Theorem phiprmpw 15481

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 15388	. . . 4 |- ( P e. Prime -> P e. NN )
2		nnnn0 11299	. . . 4 |- ( K e. NN -> K e. NN0 )
3		nnexpcl 12873	. . . 4 |- ( ( P e. NN /\ K e. NN0 ) -> ( P ^ K ) e. NN )
4	1, 2, 3	syl2an 494	. . 3 |- ( ( P e. Prime /\ K e. NN ) -> ( P ^ K ) e. NN )
5		phival 15472	. . 3 |- ( ( P ^ K ) e. NN -> ( phi ` ( P ^ K ) ) = ( # ` { x e. ( 1 ... ( P ^ K ) ) | ( x gcd ( P ^ K ) ) = 1 } ) )
6	4, 5	syl 17	. 2 |- ( ( P e. Prime /\ K e. NN ) -> ( phi ` ( P ^ K ) ) = ( # ` { x e. ( 1 ... ( P ^ K ) ) | ( x gcd ( P ^ K ) ) = 1 } ) )
7		nnm1nn0 11334	. . . . . 6 |- ( K e. NN -> ( K - 1 ) e. NN0 )
8		nnexpcl 12873	. . . . . 6 |- ( ( P e. NN /\ ( K - 1 ) e. NN0 ) -> ( P ^ ( K - 1 ) ) e. NN )
9	1, 7, 8	syl2an 494	. . . . 5 |- ( ( P e. Prime /\ K e. NN ) -> ( P ^ ( K - 1 ) ) e. NN )
10	9	nncnd 11036	. . . 4 |- ( ( P e. Prime /\ K e. NN ) -> ( P ^ ( K - 1 ) ) e. CC )
11	1	nncnd 11036	. . . . 5 |- ( P e. Prime -> P e. CC )
12	11	adantr 481	. . . 4 |- ( ( P e. Prime /\ K e. NN ) -> P e. CC )
13		ax-1cn 9994	. . . . 5 |- 1 e. CC
14		subdi 10463	. . . . 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 1413	. . . 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 693	. . 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 10057	. . . 4 |- ( ( P e. Prime /\ K e. NN ) -> ( ( P ^ ( K - 1 ) ) x. 1 ) = ( P ^ ( K - 1 ) ) )
18	17	oveq2d 6666	. . 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 3899	. . . . . . 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 12342	. . . . . . . . . . . 12 |- ( x e. ( 1 ... ( P ^ K ) ) -> x e. ZZ )
21		prmz 15389	. . . . . . . . . . . . . . . . 17 |- ( P e. Prime -> P e. ZZ )
22		rpexp 15432	. . . . . . . . . . . . . . . . 17 |- ( ( P e. ZZ /\ x e. ZZ /\ K e. NN ) -> ( ( ( P ^ K ) gcd x ) = 1 <-> ( P gcd x ) = 1 ) )
23	21, 22	syl3an1 1359	. . . . . . . . . . . . . . . 16 |- ( ( P e. Prime /\ x e. ZZ /\ K e. NN ) -> ( ( ( P ^ K ) gcd x ) = 1 <-> ( P gcd x ) = 1 ) )
24	23	3expa 1265	. . . . . . . . . . . . . . 15 |- ( ( ( P e. Prime /\ x e. ZZ ) /\ K e. NN ) -> ( ( ( P ^ K ) gcd x ) = 1 <-> ( P gcd x ) = 1 ) )
25	24	an32s 846	. . . . . . . . . . . . . 14 |- ( ( ( P e. Prime /\ K e. NN ) /\ x e. ZZ ) -> ( ( ( P ^ K ) gcd x ) = 1 <-> ( P gcd x ) = 1 ) )
26		simpr 477	. . . . . . . . . . . . . . . 16 |- ( ( ( P e. Prime /\ K e. NN ) /\ x e. ZZ ) -> x e. ZZ )
27		zexpcl 12875	. . . . . . . . . . . . . . . . . 18 |- ( ( P e. ZZ /\ K e. NN0 ) -> ( P ^ K ) e. ZZ )
28	21, 2, 27	syl2an 494	. . . . . . . . . . . . . . . . 17 |- ( ( P e. Prime /\ K e. NN ) -> ( P ^ K ) e. ZZ )
29	28	adantr 481	. . . . . . . . . . . . . . . 16 |- ( ( ( P e. Prime /\ K e. NN ) /\ x e. ZZ ) -> ( P ^ K ) e. ZZ )
