =  1  ->  -.  P  ||  ( x  -  0 ) ) )
56 imnan 691 . . . . . . . . . 10  |-  ( ( ( x  gcd  ( P ^ K ) )  =  1  ->  -.  P  ||  ( x  - 
0 ) )  <->  -.  (
( x  gcd  ( P ^ K ) )  =  1  /\  P  ||  ( x  -  0 ) ) )
5755, 56sylib 122 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  K  e.  NN )  /\  x  e.  ( 1 ... ( P ^ K ) ) )  ->  -.  (
( x  gcd  ( P ^ K ) )  =  1  /\  P  ||  ( x  -  0 ) ) )
5857ralrimiva 2563 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  A. x  e.  ( 1 ... ( P ^ K ) )  -.  ( ( x  gcd  ( P ^ K ) )  =  1  /\  P  ||  ( x  -  0
) ) )
59 rabeq0 3467 . . . . . . . 8  |-  ( { x  e.  ( 1 ... ( P ^ K ) )  |  ( ( x  gcd  ( P ^ K ) )  =  1  /\  P  ||  ( x  -  0 ) ) }  =  (/)  <->  A. x  e.  ( 1 ... ( P ^ K ) )  -.  ( ( x  gcd  ( P ^ K ) )  =  1  /\  P  ||  ( x  -  0
) ) )
6058, 59sylibr 134 . . . . . . 7  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  { x  e.  ( 1 ... ( P ^ K ) )  |  ( ( x  gcd  ( P ^ K ) )  =  1  /\  P  ||  ( x  -  0
) ) }  =  (/) )
6135, 60eqtrid 2234 . . . . . 6  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( { x  e.  (
1 ... ( P ^ K ) )  |  ( x  gcd  ( P ^ K ) )  =  1 }  i^i  { x  e.  ( 1 ... ( P ^ K ) )  |  P  ||  ( x  -  0 ) } )  =  (/) )
62 hashun 10816 . . . . . 6  |-  ( ( { x  e.  ( 1 ... ( P ^ K ) )  |  ( x  gcd  ( P ^ K ) )  =  1 }  e.  Fin  /\  {
x  e.  ( 1 ... ( P ^ K ) )  |  P  ||  ( x  -  0 ) }  e.  Fin  /\  ( { x  e.  (
1 ... ( P ^ K ) )  |  ( x  gcd  ( P ^ K ) )  =  1 }  i^i  { x  e.  ( 1 ... ( P ^ K ) )  |  P  ||  ( x  -  0 ) } )  =  (/) )  -> 
( `  ( { x  e.  ( 1 ... ( P ^ K ) )  |  ( x  gcd  ( P ^ K ) )  =  1 }  u.  { x  e.  ( 1 ... ( P ^ K ) )  |  P  ||  (
x  -  0 ) } ) )  =  ( ( `  {
x  e.  ( 1 ... ( P ^ K ) )  |  ( x  gcd  ( P ^ K ) )  =  1 } )  +  ( `  {
x  e.  ( 1 ... ( P ^ K ) )  |  P  ||  ( x  -  0 ) } ) ) )
6320, 34, 61, 62syl3anc 1249 . . . . 5  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( `  ( { x  e.  ( 1 ... ( P ^ K ) )  |  ( x  gcd  ( P ^ K ) )  =  1 }  u.  { x  e.  ( 1 ... ( P ^ K ) )  |  P  ||  (
x  -  0 ) } ) )  =  ( ( `  {
x  e.  ( 1 ... ( P ^ K ) )  |  ( x  gcd  ( P ^ K ) )  =  1 } )  +  ( `  {
x  e.  ( 1 ... ( P ^ K ) )  |  P  ||  ( x  -  0 ) } ) ) )
64 unrab 3421 . . . . . . . 8  |-  ( { x  e.  ( 1 ... ( P ^ K ) )  |  ( x  gcd  ( P ^ K ) )  =  1 }  u.  { x  e.  ( 1 ... ( P ^ K ) )  |  P  ||  ( x  -  0 ) } )  =  { x  e.  ( 1 ... ( P ^ K ) )  |  ( ( x  gcd  ( P ^ K ) )  =  1  \/  P  ||  ( x  -  0
) ) }
6554biimprd 158 . . . . . . . . . . . 12  |-  ( ( ( P  e.  Prime  /\  K  e.  NN )  /\  x  e.  ( 1 ... ( P ^ K ) ) )  ->  ( -.  P  ||  ( x  - 
0 )  ->  (
x  gcd  ( P ^ K ) )  =  1 ) )
66 con1dc 857 . . . . . . . . . . . 12  |-  (DECID  P  ||  ( x  -  0
)  ->  ( ( -.  P  ||  ( x  -  0 )  -> 
( x  gcd  ( P ^ K ) )  =  1 )  -> 
( -.  ( x  gcd  ( P ^ K ) )  =  1  ->  P  ||  (
x  -  0 ) ) ) )
