Theorem dfphi2 12913

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 9938	. 2 |- ( N e. NN <-> ( N = 1 \/ N e. ( ZZ>= ` 2 ) ) )
2		phi1 12912	. . . . 5 |- ( phi ` 1 ) = 1
3		0z 9587	. . . . . 6 |- 0 e. ZZ
4		hashsng 11159	. . . . . 6 |- ( 0 e. ZZ -> (♯ ` { 0 } ) = 1 )
5	3, 4	ax-mp 5	. . . . 5 |- (♯ ` { 0 } ) = 1
6		rabid2 2720	. . . . . . 7 |- ( { 0 } = { x e. { 0 } | ( x gcd 1 ) = 1 } <-> A. x e. { 0 } ( x gcd 1 ) = 1 )
7		elsni 3706	. . . . . . . . 9 |- ( x e. { 0 } -> x = 0 )
8	7	oveq1d 6064	. . . . . . . 8 |- ( x e. { 0 } -> ( x gcd 1 ) = ( 0 gcd 1 ) )
9		gcd1 12679	. . . . . . . . 9 |- ( 0 e. ZZ -> ( 0 gcd 1 ) = 1 )
10	3, 9	ax-mp 5	. . . . . . . 8 |- ( 0 gcd 1 ) = 1
11	8, 10	eqtrdi 2281	. . . . . . 7 |- ( x e. { 0 } -> ( x gcd 1 ) = 1 )
12	6, 11	mprgbir 2600	. . . . . 6 |- { 0 } = { x e. { 0 } | ( x gcd 1 ) = 1 }
13	12	fveq2i 5672	. . . . 5 |- (♯ ` { 0 } ) = (♯ ` { x e. { 0 } | ( x gcd 1 ) = 1 } )
14	2, 5, 13	3eqtr2i 2259	. . . 4 |- ( phi ` 1 ) = (♯ ` { x e. { 0 } | ( x gcd 1 ) = 1 } )
15		fveq2 5669	. . . 4 |- ( N = 1 -> ( phi ` N ) = ( phi ` 1 ) )
16		oveq2 6057	. . . . . . 7 |- ( N = 1 -> ( 0..^ N ) = ( 0..^ 1 ) )
17		fzo01 10560	. . . . . . 7 |- ( 0..^ 1 ) = { 0 }
18	16, 17	eqtrdi 2281	. . . . . 6 |- ( N = 1 -> ( 0..^ N ) = { 0 } )
19		oveq2 6057	. . . . . . 7 |- ( N = 1 -> ( x gcd N ) = ( x gcd 1 ) )
20	19	eqeq1d 2241	. . . . . 6 |- ( N = 1 -> ( ( x gcd N ) = 1 <-> ( x gcd 1 ) = 1 ) )
21	18, 20	rabeqbidv 2807	. . . . 5 |- ( N = 1 -> { x e. ( 0..^ N ) | ( x gcd N ) = 1 } = { x e. { 0 } | ( x gcd 1 ) = 1 } )
22	21	fveq2d 5673	. . . 4 |- ( N = 1 -> (♯ ` { x e. ( 0..^ N ) | ( x gcd N ) = 1 } ) = (♯ ` { x e. { 0 } | ( x gcd 1 ) = 1 } ) )
23	14, 15, 22	3eqtr4a 2291	. . 3 |- ( N = 1 -> ( phi ` N ) = (♯ ` { x e. ( 0..^ N ) | ( x gcd N ) = 1 } ) )
24		eluz2nn 9897	. . . . 5 |- ( N e. ( ZZ>= ` 2 ) -> N e. NN )
25		phival 12906	. . . . 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 10499	. . . . . . . . . . 11 |- ( 1..^ N ) C_ ( 1 ... N )
28	27	a1i 9	. . . . . . . . . 10 |- ( N e. ( ZZ>= ` 2 ) -> ( 1..^ N ) C_ ( 1 ... N ) )
29		sseqin2 3439	. . . . . . . . . 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 10509	. . . . . . . . . 10 |- ( 1..^ N ) C_ ( 0..^ N )
32		sseqin2 3439	. . . . . . . . . 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 2283	. . . . . . . 8 |- ( N e. ( ZZ>= ` 2 ) -> ( ( 1 ... N ) i^i ( 1..^ N ) ) = ( ( 0..^ N ) i^i ( 1..^ N ) ) )
35	34	rabeqdv 2806	. . . . . . 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 3493	. . . . . . 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 3493	. . . . . . 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 2290	. . . . . 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 12909	. . . . . . . 8 |- ( N e. ( ZZ>= ` 2 ) -> { x e. ( 1 ... N ) | ( x gcd N ) = 1 } C_ ( 1 ... ( N - 1 ) ) )
40		eluzelz 9862	. . . . . . . . 9 |- ( N e. ( ZZ>= ` 2 ) -> N e. ZZ )
41		fzoval 10481	. . . . . . . . 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 3276	. . . . . . 7 |- ( N e. ( ZZ>= ` 2 ) -> { x e. ( 1 ... N ) | ( x gcd N ) = 1 } C_ ( 1..^ N ) )
44		df-ss 3223	. . . . . . 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 12671	. . . . . . . . . . . . . . . 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 9863	. . . . . . . . . . . . . . . 16 |- ( N e. ( ZZ>= ` 2 ) -> N e. RR )
49		eluzge2nn0 9901	. . . . . . . . . . . . . . . . 17 |- ( N e. ( ZZ>= ` 2 ) -> N e. NN0 )
50	49	nn0ge0d 9555	. . . . . . . . . . . . . . . 16 |- ( N e. ( ZZ>= ` 2 ) -> 0 <_ N )
51	48, 50	absidd 11848	. . . . . . . . . . . . . . 15 |- ( N e. ( ZZ>= ` 2 ) -> ( abs ` N ) = N )
52	47, 51	eqtrd 2265	. . . . . . . . . . . . . 14 |- ( N e. ( ZZ>= ` 2 ) -> ( 0 gcd N ) = N )
53		eluz2b3 9935	. . . . . . . . . . . . . . 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 2436	. . . . . . . . . . . . 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 6064	. . . . . . . . . . . . . 14 |- ( x e. { 0 } -> ( x gcd N ) = ( 0 gcd N ) )
58	57, 17	eleq2s 2327	. . . . . . . . . . . . 13 |- ( x e. ( 0..^ 1 ) -> ( x gcd N ) = ( 0 gcd N ) )
59	58	neeq1d 2430	. . . . . . . . . . . 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 2470	. . . . . . . . . 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 9602	. . . . . . . . . . . 12 |- 1 e. ZZ
64		fzospliti 10511	. . . . . . . . . . . 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 732	. . . . . . . . . 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 2615	. . . . . . . 8 |- ( N e. ( ZZ>= ` 2 ) -> A. x e. ( 0..^ N ) ( ( x gcd N ) = 1 -> x e. ( 1..^ N ) ) )
69		rabss 3314	. . . . . . . 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 3223	. . . . . . 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 2273	. . . . 5 |- ( N e. ( ZZ>= ` 2 ) -> { x e. ( 1 ... N ) | ( x gcd N ) = 1 } = { x e. ( 0..^ N ) | ( x gcd N ) = 1 } )
74	73	fveq2d 5673	. . . 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 2265	. . 3 |- ( N e. ( ZZ>= ` 2 ) -> ( phi ` N ) = (♯ ` { x e. ( 0..^ N ) | ( x gcd N ) = 1 } ) )
76	23, 75	jaoi 724	. 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 716   = wceq 1398   e. wcel 2203   =/= wne 2412   A.wral 2520   {crab 2524   i^i cin 3209   C_ wss 3210   {csn 3688   ` cfv 5351  (class class class)co 6049   0cc0 8126   1c1 8127   - cmin 8443   NNcn 9236   2c2 9287   ZZcz 9576   ZZ>=cuz 9852   ...cfz 10341  ..^cfzo 10475  ♯chash 11136   abscabs 11678   gcd cgcd 12645   phicphi 12902

