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Theorem phiprmpw 11625
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 11519 . . . 4  |-  ( P  e.  Prime  ->  P  e.  NN )
2 nnnn0 8778 . . . 4  |-  ( K  e.  NN  ->  K  e.  NN0 )
3 nnexpcl 10083 . . . 4  |-  ( ( P  e.  NN  /\  K  e.  NN0 )  -> 
( P ^ K
)  e.  NN )
41, 2, 3syl2an 284 . . 3  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( P ^ K )  e.  NN )
5 phival 11616 . . 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 8812 . . . . . 6  |-  ( K  e.  NN  ->  ( K  -  1 )  e.  NN0 )
8 nnexpcl 10083 . . . . . 6  |-  ( ( P  e.  NN  /\  ( K  -  1
)  e.  NN0 )  ->  ( P ^ ( K  -  1 ) )  e.  NN )
91, 7, 8syl2an 284 . . . . 5  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( P ^ ( K  - 
1 ) )  e.  NN )
109nncnd 8534 . . . 4  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( P ^ ( K  - 
1 ) )  e.  CC )
111nncnd 8534 . . . . 5  |-  ( P  e.  Prime  ->  P  e.  CC )
1211adantr 271 . . . 4  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  P  e.  CC )
13 ax-1cn 7535 . . . . 5  |-  1  e.  CC
14 subdi 7960 . . . . 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 1269 . . . 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 404 . . 3  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  (
( P ^ ( K  -  1 ) )  x.  ( P  -  1 ) )  =  ( ( ( P ^ ( K  -  1 ) )  x.  P )  -  ( ( P ^
( K  -  1 ) )  x.  1 ) ) )
1710mulid1d 7602 . . . 4  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  (
( P ^ ( K  -  1 ) )  x.  1 )  =  ( P ^
( K  -  1 ) ) )
1817oveq2d 5706 . . 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 11615 . . . . . . 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 8875 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  1  e.  ZZ )
22 prmz 11520 . . . . . . . . 9  |-  ( P  e.  Prime  ->  P  e.  ZZ )
23 zexpcl 10085 . . . . . . . . 9  |-  ( ( P  e.  ZZ  /\  K  e.  NN0 )  -> 
( P ^ K
)  e.  ZZ )
2422, 2, 23syl2an 284 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( P ^ K )  e.  ZZ )
2521, 24fzfigd 9987 . . . . . . 7  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  (
1 ... ( P ^ K ) )  e. 
Fin )
2622ad2antrr 473 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  K  e.  NN )  /\  x  e.  ( 1 ... ( P ^ K ) ) )  ->  P  e.  ZZ )
27 elfzelz 9589 . . . . . . . . . . 11  |-  ( x  e.  ( 1 ... ( P ^ K
) )  ->  x  e.  ZZ )
2827adantl 272 . . . . . . . . . 10  |-  ( ( ( P  e.  Prime  /\  K  e.  NN )  /\  x  e.  ( 1 ... ( P ^ K ) ) )  ->  x  e.  ZZ )
29 0zd 8860 . . . . . . . . . 10  |-  ( ( ( P  e.  Prime  /\  K  e.  NN )  /\  x  e.  ( 1 ... ( P ^ K ) ) )  ->  0  e.  ZZ )
3028, 29zsubcld 8972 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  K  e.  NN )  /\  x  e.  ( 1 ... ( P ^ K ) ) )  ->  ( x  -  0 )  e.  ZZ )
31 zdvdsdc 11244 . . . . . . . . 9  |-  ( ( P  e.  ZZ  /\  ( x  -  0
)  e.  ZZ )  -> DECID 
P  ||  ( x  -  0 ) )
3226, 30, 31syl2anc 404 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  K  e.  NN )  /\  x  e.  ( 1 ... ( P ^ K ) ) )  -> DECID  P  ||  ( x  -  0 ) )
