Theorem dfphi2 12728

Description: Alternate definition of the Euler phi function. (Contributed by Mario Carneiro, 23-Feb-2014.) (Revised by Mario Carneiro, 2-May-2016.)

Assertion

Ref	Expression
dfphi2	|- ( N e. NN -> ( phi ` N ) = (♯ ` { x e. ( 0..^ N ) | ( x gcd N ) = 1 } ) )

Distinct variable group:   x, N

Proof of Theorem dfphi2

Step	Hyp	Ref	Expression
1		elnn1uz2 9790	. 2 |- ( N e. NN <-> ( N = 1 \/ N e. ( ZZ>= ` 2 ) ) )
2		phi1 12727	. . . . 5 |- ( phi ` 1 ) = 1
3		0z 9445	. . . . . 6 |- 0 e. ZZ
4		hashsng 11007	. . . . . 6 |- ( 0 e. ZZ -> (♯ ` { 0 } ) = 1 )
5	3, 4	ax-mp 5	. . . . 5 |- (♯ ` { 0 } ) = 1
6		rabid2 2708	. . . . . . 7 |- ( { 0 } = { x e. { 0 } | ( x gcd 1 ) = 1 } <-> A. x e. { 0 } ( x gcd 1 ) = 1 )
7		elsni 3684	. . . . . . . . 9 |- ( x e. { 0 } -> x = 0 )
8	7	oveq1d 6009	. . . . . . . 8 |- ( x e. { 0 } -> ( x gcd 1 ) = ( 0 gcd 1 ) )
9		gcd1 12494	. . . . . . . . 9 |- ( 0 e. ZZ -> ( 0 gcd 1 ) = 1 )
10	3, 9	ax-mp 5	. . . . . . . 8 |- ( 0 gcd 1 ) = 1
11	8, 10	eqtrdi 2278	. . . . . . 7 |- ( x e. { 0 } -> ( x gcd 1 ) = 1 )
12	6, 11	mprgbir 2588	. . . . . 6 |- { 0 } = { x e. { 0 } | ( x gcd 1 ) = 1 }
13	12	fveq2i 5626	. . . . 5 |- (♯ ` { 0 } ) = (♯ ` { x e. { 0 } | ( x gcd 1 ) = 1 } )
14	2, 5, 13	3eqtr2i 2256	. . . 4 |- ( phi ` 1 ) = (♯ ` { x e. { 0 } | ( x gcd 1 ) = 1 } )
15		fveq2 5623	. . . 4 |- ( N = 1 -> ( phi ` N ) = ( phi ` 1 ) )
16		oveq2 6002	. . . . . . 7 |- ( N = 1 -> ( 0..^ N ) = ( 0..^ 1 ) )
17		fzo01 10409	. . . . . . 7 |- ( 0..^ 1 ) = { 0 }
18	16, 17	eqtrdi 2278	. . . . . 6 |- ( N = 1 -> ( 0..^ N ) = { 0 } )
19		oveq2 6002	. . . . . . 7 |- ( N = 1 -> ( x gcd N ) = ( x gcd 1 ) )
20	19	eqeq1d 2238	. . . . . 6 |- ( N = 1 -> ( ( x gcd N ) = 1 <-> ( x gcd 1 ) = 1 ) )
21	18, 20	rabeqbidv 2794	. . . . 5 |- ( N = 1 -> { x e. ( 0..^ N ) | ( x gcd N ) = 1 } = { x e. { 0 } | ( x gcd 1 ) = 1 } )
22	21	fveq2d 5627	. . . 4 |- ( N = 1 -> (♯ ` { x e. ( 0..^ N ) | ( x gcd N ) = 1 } ) = (♯ ` { x e. { 0 } | ( x gcd 1 ) = 1 } ) )
23	14, 15, 22	3eqtr4a 2288	. . 3 |- ( N = 1 -> ( phi ` N ) = (♯ ` { x e. ( 0..^ N ) | ( x gcd N ) = 1 } ) )
24		eluz2nn 9749	. . . . 5 |- ( N e. ( ZZ>= ` 2 ) -> N e. NN )
25		phival 12721	. . . . 5 |- ( N e. NN -> ( phi ` N ) = (♯ ` { x e. ( 1 ... N ) | ( x gcd N ) = 1 } ) )
26	24, 25	syl 14	. . . 4 |- ( N e. ( ZZ>= ` 2 ) -> ( phi ` N ) = (♯ ` { x e. ( 1 ... N ) | ( x gcd N ) = 1 } ) )
27		fzossfz 10350	. . . . . . . . . . 11 |- ( 1..^ N ) C_ ( 1 ... N )
28	27	a1i 9	. . . . . . . . . 10 |- ( N e. ( ZZ>= ` 2 ) -> ( 1..^ N ) C_ ( 1 ... N ) )
29		sseqin2 3423	. . . . . . . . . 10 |- ( ( 1..^ N ) C_ ( 1 ... N ) <-> ( ( 1 ... N ) i^i ( 1..^ N ) ) = ( 1..^ N ) )
30	28, 29	sylib 122	. . . . . . . . 9 |- ( N e. ( ZZ>= ` 2 ) -> ( ( 1 ... N ) i^i ( 1..^ N ) ) = ( 1..^ N ) )
31		fzo0ss1 10360	. . . . . . . . . 10 |- ( 1..^ N ) C_ ( 0..^ N )
32		sseqin2 3423	. . . . . . . . . 10 |- ( ( 1..^ N ) C_ ( 0..^ N ) <-> ( ( 0..^ N ) i^i ( 1..^ N ) ) = ( 1..^ N ) )
