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Theorem phiprmpw 14390
Description: Value of the Euler  phi function at a prime power. (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 14304 . . . 4  |-  ( P  e.  Prime  ->  P  e.  NN )
2 nnnn0 10798 . . . 4  |-  ( K  e.  NN  ->  K  e.  NN0 )
3 nnexpcl 12161 . . . 4  |-  ( ( P  e.  NN  /\  K  e.  NN0 )  -> 
( P ^ K
)  e.  NN )
41, 2, 3syl2an 475 . . 3  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( P ^ K )  e.  NN )
5 phival 14381 . . 3  |-  ( ( P ^ K )  e.  NN  ->  ( phi `  ( P ^ K ) )  =  ( # `  {
x  e.  ( 1 ... ( P ^ K ) )  |  ( x  gcd  ( P ^ K ) )  =  1 } ) )
64, 5syl 16 . 2  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( phi `  ( P ^ K ) )  =  ( # `  {
x  e.  ( 1 ... ( P ^ K ) )  |  ( x  gcd  ( P ^ K ) )  =  1 } ) )
7 nnm1nn0 10833 . . . . . 6  |-  ( K  e.  NN  ->  ( K  -  1 )  e.  NN0 )
8 nnexpcl 12161 . . . . . 6  |-  ( ( P  e.  NN  /\  ( K  -  1
)  e.  NN0 )  ->  ( P ^ ( K  -  1 ) )  e.  NN )
91, 7, 8syl2an 475 . . . . 5  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( P ^ ( K  - 
1 ) )  e.  NN )
109nncnd 10547 . . . 4  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( P ^ ( K  - 
1 ) )  e.  CC )
111nncnd 10547 . . . . 5  |-  ( P  e.  Prime  ->  P  e.  CC )
1211adantr 463 . . . 4  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  P  e.  CC )
13 ax-1cn 9539 . . . . 5  |-  1  e.  CC
14 subdi 9986 . . . . 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 1311 . . . 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 659 . . 3  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  (
( P ^ ( K  -  1 ) )  x.  ( P  -  1 ) )  =  ( ( ( P ^ ( K  -  1 ) )  x.  P )  -  ( ( P ^
( K  -  1 ) )  x.  1 ) ) )
1710mulid1d 9602 . . . 4  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  (
( P ^ ( K  -  1 ) )  x.  1 )  =  ( P ^
( K  -  1 ) ) )
1817oveq2d 6286 . . 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 inrab 3767 . . . . . . 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
) ) }
20 elfzelz 11691 . . . . . . . . . . . 12  |-  ( x  e.  ( 1 ... ( P ^ K
) )  ->  x  e.  ZZ )
21 prmz 14305 . . . . . . . . . . . . . . . . 17  |-  ( P  e.  Prime  ->  P  e.  ZZ )
22 rpexp 14345 . . . . . . . . . . . . . . . . 17  |-  ( ( P  e.  ZZ  /\  x  e.  ZZ  /\  K  e.  NN )  ->  (
( ( P ^ K )  gcd  x
)  =  1  <->  ( P  gcd  x )  =  1 ) )
2321, 22syl3an1 1259 . . . . . . . . . . . . . . . 16  |-  ( ( P  e.  Prime  /\  x  e.  ZZ  /\  K  e.  NN )  ->  (
( ( P ^ K )  gcd  x
)  =  1  <->  ( P  gcd  x )  =  1 ) )
24233expa 1194 . . . . . . . . . . . . . . 15  |-  ( ( ( P  e.  Prime  /\  x  e.  ZZ )  /\  K  e.  NN )  ->  ( ( ( P ^ K )  gcd  x )  =  1  <->  ( P  gcd  x )  =  1 ) )
2524an32s 802 . . . . . . . . . . . . . 14  |-  ( ( ( P  e.  Prime  /\  K  e.  NN )  /\  x  e.  ZZ )  ->  ( ( ( P ^ K )  gcd  x )  =  1  <->  ( P  gcd  x )  =  1 ) )
