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.

Step	Hyp	Ref	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 )
4	1, 2, 3	syl2an 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 } ) )
6	4, 5	syl 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 )
9	1, 7, 8	syl2an 289	. . . . 5 |- ( ( P e. Prime /\ K e. NN ) -> ( P ^ ( K - 1 ) ) e. NN )
10	9	nncnd 8962	. . . 4 |- ( ( P e. Prime /\ K e. NN ) -> ( P ^ ( K - 1 ) ) e. CC )
11	1	nncnd 8962	. . . . 5 |- ( P e. Prime -> P e. CC )
12	11	adantr 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 ) ) )
15	13, 14	mp3an3 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 ) ) )
16	10, 12, 15	syl2anc 411	. . 3 |- ( ( P e. Prime /\ K e. NN ) -> ( ( P ^ ( K - 1 ) ) x. ( P - 1 ) ) = ( ( ( P ^ ( K - 1 ) ) x. P ) - ( ( P ^ ( K - 1 ) ) x. 1 ) ) )
17	10	mulridd 8003	. . . 4 |- ( ( P e. Prime /\ K e. NN ) -> ( ( P ^ ( K - 1 ) ) x. 1 ) = ( P ^ ( K - 1 ) ) )
18	17	oveq2d 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 )
20	4, 19	syl 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 )
24	22, 2, 23	syl2an 289	. . . . . . . 8 |- ( ( P e. Prime /\ K e. NN ) -> ( P ^ K ) e. ZZ )
25	21, 24	fzfigd 10461	. . . . . . 7 |- ( ( P e. Prime /\ K e. NN ) -> ( 1 ... ( P ^ K ) ) e. Fin )
26	22	ad2antrr 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 )
28	27	adantl 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 )
30	28, 29	zsubcld 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 ) )
32	26, 30, 31	syl2anc 411	. . . . . . . 8 |- ( ( ( P e. Prime /\ K e. NN ) /\ x e. ( 1 ... ( P ^ K ) ) ) -> DECID P || ( x - 0 ) )
33	32	ralrimiva 2563	. . . . . . 7 |- ( ( P e. Prime /\ K e. NN ) -> A. x e. ( 1 ... ( P ^ K ) )DECID P || ( x - 0 ) )
34	25, 33	ssfirab 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 ) )
37	22, 36	syl3an1 1282	. . . . . . . . . . . . . . . 16 |- ( ( P e. Prime /\ x e. ZZ /\ K e. NN ) -> ( ( ( P ^ K ) gcd x ) = 1 <-> ( P gcd x ) = 1 ) )
38	37	3expa 1205	. . . . . . . . . . . . . . 15 |- ( ( ( P e. Prime /\ x e. ZZ ) /\ K e. NN ) -> ( ( ( P ^ K ) gcd x ) = 1 <-> ( P gcd x ) = 1 ) )
39	38	an32s 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 )
41	24	adantr 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 ) )
43	40, 41, 42	syl2anc 411	. . . . . . . . . . . . . . 15 |- ( ( ( P e. Prime /\ K e. NN ) /\ x e. ZZ ) -> ( x gcd ( P ^ K ) ) = ( ( P ^ K ) gcd x ) )
44	43	eqeq1d 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 ) )
46	45	adantlr 477	. . . . . . . . . . . . . 14 |- ( ( ( P e. Prime /\ K e. NN ) /\ x e. ZZ ) -> ( -. P || x <-> ( P gcd x ) = 1 ) )
47	39, 44, 46	3bitr4d 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 )
49	48	adantl 277	. . . . . . . . . . . . . . . 16 |- ( ( ( P e. Prime /\ K e. NN ) /\ x e. ZZ ) -> x e. CC )
50	49	subid1d 8286	. . . . . . . . . . . . . . 15 |- ( ( ( P e. Prime /\ K e. NN ) /\ x e. ZZ ) -> ( x - 0 ) = x )
51	50	breq2d 4030	. . . . . . . . . . . . . 14 |- ( ( ( P e. Prime /\ K e. NN ) /\ x e. ZZ ) -> ( P || ( x - 0 ) <-> P || x ) )
52	51	notbid 668	. . . . . . . . . . . . 13 |- ( ( ( P e. Prime /\ K e. NN ) /\ x e. ZZ ) -> ( -. P || ( x - 0 ) <-> -. P || x ) )
53	47, 52	bitr4d 191	. . . . . . . . . . . 12 |- ( ( ( P e. Prime /\ K e. NN ) /\ x e. ZZ ) -> ( ( x gcd ( P ^ K ) ) = 1 <-> -. P || ( x - 0 ) ) )