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-iinf 4709  ax-cnex 8217  ax-resscn 8218  ax-1cn 8219  ax-1re 8220  ax-icn 8221  ax-addcl 8222  ax-addrcl 8223  ax-mulcl 8224  ax-mulrcl 8225  ax-addcom 8226  ax-mulcom 8227  ax-addass 8228  ax-mulass 8229  ax-distr 8230  ax-i2m1 8231  ax-0lt1 8232  ax-1rid 8233  ax-0id 8234  ax-rnegex 8235  ax-precex 8236  ax-cnre 8237  ax-pre-ltirr 8238  ax-pre-ltwlin 8239  ax-pre-lttrn 8240  ax-pre-apti 8241  ax-pre-ltadd 8242  ax-pre-mulgt0 8243  ax-pre-mulext 8244  ax-arch 8245  ax-caucvg 8246

This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-if 3620  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-id 4413  df-po 4416  df-iso 4417  df-iord 4486  df-on 4488  df-ilim 4489  df-suc 4491  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-recs 6535  df-frec 6621  df-1o 6646  df-er 6766  df-en 6975  df-dom 6976  df-fin 6977  df-sup 7274  df-pnf 8309  df-mnf 8310  df-xr 8311  df-ltxr 8312  df-le 8313  df-sub 8445  df-neg 8446  df-reap 8848  df-ap 8855  df-div 8946  df-inn 9237  df-2 9295  df-3 9296  df-4 9297  df-n0 9496  df-z 9577  df-uz 9853  df-q 9951  df-rp 9986  df-fz 10342  df-fzo 10476  df-fl 10629  df-mod 10684  df-seqfrec 10809  df-exp 10900  df-ihash 11137  df-cj 11523  df-re 11524  df-im 11525  df-rsqrt 11679  df-abs 11680  df-dvds 12470  df-gcd 12646  df-phi 12904

This theorem is referenced by:  phimullem  12918  eulerth  12926  hashgcdeq  12933