6732, 65, 66sylc 62 . . . . . . . . . . 11  |-  ( ( ( P  e.  Prime  /\  K  e.  NN )  /\  x  e.  ( 1 ... ( P ^ K ) ) )  ->  ( -.  ( x  gcd  ( P ^ K ) )  =  1  ->  P  ||  ( x  -  0 ) ) )
6824adantr 276 . . . . . . . . . . . . . . 15  |-  ( ( ( P  e.  Prime  /\  K  e.  NN )  /\  x  e.  ( 1 ... ( P ^ K ) ) )  ->  ( P ^ K )  e.  ZZ )
6928, 68gcdcld 12000 . . . . . . . . . . . . . 14  |-  ( ( ( P  e.  Prime  /\  K  e.  NN )  /\  x  e.  ( 1 ... ( P ^ K ) ) )  ->  ( x  gcd  ( P ^ K
) )  e.  NN0 )
7069nn0zd 9402 . . . . . . . . . . . . 13  |-  ( ( ( P  e.  Prime  /\  K  e.  NN )  /\  x  e.  ( 1 ... ( P ^ K ) ) )  ->  ( x  gcd  ( P ^ K
) )  e.  ZZ )
71 1zzd 9309 . . . . . . . . . . . . 13  |-  ( ( ( P  e.  Prime  /\  K  e.  NN )  /\  x  e.  ( 1 ... ( P ^ K ) ) )  ->  1  e.  ZZ )
72 zdceq 9357 . . . . . . . . . . . . 13  |-  ( ( ( x  gcd  ( P ^ K ) )  e.  ZZ  /\  1  e.  ZZ )  -> DECID  ( x  gcd  ( P ^ K ) )  =  1 )
7370, 71, 72syl2anc 411 . . . . . . . . . . . 12  |-  ( ( ( P  e.  Prime  /\  K  e.  NN )  /\  x  e.  ( 1 ... ( P ^ K ) ) )  -> DECID  ( x  gcd  ( P ^ K ) )  =  1 )
74 dfordc 893 . . . . . . . . . . . 12  |-  (DECID  ( x  gcd  ( P ^ K ) )  =  1  ->  ( (
( x  gcd  ( P ^ K ) )  =  1  \/  P  ||  ( x  -  0 ) )  <->  ( -.  ( x  gcd  ( P ^ K ) )  =  1  ->  P  ||  ( x  -  0 ) ) ) )
7573, 74syl 14 . . . . . . . . . . 11  |-  ( ( ( P  e.  Prime  /\  K  e.  NN )  /\  x  e.  ( 1 ... ( P ^ K ) ) )  ->  ( (
( x  gcd  ( P ^ K ) )  =  1  \/  P  ||  ( x  -  0 ) )  <->  ( -.  ( x  gcd  ( P ^ K ) )  =  1  ->  P  ||  ( x  -  0 ) ) ) )
7667, 75mpbird 167 . . . . . . . . . 10  |-  ( ( ( P  e.  Prime  /\  K  e.  NN )  /\  x  e.  ( 1 ... ( P ^ K ) ) )  ->  ( (
x  gcd  ( P ^ K ) )  =  1  \/  P  ||  ( x  -  0
) ) )
7776ralrimiva 2563 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  A. x  e.  ( 1 ... ( P ^ K ) ) ( ( x  gcd  ( P ^ K ) )  =  1  \/  P  ||  ( x  -  0 ) ) )
78 rabid2 2667 . . . . . . . . 9  |-  ( ( 1 ... ( P ^ K ) )  =  { x  e.  ( 1 ... ( P ^ K ) )  |  ( ( x  gcd  ( P ^ K ) )  =  1  \/  P  ||  ( x  -  0
) ) }  <->  A. x  e.  ( 1 ... ( P ^ K ) ) ( ( x  gcd  ( P ^ K ) )  =  1  \/  P  ||  ( x  -  0 ) ) )
7977, 78sylibr 134 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  (
1 ... ( P ^ K ) )  =  { x  e.  ( 1 ... ( P ^ K ) )  |  ( ( x  gcd  ( P ^ K ) )  =  1  \/  P  ||  ( x  -  0
) ) } )
8064, 79eqtr4id 2241 . . . . . . 7  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( { x  e.  (
1 ... ( P ^ K ) )  |  ( x  gcd  ( P ^ K ) )  =  1 }  u.  { x  e.  ( 1 ... ( P ^ K ) )  |  P  ||  ( x  -  0 ) } )  =  ( 1 ... ( P ^ K ) ) )
8180fveq2d 5538 . . . . . 6  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( `  ( { x  e.  ( 1 ... ( P ^ K ) )  |  ( x  gcd  ( P ^ K ) )  =  1 }  u.  { x  e.  ( 1 ... ( P ^ K ) )  |  P  ||  (
x  -  0 ) } ) )  =  ( `  ( 1 ... ( P ^ K
) ) ) )
824nnnn0d 9258 . . . . . . 7  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( P ^ K )  e. 