54	27, 53	sylan2 286	. . . . . . . . . . 11 |- ( ( ( P e. Prime /\ K e. NN ) /\ x e. ( 1 ... ( P ^ K ) ) ) -> ( ( x gcd ( P ^ K ) ) = 1 <-> -. P || ( x - 0 ) ) )
55	54	biimpd 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 ) ) )
57	55, 56	sylib 122	. . . . . . . . 9 |- ( ( ( P e. Prime /\ K e. NN ) /\ x e. ( 1 ... ( P ^ K ) ) ) -> -. ( ( x gcd ( P ^ K ) ) = 1 /\ P || ( x - 0 ) ) )
58	57	ralrimiva 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 ) ) )
60	58, 59	sylibr 134	. . . . . . 7 |- ( ( P e. Prime /\ K e. NN ) -> { x e. ( 1 ... ( P ^ K ) ) | ( ( x gcd ( P ^ K ) ) = 1 /\ P || ( x - 0 ) ) } = (/) )
61	35, 60	eqtrid 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 ) } ) ) )
63	20, 34, 61, 62	syl3anc 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 ) ) }
65	54	biimprd 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 ) ) ) )
67	32, 65, 66	sylc 62	. . . . . . . . . . 11 |- ( ( ( P e. Prime /\ K e. NN ) /\ x e. ( 1 ... ( P ^ K ) ) ) -> ( -. ( x gcd ( P ^ K ) ) = 1 -> P || ( x - 0 ) ) )
68	24	adantr 276	. . . . . . . . . . . . . . 15 |- ( ( ( P e. Prime /\ K e. NN ) /\ x e. ( 1 ... ( P ^ K ) ) ) -> ( P ^ K ) e. ZZ )
69	28, 68	gcdcld 12000	. . . . . . . . . . . . . 14 |- ( ( ( P e. Prime /\ K e. NN ) /\ x e. ( 1 ... ( P ^ K ) ) ) -> ( x gcd ( P ^ K ) ) e. NN0 )
70	69	nn0zd 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 )
73	70, 71, 72	syl2anc 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 ) ) ) )
75	73, 74	syl 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 ) ) ) )
76	67, 75	mpbird 167	. . . . . . . . . 10 |- ( ( ( P e. Prime /\ K e. NN ) /\ x e. ( 1 ... ( P ^ K ) ) ) -> ( ( x gcd ( P ^ K ) ) = 1 \/ P || ( x - 0 ) ) )
77	76	ralrimiva 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 ) ) )
79	77, 78	sylibr 134	. . . . . . . 8 |- ( ( P e. Prime /\ K e. NN ) -> ( 1 ... ( P ^ K ) ) = { x e. ( 1 ... ( P ^ K ) ) | ( ( x gcd ( P ^ K ) ) = 1 \/ P || ( x - 0 ) ) } )
80	64, 79	eqtr4id 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 ) ) )
81	80	fveq2d 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 ) ) ) )
82	4	nnnn0d 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 ) )
84	82, 83	syl 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 ) )
86	11, 85	sylan 283	. . . . . 6 |- ( ( P e. Prime /\ K e. NN ) -> ( P ^ K ) = ( ( P ^ ( K - 1 ) ) x. P ) )
87	81, 84, 86	3eqtrd 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 )
89	20, 88	syl 14	. . . . . . 7 |- ( ( P e. Prime /\ K e. NN ) -> (♯ ` { x e. ( 1 ... ( P ^ K ) ) | ( x gcd ( P ^ K ) ) = 1 } ) e. NN0 )
90	89	nn0cnd 9260	. . . . . 6 |- ( ( P e. Prime /\ K e. NN ) -> (♯ ` { x e. ( 1 ... ( P ^ K ) ) | ( x gcd ( P ^ K ) ) = 1 } ) e. CC )
91	1	adantr 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
94	93	fveq2i 5537	. . . . . . . . . . 11 |- ( ZZ>= ` ( 1 - 1 ) ) = ( ZZ>= ` 0 )
95	92, 94	eqtr4i 2213	. . . . . . . . . 10 |- NN0 = ( ZZ>= ` ( 1 - 1 ) )
96	82, 95	eleqtrdi 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 )
98	91, 21, 96, 97	hashdvds 12252	. . . . . . . 8 |- ( ( P e. Prime /\ K e. NN ) -> (♯ ` { x e. ( 1 ... ( P ^ K ) ) | P || ( x - 0 ) } ) = ( ( |_ ` ( ( ( P ^ K ) - 0 ) / P ) ) - ( |_ ` ( ( ( 1 - 1 ) - 0 ) / P ) ) ) )
99	4	nncnd 8962	. . . . . . . . . . . . . 14 |- ( ( P e. Prime /\ K e. NN ) -> ( P ^ K ) e. CC )