26 simpr 459 . . . . . . . . . . . . . . . 16  |-  ( ( ( P  e.  Prime  /\  K  e.  NN )  /\  x  e.  ZZ )  ->  x  e.  ZZ )
27 zexpcl 12163 . . . . . . . . . . . . . . . . . 18  |-  ( ( P  e.  ZZ  /\  K  e.  NN0 )  -> 
( P ^ K
)  e.  ZZ )
2821, 2, 27syl2an 475 . . . . . . . . . . . . . . . . 17  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( P ^ K )  e.  ZZ )
2928adantr 463 . . . . . . . . . . . . . . . 16  |-  ( ( ( P  e.  Prime  /\  K  e.  NN )  /\  x  e.  ZZ )  ->  ( P ^ K )  e.  ZZ )
30 gcdcom 14242 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  ZZ  /\  ( P ^ K )  e.  ZZ )  -> 
( x  gcd  ( P ^ K ) )  =  ( ( P ^ K )  gcd  x ) )
3126, 29, 30syl2anc 659 . . . . . . . . . . . . . . 15  |-  ( ( ( P  e.  Prime  /\  K  e.  NN )  /\  x  e.  ZZ )  ->  ( x  gcd  ( P ^ K ) )  =  ( ( P ^ K )  gcd  x ) )
3231eqeq1d 2456 . . . . . . . . . . . . . 14  |-  ( ( ( P  e.  Prime  /\  K  e.  NN )  /\  x  e.  ZZ )  ->  ( ( x  gcd  ( P ^ K ) )  =  1  <->  ( ( P ^ K )  gcd  x )  =  1 ) )
33 coprm 14325 . . . . . . . . . . . . . . 15  |-  ( ( P  e.  Prime  /\  x  e.  ZZ )  ->  ( -.  P  ||  x  <->  ( P  gcd  x )  =  1 ) )
3433adantlr 712 . . . . . . . . . . . . . 14  |-  ( ( ( P  e.  Prime  /\  K  e.  NN )  /\  x  e.  ZZ )  ->  ( -.  P  ||  x  <->  ( P  gcd  x )  =  1 ) )
3525, 32, 343bitr4d 285 . . . . . . . . . . . . 13  |-  ( ( ( P  e.  Prime  /\  K  e.  NN )  /\  x  e.  ZZ )  ->  ( ( x  gcd  ( P ^ K ) )  =  1  <->  -.  P  ||  x
) )
36 zcn 10865 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  ZZ  ->  x  e.  CC )
3736adantl 464 . . . . . . . . . . . . . . . 16  |-  ( ( ( P  e.  Prime  /\  K  e.  NN )  /\  x  e.  ZZ )  ->  x  e.  CC )
3837subid1d 9911 . . . . . . . . . . . . . . 15  |-  ( ( ( P  e.  Prime  /\  K  e.  NN )  /\  x  e.  ZZ )  ->  ( x  - 
0 )  =  x )
3938breq2d 4451 . . . . . . . . . . . . . 14  |-  ( ( ( P  e.  Prime  /\  K  e.  NN )  /\  x  e.  ZZ )  ->  ( P  ||  ( x  -  0
)  <->  P  ||  x ) )
4039notbid 292 . . . . . . . . . . . . 13  |-  ( ( ( P  e.  Prime  /\  K  e.  NN )  /\  x  e.  ZZ )  ->  ( -.  P  ||  ( x  -  0 )  <->  -.  P  ||  x
) )
4135, 40bitr4d 256 . . . . . . . . . . . 12  |-  ( ( ( P  e.  Prime  /\  K  e.  NN )  /\  x  e.  ZZ )  ->  ( ( x  gcd  ( P ^ K ) )  =  1  <->  -.  P  ||  (
x  -  0 ) ) )
4220, 41sylan2 472 . . . . . . . . . . 11  |-  ( ( ( P  e.  Prime  /\  K  e.  NN )  /\  x  e.  ( 1 ... ( P ^ K ) ) )  ->  ( (
x  gcd  ( P ^ K ) )  =  1  <->  -.  P  ||  (
x  -  0 ) ) )
4342biimpd 207 . . . . . . . . . 10  |-  ( ( ( P  e.  Prime  /\  K  e.  NN )  /\  x  e.  ( 1 ... ( P ^ K ) ) )  ->  ( (
x  gcd  ( P ^ K ) )  =  1  ->  -.  P  ||  ( x  -  0 ) ) )
44 imnan 420 . . . . . . . . . 10  |-  ( ( ( x  gcd  ( P ^ K ) )  =  1  ->  -.  P  ||  ( x  - 
0 ) )  <->  -.  (