3332ralrimiva 2458 . . . . . . 7  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  A. x  e.  ( 1 ... ( P ^ K ) )DECID  P 
||  ( x  - 
0 ) )
3425, 33ssfirab 6723 . . . . . 6  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  { x  e.  ( 1 ... ( P ^ K ) )  |  P  ||  (
x  -  0 ) }  e.  Fin )
35 inrab 3287 . . . . . . 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 11559 . . . . . . . . . . . . . . . . 17  |-  ( ( P  e.  ZZ  /\  x  e.  ZZ  /\  K  e.  NN )  ->  (
( ( P ^ K )  gcd  x
)  =  1  <->  ( P  gcd  x )  =  1 ) )
3722, 36syl3an1 1214 . . . . . . . . . . . . . . . 16  |-  ( ( P  e.  Prime  /\  x  e.  ZZ  /\  K  e.  NN )  ->  (
( ( P ^ K )  gcd  x
)  =  1  <->  ( P  gcd  x )  =  1 ) )
38373expa 1146 . . . . . . . . . . . . . . 15  |-  ( ( ( P  e.  Prime  /\  x  e.  ZZ )  /\  K  e.  NN )  ->  ( ( ( P ^ K )  gcd  x )  =  1  <->  ( P  gcd  x )  =  1 ) )
3938an32s 536 . . . . . . . . . . . . . 14  |-  ( ( ( P  e.  Prime  /\  K  e.  NN )  /\  x  e.  ZZ )  ->  ( ( ( P ^ K )  gcd  x )  =  1  <->  ( P  gcd  x )  =  1 ) )
40 simpr 109 . . . . . . . . . . . . . . . 16  |-  ( ( ( P  e.  Prime  /\  K  e.  NN )  /\  x  e.  ZZ )  ->  x  e.  ZZ )
4124adantr 271 . . . . . . . . . . . . . . . 16  |-  ( ( ( P  e.  Prime  /\  K  e.  NN )  /\  x  e.  ZZ )  ->  ( P ^ K )  e.  ZZ )
42 gcdcom 11392 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  ZZ  /\  ( P ^ K )  e.  ZZ )  -> 
( x  gcd  ( P ^ K ) )  =  ( ( P ^ K )  gcd  x ) )
4340, 41, 42syl2anc 404 . . . . . . . . . . . . . . 15  |-  ( ( ( P  e.  Prime  /\  K  e.  NN )  /\  x  e.  ZZ )  ->  ( x  gcd  ( P ^ K ) )  =  ( ( P ^ K )  gcd  x ) )
4443eqeq1d 2103 . . . . . . . . . . . . . 14  |-  ( ( ( P  e.  Prime  /\  K  e.  NN )  /\  x  e.  ZZ )  ->  ( ( x  gcd  ( P ^ K ) )  =  1  <->  ( ( P ^ K )  gcd  x )  =  1 ) )
45 coprm 11550 . . . . . . . . . . . . . . 15  |-  ( ( P  e.  Prime  /\  x  e.  ZZ )  ->  ( -.  P  ||  x  <->  ( P  gcd  x )  =  1 ) )
4645adantlr 462 . . . . . . . . . . . . . 14  |-  ( ( ( P  e.  Prime  /\  K  e.  NN )  /\  x  e.  ZZ )  ->  ( -.  P  ||  x  <->  ( P  gcd  x )  =  1 ) )
4739, 44, 463bitr4d 219 . . . . . . . . . . . . 13  |-  ( ( ( P  e.  Prime  /\  K  e.  NN )  /\  x  e.  ZZ )  ->  ( ( x  gcd  ( P ^ K ) )  =  1  <->  -.  P  ||  x
) )
48 zcn 8853 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  ZZ  ->  x  e.  CC )
4948adantl 272 . . . . . . . . . . . . . . . 16  |-  ( ( ( P  e.  Prime  /\  K  e.  NN )  /\  x  e.  ZZ )  ->  x  e.  CC )
5049subid1d 7879 . . . . . . . . . . . . . . 15  |-  ( ( ( P  e.  Prime  /\  K  e.  NN )  /\  x  e.  ZZ )  ->  ( x  - 
0 )  =  x )
5150breq2d 3879 . . . . . . . . . . . . . 14  |-  ( ( ( P  e.  Prime  /\  K  e.  NN )  /\  x  e.  ZZ )  ->  ( P  ||  ( x  -  0
)  <->  P  ||  x ) )
5251notbid 630 . . . . . . . . . . . . 13  |-  ( ( ( P  e.  Prime  /\  K  e.  NN )  /\  x  e.  ZZ )  ->  ( -.  P  ||  ( x  -  0 )  <->  -.  P  ||  x
) )