30		gcdcom 15235	. . . . . . . . . . . . . . . 16 |- ( ( x e. ZZ /\ ( P ^ K ) e. ZZ ) -> ( x gcd ( P ^ K ) ) = ( ( P ^ K ) gcd x ) )
31	26, 29, 30	syl2anc 693	. . . . . . . . . . . . . . 15 |- ( ( ( P e. Prime /\ K e. NN ) /\ x e. ZZ ) -> ( x gcd ( P ^ K ) ) = ( ( P ^ K ) gcd x ) )
32	31	eqeq1d 2624	. . . . . . . . . . . . . 14 |- ( ( ( P e. Prime /\ K e. NN ) /\ x e. ZZ ) -> ( ( x gcd ( P ^ K ) ) = 1 <-> ( ( P ^ K ) gcd x ) = 1 ) )
33		coprm 15423	. . . . . . . . . . . . . . 15 |- ( ( P e. Prime /\ x e. ZZ ) -> ( -. P || x <-> ( P gcd x ) = 1 ) )
34	33	adantlr 751	. . . . . . . . . . . . . 14 |- ( ( ( P e. Prime /\ K e. NN ) /\ x e. ZZ ) -> ( -. P || x <-> ( P gcd x ) = 1 ) )
35	25, 32, 34	3bitr4d 300	. . . . . . . . . . . . 13 |- ( ( ( P e. Prime /\ K e. NN ) /\ x e. ZZ ) -> ( ( x gcd ( P ^ K ) ) = 1 <-> -. P || x ) )
36		zcn 11382	. . . . . . . . . . . . . . . . 17 |- ( x e. ZZ -> x e. CC )
37	36	adantl 482	. . . . . . . . . . . . . . . 16 |- ( ( ( P e. Prime /\ K e. NN ) /\ x e. ZZ ) -> x e. CC )
38	37	subid1d 10381	. . . . . . . . . . . . . . 15 |- ( ( ( P e. Prime /\ K e. NN ) /\ x e. ZZ ) -> ( x - 0 ) = x )
39	38	breq2d 4665	. . . . . . . . . . . . . 14 |- ( ( ( P e. Prime /\ K e. NN ) /\ x e. ZZ ) -> ( P || ( x - 0 ) <-> P || x ) )
40	39	notbid 308	. . . . . . . . . . . . 13 |- ( ( ( P e. Prime /\ K e. NN ) /\ x e. ZZ ) -> ( -. P || ( x - 0 ) <-> -. P || x ) )
41	35, 40	bitr4d 271	. . . . . . . . . . . 12 |- ( ( ( P e. Prime /\ K e. NN ) /\ x e. ZZ ) -> ( ( x gcd ( P ^ K ) ) = 1 <-> -. P || ( x - 0 ) ) )
42	20, 41	sylan2 491	. . . . . . . . . . 11 |- ( ( ( P e. Prime /\ K e. NN ) /\ x e. ( 1 ... ( P ^ K ) ) ) -> ( ( x gcd ( P ^ K ) ) = 1 <-> -. P || ( x - 0 ) ) )
43	42	biimpd 219	. . . . . . . . . 10 |- ( ( ( P e. Prime /\ K e. NN ) /\ x e. ( 1 ... ( P ^ K ) ) ) -> ( ( x gcd ( P ^ K ) ) = 1 -> -. P || ( x - 0 ) ) )
44		imnan 438	. . . . . . . . . 10 |- ( ( ( x gcd ( P ^ K ) ) = 1 -> -. P || ( x - 0 ) ) <-> -. ( ( x gcd ( P ^ K ) ) = 1 /\ P || ( x - 0 ) ) )
45	43, 44	sylib 208	. . . . . . . . 9 |- ( ( ( P e. Prime /\ K e. NN ) /\ x e. ( 1 ... ( P ^ K ) ) ) -> -. ( ( x gcd ( P ^ K ) ) = 1 /\ P || ( x - 0 ) ) )
46	45	ralrimiva 2966	. . . . . . . 8 |- ( ( P e. Prime /\ K e. NN ) -> A. x e. ( 1 ... ( P ^ K ) ) -. ( ( x gcd ( P ^ K ) ) = 1 /\ P || ( x - 0 ) ) )
47		rabeq0 3957	. . . . . . . 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 ) ) )
48	46, 47	sylibr 224	. . . . . . 7 |- ( ( P e. Prime /\ K e. NN ) -> { x e. ( 1 ... ( P ^ K ) ) | ( ( x gcd ( P ^ K ) ) = 1 /\ P || ( x - 0 ) ) } = (/) )
49	19, 48	syl5eq 2668	. . . . . 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 12771	. . . . . . . 8 |- ( 1 ... ( P ^ K ) ) e. Fin
51		ssrab2 3687	. . . . . . . 8 |- { x e. ( 1 ... ( P ^ K ) ) | ( x gcd ( P ^ K ) ) = 1 } C_ ( 1 ... ( P ^ K ) )