( x  gcd  ( P ^ K ) )  =  1  /\  P  ||  ( x  -  0 ) ) )
4543, 44sylib 196 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  K  e.  NN )  /\  x  e.  ( 1 ... ( P ^ K ) ) )  ->  -.  (
( x  gcd  ( P ^ K ) )  =  1  /\  P  ||  ( x  -  0 ) ) )
4645ralrimiva 2868 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  A. x  e.  ( 1 ... ( P ^ K ) )  -.  ( ( x  gcd  ( P ^ K ) )  =  1  /\  P  ||  ( x  -  0
) ) )
47 rabeq0 3806 . . . . . . . 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
) ) )
4846, 47sylibr 212 . . . . . . 7  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  { x  e.  ( 1 ... ( P ^ K ) )  |  ( ( x  gcd  ( P ^ K ) )  =  1  /\  P  ||  ( x  -  0
) ) }  =  (/) )
4919, 48syl5eq 2507 . . . . . 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 ) } )  =  (/) )
50 fzfi 12064 . . . . . . . 8  |-  ( 1 ... ( P ^ K ) )  e. 
Fin
51 ssrab2 3571 . . . . . . . 8  |-  { x  e.  ( 1 ... ( P ^ K ) )  |  ( x  gcd  ( P ^ K ) )  =  1 } 
C_  ( 1 ... ( P ^ K
) )
52 ssfi 7733 . . . . . . . 8  |-  ( ( ( 1 ... ( P ^ K ) )  e.  Fin  /\  {
x  e.  ( 1 ... ( P ^ K ) )  |  ( x  gcd  ( P ^ K ) )  =  1 }  C_  ( 1 ... ( P ^ K ) ) )  ->  { x  e.  ( 1 ... ( P ^ K ) )  |  ( x  gcd  ( P ^ K ) )  =  1 }  e.  Fin )
5350, 51, 52mp2an 670 . . . . . . 7  |-  { x  e.  ( 1 ... ( P ^ K ) )  |  ( x  gcd  ( P ^ K ) )  =  1 }  e.  Fin
54 ssrab2 3571 . . . . . . . 8  |-  { x  e.  ( 1 ... ( P ^ K ) )  |  P  ||  (
x  -  0 ) }  C_  ( 1 ... ( P ^ K ) )
55 ssfi 7733 . . . . . . . 8  |-  ( ( ( 1 ... ( P ^ K ) )  e.  Fin  /\  {
x  e.  ( 1 ... ( P ^ K ) )  |  P  ||  ( x  -  0 ) } 
C_  ( 1 ... ( P ^ K
) ) )  ->  { x  e.  (
1 ... ( P ^ K ) )  |  P  ||  ( x  -  0 ) }  e.  Fin )
5650, 54, 55mp2an 670 . . . . . . 7  |-  { x  e.  ( 1 ... ( P ^ K ) )  |  P  ||  (
x  -  0 ) }  e.  Fin
57 hashun 12433 . . . . . . 7  |-  ( ( { 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 ) } ) ) )
5853, 56, 57mp3an12 1312 . . . . . 6  |-  ( ( { 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 ) } ) ) )
5949, 58syl 16 . . . . 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 ) } ) ) )
6042biimprd 223 . . . . . . . . . . . 12  |-  ( ( ( P  e.  Prime  /\  K  e.  NN )  /\  x  e.  ( 1 ... ( P ^ K ) ) )  ->  ( -.  P  ||  ( x  - 
0 )  ->  (
x  gcd  ( P ^ K ) )  =  1 ) )
6160con1d 124 . . . . . . . . . . 11  |-  ( ( ( P  e.  Prime  /\  K  e.  NN )  /\  x  e.  ( 1 ... ( P ^ K ) ) )  ->  ( -.  ( x  gcd  ( P ^ K ) )  =  1  ->  P  ||  ( x  -  0 ) ) )
6261orrd 376 . . . . . . . . . 10  |-  ( ( ( P  e.  Prime  /\  K  e.  NN )  /\  x  e.  ( 1 ... ( P ^ K ) ) )  ->  ( (
x  gcd  ( P ^ K ) )  =  1  \/  P  ||  ( x  -  0
) ) )
6362ralrimiva 2868 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  A. x  e.  ( 1 ... ( P ^ K ) ) ( ( x  gcd  ( P ^ K ) )  =  1  \/  P  ||  ( x  -  0 ) ) )
64 rabid2 3032 . . . . . . . . 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 ) ) )