5347, 52bitr4d 190 . . . . . . . . . . . 12  |-  ( ( ( P  e.  Prime  /\  K  e.  NN )  /\  x  e.  ZZ )  ->  ( ( x  gcd  ( P ^ K ) )  =  1  <->  -.  P  ||  (
x  -  0 ) ) )
5427, 53sylan2 281 . . . . . . . . . . 11  |-  ( ( ( P  e.  Prime  /\  K  e.  NN )  /\  x  e.  ( 1 ... ( P ^ K ) ) )  ->  ( (
x  gcd  ( P ^ K ) )  =  1  <->  -.  P  ||  (
x  -  0 ) ) )
5554biimpd 143 . . . . . . . . . 10  |-  ( ( ( P  e.  Prime  /\  K  e.  NN )  /\  x  e.  ( 1 ... ( P ^ K ) ) )  ->  ( (
x  gcd  ( P ^ K ) )  =  1  ->  -.  P  ||  ( x  -  0 ) ) )
56 imnan 662 . . . . . . . . . 10  |-  ( ( ( x  gcd  ( P ^ K ) )  =  1  ->  -.  P  ||  ( x  - 
0 ) )  <->  -.  (
( x  gcd  ( P ^ K ) )  =  1  /\  P  ||  ( x  -  0 ) ) )
5755, 56sylib 121 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  K  e.  NN )  /\  x  e.  ( 1 ... ( P ^ K ) ) )  ->  -.  (
( x  gcd  ( P ^ K ) )  =  1  /\  P  ||  ( x  -  0 ) ) )
5857ralrimiva 2458 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  A. x  e.  ( 1 ... ( P ^ K ) )  -.  ( ( x  gcd  ( P ^ K ) )  =  1  /\  P  ||  ( x  -  0
) ) )
59 rabeq0 3331 . . . . . . . 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 133 . . . . . . 7  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  { x  e.  ( 1 ... ( P ^ K ) )  |  ( ( x  gcd  ( P ^ K ) )  =  1  /\  P  ||  ( x  -  0
) ) }  =  (/) )
6135, 60syl5eq 2139 . . . . . 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 10328 . . . . . 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 1181 . . . . 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 ) } ) ) )
6454biimprd 157 . . . . . . . . . . . 12  |-  ( ( ( P  e.  Prime  /\  K  e.  NN )  /\  x  e.  ( 1 ... ( P ^ K ) ) )  ->  ( -.  P  ||  ( x  - 
0 )  ->  (
x  gcd  ( P ^ K ) )  =  1 ) )
65 con1dc 794 . . . . . . . . . . . 12  |-  (DECID  P  ||  ( x  -  0
)  ->  ( ( -.  P  ||  ( x  -  0 )  -> 
( x  gcd  ( P ^ K ) )  =  1 )  -> 
( -.  ( x  gcd  ( P ^ K ) )  =  1  ->  P  ||  (
x  -  0 ) ) ) )
6632, 64, 65sylc 62 . . . . . . . . . . 11  |-  ( ( ( P  e.  Prime  /\  K  e.  NN )  /\  x  e.  ( 1 ... ( P ^ K ) ) )  ->  ( -.  ( x  gcd  ( P ^ K ) )  =  1  ->  P  ||  ( x  -  0 ) ) )
6724adantr 271 . . . . . . . . . . . . . . 15  |-  ( ( ( P  e.  Prime  /\  K  e.  NN )  /\  x  e.  ( 1 ... ( P ^ K ) ) )  ->  ( P ^ K )  e.  ZZ )
6828, 67gcdcld 11387 . . . . . . . . . . . . . 14  |-  ( ( ( P  e.  Prime  /\  K  e.  NN )  /\  x  e.  ( 1 ... ( P ^ K ) ) )  ->  ( x  gcd  ( P ^ K
) )  e.  NN0 )
6968nn0zd 8965 . . . . . . . . . . . . 13  |-  ( ( ( P  e.  Prime  /\  K  e.  NN )  /\  x  e.  ( 1 ... ( P ^ K ) ) )  ->  ( x  gcd  ( P ^ K
) )  e.  ZZ )
70 1zzd 8875 . . . . . . . . . . . . 13  |-  ( ( ( P  e.  Prime  /\  K  e.  NN )  /\  x  e.  ( 1 ... ( P ^ K ) ) )  ->  1  e.  ZZ )
71 zdceq 8920 . . . . . . . . . . . . 13  |-  ( ( ( x  gcd  ( P ^ K ) )  e.  ZZ  /\  1  e.  ZZ )  -> DECID  ( x  gcd  ( P ^ K ) )  =  1 )
7269, 70, 71syl2anc 404 . . . . . . . . . . . 12  |-  ( ( ( P  e.  Prime  /\  K  e.  NN )  /\  x  e.  ( 1 ... ( P ^ K ) ) )  -> DECID  ( x  gcd  ( P ^ K ) )  =  1 )
