Theorem phiprmpw 10976

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 10870	. . . 4 |- ( P e. Prime -> P e. NN )
2		nnnn0 8570	. . . 4 |- ( K e. NN -> K e. NN0 )
3		nnexpcl 9803	. . . 4 |- ( ( P e. NN /\ K e. NN0 ) -> ( P ^ K ) e. NN )
4	1, 2, 3	syl2an 283	. . 3 |- ( ( P e. Prime /\ K e. NN ) -> ( P ^ K ) e. NN )
5		phival 10967	. . 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 8604	. . . . . 6 |- ( K e. NN -> ( K - 1 ) e. NN0 )
8		nnexpcl 9803	. . . . . 6 |- ( ( P e. NN /\ ( K - 1 ) e. NN0 ) -> ( P ^ ( K - 1 ) ) e. NN )
9	1, 7, 8	syl2an 283	. . . . 5 |- ( ( P e. Prime /\ K e. NN ) -> ( P ^ ( K - 1 ) ) e. NN )
10	9	nncnd 8328	. . . 4 |- ( ( P e. Prime /\ K e. NN ) -> ( P ^ ( K - 1 ) ) e. CC )
11	1	nncnd 8328	. . . . 5 |- ( P e. Prime -> P e. CC )
12	11	adantr 270	. . . 4 |- ( ( P e. Prime /\ K e. NN ) -> P e. CC )
13		ax-1cn 7339	. . . . 5 |- 1 e. CC
14		subdi 7764	. . . . 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 1258	. . . 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 403	. . 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	mulid1d 7406	. . . 4 |- ( ( P e. Prime /\ K e. NN ) -> ( ( P ^ ( K - 1 ) ) x. 1 ) = ( P ^ ( K - 1 ) ) )
18	17	oveq2d 5605	. . 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 10966	. . . . . . 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 8671	. . . . . . . 8 |- ( ( P e. Prime /\ K e. NN ) -> 1 e. ZZ )
22		prmz 10871	. . . . . . . . 9 |- ( P e. Prime -> P e. ZZ )
23		zexpcl 9805	. . . . . . . . 9 |- ( ( P e. ZZ /\ K e. NN0 ) -> ( P ^ K ) e. ZZ )
24	22, 2, 23	syl2an 283	. . . . . . . 8 |- ( ( P e. Prime /\ K e. NN ) -> ( P ^ K ) e. ZZ )
25	21, 24	fzfigd 9725	. . . . . . 7 |- ( ( P e. Prime /\ K e. NN ) -> ( 1 ... ( P ^ K ) ) e. Fin )
26	22	ad2antrr 472	. . . . . . . . 9 |- ( ( ( P e. Prime /\ K e. NN ) /\ x e. ( 1 ... ( P ^ K ) ) ) -> P e. ZZ )
27		elfzelz 9333	. . . . . . . . . . 11 |- ( x e. ( 1 ... ( P ^ K ) ) -> x e. ZZ )
28	27	adantl 271	. . . . . . . . . 10 |- ( ( ( P e. Prime /\ K e. NN ) /\ x e. ( 1 ... ( P ^ K ) ) ) -> x e. ZZ )
29		0zd 8656	. . . . . . . . . 10 |- ( ( ( P e. Prime /\ K e. NN ) /\ x e. ( 1 ... ( P ^ K ) ) ) -> 0 e. ZZ )
30	28, 29	zsubcld 8767	. . . . . . . . 9 |- ( ( ( P e. Prime /\ K e. NN ) /\ x e. ( 1 ... ( P ^ K ) ) ) -> ( x - 0 ) e. ZZ )
31		zdvdsdc 10595	. . . . . . . . 9 |- ( ( P e. ZZ /\ ( x - 0 ) e. ZZ ) -> DECID P || ( x - 0 ) )
32	26, 30, 31	syl2anc 403	. . . . . . . 8 |- ( ( ( P e. Prime /\ K e. NN ) /\ x e. ( 1 ... ( P ^ K ) ) ) -> DECID P || ( x - 0 ) )
33	32	ralrimiva 2440	. . . . . . 7 |- ( ( P e. Prime /\ K e. NN ) -> A. x e. ( 1 ... ( P ^ K ) )DECID P || ( x - 0 ) )