NN0 )
83 hashfz1 10794 . . . . . . 7  |-  ( ( P ^ K )  e.  NN0  ->  ( `  (
1 ... ( P ^ K ) ) )  =  ( P ^ K ) )
8482, 83syl 14 . . . . . 6  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( `  ( 1 ... ( P ^ K ) ) )  =  ( P ^ K ) )
85 expm1t 10578 . . . . . . 7  |-  ( ( P  e.  CC  /\  K  e.  NN )  ->  ( P ^ K
)  =  ( ( P ^ ( K  -  1 ) )  x.  P ) )
8611, 85sylan 283 . . . . . 6  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( P ^ K )  =  ( ( P ^
( K  -  1 ) )  x.  P
) )
8781, 84, 863eqtrd 2226 . . . . 5  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( `  ( { x  e.  ( 1 ... ( P ^ K ) )  |  ( x  gcd  ( P ^ K ) )  =  1 }  u.  { x  e.  ( 1 ... ( P ^ K ) )  |  P  ||  (
x  -  0 ) } ) )  =  ( ( P ^
( K  -  1 ) )  x.  P
) )
88 hashcl 10792 . . . . . . . 8  |-  ( { x  e.  ( 1 ... ( P ^ K ) )  |  ( x  gcd  ( P ^ K ) )  =  1 }  e.  Fin  ->  ( `  { x  e.  ( 1 ... ( P ^ K ) )  |  ( x  gcd  ( P ^ K ) )  =  1 } )  e.  NN0 )
8920, 88syl 14 . . . . . . 7  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( `  { x  e.  ( 1 ... ( P ^ K ) )  |  ( x  gcd  ( P ^ K ) )  =  1 } )  e.  NN0 )
9089nn0cnd 9260 . . . . . 6  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( `  { x  e.  ( 1 ... ( P ^ K ) )  |  ( x  gcd  ( P ^ K ) )  =  1 } )  e.  CC )
911adantr 276 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  P  e.  NN )
92 nn0uz 9591 . . . . . . . . . . 11  |-  NN0  =  ( ZZ>= `  0 )
93 1m1e0 9017 . . . . . . . . . . . 12  |-  ( 1  -  1 )  =  0
9493fveq2i 5537 . . . . . . . . . . 11  |-  ( ZZ>= `  ( 1  -  1 ) )  =  (
ZZ>= `  0 )
9592, 94eqtr4i 2213 . . . . . . . . . 10  |-  NN0  =  ( ZZ>= `  ( 1  -  1 ) )
9682, 95eleqtrdi 2282 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( P ^ K )  e.  ( ZZ>= `  ( 1  -  1 ) ) )
97 0zd 9294 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  0  e.  ZZ )
9891, 21, 96, 97hashdvds 12252 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( `  { x  e.  ( 1 ... ( P ^ K ) )  |  P  ||  (
x  -  0 ) } )  =  ( ( |_ `  (
( ( P ^ K )  -  0 )  /  P ) )  -  ( |_
`  ( ( ( 1  -  1 )  -  0 )  /  P ) ) ) )
994nncnd 8962 . . . . . . . . . . . . . 14  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( P ^ K )  e.  CC )
10099subid1d 8286 . . . . . . . . . . . . 13  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  (
( P ^ K
)  -  0 )  =  ( P ^ K ) )
101100oveq1d 5910 . . . . . . . . . . . 12  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  (
( ( P ^ K )  -  0 )  /  P )  =  ( ( P ^ K )  /  P ) )
10291nnap0d 8994 . . . . . . . . . . . . 13  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  P #  0 )
103 nnz 9301 . . . . . . . . . . . . . 14  |-  ( K  e.  NN  ->  K  e.  ZZ )
104103adantl 277 . . . . . . . . . . . . 13  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  K  e.  ZZ )
10512, 102, 104expm1apd 10694 . . . . . . . . . . . 12  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( P ^ ( K  - 
1 ) )  =  ( ( P ^ K )  /  P
) )
106101, 105eqtr4d 2225 . . . . . . . . . . 11  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  (
( ( P ^ K )  -  0 )  /  P )  =  ( P ^
( K  -  1 ) ) )