33	31, 32	mpbi 145	. . . . . . . . 9 |- ( ( 0..^ N ) i^i ( 1..^ N ) ) = ( 1..^ N )
34	30, 33	eqtr4di 2280	. . . . . . . 8 |- ( N e. ( ZZ>= ` 2 ) -> ( ( 1 ... N ) i^i ( 1..^ N ) ) = ( ( 0..^ N ) i^i ( 1..^ N ) ) )
35	34	rabeqdv 2793	. . . . . . 7 |- ( N e. ( ZZ>= ` 2 ) -> { x e. ( ( 1 ... N ) i^i ( 1..^ N ) ) | ( x gcd N ) = 1 } = { x e. ( ( 0..^ N ) i^i ( 1..^ N ) ) | ( x gcd N ) = 1 } )
36		inrab2 3477	. . . . . . 7 |- ( { x e. ( 1 ... N ) | ( x gcd N ) = 1 } i^i ( 1..^ N ) ) = { x e. ( ( 1 ... N ) i^i ( 1..^ N ) ) | ( x gcd N ) = 1 }
37		inrab2 3477	. . . . . . 7 |- ( { x e. ( 0..^ N ) | ( x gcd N ) = 1 } i^i ( 1..^ N ) ) = { x e. ( ( 0..^ N ) i^i ( 1..^ N ) ) | ( x gcd N ) = 1 }
38	35, 36, 37	3eqtr4g 2287	. . . . . 6 |- ( N e. ( ZZ>= ` 2 ) -> ( { x e. ( 1 ... N ) | ( x gcd N ) = 1 } i^i ( 1..^ N ) ) = ( { x e. ( 0..^ N ) | ( x gcd N ) = 1 } i^i ( 1..^ N ) ) )
39		phibndlem 12724	. . . . . . . 8 |- ( N e. ( ZZ>= ` 2 ) -> { x e. ( 1 ... N ) | ( x gcd N ) = 1 } C_ ( 1 ... ( N - 1 ) ) )
40		eluzelz 9719	. . . . . . . . 9 |- ( N e. ( ZZ>= ` 2 ) -> N e. ZZ )
41		fzoval 10332	. . . . . . . . 9 |- ( N e. ZZ -> ( 1..^ N ) = ( 1 ... ( N - 1 ) ) )
42	40, 41	syl 14	. . . . . . . 8 |- ( N e. ( ZZ>= ` 2 ) -> ( 1..^ N ) = ( 1 ... ( N - 1 ) ) )
43	39, 42	sseqtrrd 3263	. . . . . . 7 |- ( N e. ( ZZ>= ` 2 ) -> { x e. ( 1 ... N ) | ( x gcd N ) = 1 } C_ ( 1..^ N ) )
44		df-ss 3210	. . . . . . 7 |- ( { x e. ( 1 ... N ) | ( x gcd N ) = 1 } C_ ( 1..^ N ) <-> ( { x e. ( 1 ... N ) | ( x gcd N ) = 1 } i^i ( 1..^ N ) ) = { x e. ( 1 ... N ) | ( x gcd N ) = 1 } )
45	43, 44	sylib 122	. . . . . 6 |- ( N e. ( ZZ>= ` 2 ) -> ( { x e. ( 1 ... N ) | ( x gcd N ) = 1 } i^i ( 1..^ N ) ) = { x e. ( 1 ... N ) | ( x gcd N ) = 1 } )
46		gcd0id 12486	. . . . . . . . . . . . . . . 16 |- ( N e. ZZ -> ( 0 gcd N ) = ( abs ` N ) )
47	40, 46	syl 14	. . . . . . . . . . . . . . 15 |- ( N e. ( ZZ>= ` 2 ) -> ( 0 gcd N ) = ( abs ` N ) )
48		eluzelre 9720	. . . . . . . . . . . . . . . 16 |- ( N e. ( ZZ>= ` 2 ) -> N e. RR )
49		eluzge2nn0 9752	. . . . . . . . . . . . . . . . 17 |- ( N e. ( ZZ>= ` 2 ) -> N e. NN0 )
50	49	nn0ge0d 9413	. . . . . . . . . . . . . . . 16 |- ( N e. ( ZZ>= ` 2 ) -> 0 <_ N )
51	48, 50	absidd 11664	. . . . . . . . . . . . . . 15 |- ( N e. ( ZZ>= ` 2 ) -> ( abs ` N ) = N )
52	47, 51	eqtrd 2262	. . . . . . . . . . . . . 14 |- ( N e. ( ZZ>= ` 2 ) -> ( 0 gcd N ) = N )
53		eluz2b3 9787	. . . . . . . . . . . . . . 15 |- ( N e. ( ZZ>= ` 2 ) <-> ( N e. NN /\ N =/= 1 ) )
54	53	simprbi 275	. . . . . . . . . . . . . 14 |- ( N e. ( ZZ>= ` 2 ) -> N =/= 1 )
55	52, 54	eqnetrd 2424	. . . . . . . . . . . . 13 |- ( N e. ( ZZ>= ` 2 ) -> ( 0 gcd N ) =/= 1 )
56	55	adantr 276	. . . . . . . . . . . 12 |- ( ( N e. ( ZZ>= ` 2 ) /\ x e. ( 0..^ N ) ) -> ( 0 gcd N ) =/= 1 )
57	7	oveq1d 6009	. . . . . . . . . . . . . 14 |- ( x e. { 0 } -> ( x gcd N ) = ( 0 gcd N ) )