34	25, 33	ssfirab 6566	. . . . . 6 |- ( ( P e. Prime /\ K e. NN ) -> { x e. ( 1 ... ( P ^ K ) ) | P || ( x - 0 ) } e. Fin )
35		inrab 3254	. . . . . . 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 10910	. . . . . . . . . . . . . . . . 17 |- ( ( P e. ZZ /\ x e. ZZ /\ K e. NN ) -> ( ( ( P ^ K ) gcd x ) = 1 <-> ( P gcd x ) = 1 ) )
37	22, 36	syl3an1 1203	. . . . . . . . . . . . . . . 16 |- ( ( P e. Prime /\ x e. ZZ /\ K e. NN ) -> ( ( ( P ^ K ) gcd x ) = 1 <-> ( P gcd x ) = 1 ) )
38	37	3expa 1139	. . . . . . . . . . . . . . 15 |- ( ( ( P e. Prime /\ x e. ZZ ) /\ K e. NN ) -> ( ( ( P ^ K ) gcd x ) = 1 <-> ( P gcd x ) = 1 ) )
39	38	an32s 533	. . . . . . . . . . . . . 14 |- ( ( ( P e. Prime /\ K e. NN ) /\ x e. ZZ ) -> ( ( ( P ^ K ) gcd x ) = 1 <-> ( P gcd x ) = 1 ) )
40		simpr 108	. . . . . . . . . . . . . . . 16 |- ( ( ( P e. Prime /\ K e. NN ) /\ x e. ZZ ) -> x e. ZZ )
41	24	adantr 270	. . . . . . . . . . . . . . . 16 |- ( ( ( P e. Prime /\ K e. NN ) /\ x e. ZZ ) -> ( P ^ K ) e. ZZ )
42		gcdcom 10743	. . . . . . . . . . . . . . . 16 |- ( ( x e. ZZ /\ ( P ^ K ) e. ZZ ) -> ( x gcd ( P ^ K ) ) = ( ( P ^ K ) gcd x ) )
43	40, 41, 42	syl2anc 403	. . . . . . . . . . . . . . 15 |- ( ( ( P e. Prime /\ K e. NN ) /\ x e. ZZ ) -> ( x gcd ( P ^ K ) ) = ( ( P ^ K ) gcd x ) )
44	43	eqeq1d 2091	. . . . . . . . . . . . . 14 |- ( ( ( P e. Prime /\ K e. NN ) /\ x e. ZZ ) -> ( ( x gcd ( P ^ K ) ) = 1 <-> ( ( P ^ K ) gcd x ) = 1 ) )
45		coprm 10901	. . . . . . . . . . . . . . 15 |- ( ( P e. Prime /\ x e. ZZ ) -> ( -. P || x <-> ( P gcd x ) = 1 ) )
46	45	adantlr 461	. . . . . . . . . . . . . 14 |- ( ( ( P e. Prime /\ K e. NN ) /\ x e. ZZ ) -> ( -. P || x <-> ( P gcd x ) = 1 ) )
47	39, 44, 46	3bitr4d 218	. . . . . . . . . . . . 13 |- ( ( ( P e. Prime /\ K e. NN ) /\ x e. ZZ ) -> ( ( x gcd ( P ^ K ) ) = 1 <-> -. P || x ) )
48		zcn 8649	. . . . . . . . . . . . . . . . 17 |- ( x e. ZZ -> x e. CC )
49	48	adantl 271	. . . . . . . . . . . . . . . 16 |- ( ( ( P e. Prime /\ K e. NN ) /\ x e. ZZ ) -> x e. CC )
50	49	subid1d 7683	. . . . . . . . . . . . . . 15 |- ( ( ( P e. Prime /\ K e. NN ) /\ x e. ZZ ) -> ( x - 0 ) = x )
51	50	breq2d 3823	. . . . . . . . . . . . . 14 |- ( ( ( P e. Prime /\ K e. NN ) /\ x e. ZZ ) -> ( P || ( x - 0 ) <-> P || x ) )
52	51	notbid 625	. . . . . . . . . . . . 13 |- ( ( ( P e. Prime /\ K e. NN ) /\ x e. ZZ ) -> ( -. P || ( x - 0 ) <-> -. P || x ) )
53	47, 52	bitr4d 189	. . . . . . . . . . . 12 |- ( ( ( P e. Prime /\ K e. NN ) /\ x e. ZZ ) -> ( ( x gcd ( P ^ K ) ) = 1 <-> -. P || ( x - 0 ) ) )