73 dfordc 832 . . . . . . . . . . . 12  |-  (DECID  ( x  gcd  ( P ^ K ) )  =  1  ->  ( (
( x  gcd  ( P ^ K ) )  =  1  \/  P  ||  ( x  -  0 ) )  <->  ( -.  ( x  gcd  ( P ^ K ) )  =  1  ->  P  ||  ( x  -  0 ) ) ) )
7472, 73syl 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 ) ) ) )
7566, 74mpbird 166 . . . . . . . . . 10  |-  ( ( ( P  e.  Prime  /\  K  e.  NN )  /\  x  e.  ( 1 ... ( P ^ K ) ) )  ->  ( (
x  gcd  ( P ^ K ) )  =  1  \/  P  ||  ( x  -  0
) ) )
7675ralrimiva 2458 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  A. x  e.  ( 1 ... ( P ^ K ) ) ( ( x  gcd  ( P ^ K ) )  =  1  \/  P  ||  ( x  -  0 ) ) )
77 rabid2 2557 . . . . . . . . 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 ) ) )
7876, 77sylibr 133 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  (
1 ... ( P ^ K ) )  =  { x  e.  ( 1 ... ( P ^ K ) )  |  ( ( x  gcd  ( P ^ K ) )  =  1  \/  P  ||  ( x  -  0
) ) } )
79 unrab 3286 . . . . . . . 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
) ) }
8078, 79syl6reqr 2146 . . . . . . 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 5344 . . . . . 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 8824 . . . . . . 7  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( P ^ K )  e. 
NN0 )
83 hashfz1 10306 . . . . . . 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 10098 . . . . . . 7  |-  ( ( P  e.  CC  /\  K  e.  NN )  ->  ( P ^ K
)  =  ( ( P ^ ( K  -  1 ) )  x.  P ) )
8611, 85sylan 278 . . . . . 6  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( P ^ K )  =  ( ( P ^
( K  -  1 ) )  x.  P
) )
8781, 84, 863eqtrd 2131 . . . . 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 10304 . . . . . . . 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 8826 . . . . . 6  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( `  { x  e.  ( 1 ... ( P ^ K ) )  |  ( x  gcd  ( P ^ K ) )  =  1 } )  e.  CC )
911adantr 271 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  P  e.  NN )
92 nn0uz 9152 . . . . . . . . . . 11  |-  NN0  =  ( ZZ>= `  0 )
93 1m1e0 8589 . . . . . . . . . . . 12  |-  ( 1  -  1 )  =  0
9493fveq2i 5343 . . . . . . . . . . 11  |-  ( ZZ>= `  ( 1  -  1 ) )  =  (
ZZ>= `  0 )
9592, 94eqtr4i 2118 . . . . . . . . . 10  |-  NN0  =  ( ZZ>= `  ( 1  -  1 ) )
9682, 95syl6eleq 2187 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( P ^ K )  e.  ( ZZ>= `  ( 1  -  1 ) ) )
97 0zd 8860 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  0  e.  ZZ )
9891, 21, 96, 97hashdvds 11624 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( `  { x  e.  ( 1 ... ( P ^ K ) )  |  P  ||  (
x  -  0 ) } )  =  ( ( |_ `  (
( ( P ^ K )  -  0 )  /  P ) )  -  ( |_
`  ( ( ( 1  -  1 )  -  0 )  /  P ) ) ) )
994nncnd 8534 . . . . . . . . . . . . . 14  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( P ^ K )  e.  CC )
10099subid1d 7879 . . . . . . . . . . . . 13  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  (
( P ^ K
)  -  0 )  =  ( P ^ K ) )