52		ssfi 8180	. . . . . . . 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 )
53	50, 51, 52	mp2an 708	. . . . . . 7 |- { x e. ( 1 ... ( P ^ K ) ) | ( x gcd ( P ^ K ) ) = 1 } e. Fin
54		ssrab2 3687	. . . . . . . 8 |- { x e. ( 1 ... ( P ^ K ) ) | P || ( x - 0 ) } C_ ( 1 ... ( P ^ K ) )
55		ssfi 8180	. . . . . . . 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 )
56	50, 54, 55	mp2an 708	. . . . . . 7 |- { x e. ( 1 ... ( P ^ K ) ) | P || ( x - 0 ) } e. Fin
57		hashun 13171	. . . . . . 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 ) } ) ) )
58	53, 56, 57	mp3an12 1414	. . . . . 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 ) } ) ) )
59	49, 58	syl 17	. . . . 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 ) } ) ) )
60	42	biimprd 238	. . . . . . . . . . . 12 |- ( ( ( P e. Prime /\ K e. NN ) /\ x e. ( 1 ... ( P ^ K ) ) ) -> ( -. P || ( x - 0 ) -> ( x gcd ( P ^ K ) ) = 1 ) )
61	60	con1d 139	. . . . . . . . . . 11 |- ( ( ( P e. Prime /\ K e. NN ) /\ x e. ( 1 ... ( P ^ K ) ) ) -> ( -. ( x gcd ( P ^ K ) ) = 1 -> P || ( x - 0 ) ) )
62	61	orrd 393	. . . . . . . . . 10 |- ( ( ( P e. Prime /\ K e. NN ) /\ x e. ( 1 ... ( P ^ K ) ) ) -> ( ( x gcd ( P ^ K ) ) = 1 \/ P || ( x - 0 ) ) )
63	62	ralrimiva 2966	. . . . . . . . 9 |- ( ( P e. Prime /\ K e. NN ) -> A. x e. ( 1 ... ( P ^ K ) ) ( ( x gcd ( P ^ K ) ) = 1 \/ P || ( x - 0 ) ) )
64		rabid2 3118	. . . . . . . . 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 ) ) )
65	63, 64	sylibr 224	. . . . . . . 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 3898	. . . . . . . 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 ) ) }
67	65, 66	syl6reqr 2675	. . . . . . 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 ) ) )
68	67	fveq2d 6195	. . . . . 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 ) ) ) )
69	4	nnnn0d 11351	. . . . . . 7 |- ( ( P e. Prime /\ K e. NN ) -> ( P ^ K ) e. NN0 )
70		hashfz1 13134	. . . . . . 7 |- ( ( P ^ K ) e. NN0 -> ( # ` ( 1 ... ( P ^ K ) ) ) = ( P ^ K ) )
71	69, 70	syl 17	. . . . . 6 |- ( ( P e. Prime /\ K e. NN ) -> ( # ` ( 1 ... ( P ^ K ) ) ) = ( P ^ K ) )
72		expm1t 12888	. . . . . . 7 |- ( ( P e. CC /\ K e. NN ) -> ( P ^ K ) = ( ( P ^ ( K - 1 ) ) x. P ) )
73	11, 72	sylan 488	. . . . . 6 |- ( ( P e. Prime /\ K e. NN ) -> ( P ^ K ) = ( ( P ^ ( K - 1 ) ) x. P ) )
74	68, 71, 73	3eqtrd 2660	. . . . 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 ) )
75	1	adantr 481	. . . . . . . . 9 |- ( ( P e. Prime /\ K e. NN ) -> P e. NN )
76		1zzd 11408	. . . . . . . . 9 |- ( ( P e. Prime /\ K e. NN ) -> 1 e. ZZ )
77		nn0uz 11722	. . . . . . . . . . 11 |- NN0 = ( ZZ>= ` 0 )
78		1m1e0 11089	. . . . . . . . . . . 12 |- ( 1 - 1 ) = 0