54	27, 53	sylan2 280	. . . . . . . . . . 11 |- ( ( ( P e. Prime /\ K e. NN ) /\ x e. ( 1 ... ( P ^ K ) ) ) -> ( ( x gcd ( P ^ K ) ) = 1 <-> -. P || ( x - 0 ) ) )
55	54	biimpd 142	. . . . . . . . . 10 |- ( ( ( P e. Prime /\ K e. NN ) /\ x e. ( 1 ... ( P ^ K ) ) ) -> ( ( x gcd ( P ^ K ) ) = 1 -> -. P || ( x - 0 ) ) )
56		imnan 657	. . . . . . . . . 10 |- ( ( ( x gcd ( P ^ K ) ) = 1 -> -. P || ( x - 0 ) ) <-> -. ( ( x gcd ( P ^ K ) ) = 1 /\ P || ( x - 0 ) ) )
57	55, 56	sylib 120	. . . . . . . . 9 |- ( ( ( P e. Prime /\ K e. NN ) /\ x e. ( 1 ... ( P ^ K ) ) ) -> -. ( ( x gcd ( P ^ K ) ) = 1 /\ P || ( x - 0 ) ) )
58	57	ralrimiva 2440	. . . . . . . 8 |- ( ( P e. Prime /\ K e. NN ) -> A. x e. ( 1 ... ( P ^ K ) ) -. ( ( x gcd ( P ^ K ) ) = 1 /\ P || ( x - 0 ) ) )
59		rabeq0 3295	. . . . . . . 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 132	. . . . . . 7 |- ( ( P e. Prime /\ K e. NN ) -> { x e. ( 1 ... ( P ^ K ) ) | ( ( x gcd ( P ^ K ) ) = 1 /\ P || ( x - 0 ) ) } = (/) )
61	35, 60	syl5eq 2127	. . . . . 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 10046	. . . . . 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 1170	. . . . 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	54	biimprd 156	. . . . . . . . . . . 12 |- ( ( ( P e. Prime /\ K e. NN ) /\ x e. ( 1 ... ( P ^ K ) ) ) -> ( -. P || ( x - 0 ) -> ( x gcd ( P ^ K ) ) = 1 ) )
65		con1dc 787	. . . . . . . . . . . 12 |- (DECID P || ( x - 0 ) -> ( ( -. P || ( x - 0 ) -> ( x gcd ( P ^ K ) ) = 1 ) -> ( -. ( x gcd ( P ^ K ) ) = 1 -> P || ( x - 0 ) ) ) )
66	32, 64, 65	sylc 61	. . . . . . . . . . 11 |- ( ( ( P e. Prime /\ K e. NN ) /\ x e. ( 1 ... ( P ^ K ) ) ) -> ( -. ( x gcd ( P ^ K ) ) = 1 -> P || ( x - 0 ) ) )
67	24	adantr 270	. . . . . . . . . . . . . . 15 |- ( ( ( P e. Prime /\ K e. NN ) /\ x e. ( 1 ... ( P ^ K ) ) ) -> ( P ^ K ) e. ZZ )
68	28, 67	gcdcld 10738	. . . . . . . . . . . . . 14 |- ( ( ( P e. Prime /\ K e. NN ) /\ x e. ( 1 ... ( P ^ K ) ) ) -> ( x gcd ( P ^ K ) ) e. NN0 )
69	68	nn0zd 8760	. . . . . . . . . . . . 13 |- ( ( ( P e. Prime /\ K e. NN ) /\ x e. ( 1 ... ( P ^ K ) ) ) -> ( x gcd ( P ^ K ) ) e. ZZ )
70		1zzd 8671	. . . . . . . . . . . . 13 |- ( ( ( P e. Prime /\ K e. NN ) /\ x e. ( 1 ... ( P ^ K ) ) ) -> 1 e. ZZ )
71		zdceq 8716	. . . . . . . . . . . . 13 |- ( ( ( x gcd ( P ^ K ) ) e. ZZ /\ 1 e. ZZ ) -> DECID ( x gcd ( P ^ K ) ) = 1 )
72	69, 70, 71	syl2anc 403	. . . . . . . . . . . 12 |- ( ( ( P e. Prime /\ K e. NN ) /\ x e. ( 1 ... ( P ^ K ) ) ) -> DECID ( x gcd ( P ^ K ) ) = 1 )
73		dfordc 825	. . . . . . . . . . . 12 |- (DECID ( x gcd ( P ^ K ) ) = 1 -> ( ( ( x gcd ( P ^ K ) ) = 1 \/ P || ( x - 0 ) ) <-> ( -. ( x gcd ( P ^ K ) ) = 1 -> P || ( x - 0 ) ) ) )
