Theorem phiprmpw 12739

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 12627	. . . 4 |- ( P e. Prime -> P e. NN )
2		nnnn0 9372	. . . 4 |- ( K e. NN -> K e. NN0 )
3		nnexpcl 10769	. . . 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 12730	. . 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 9406	. . . . . 6 |- ( K e. NN -> ( K - 1 ) e. NN0 )
8		nnexpcl 10769	. . . . . 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 9120	. . . 4 |- ( ( P e. Prime /\ K e. NN ) -> ( P ^ ( K - 1 ) ) e. CC )
11	1	nncnd 9120	. . . . 5 |- ( P e. Prime -> P e. CC )
12	11	adantr 276	. . . 4 |- ( ( P e. Prime /\ K e. NN ) -> P e. CC )
13		ax-1cn 8088	. . . . 5 |- 1 e. CC
14		subdi 8527	. . . . 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 1360	. . . 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 8159	. . . 4 |- ( ( P e. Prime /\ K e. NN ) -> ( ( P ^ ( K - 1 ) ) x. 1 ) = ( P ^ ( K - 1 ) ) )
18	17	oveq2d 6016	. . 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 12729	. . . . . . 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 9469	. . . . . . . 8 |- ( ( P e. Prime /\ K e. NN ) -> 1 e. ZZ )
22		prmz 12628	. . . . . . . . 9 |- ( P e. Prime -> P e. ZZ )
23		zexpcl 10771	. . . . . . . . 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 10648	. . . . . . 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 10217	. . . . . . . . . . 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 9454	. . . . . . . . . 10 |- ( ( ( P e. Prime /\ K e. NN ) /\ x e. ( 1 ... ( P ^ K ) ) ) -> 0 e. ZZ )
30	28, 29	zsubcld 9570	. . . . . . . . 9 |- ( ( ( P e. Prime /\ K e. NN ) /\ x e. ( 1 ... ( P ^ K ) ) ) -> ( x - 0 ) e. ZZ )
31		zdvdsdc 12318	. . . . . . . . 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 2603	. . . . . . 7 |- ( ( P e. Prime /\ K e. NN ) -> A. x e. ( 1 ... ( P ^ K ) )DECID P || ( x - 0 ) )
34	25, 33	ssfirab 7094	. . . . . 6 |- ( ( P e. Prime /\ K e. NN ) -> { x e. ( 1 ... ( P ^ K ) ) | P || ( x - 0 ) } e. Fin )
35		inrab 3476	. . . . . . 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 12670	. . . . . . . . . . . . . . . . 17 |- ( ( P e. ZZ /\ x e. ZZ /\ K e. NN ) -> ( ( ( P ^ K ) gcd x ) = 1 <-> ( P gcd x ) = 1 ) )
37	22, 36	syl3an1 1304	. . . . . . . . . . . . . . . 16 |- ( ( P e. Prime /\ x e. ZZ /\ K e. NN ) -> ( ( ( P ^ K ) gcd x ) = 1 <-> ( P gcd x ) = 1 ) )
38	37	3expa 1227	. . . . . . . . . . . . . . 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 12489	. . . . . . . . . . . . . . . 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 2238	. . . . . . . . . . . . . 14 |- ( ( ( P e. Prime /\ K e. NN ) /\ x e. ZZ ) -> ( ( x gcd ( P ^ K ) ) = 1 <-> ( ( P ^ K ) gcd x ) = 1 ) )
45		coprm 12661	. . . . . . . . . . . . . . 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 9447	. . . . . . . . . . . . . . . . 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 8442	. . . . . . . . . . . . . . 15 |- ( ( ( P e. Prime /\ K e. NN ) /\ x e. ZZ ) -> ( x - 0 ) = x )
51	50	breq2d 4094	. . . . . . . . . . . . . 14 |- ( ( ( P e. Prime /\ K e. NN ) /\ x e. ZZ ) -> ( P || ( x - 0 ) <-> P || x ) )
52	51	notbid 671	. . . . . . . . . . . . 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 694	. . . . . . . . . 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 2603	. . . . . . . 8 |- ( ( P e. Prime /\ K e. NN ) -> A. x e. ( 1 ... ( P ^ K ) ) -. ( ( x gcd ( P ^ K ) ) = 1 /\ P || ( x - 0 ) ) )