6563, 64sylibr 212 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  (
1 ... ( P ^ K ) )  =  { x  e.  ( 1 ... ( P ^ K ) )  |  ( ( x  gcd  ( P ^ K ) )  =  1  \/  P  ||  ( x  -  0
) ) } )
66 unrab 3766 . . . . . . . 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
) ) }
6765, 66syl6reqr 2514 . . . . . . 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 ) ) )
6867fveq2d 5852 . . . . . 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 ) ) ) )
694nnnn0d 10848 . . . . . . 7  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( P ^ K )  e. 
NN0 )
70 hashfz1 12401 . . . . . . 7  |-  ( ( P ^ K )  e.  NN0  ->  ( # `  ( 1 ... ( P ^ K ) ) )  =  ( P ^ K ) )
7169, 70syl 16 . . . . . 6  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( # `
 ( 1 ... ( P ^ K
) ) )  =  ( P ^ K
) )
72 expm1t 12176 . . . . . . 7  |-  ( ( P  e.  CC  /\  K  e.  NN )  ->  ( P ^ K
)  =  ( ( P ^ ( K  -  1 ) )  x.  P ) )
7311, 72sylan 469 . . . . . 6  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( P ^ K )  =  ( ( P ^
( K  -  1 ) )  x.  P
) )
7468, 71, 733eqtrd 2499 . . . . 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
) )
751adantr 463 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  P  e.  NN )
76 1zzd 10891 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  1  e.  ZZ )
77 nn0uz 11116 . . . . . . . . . . 11  |-  NN0  =  ( ZZ>= `  0 )
78 1m1e0 10600 . . . . . . . . . . . 12  |-  ( 1  -  1 )  =  0
7978fveq2i 5851 . . . . . . . . . . 11  |-  ( ZZ>= `  ( 1  -  1 ) )  =  (
ZZ>= `  0 )
8077, 79eqtr4i 2486 . . . . . . . . . 10  |-  NN0  =  ( ZZ>= `  ( 1  -  1 ) )
8169, 80syl6eleq 2552 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( P ^ K )  e.  ( ZZ>= `  ( 1  -  1 ) ) )
82 0zd 10872 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  0  e.  ZZ )
8375, 76, 81, 82hashdvds 14389 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( # `
 { x  e.  ( 1 ... ( P ^ K ) )  |  P  ||  (
x  -  0 ) } )  =  ( ( |_ `  (
( ( P ^ K )  -  0 )  /  P ) )  -  ( |_
`  ( ( ( 1  -  1 )  -  0 )  /  P ) ) ) )
844nncnd 10547 . . . . . . . . . . . . . 14  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( P ^ K )  e.  CC )
8584subid1d 9911 . . . . . . . . . . . . 13  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  (
( P ^ K
)  -  0 )  =  ( P ^ K ) )
8685oveq1d 6285 . . . . . . . . . . . 12  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  (
( ( P ^ K )  -  0 )  /  P )  =  ( ( P ^ K )  /  P ) )
8775nnne0d 10576 . . . . . . . . . . . . 13  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  P  =/=  0 )
88 nnz 10882 . . . . . . . . . . . . . 14  |-  ( K  e.  NN  ->  K  e.  ZZ )
8988adantl 464 . . . . . . . . . . . . 13  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  K  e.  ZZ )
9012, 87, 89expm1d 12302 . . . . . . . . . . . 12  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( P ^ ( K  - 