101100oveq1d 5705 . . . . . . . . . . . 12  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  (
( ( P ^ K )  -  0 )  /  P )  =  ( ( P ^ K )  /  P ) )
10291nnap0d 8566 . . . . . . . . . . . . 13  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  P #  0 )
103 nnz 8867 . . . . . . . . . . . . . 14  |-  ( K  e.  NN  ->  K  e.  ZZ )
104103adantl 272 . . . . . . . . . . . . 13  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  K  e.  ZZ )
10512, 102, 104expm1apd 10211 . . . . . . . . . . . 12  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( P ^ ( K  - 
1 ) )  =  ( ( P ^ K )  /  P
) )
106101, 105eqtr4d 2130 . . . . . . . . . . 11  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  (
( ( P ^ K )  -  0 )  /  P )  =  ( P ^
( K  -  1 ) ) )
107106fveq2d 5344 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( |_ `  ( ( ( P ^ K )  -  0 )  /  P ) )  =  ( |_ `  ( P ^ ( K  - 
1 ) ) ) )
1089nnzd 8966 . . . . . . . . . . 11  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( P ^ ( K  - 
1 ) )  e.  ZZ )
109 flid 9840 . . . . . . . . . . 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 2127 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( |_ `  ( ( ( P ^ K )  -  0 )  /  P ) )  =  ( P ^ ( K  -  1 ) ) )
11293oveq1i 5700 . . . . . . . . . . . . . 14  |-  ( ( 1  -  1 )  -  0 )  =  ( 0  -  0 )
113 0m0e0 8632 . . . . . . . . . . . . . 14  |-  ( 0  -  0 )  =  0
114112, 113eqtri 2115 . . . . . . . . . . . . 13  |-  ( ( 1  -  1 )  -  0 )  =  0
115114oveq1i 5700 . . . . . . . . . . . 12  |-  ( ( ( 1  -  1 )  -  0 )  /  P )  =  ( 0  /  P
)
11612, 102div0apd 8351 . . . . . . . . . . . 12  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  (
0  /  P )  =  0 )
117115, 116syl5eq 2139 . . . . . . . . . . 11  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  (
( ( 1  -  1 )  -  0 )  /  P )  =  0 )
118117fveq2d 5344 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( |_ `  ( ( ( 1  -  1 )  -  0 )  /  P ) )  =  ( |_ `  0
) )
119 0z 8859 . . . . . . . . . . 11  |-  0  e.  ZZ
120 flid 9840 . . . . . . . . . . 11  |-  ( 0  e.  ZZ  ->  ( |_ `  0 )  =  0 )
121119, 120ax-mp 7 . . . . . . . . . 10  |-  ( |_
`  0 )  =  0
122118, 121syl6eq 2143 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( |_ `  ( ( ( 1  -  1 )  -  0 )  /  P ) )  =  0 )
123111, 122oveq12d 5708 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  (
( |_ `  (
( ( P ^ K )  -  0 )  /  P ) )  -  ( |_
`  ( ( ( 1  -  1 )  -  0 )  /  P ) ) )  =  ( ( P ^ ( K  - 
1 ) )  - 
0 ) )
12410subid1d 7879 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  (
( P ^ ( K  -  1 ) )  -  0 )  =  ( P ^
( K  -  1 ) ) )
12598, 123, 1243eqtrd 2131 . . . . . . 7  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( `  { x  e.  ( 1 ... ( P ^ K ) )  |  P  ||  (
x  -  0 ) } )  =  ( P ^ ( K  -  1 ) ) )
126125oveq2d 5706 . . . . . 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 7736 . . . . 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 2136 . . . 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 7605 . . . . 5  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  (