58	57, 17	eleq2s 2324	. . . . . . . . . . . . 13 |- ( x e. ( 0..^ 1 ) -> ( x gcd N ) = ( 0 gcd N ) )
59	58	neeq1d 2418	. . . . . . . . . . . 12 |- ( x e. ( 0..^ 1 ) -> ( ( x gcd N ) =/= 1 <-> ( 0 gcd N ) =/= 1 ) )
60	56, 59	syl5ibrcom 157	. . . . . . . . . . 11 |- ( ( N e. ( ZZ>= ` 2 ) /\ x e. ( 0..^ N ) ) -> ( x e. ( 0..^ 1 ) -> ( x gcd N ) =/= 1 ) )
61	60	necon2bd 2458	. . . . . . . . . 10 |- ( ( N e. ( ZZ>= ` 2 ) /\ x e. ( 0..^ N ) ) -> ( ( x gcd N ) = 1 -> -. x e. ( 0..^ 1 ) ) )
62		simpr 110	. . . . . . . . . . . 12 |- ( ( N e. ( ZZ>= ` 2 ) /\ x e. ( 0..^ N ) ) -> x e. ( 0..^ N ) )
63		1z 9460	. . . . . . . . . . . 12 |- 1 e. ZZ
64		fzospliti 10362	. . . . . . . . . . . 12 |- ( ( x e. ( 0..^ N ) /\ 1 e. ZZ ) -> ( x e. ( 0..^ 1 ) \/ x e. ( 1..^ N ) ) )
65	62, 63, 64	sylancl 413	. . . . . . . . . . 11 |- ( ( N e. ( ZZ>= ` 2 ) /\ x e. ( 0..^ N ) ) -> ( x e. ( 0..^ 1 ) \/ x e. ( 1..^ N ) ) )
66	65	ord 729	. . . . . . . . . 10 |- ( ( N e. ( ZZ>= ` 2 ) /\ x e. ( 0..^ N ) ) -> ( -. x e. ( 0..^ 1 ) -> x e. ( 1..^ N ) ) )
67	61, 66	syld 45	. . . . . . . . 9 |- ( ( N e. ( ZZ>= ` 2 ) /\ x e. ( 0..^ N ) ) -> ( ( x gcd N ) = 1 -> x e. ( 1..^ N ) ) )
68	67	ralrimiva 2603	. . . . . . . 8 |- ( N e. ( ZZ>= ` 2 ) -> A. x e. ( 0..^ N ) ( ( x gcd N ) = 1 -> x e. ( 1..^ N ) ) )
69		rabss 3301	. . . . . . . 8 |- ( { x e. ( 0..^ N ) | ( x gcd N ) = 1 } C_ ( 1..^ N ) <-> A. x e. ( 0..^ N ) ( ( x gcd N ) = 1 -> x e. ( 1..^ N ) ) )
70	68, 69	sylibr 134	. . . . . . 7 |- ( N e. ( ZZ>= ` 2 ) -> { x e. ( 0..^ N ) | ( x gcd N ) = 1 } C_ ( 1..^ N ) )
71		df-ss 3210	. . . . . . 7 |- ( { x e. ( 0..^ N ) | ( x gcd N ) = 1 } C_ ( 1..^ N ) <-> ( { x e. ( 0..^ N ) | ( x gcd N ) = 1 } i^i ( 1..^ N ) ) = { x e. ( 0..^ N ) | ( x gcd N ) = 1 } )
72	70, 71	sylib 122	. . . . . 6 |- ( N e. ( ZZ>= ` 2 ) -> ( { x e. ( 0..^ N ) | ( x gcd N ) = 1 } i^i ( 1..^ N ) ) = { x e. ( 0..^ N ) | ( x gcd N ) = 1 } )
73	38, 45, 72	3eqtr3d 2270	. . . . 5 |- ( N e. ( ZZ>= ` 2 ) -> { x e. ( 1 ... N ) | ( x gcd N ) = 1 } = { x e. ( 0..^ N ) | ( x gcd N ) = 1 } )
74	73	fveq2d 5627	. . . 4 |- ( N e. ( ZZ>= ` 2 ) -> (♯ ` { x e. ( 1 ... N ) | ( x gcd N ) = 1 } ) = (♯ ` { x e. ( 0..^ N ) | ( x gcd N ) = 1 } ) )
75	26, 74	eqtrd 2262	. . 3 |- ( N e. ( ZZ>= ` 2 ) -> ( phi ` N ) = (♯ ` { x e. ( 0..^ N ) | ( x gcd N ) = 1 } ) )
76	23, 75	jaoi 721	. 2 |- ( ( N = 1 \/ N e. ( ZZ>= ` 2 ) ) -> ( phi ` N ) = (♯ ` { x e. ( 0..^ N ) | ( x gcd N ) = 1 } ) )
77	1, 76	sylbi 121	1 |- ( N e. NN -> ( phi ` N ) = (♯ ` { x e. ( 0..^ N ) | ( x gcd N ) = 1 } ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 713   = wceq 1395   e. wcel 2200   =/= wne 2400   A.wral 2508   {crab 2512   i^i cin 3196   C_ wss 3197   {csn 3666   ` cfv 5314  (class class class)co 5994   0cc0 7987   1c1 7988   - cmin 8305   NNcn 9098   2c2 9149   ZZcz 9434   ZZ>=cuz 9710   ...cfz 10192  ..^cfzo 10326  ♯chash 10984   abscabs 11494   gcd cgcd 12460   phicphi 12717