1 ) )  =  ( ( P ^ K )  /  P
) )
9186, 90eqtr4d 2498 . . . . . . . . . . 11  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  (
( ( P ^ K )  -  0 )  /  P )  =  ( P ^
( K  -  1 ) ) )
9291fveq2d 5852 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( |_ `  ( ( ( P ^ K )  -  0 )  /  P ) )  =  ( |_ `  ( P ^ ( K  - 
1 ) ) ) )
939nnzd 10964 . . . . . . . . . . 11  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( P ^ ( K  - 
1 ) )  e.  ZZ )
94 flid 11926 . . . . . . . . . . 11  |-  ( ( P ^ ( K  -  1 ) )  e.  ZZ  ->  ( |_ `  ( P ^
( K  -  1 ) ) )  =  ( P ^ ( K  -  1 ) ) )
9593, 94syl 16 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( |_ `  ( P ^
( K  -  1 ) ) )  =  ( P ^ ( K  -  1 ) ) )
9692, 95eqtrd 2495 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( |_ `  ( ( ( P ^ K )  -  0 )  /  P ) )  =  ( P ^ ( K  -  1 ) ) )
9778oveq1i 6280 . . . . . . . . . . . . . 14  |-  ( ( 1  -  1 )  -  0 )  =  ( 0  -  0 )
98 0m0e0 10641 . . . . . . . . . . . . . 14  |-  ( 0  -  0 )  =  0
9997, 98eqtri 2483 . . . . . . . . . . . . 13  |-  ( ( 1  -  1 )  -  0 )  =  0
10099oveq1i 6280 . . . . . . . . . . . 12  |-  ( ( ( 1  -  1 )  -  0 )  /  P )  =  ( 0  /  P
)
10112, 87div0d 10315 . . . . . . . . . . . 12  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  (
0  /  P )  =  0 )
102100, 101syl5eq 2507 . . . . . . . . . . 11  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  (
( ( 1  -  1 )  -  0 )  /  P )  =  0 )
103102fveq2d 5852 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( |_ `  ( ( ( 1  -  1 )  -  0 )  /  P ) )  =  ( |_ `  0
) )
104 0z 10871 . . . . . . . . . . 11  |-  0  e.  ZZ
105 flid 11926 . . . . . . . . . . 11  |-  ( 0  e.  ZZ  ->  ( |_ `  0 )  =  0 )
106104, 105ax-mp 5 . . . . . . . . . 10  |-  ( |_
`  0 )  =  0
107103, 106syl6eq 2511 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( |_ `  ( ( ( 1  -  1 )  -  0 )  /  P ) )  =  0 )
10896, 107oveq12d 6288 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  (
( |_ `  (
( ( P ^ K )  -  0 )  /  P ) )  -  ( |_
`  ( ( ( 1  -  1 )  -  0 )  /  P ) ) )  =  ( ( P ^ ( K  - 
1 ) )  - 
0 ) )
10910subid1d 9911 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  (
( P ^ ( K  -  1 ) )  -  0 )  =  ( P ^
( K  -  1 ) ) )
11083, 108, 1093eqtrd 2499 . . . . . . 7  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( # `
 { x  e.  ( 1 ... ( P ^ K ) )  |  P  ||  (
x  -  0 ) } )  =  ( P ^ ( K  -  1 ) ) )
111110oveq2d 6286 . . . . . 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 ) ) ) )
112 hashcl 12410 . . . . . . . . 9  |-  ( { x  e.  ( 1 ... ( P ^ K ) )  |  ( x  gcd  ( P ^ K ) )  =  1 }  e.  Fin  ->  ( # `  {
x  e.  ( 1 ... ( P ^ K ) )  |  ( x  gcd  ( P ^ K ) )  =  1 } )  e.  NN0 )
11353, 112ax-mp 5 . . . . . . . 8  |-  ( # `  { x  e.  ( 1 ... ( P ^ K ) )  |  ( x  gcd  ( P ^ K ) )  =  1 } )  e.  NN0