79	78	fveq2i 6194	. . . . . . . . . . 11 |- ( ZZ>= ` ( 1 - 1 ) ) = ( ZZ>= ` 0 )
80	77, 79	eqtr4i 2647	. . . . . . . . . 10 |- NN0 = ( ZZ>= ` ( 1 - 1 ) )
81	69, 80	syl6eleq 2711	. . . . . . . . 9 |- ( ( P e. Prime /\ K e. NN ) -> ( P ^ K ) e. ( ZZ>= ` ( 1 - 1 ) ) )
82		0zd 11389	. . . . . . . . 9 |- ( ( P e. Prime /\ K e. NN ) -> 0 e. ZZ )
83	75, 76, 81, 82	hashdvds 15480	. . . . . . . 8 |- ( ( P e. Prime /\ K e. NN ) -> ( # ` { x e. ( 1 ... ( P ^ K ) ) | P || ( x - 0 ) } ) = ( ( |_ ` ( ( ( P ^ K ) - 0 ) / P ) ) - ( |_ ` ( ( ( 1 - 1 ) - 0 ) / P ) ) ) )
84	4	nncnd 11036	. . . . . . . . . . . . . 14 |- ( ( P e. Prime /\ K e. NN ) -> ( P ^ K ) e. CC )
85	84	subid1d 10381	. . . . . . . . . . . . 13 |- ( ( P e. Prime /\ K e. NN ) -> ( ( P ^ K ) - 0 ) = ( P ^ K ) )
86	85	oveq1d 6665	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ K e. NN ) -> ( ( ( P ^ K ) - 0 ) / P ) = ( ( P ^ K ) / P ) )
87	75	nnne0d 11065	. . . . . . . . . . . . 13 |- ( ( P e. Prime /\ K e. NN ) -> P =/= 0 )
88		nnz 11399	. . . . . . . . . . . . . 14 |- ( K e. NN -> K e. ZZ )
89	88	adantl 482	. . . . . . . . . . . . 13 |- ( ( P e. Prime /\ K e. NN ) -> K e. ZZ )
90	12, 87, 89	expm1d 13018	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ K e. NN ) -> ( P ^ ( K - 1 ) ) = ( ( P ^ K ) / P ) )
91	86, 90	eqtr4d 2659	. . . . . . . . . . 11 |- ( ( P e. Prime /\ K e. NN ) -> ( ( ( P ^ K ) - 0 ) / P ) = ( P ^ ( K - 1 ) ) )
92	91	fveq2d 6195	. . . . . . . . . 10 |- ( ( P e. Prime /\ K e. NN ) -> ( |_ ` ( ( ( P ^ K ) - 0 ) / P ) ) = ( |_ ` ( P ^ ( K - 1 ) ) ) )
93	9	nnzd 11481	. . . . . . . . . . 11 |- ( ( P e. Prime /\ K e. NN ) -> ( P ^ ( K - 1 ) ) e. ZZ )
94		flid 12609	. . . . . . . . . . 11 |- ( ( P ^ ( K - 1 ) ) e. ZZ -> ( |_ ` ( P ^ ( K - 1 ) ) ) = ( P ^ ( K - 1 ) ) )
95	93, 94	syl 17	. . . . . . . . . 10 |- ( ( P e. Prime /\ K e. NN ) -> ( |_ ` ( P ^ ( K - 1 ) ) ) = ( P ^ ( K - 1 ) ) )
96	92, 95	eqtrd 2656	. . . . . . . . 9 |- ( ( P e. Prime /\ K e. NN ) -> ( |_ ` ( ( ( P ^ K ) - 0 ) / P ) ) = ( P ^ ( K - 1 ) ) )
97	78	oveq1i 6660	. . . . . . . . . . . . . 14 |- ( ( 1 - 1 ) - 0 ) = ( 0 - 0 )
98		0m0e0 11130	. . . . . . . . . . . . . 14 |- ( 0 - 0 ) = 0
99	97, 98	eqtri 2644	. . . . . . . . . . . . 13 |- ( ( 1 - 1 ) - 0 ) = 0
100	99	oveq1i 6660	. . . . . . . . . . . 12 |- ( ( ( 1 - 1 ) - 0 ) / P ) = ( 0 / P )
101	12, 87	div0d 10800	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ K e. NN ) -> ( 0 / P ) = 0 )
102	100, 101	syl5eq 2668	. . . . . . . . . . 11 |- ( ( P e. Prime /\ K e. NN ) -> ( ( ( 1 - 1 ) - 0 ) / P ) = 0 )
103	102	fveq2d 6195	. . . . . . . . . 10 |- ( ( P e. Prime /\ K e. NN ) -> ( |_ ` ( ( ( 1 - 1 ) - 0 ) / P ) ) = ( |_ ` 0 ) )
104		0z 11388	. . . . . . . . . . 11 |- 0 e. ZZ
105		flid 12609	. . . . . . . . . . 11 |- ( 0 e. ZZ -> ( |_ ` 0 ) = 0 )