74	72, 73	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 ) ) ) )
75	66, 74	mpbird 165	. . . . . . . . . 10 |- ( ( ( P e. Prime /\ K e. NN ) /\ x e. ( 1 ... ( P ^ K ) ) ) -> ( ( x gcd ( P ^ K ) ) = 1 \/ P || ( x - 0 ) ) )
76	75	ralrimiva 2440	. . . . . . . . 9 |- ( ( P e. Prime /\ K e. NN ) -> A. x e. ( 1 ... ( P ^ K ) ) ( ( x gcd ( P ^ K ) ) = 1 \/ P || ( x - 0 ) ) )
77		rabid2 2536	. . . . . . . . 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 ) ) )
78	76, 77	sylibr 132	. . . . . . . 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 3253	. . . . . . . 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 ) ) }
80	78, 79	syl6reqr 2134	. . . . . . 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 5255	. . . . . 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 8616	. . . . . . 7 |- ( ( P e. Prime /\ K e. NN ) -> ( P ^ K ) e. NN0 )
83		hashfz1 10024	. . . . . . 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 9818	. . . . . . 7 |- ( ( P e. CC /\ K e. NN ) -> ( P ^ K ) = ( ( P ^ ( K - 1 ) ) x. P ) )
86	11, 85	sylan 277	. . . . . 6 |- ( ( P e. Prime /\ K e. NN ) -> ( P ^ K ) = ( ( P ^ ( K - 1 ) ) x. P ) )
87	81, 84, 86	3eqtrd 2119	. . . . 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 10022	. . . . . . . 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 8618	. . . . . 6 |- ( ( P e. Prime /\ K e. NN ) -> (♯ ` { x e. ( 1 ... ( P ^ K ) ) | ( x gcd ( P ^ K ) ) = 1 } ) e. CC )
91	1	adantr 270	. . . . . . . . 9 |- ( ( P e. Prime /\ K e. NN ) -> P e. NN )
92		nn0uz 8946	. . . . . . . . . . 11 |- NN0 = ( ZZ>= ` 0 )
93		1m1e0 8383	. . . . . . . . . . . 12 |- ( 1 - 1 ) = 0
94	93	fveq2i 5254	. . . . . . . . . . 11 |- ( ZZ>= ` ( 1 - 1 ) ) = ( ZZ>= ` 0 )
95	92, 94	eqtr4i 2106	. . . . . . . . . 10 |- NN0 = ( ZZ>= ` ( 1 - 1 ) )
96	82, 95	syl6eleq 2175	. . . . . . . . 9 |- ( ( P e. Prime /\ K e. NN ) -> ( P ^ K ) e. ( ZZ>= ` ( 1 - 1 ) ) )
97		0zd 8656	. . . . . . . . 9 |- ( ( P e. Prime /\ K e. NN ) -> 0 e. ZZ )
98	91, 21, 96, 97	hashdvds 10975	. . . . . . . 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 8328	. . . . . . . . . . . . . 14 |- ( ( P e. Prime /\ K e. NN ) -> ( P ^ K ) e. CC )
100	99	subid1d 7683	. . . . . . . . . . . . 13 |- ( ( P e. Prime /\ K e. NN ) -> ( ( P ^ K ) - 0 ) = ( P ^ K ) )
101	100	oveq1d 5604	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ K e. NN ) -> ( ( ( P ^ K ) - 0 ) / P ) = ( ( P ^ K ) / P ) )
102	91	nnap0d 8359	. . . . . . . . . . . . 13 |- ( ( P e. Prime /\ K e. NN ) -> P # 0 )
103		nnz 8663	. . . . . . . . . . . . . 14 |- ( K e. NN -> K e. ZZ )
104	103	adantl 271	. . . . . . . . . . . . 13 |- ( ( P e. Prime /\ K e. NN ) -> K e. ZZ )