114113nn0cni 10803 . . . . . . 7  |-  ( # `  { x  e.  ( 1 ... ( P ^ K ) )  |  ( x  gcd  ( P ^ K ) )  =  1 } )  e.  CC
115 addcom 9755 . . . . . . 7  |-  ( ( ( # `  {
x  e.  ( 1 ... ( P ^ K ) )  |  ( x  gcd  ( P ^ K ) )  =  1 } )  e.  CC  /\  ( P ^ ( K  - 
1 ) )  e.  CC )  ->  (
( # `  { x  e.  ( 1 ... ( P ^ K ) )  |  ( x  gcd  ( P ^ K ) )  =  1 } )  +  ( P ^ ( K  - 
1 ) ) )  =  ( ( P ^ ( K  - 
1 ) )  +  ( # `  {
x  e.  ( 1 ... ( P ^ K ) )  |  ( x  gcd  ( P ^ K ) )  =  1 } ) ) )
116114, 10, 115sylancr 661 . . . . . 6  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  (
( # `  { x  e.  ( 1 ... ( P ^ K ) )  |  ( x  gcd  ( P ^ K ) )  =  1 } )  +  ( P ^ ( K  - 
1 ) ) )  =  ( ( P ^ ( K  - 
1 ) )  +  ( # `  {
x  e.  ( 1 ... ( P ^ K ) )  |  ( x  gcd  ( P ^ K ) )  =  1 } ) ) )
117111, 116eqtrd 2495 . . . . 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 } ) ) )
11859, 74, 1173eqtr3rd 2504 . . . 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 ) )
11910, 12mulcld 9605 . . . . 5  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  (
( P ^ ( K  -  1 ) )  x.  P )  e.  CC )
120114a1i 11 . . . . 5  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( # `
 { x  e.  ( 1 ... ( P ^ K ) )  |  ( x  gcd  ( P ^ K ) )  =  1 } )  e.  CC )
121119, 10, 120subaddd 9940 . . . 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 ) ) )
122118, 121mpbird 232 . . 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 } ) )
12316, 18, 1223eqtrrd 2500 . 2  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( # `
 { x  e.  ( 1 ... ( P ^ K ) )  |  ( x  gcd  ( P ^ K ) )  =  1 } )  =  ( ( P ^ ( K  -  1 ) )  x.  ( P  - 
1 ) ) )
1246, 123eqtrd 2495 1  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( phi `  ( P ^ K ) )  =  ( ( P ^
( K  -  1 ) )  x.  ( P  -  1 ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 366    /\ wa 367    = wceq 1398    e. wcel 1823   A.wral 2804   {crab 2808    u. cun 3459    i^i cin 3460    C_ wss 3461   (/)c0 3783   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   Fincfn 7509   CCcc 9479   0cc0 9481   1c1 9482    + caddc 9484    x. cmul 9486    - cmin 9796    / cdiv 10202   NNcn 10531   NN0cn0 10791   ZZcz 10860   ZZ>=cuz 11082   ...cfz 11675   |_cfl 11908   ^cexp 12148   #chash 12387    || cdvds 14070    gcd cgcd 14228   Primecprime 14301   phicphi 14378
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-sup 7893  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11083  df-rp 11222  df-fz 11676  df-fl 11910  df-mod 11979  df-seq 12090  df-exp 12149  df-hash 12388  df-cj 13014  df-re 13015  df-im 13016  df-sqrt 13150  df-abs 13151  df-dvds 14071  df-gcd 14229  df-prm 14302  df-phi 14380
This theorem is referenced by:  phiprm  14391
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