107106fveq2d 5538 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( |_ `  ( ( ( P ^ K )  -  0 )  /  P ) )  =  ( |_ `  ( P ^ ( K  - 
1 ) ) ) )
1089nnzd 9403 . . . . . . . . . . 11  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( P ^ ( K  - 
1 ) )  e.  ZZ )
109 flid 10314 . . . . . . . . . . 11  |-  ( ( P ^ ( K  -  1 ) )  e.  ZZ  ->  ( |_ `  ( P ^
( K  -  1 ) ) )  =  ( P ^ ( K  -  1 ) ) )
110108, 109syl 14 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( |_ `  ( P ^
( K  -  1 ) ) )  =  ( P ^ ( K  -  1 ) ) )
111107, 110eqtrd 2222 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( |_ `  ( ( ( P ^ K )  -  0 )  /  P ) )  =  ( P ^ ( K  -  1 ) ) )
11293oveq1i 5905 . . . . . . . . . . . . . 14  |-  ( ( 1  -  1 )  -  0 )  =  ( 0  -  0 )
113 0m0e0 9060 . . . . . . . . . . . . . 14  |-  ( 0  -  0 )  =  0
114112, 113eqtri 2210 . . . . . . . . . . . . 13  |-  ( ( 1  -  1 )  -  0 )  =  0
115114oveq1i 5905 . . . . . . . . . . . 12  |-  ( ( ( 1  -  1 )  -  0 )  /  P )  =  ( 0  /  P
)
11612, 102div0apd 8773 . . . . . . . . . . . 12  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  (
0  /  P )  =  0 )
117115, 116eqtrid 2234 . . . . . . . . . . 11  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  (
( ( 1  -  1 )  -  0 )  /  P )  =  0 )
118117fveq2d 5538 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( |_ `  ( ( ( 1  -  1 )  -  0 )  /  P ) )  =  ( |_ `  0
) )
119 0z 9293 . . . . . . . . . . 11  |-  0  e.  ZZ
120 flid 10314 . . . . . . . . . . 11  |-  ( 0  e.  ZZ  ->  ( |_ `  0 )  =  0 )
121119, 120ax-mp 5 . . . . . . . . . 10  |-  ( |_
`  0 )  =  0
122118, 121eqtrdi 2238 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( |_ `  ( ( ( 1  -  1 )  -  0 )  /  P ) )  =  0 )
123111, 122oveq12d 5913 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  (
( |_ `  (
( ( P ^ K )  -  0 )  /  P ) )  -  ( |_
`  ( ( ( 1  -  1 )  -  0 )  /  P ) ) )  =  ( ( P ^ ( K  - 
1 ) )  - 
0 ) )
12410subid1d 8286 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  (
( P ^ ( K  -  1 ) )  -  0 )  =  ( P ^
( K  -  1 ) ) )
12598, 123, 1243eqtrd 2226 . . . . . . 7  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( `  { x  e.  ( 1 ... ( P ^ K ) )  |  P  ||  (
x  -  0 ) } )  =  ( P ^ ( K  -  1 ) ) )
126125oveq2d 5911 . . . . . 6  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  (
( `  { x  e.  ( 1 ... ( P ^ K ) )  |  ( x  gcd  ( P ^ K ) )  =  1 } )  +  ( `  {
x  e.  ( 1 ... ( P ^ K ) )  |  P  ||  ( x  -  0 ) } ) )  =  ( ( `  { x  e.  ( 1 ... ( P ^ K ) )  |  ( x  gcd  ( P ^ K ) )  =  1 } )  +  ( P ^ ( K  - 
1 ) ) ) )
12790, 10, 126comraddd 8143 . . . . 5  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  (
( `  { x  e.  ( 1 ... ( P ^ K ) )  |  ( x  gcd  ( P ^ K ) )  =  1 } )  +  ( `  {
x  e.  ( 1 ... ( P ^ K ) )  |  P  ||  ( x  -  0 ) } ) )  =  ( ( P ^ ( K  -  1 ) )  +  ( `  {
x  e.  ( 1 ... ( P ^ K ) )  |  ( x  gcd  ( P ^ K ) )  =  1 } ) ) )
12863, 87, 1273eqtr3rd 2231 . . . 4  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  (
( P ^ ( K  -  1 ) )  +  ( `  {
x  e.  ( 1 ... ( P ^ K ) )  |  ( x  gcd  ( P ^ K ) )  =  1 } ) )  =  ( ( P ^ ( K  -  1 ) )  x.  P ) )