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 4521  ax-setind 4626  ax-iinf 4677  ax-cnex 8078  ax-resscn 8079  ax-1cn 8080  ax-1re 8081  ax-icn 8082  ax-addcl 8083  ax-addrcl 8084  ax-mulcl 8085  ax-mulrcl 8086  ax-addcom 8087  ax-mulcom 8088  ax-addass 8089  ax-mulass 8090  ax-distr 8091  ax-i2m1 8092  ax-0lt1 8093  ax-1rid 8094  ax-0id 8095  ax-rnegex 8096  ax-precex 8097  ax-cnre 8098  ax-pre-ltirr 8099  ax-pre-ltwlin 8100  ax-pre-lttrn 8101  ax-pre-apti 8102  ax-pre-ltadd 8103  ax-pre-mulgt0 8104  ax-pre-mulext 8105  ax-arch 8106  ax-caucvg 8107

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 4381  df-po 4384  df-iso 4385  df-iord 4454  df-on 4456  df-ilim 4457  df-suc 4459  df-iom 4680  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-rn 4727  df-res 4728  df-ima 4729  df-iota 5274  df-fun 5316  df-fn 5317  df-f 5318  df-f1 5319  df-fo 5320  df-f1o 5321  df-fv 5322  df-riota 5947  df-ov 5997  df-oprab 5998  df-mpo 5999  df-1st 6276  df-2nd 6277  df-recs 6441  df-frec 6527  df-1o 6552  df-er 6670  df-en 6878  df-dom 6879  df-fin 6880  df-sup 7139  df-pnf 8171  df-mnf 8172  df-xr 8173  df-ltxr 8174  df-le 8175  df-sub 8307  df-neg 8308  df-reap 8710  df-ap 8717  df-div 8808  df-inn 9099  df-2 9157  df-3 9158  df-4 9159  df-n0 9358  df-z 9435  df-uz 9711  df-q 9803  df-rp 9838  df-fz 10193  df-fzo 10327  df-fl 10477  df-mod 10532  df-seqfrec 10657  df-exp 10748  df-ihash 10985  df-cj 11339  df-re 11340  df-im 11341  df-rsqrt 11495  df-abs 11496  df-dvds 12285  df-gcd 12461  df-phi 12719

This theorem is referenced by:  phimullem  12733  eulerth  12741  hashgcdeq  12748