( P ^ ( K  -  1 ) )  x.  P )  e.  CC )
130129, 10, 90subaddd 7908 . . . 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 166 . . 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 2132 . 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 2127 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 103    <-> wb 104    \/ wo 667  DECID wdc 783    = wceq 1296    e. wcel 1445   A.wral 2370   {crab 2374    u. cun 3011    i^i cin 3012   (/)c0 3302   class class class wbr 3867   ` cfv 5049  (class class class)co 5690   Fincfn 6537   CCcc 7445   0cc0 7447   1c1 7448    + caddc 7450    x. cmul 7452    - cmin 7750    / cdiv 8236   NNcn 8520   NN0cn0 8771   ZZcz 8848   ZZ>=cuz 9118   ...cfz 9573   |_cfl 9824   ^cexp 10069  ♯chash 10298    || cdvds 11223    gcd cgcd 11365   Primecprime 11516   phicphi 11613
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-coll 3975  ax-sep 3978  ax-nul 3986  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-setind 4381  ax-iinf 4431  ax-cnex 7533  ax-resscn 7534  ax-1cn 7535  ax-1re 7536  ax-icn 7537  ax-addcl 7538  ax-addrcl 7539  ax-mulcl 7540  ax-mulrcl 7541  ax-addcom 7542  ax-mulcom 7543  ax-addass 7544  ax-mulass 7545  ax-distr 7546  ax-i2m1 7547  ax-0lt1 7548  ax-1rid 7549  ax-0id 7550  ax-rnegex 7551  ax-precex 7552  ax-cnre 7553  ax-pre-ltirr 7554  ax-pre-ltwlin 7555  ax-pre-lttrn 7556  ax-pre-apti 7557  ax-pre-ltadd 7558  ax-pre-mulgt0 7559  ax-pre-mulext 7560  ax-arch 7561  ax-caucvg 7562
This theorem depends on definitions:  df-bi 116  df-dc 784  df-3or 928  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-nel 2358  df-ral 2375  df-rex 2376  df-reu 2377  df-rmo 2378  df-rab 2379  df-v 2635  df-sbc 2855  df-csb 2948  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303  df-if 3414  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-int 3711  df-iun 3754  df-br 3868  df-opab 3922  df-mpt 3923  df-tr 3959  df-id 4144  df-po 4147  df-iso 4148  df-iord 4217  df-on 4219  df-ilim 4220  df-suc 4222  df-iom 4434  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-f1 5054  df-fo 5055  df-f1o 5056  df-fv 5057  df-riota 5646  df-ov 5693  df-oprab 5694  df-mpt2 5695  df-1st 5949  df-2nd 5950  df-recs 6108  df-irdg 6173  df-frec 6194  df-1o 6219  df-2o 6220  df-oadd 6223  df-er 6332  df-en 6538  df-dom 6539  df-fin 6540  df-sup 6759  df-pnf 7621  df-mnf 7622  df-xr 7623  df-ltxr 7624  df-le 7625  df-sub 7752  df-neg 7753  df-reap 8149  df-ap 8156  df-div 8237  df-inn 8521  df-2 8579  df-3 8580  df-4 8581  df-n0 8772  df-z 8849  df-uz 9119  df-q 9204  df-rp 9234  df-fz 9574  df-fzo 9703  df-fl 9826  df-mod 9879  df-seqfrec 10001  df-exp 10070  df-ihash 10299  df-cj 10391  df-re 10392  df-im 10393  df-rsqrt 10546  df-abs 10547  df-dvds 11224  df-gcd 11366  df-prm 11517  df-phi 11614
This theorem is referenced by:  phiprm  11626
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