59		rabeq0 3521	. . . . . . . 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 2274	. . . . . 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 11022	. . . . . 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 1271	. . . . 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 3475	. . . . . . . 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 861	. . . . . . . . . . . 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 12484	. . . . . . . . . . . . . 14 |- ( ( ( P e. Prime /\ K e. NN ) /\ x e. ( 1 ... ( P ^ K ) ) ) -> ( x gcd ( P ^ K ) ) e. NN0 )
70	69	nn0zd 9563	. . . . . . . . . . . . 13 |- ( ( ( P e. Prime /\ K e. NN ) /\ x e. ( 1 ... ( P ^ K ) ) ) -> ( x gcd ( P ^ K ) ) e. ZZ )
71		1zzd 9469	. . . . . . . . . . . . 13 |- ( ( ( P e. Prime /\ K e. NN ) /\ x e. ( 1 ... ( P ^ K ) ) ) -> 1 e. ZZ )
72		zdceq 9518	. . . . . . . . . . . . 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 897	. . . . . . . . . . . 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 2603	. . . . . . . . 9 |- ( ( P e. Prime /\ K e. NN ) -> A. x e. ( 1 ... ( P ^ K ) ) ( ( x gcd ( P ^ K ) ) = 1 \/ P || ( x - 0 ) ) )
78		rabid2 2708	. . . . . . . . 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 2281	. . . . . . 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 5630	. . . . . 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 9418	. . . . . . 7 |- ( ( P e. Prime /\ K e. NN ) -> ( P ^ K ) e. NN0 )
83		hashfz1 11000	. . . . . . 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 10784	. . . . . . 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 2266	. . . . 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 10998	. . . . . . . 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 9420	. . . . . 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 9753	. . . . . . . . . . 11 |- NN0 = ( ZZ>= ` 0 )
93		1m1e0 9175	. . . . . . . . . . . 12 |- ( 1 - 1 ) = 0
94	93	fveq2i 5629	. . . . . . . . . . 11 |- ( ZZ>= ` ( 1 - 1 ) ) = ( ZZ>= ` 0 )
95	92, 94	eqtr4i 2253	. . . . . . . . . 10 |- NN0 = ( ZZ>= ` ( 1 - 1 ) )
96	82, 95	eleqtrdi 2322	. . . . . . . . 9 |- ( ( P e. Prime /\ K e. NN ) -> ( P ^ K ) e. ( ZZ>= ` ( 1 - 1 ) ) )
97		0zd 9454	. . . . . . . . 9 |- ( ( P e. Prime /\ K e. NN ) -> 0 e. ZZ )
98	91, 21, 96, 97	hashdvds 12738	. . . . . . . 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 9120	. . . . . . . . . . . . . 14 |- ( ( P e. Prime /\ K e. NN ) -> ( P ^ K ) e. CC )
100	99	subid1d 8442	. . . . . . . . . . . . 13 |- ( ( P e. Prime /\ K e. NN ) -> ( ( P ^ K ) - 0 ) = ( P ^ K ) )
101	100	oveq1d 6015	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ K e. NN ) -> ( ( ( P ^ K ) - 0 ) / P ) = ( ( P ^ K ) / P ) )
102	91	nnap0d 9152	. . . . . . . . . . . . 13 |- ( ( P e. Prime /\ K e. NN ) -> P # 0 )
103		nnz 9461	. . . . . . . . . . . . . 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 10900	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ K e. NN ) -> ( P ^ ( K - 1 ) ) = ( ( P ^ K ) / P ) )
106	101, 105	eqtr4d 2265	. . . . . . . . . . 11 |- ( ( P e. Prime /\ K e. NN ) -> ( ( ( P ^ K ) - 0 ) / P ) = ( P ^ ( K - 1 ) ) )
107	106	fveq2d 5630	. . . . . . . . . 10 |- ( ( P e. Prime /\ K e. NN ) -> ( |_ ` ( ( ( P ^ K ) - 0 ) / P ) ) = ( |_ ` ( P ^ ( K - 1 ) ) ) )
108	9	nnzd 9564	. . . . . . . . . . 11 |- ( ( P e. Prime /\ K e. NN ) -> ( P ^ ( K - 1 ) ) e. ZZ )
109		flid 10499	. . . . . . . . . . 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 2262	. . . . . . . . 9 |- ( ( P e. Prime /\ K e. NN ) -> ( |_ ` ( ( ( P ^ K ) - 0 ) / P ) ) = ( P ^ ( K - 1 ) ) )
112	93	oveq1i 6010	. . . . . . . . . . . . . 14 |- ( ( 1 - 1 ) - 0 ) = ( 0 - 0 )
113		0m0e0 9218	. . . . . . . . . . . . . 14 |- ( 0 - 0 ) = 0