100	99	subid1d 8286	. . . . . . . . . . . . 13 |- ( ( P e. Prime /\ K e. NN ) -> ( ( P ^ K ) - 0 ) = ( P ^ K ) )
101	100	oveq1d 5910	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ K e. NN ) -> ( ( ( P ^ K ) - 0 ) / P ) = ( ( P ^ K ) / P ) )
102	91	nnap0d 8994	. . . . . . . . . . . . 13 |- ( ( P e. Prime /\ K e. NN ) -> P # 0 )
103		nnz 9301	. . . . . . . . . . . . . 14 |- ( K e. NN -> K e. ZZ )
104	103	adantl 277	. . . . . . . . . . . . 13 |- ( ( P e. Prime /\ K e. NN ) -> K e. ZZ )
105	12, 102, 104	expm1apd 10694	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ K e. NN ) -> ( P ^ ( K - 1 ) ) = ( ( P ^ K ) / P ) )
106	101, 105	eqtr4d 2225	. . . . . . . . . . 11 |- ( ( P e. Prime /\ K e. NN ) -> ( ( ( P ^ K ) - 0 ) / P ) = ( P ^ ( K - 1 ) ) )
107	106	fveq2d 5538	. . . . . . . . . 10 |- ( ( P e. Prime /\ K e. NN ) -> ( |_ ` ( ( ( P ^ K ) - 0 ) / P ) ) = ( |_ ` ( P ^ ( K - 1 ) ) ) )
108	9	nnzd 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 ) ) )
110	108, 109	syl 14	. . . . . . . . . 10 |- ( ( P e. Prime /\ K e. NN ) -> ( |_ ` ( P ^ ( K - 1 ) ) ) = ( P ^ ( K - 1 ) ) )
111	107, 110	eqtrd 2222	. . . . . . . . 9 |- ( ( P e. Prime /\ K e. NN ) -> ( |_ ` ( ( ( P ^ K ) - 0 ) / P ) ) = ( P ^ ( K - 1 ) ) )
112	93	oveq1i 5905	. . . . . . . . . . . . . 14 |- ( ( 1 - 1 ) - 0 ) = ( 0 - 0 )
113		0m0e0 9060	. . . . . . . . . . . . . 14 |- ( 0 - 0 ) = 0
114	112, 113	eqtri 2210	. . . . . . . . . . . . 13 |- ( ( 1 - 1 ) - 0 ) = 0
115	114	oveq1i 5905	. . . . . . . . . . . 12 |- ( ( ( 1 - 1 ) - 0 ) / P ) = ( 0 / P )
116	12, 102	div0apd 8773	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ K e. NN ) -> ( 0 / P ) = 0 )
117	115, 116	eqtrid 2234	. . . . . . . . . . 11 |- ( ( P e. Prime /\ K e. NN ) -> ( ( ( 1 - 1 ) - 0 ) / P ) = 0 )
118	117	fveq2d 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 )
121	119, 120	ax-mp 5	. . . . . . . . . 10 |- ( |_ ` 0 ) = 0
122	118, 121	eqtrdi 2238	. . . . . . . . 9 |- ( ( P e. Prime /\ K e. NN ) -> ( |_ ` ( ( ( 1 - 1 ) - 0 ) / P ) ) = 0 )
123	111, 122	oveq12d 5913	. . . . . . . 8 |- ( ( P e. Prime /\ K e. NN ) -> ( ( |_ ` ( ( ( P ^ K ) - 0 ) / P ) ) - ( |_ ` ( ( ( 1 - 1 ) - 0 ) / P ) ) ) = ( ( P ^ ( K - 1 ) ) - 0 ) )
124	10	subid1d 8286	. . . . . . . 8 |- ( ( P e. Prime /\ K e. NN ) -> ( ( P ^ ( K - 1 ) ) - 0 ) = ( P ^ ( K - 1 ) ) )
125	98, 123, 124	3eqtrd 2226	. . . . . . 7 |- ( ( P e. Prime /\ K e. NN ) -> (♯ ` { x e. ( 1 ... ( P ^ K ) ) | P || ( x - 0 ) } ) = ( P ^ ( K - 1 ) ) )
126	125	oveq2d 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 ) ) ) )
127	90, 10, 126	comraddd 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 } ) ) )
128	63, 87, 127	3eqtr3rd 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 ) )
129	10, 12	mulcld 8007	. . . . 5 |- ( ( P e. Prime /\ K e. NN ) -> ( ( P ^ ( K - 1 ) ) x. P ) e. CC )
130	129, 10, 90	subaddd 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 ) ) )
131	128, 130	mpbird 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 } ) )
132	16, 18, 131	3eqtrrd 2227	. 2 |- ( ( P e. Prime /\ K e. NN ) -> (♯ ` { x e. ( 1 ... ( P ^ K ) ) | ( x gcd ( P ^ K ) ) = 1 } ) = ( ( P ^ ( K - 1 ) ) x. ( P - 1 ) ) )
133	6, 132	eqtrd 2222	1 |- ( ( P e. Prime /\ K e. NN ) -> ( phi ` ( P ^ K ) ) = ( ( P ^ ( K - 1 ) ) x. ( P - 1 ) ) )

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