105	12, 102, 104	expm1apd 9929	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ K e. NN ) -> ( P ^ ( K - 1 ) ) = ( ( P ^ K ) / P ) )
106	101, 105	eqtr4d 2118	. . . . . . . . . . 11 |- ( ( P e. Prime /\ K e. NN ) -> ( ( ( P ^ K ) - 0 ) / P ) = ( P ^ ( K - 1 ) ) )
107	106	fveq2d 5255	. . . . . . . . . 10 |- ( ( P e. Prime /\ K e. NN ) -> ( |_ ` ( ( ( P ^ K ) - 0 ) / P ) ) = ( |_ ` ( P ^ ( K - 1 ) ) ) )
108	9	nnzd 8761	. . . . . . . . . . 11 |- ( ( P e. Prime /\ K e. NN ) -> ( P ^ ( K - 1 ) ) e. ZZ )
109		flid 9578	. . . . . . . . . . 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 2115	. . . . . . . . 9 |- ( ( P e. Prime /\ K e. NN ) -> ( |_ ` ( ( ( P ^ K ) - 0 ) / P ) ) = ( P ^ ( K - 1 ) ) )
112	93	oveq1i 5599	. . . . . . . . . . . . . 14 |- ( ( 1 - 1 ) - 0 ) = ( 0 - 0 )
113		0m0e0 8426	. . . . . . . . . . . . . 14 |- ( 0 - 0 ) = 0
114	112, 113	eqtri 2103	. . . . . . . . . . . . 13 |- ( ( 1 - 1 ) - 0 ) = 0
115	114	oveq1i 5599	. . . . . . . . . . . 12 |- ( ( ( 1 - 1 ) - 0 ) / P ) = ( 0 / P )
116	12, 102	div0apd 8150	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ K e. NN ) -> ( 0 / P ) = 0 )
117	115, 116	syl5eq 2127	. . . . . . . . . . 11 |- ( ( P e. Prime /\ K e. NN ) -> ( ( ( 1 - 1 ) - 0 ) / P ) = 0 )
118	117	fveq2d 5255	. . . . . . . . . 10 |- ( ( P e. Prime /\ K e. NN ) -> ( |_ ` ( ( ( 1 - 1 ) - 0 ) / P ) ) = ( |_ ` 0 ) )
119		0z 8655	. . . . . . . . . . 11 |- 0 e. ZZ
120		flid 9578	. . . . . . . . . . 11 |- ( 0 e. ZZ -> ( |_ ` 0 ) = 0 )
121	119, 120	ax-mp 7	. . . . . . . . . 10 |- ( |_ ` 0 ) = 0
122	118, 121	syl6eq 2131	. . . . . . . . 9 |- ( ( P e. Prime /\ K e. NN ) -> ( |_ ` ( ( ( 1 - 1 ) - 0 ) / P ) ) = 0 )
123	111, 122	oveq12d 5607	. . . . . . . 8 |- ( ( P e. Prime /\ K e. NN ) -> ( ( |_ ` ( ( ( P ^ K ) - 0 ) / P ) ) - ( |_ ` ( ( ( 1 - 1 ) - 0 ) / P ) ) ) = ( ( P ^ ( K - 1 ) ) - 0 ) )
124	10	subid1d 7683	. . . . . . . 8 |- ( ( P e. Prime /\ K e. NN ) -> ( ( P ^ ( K - 1 ) ) - 0 ) = ( P ^ ( K - 1 ) ) )
125	98, 123, 124	3eqtrd 2119	. . . . . . 7 |- ( ( P e. Prime /\ K e. NN ) -> (♯ ` { x e. ( 1 ... ( P ^ K ) ) | P || ( x - 0 ) } ) = ( P ^ ( K - 1 ) ) )
126	125	oveq2d 5605	. . . . . 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 7540	. . . . 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 2124	. . . 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 7409	. . . . 5 |- ( ( P e. Prime /\ K e. NN ) -> ( ( P ^ ( K - 1 ) ) x. P ) e. CC )