106	104, 105	ax-mp 5	. . . . . . . . . 10 |- ( |_ ` 0 ) = 0
107	103, 106	syl6eq 2672	. . . . . . . . 9 |- ( ( P e. Prime /\ K e. NN ) -> ( |_ ` ( ( ( 1 - 1 ) - 0 ) / P ) ) = 0 )
108	96, 107	oveq12d 6668	. . . . . . . 8 |- ( ( P e. Prime /\ K e. NN ) -> ( ( |_ ` ( ( ( P ^ K ) - 0 ) / P ) ) - ( |_ ` ( ( ( 1 - 1 ) - 0 ) / P ) ) ) = ( ( P ^ ( K - 1 ) ) - 0 ) )
109	10	subid1d 10381	. . . . . . . 8 |- ( ( P e. Prime /\ K e. NN ) -> ( ( P ^ ( K - 1 ) ) - 0 ) = ( P ^ ( K - 1 ) ) )
110	83, 108, 109	3eqtrd 2660	. . . . . . 7 |- ( ( P e. Prime /\ K e. NN ) -> ( # ` { x e. ( 1 ... ( P ^ K ) ) | P || ( x - 0 ) } ) = ( P ^ ( K - 1 ) ) )
111	110	oveq2d 6666	. . . . . 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 13147	. . . . . . . . 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 )
113	53, 112	ax-mp 5	. . . . . . . 8 |- ( # ` { x e. ( 1 ... ( P ^ K ) ) | ( x gcd ( P ^ K ) ) = 1 } ) e. NN0
114	113	nn0cni 11304	. . . . . . 7 |- ( # ` { x e. ( 1 ... ( P ^ K ) ) | ( x gcd ( P ^ K ) ) = 1 } ) e. CC
115		addcom 10222	. . . . . . 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 } ) ) )
116	114, 10, 115	sylancr 695	. . . . . 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 } ) ) )
117	111, 116	eqtrd 2656	. . . . 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 } ) ) )
118	59, 74, 117	3eqtr3rd 2665	. . . 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 ) )
119	10, 12	mulcld 10060	. . . . 5 |- ( ( P e. Prime /\ K e. NN ) -> ( ( P ^ ( K - 1 ) ) x. P ) e. CC )
120	114	a1i 11	. . . . 5 |- ( ( P e. Prime /\ K e. NN ) -> ( # ` { x e. ( 1 ... ( P ^ K ) ) | ( x gcd ( P ^ K ) ) = 1 } ) e. CC )
121	119, 10, 120	subaddd 10410	. . . 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 ) ) )
122	118, 121	mpbird 247	. . 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 } ) )
123	16, 18, 122	3eqtrrd 2661	. 2 |- ( ( P e. Prime /\ K e. NN ) -> ( # ` { x e. ( 1 ... ( P ^ K ) ) | ( x gcd ( P ^ K ) ) = 1 } ) = ( ( P ^ ( K - 1 ) ) x. ( P - 1 ) ) )
124	6, 123	eqtrd 2656	1 |- ( ( P e. Prime /\ K e. NN ) -> ( phi ` ( P ^ K ) ) = ( ( P ^ ( K - 1 ) ) x. ( P - 1 ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   u. cun 3572   i^i cin 3573   C_ wss 3574   (/)c0 3915   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Fincfn 7955   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   - cmin 10266   / cdiv 10684   NNcn 11020   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687   ...cfz 12326   |_cfl 12591   ^cexp 12860   #chash 13117   || cdvds 14983   gcd cgcd 15216   Primecprime 15385   phicphi 15469

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-dvds 14984  df-gcd 15217  df-prm 15386  df-phi 15471

This theorem is referenced by:  phiprm  15482