114	112, 113	eqtri 2250	. . . . . . . . . . . . 13 |- ( ( 1 - 1 ) - 0 ) = 0
115	114	oveq1i 6010	. . . . . . . . . . . 12 |- ( ( ( 1 - 1 ) - 0 ) / P ) = ( 0 / P )
116	12, 102	div0apd 8930	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ K e. NN ) -> ( 0 / P ) = 0 )
117	115, 116	eqtrid 2274	. . . . . . . . . . 11 |- ( ( P e. Prime /\ K e. NN ) -> ( ( ( 1 - 1 ) - 0 ) / P ) = 0 )
118	117	fveq2d 5630	. . . . . . . . . 10 |- ( ( P e. Prime /\ K e. NN ) -> ( |_ ` ( ( ( 1 - 1 ) - 0 ) / P ) ) = ( |_ ` 0 ) )
119		0z 9453	. . . . . . . . . . 11 |- 0 e. ZZ
120		flid 10499	. . . . . . . . . . 11 |- ( 0 e. ZZ -> ( |_ ` 0 ) = 0 )
121	119, 120	ax-mp 5	. . . . . . . . . 10 |- ( |_ ` 0 ) = 0
122	118, 121	eqtrdi 2278	. . . . . . . . 9 |- ( ( P e. Prime /\ K e. NN ) -> ( |_ ` ( ( ( 1 - 1 ) - 0 ) / P ) ) = 0 )
123	111, 122	oveq12d 6018	. . . . . . . 8 |- ( ( P e. Prime /\ K e. NN ) -> ( ( |_ ` ( ( ( P ^ K ) - 0 ) / P ) ) - ( |_ ` ( ( ( 1 - 1 ) - 0 ) / P ) ) ) = ( ( P ^ ( K - 1 ) ) - 0 ) )
124	10	subid1d 8442	. . . . . . . 8 |- ( ( P e. Prime /\ K e. NN ) -> ( ( P ^ ( K - 1 ) ) - 0 ) = ( P ^ ( K - 1 ) ) )
125	98, 123, 124	3eqtrd 2266	. . . . . . 7 |- ( ( P e. Prime /\ K e. NN ) -> (♯ ` { x e. ( 1 ... ( P ^ K ) ) | P || ( x - 0 ) } ) = ( P ^ ( K - 1 ) ) )
126	125	oveq2d 6016	. . . . . 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 8299	. . . . 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 2271	. . . 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 8163	. . . . 5 |- ( ( P e. Prime /\ K e. NN ) -> ( ( P ^ ( K - 1 ) ) x. P ) e. CC )
130	129, 10, 90	subaddd 8471	. . . 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 2267	. 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 2262	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 713  DECID wdc 839   = wceq 1395   e. wcel 2200   A.wral 2508   {crab 2512   u. cun 3195   i^i cin 3196   (/)c0 3491   class class class wbr 4082   ` cfv 5317  (class class class)co 6000   Fincfn 6885   CCcc 7993   0cc0 7995   1c1 7996   + caddc 7998   x. cmul 8000   - cmin 8313   / cdiv 8815   NNcn 9106   NN0cn0 9365   ZZcz 9442   ZZ>=cuz 9718   ...cfz 10200   |_cfl 10483   ^cexp 10755  ♯chash 10992   || cdvds 12293   gcd cgcd 12469   Primecprime 12624   phicphi 12726

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-mulrcl 8094  ax-addcom 8095  ax-mulcom 8096  ax-addass 8097  ax-mulass 8098  ax-distr 8099  ax-i2m1 8100  ax-0lt1 8101  ax-1rid 8102  ax-0id 8103  ax-rnegex 8104  ax-precex 8105  ax-cnre 8106  ax-pre-ltirr 8107  ax-pre-ltwlin 8108  ax-pre-lttrn 8109  ax-pre-apti 8110  ax-pre-ltadd 8111  ax-pre-mulgt0 8112  ax-pre-mulext 8113  ax-arch 8114  ax-caucvg 8115

This theorem depends on definitions:  df-bi 117  df-stab 836  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4383  df-po 4386  df-iso 4387  df-iord 4456  df-on 4458  df-ilim 4459  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-recs 6449  df-irdg 6514  df-frec 6535  df-1o 6560  df-2o 6561  df-oadd 6564  df-er 6678  df-en 6886  df-dom 6887  df-fin 6888  df-sup 7147  df-pnf 8179  df-mnf 8180  df-xr 8181  df-ltxr 8182  df-le 8183  df-sub 8315  df-neg 8316  df-reap 8718  df-ap 8725  df-div 8816  df-inn 9107  df-2 9165  df-3 9166  df-4 9167  df-n0 9366  df-z 9443  df-uz 9719  df-q 9811  df-rp 9846  df-fz 10201  df-fzo 10335  df-fl 10485  df-mod 10540  df-seqfrec 10665  df-exp 10756  df-ihash 10993  df-cj 11348  df-re 11349  df-im 11350  df-rsqrt 11504  df-abs 11505  df-dvds 12294  df-gcd 12470  df-prm 12625  df-phi 12728

This theorem is referenced by:  phiprm  12740