130	129, 10, 90	subaddd 7712	. . . 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 165	. . 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 2120	. 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 2115	1 |- ( ( P e. Prime /\ K e. NN ) -> ( phi ` ( P ^ K ) ) = ( ( P ^ ( K - 1 ) ) x. ( P - 1 ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   <-> wb 103   \/ wo 662  DECID wdc 776   = wceq 1285   e. wcel 1434   A.wral 2353   {crab 2357   u. cun 2982   i^i cin 2983   (/)c0 3269   class class class wbr 3811   ` cfv 4967  (class class class)co 5589   Fincfn 6385   CCcc 7249   0cc0 7251   1c1 7252   + caddc 7254   x. cmul 7256   - cmin 7554   / cdiv 8035   NNcn 8314   NN0cn0 8563   ZZcz 8644   ZZ>=cuz 8912   ...cfz 9317   |_cfl 9562   ^cexp 9789  ♯chash 10016   || cdvds 10574   gcd cgcd 10716   Primecprime 10867   phicphi 10964

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3919  ax-sep 3922  ax-nul 3930  ax-pow 3974  ax-pr 3999  ax-un 4223  ax-setind 4315  ax-iinf 4365  ax-cnex 7337  ax-resscn 7338  ax-1cn 7339  ax-1re 7340  ax-icn 7341  ax-addcl 7342  ax-addrcl 7343  ax-mulcl 7344  ax-mulrcl 7345  ax-addcom 7346  ax-mulcom 7347  ax-addass 7348  ax-mulass 7349  ax-distr 7350  ax-i2m1 7351  ax-0lt1 7352  ax-1rid 7353  ax-0id 7354  ax-rnegex 7355  ax-precex 7356  ax-cnre 7357  ax-pre-ltirr 7358  ax-pre-ltwlin 7359  ax-pre-lttrn 7360  ax-pre-apti 7361  ax-pre-ltadd 7362  ax-pre-mulgt0 7363  ax-pre-mulext 7364  ax-arch 7365  ax-caucvg 7366

This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rmo 2361  df-rab 2362  df-v 2614  df-sbc 2827  df-csb 2920  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-if 3374  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-int 3663  df-iun 3706  df-br 3812  df-opab 3866  df-mpt 3867  df-tr 3902  df-id 4083  df-po 4086  df-iso 4087  df-iord 4156  df-on 4158  df-ilim 4159  df-suc 4161  df-iom 4368  df-xp 4405  df-rel 4406  df-cnv 4407  df-co 4408  df-dm 4409  df-rn 4410  df-res 4411  df-ima 4412  df-iota 4932  df-fun 4969  df-fn 4970  df-f 4971  df-f1 4972  df-fo 4973  df-f1o 4974  df-fv 4975  df-riota 5545  df-ov 5592  df-oprab 5593  df-mpt2 5594  df-1st 5844  df-2nd 5845  df-recs 6000  df-irdg 6065  df-frec 6086  df-1o 6111  df-2o 6112  df-oadd 6115  df-er 6220  df-en 6386  df-dom 6387  df-fin 6388  df-sup 6584  df-pnf 7425  df-mnf 7426  df-xr 7427  df-ltxr 7428  df-le 7429  df-sub 7556  df-neg 7557  df-reap 7950  df-ap 7957  df-div 8036  df-inn 8315  df-2 8373  df-3 8374  df-4 8375  df-n0 8564  df-z 8645  df-uz 8913  df-q 8998  df-rp 9028  df-fz 9318  df-fzo 9442  df-fl 9564  df-mod 9617  df-iseq 9739  df-iexp 9790  df-ihash 10017  df-cj 10101  df-re 10102  df-im 10103  df-rsqrt 10256  df-abs 10257  df-dvds 10575  df-gcd 10717  df-prm 10868  df-phi 10965

This theorem is referenced by:  phiprm  10977