12910, 12mulcld 8007 . . . . 5  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  (
( P ^ ( K  -  1 ) )  x.  P )  e.  CC )
130129, 10, 90subaddd 8315 . . . 4  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  (
( ( ( P ^ ( K  - 
1 ) )  x.  P )  -  ( P ^ ( K  - 
1 ) ) )  =  ( `  {
x  e.  ( 1 ... ( P ^ K ) )  |  ( x  gcd  ( P ^ K ) )  =  1 } )  <-> 
( ( P ^
( K  -  1 ) )  +  ( `  { x  e.  ( 1 ... ( P ^ K ) )  |  ( x  gcd  ( P ^ K ) )  =  1 } ) )  =  ( ( P ^ ( K  -  1 ) )  x.  P ) ) )
131128, 130mpbird 167 . . 3  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  (
( ( P ^
( K  -  1 ) )  x.  P
)  -  ( P ^ ( K  - 
1 ) ) )  =  ( `  {
x  e.  ( 1 ... ( P ^ K ) )  |  ( x  gcd  ( P ^ K ) )  =  1 } ) )
13216, 18, 1313eqtrrd 2227 . 2  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( `  { x  e.  ( 1 ... ( P ^ K ) )  |  ( x  gcd  ( P ^ K ) )  =  1 } )  =  ( ( P ^ ( K  -  1 ) )  x.  ( P  - 
1 ) ) )
1336, 132eqtrd 2222 1  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( phi `  ( P ^ K ) )  =  ( ( P ^
( K  -  1 ) )  x.  ( P  -  1 ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    <-> wb 105    \/ wo 709  DECID wdc 835    = wceq 1364    e. wcel 2160   A.wral 2468   {crab 2472    u. cun 3142    i^i cin 3143   (/)c0 3437   class class class wbr 4018   ` cfv 5235  (class class class)co 5895   Fincfn 6765   CCcc 7838   0cc0 7840   1c1 7841    + caddc 7843    x. cmul 7845    - cmin 8157    / cdiv 8658   NNcn 8948   NN0cn0 9205   ZZcz 9282   ZZ>=cuz 9557   ...cfz 10037   |_cfl 10298   ^cexp 10549  ♯chash 10786    || cdvds 11825    gcd cgcd 11974   Primecprime 12138   phicphi 12240
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605  ax-cnex 7931  ax-resscn 7932  ax-1cn 7933  ax-1re 7934  ax-icn 7935  ax-addcl 7936  ax-addrcl 7937  ax-mulcl 7938  ax-mulrcl 7939  ax-addcom 7940  ax-mulcom 7941  ax-addass 7942  ax-mulass 7943  ax-distr 7944  ax-i2m1 7945  ax-0lt1 7946  ax-1rid 7947  ax-0id 7948  ax-rnegex 7949  ax-precex 7950  ax-cnre 7951  ax-pre-ltirr 7952  ax-pre-ltwlin 7953  ax-pre-lttrn 7954  ax-pre-apti 7955  ax-pre-ltadd 7956  ax-pre-mulgt0 7957  ax-pre-mulext 7958  ax-arch 7959  ax-caucvg 7960
This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-po 4314  df-iso 4315  df-iord 4384  df-on 4386  df-ilim 4387  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-riota 5851  df-ov 5898  df-oprab 5899  df-mpo 5900  df-1st 6164  df-2nd 6165  df-recs 6329  df-irdg 6394  df-frec 6415  df-1o 6440  df-2o 6441  df-oadd 6444  df-er 6558  df-en 6766  df-dom 6767  df-fin 6768  df-sup 7012  df-pnf 8023  df-mnf 8024  df-xr 8025  df-ltxr 8026  df-le 8027  df-sub 8159  df-neg 8160  df-reap 8561  df-ap 8568  df-div 8659  df-inn 8949  df-2 9007  df-3 9008  df-4 9009  df-n0 9206  df-z 9283  df-uz 9558  df-q 9649  df-rp 9683  df-fz 10038  df-fzo 10172  df-fl 10300  df-mod 10353  df-seqfrec 10476  df-exp 10550  df-ihash 10787  df-cj 10882  df-re 10883  df-im 10884  df-rsqrt 11038  df-abs 11039  df-dvds 11826  df-gcd 11975  df-prm 12139  df-phi 12242
This theorem is referenced by:  phiprm  12254
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