dfphi2 - Metamath Proof Explorer

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Theorem dfphi2 16750

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

**Assertion**
Ref	Expression
dfphi2	⊢ (𝑁 ∈ ℕ → (ϕ‘𝑁) = (♯‘{𝑥 ∈ (0..^𝑁) ∣ (𝑥 gcd 𝑁) = 1}))

Distinct variable group: 𝑥,𝑁

**Proof of Theorem dfphi2**
Step	Hyp	Ref	Expression
1		elnn1uz2 12890	. 2 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 = 1 ∨ 𝑁 ∈ (ℤ_≥‘2)))
2		phi1 16749	. . . . 5 ⊢ (ϕ‘1) = 1
3		0z 12546	. . . . . 6 ⊢ 0 ∈ ℤ
4		hashsng 14340	. . . . . 6 ⊢ (0 ∈ ℤ → (♯‘{0}) = 1)
5	3, 4	ax-mp 5	. . . . 5 ⊢ (♯‘{0}) = 1
6		rabid2 3442	. . . . . . 7 ⊢ ({0} = {𝑥 ∈ {0} ∣ (𝑥 gcd 1) = 1} ↔ ∀𝑥 ∈ {0} (𝑥 gcd 1) = 1)
7		elsni 4608	. . . . . . . . 9 ⊢ (𝑥 ∈ {0} → 𝑥 = 0)
8	7	oveq1d 7404	. . . . . . . 8 ⊢ (𝑥 ∈ {0} → (𝑥 gcd 1) = (0 gcd 1))
9		gcd1 16504	. . . . . . . . 9 ⊢ (0 ∈ ℤ → (0 gcd 1) = 1)
10	3, 9	ax-mp 5	. . . . . . . 8 ⊢ (0 gcd 1) = 1
11	8, 10	eqtrdi 2781	. . . . . . 7 ⊢ (𝑥 ∈ {0} → (𝑥 gcd 1) = 1)
12	6, 11	mprgbir 3052	. . . . . 6 ⊢ {0} = {𝑥 ∈ {0} ∣ (𝑥 gcd 1) = 1}
13	12	fveq2i 6863	. . . . 5 ⊢ (♯‘{0}) = (♯‘{𝑥 ∈ {0} ∣ (𝑥 gcd 1) = 1})
14	2, 5, 13	3eqtr2i 2759	. . . 4 ⊢ (ϕ‘1) = (♯‘{𝑥 ∈ {0} ∣ (𝑥 gcd 1) = 1})
15		fveq2 6860	. . . 4 ⊢ (𝑁 = 1 → (ϕ‘𝑁) = (ϕ‘1))
16		oveq2 7397	. . . . . . 7 ⊢ (𝑁 = 1 → (0..^𝑁) = (0..^1))
17		fzo01 13714	. . . . . . 7 ⊢ (0..^1) = {0}
18	16, 17	eqtrdi 2781	. . . . . 6 ⊢ (𝑁 = 1 → (0..^𝑁) = {0})
19		oveq2 7397	. . . . . . 7 ⊢ (𝑁 = 1 → (𝑥 gcd 𝑁) = (𝑥 gcd 1))
20	19	eqeq1d 2732	. . . . . 6 ⊢ (𝑁 = 1 → ((𝑥 gcd 𝑁) = 1 ↔ (𝑥 gcd 1) = 1))
21	18, 20	rabeqbidv 3427	. . . . 5 ⊢ (𝑁 = 1 → {𝑥 ∈ (0..^𝑁) ∣ (𝑥 gcd 𝑁) = 1} = {𝑥 ∈ {0} ∣ (𝑥 gcd 1) = 1})
22	21	fveq2d 6864	. . . 4 ⊢ (𝑁 = 1 → (♯‘{𝑥 ∈ (0..^𝑁) ∣ (𝑥 gcd 𝑁) = 1}) = (♯‘{𝑥 ∈ {0} ∣ (𝑥 gcd 1) = 1}))
23	14, 15, 22	3eqtr4a 2791	. . 3 ⊢ (𝑁 = 1 → (ϕ‘𝑁) = (♯‘{𝑥 ∈ (0..^𝑁) ∣ (𝑥 gcd 𝑁) = 1}))
24		eluz2nn 12853	. . . . 5 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 𝑁 ∈ ℕ)
25		phival 16743	. . . . 5 ⊢ (𝑁 ∈ ℕ → (ϕ‘𝑁) = (♯‘{𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1}))
26	24, 25	syl 17	. . . 4 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (ϕ‘𝑁) = (♯‘{𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1}))
27		fzossfz 13645	. . . . . . . . . . 11 ⊢ (1..^𝑁) ⊆ (1...𝑁)
28	27	a1i 11	. . . . . . . . . 10 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (1..^𝑁) ⊆ (1...𝑁))
29		sseqin2 4188	. . . . . . . . . 10 ⊢ ((1..^𝑁) ⊆ (1...𝑁) ↔ ((1...𝑁) ∩ (1..^𝑁)) = (1..^𝑁))
30	28, 29	sylib 218	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ_≥‘2) → ((1...𝑁) ∩ (1..^𝑁)) = (1..^𝑁))
31		fzo0ss1 13656	. . . . . . . . . 10 ⊢ (1..^𝑁) ⊆ (0..^𝑁)
32		sseqin2 4188	. . . . . . . . . 10 ⊢ ((1..^𝑁) ⊆ (0..^𝑁) ↔ ((0..^𝑁) ∩ (1..^𝑁)) = (1..^𝑁))
33	31, 32	mpbi 230	. . . . . . . . 9 ⊢ ((0..^𝑁) ∩ (1..^𝑁)) = (1..^𝑁)
34	30, 33	eqtr4di 2783	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘2) → ((1...𝑁) ∩ (1..^𝑁)) = ((0..^𝑁) ∩ (1..^𝑁)))
35	34	rabeqdv 3424	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘2) → {𝑥 ∈ ((1...𝑁) ∩ (1..^𝑁)) ∣ (𝑥 gcd 𝑁) = 1} = {𝑥 ∈ ((0..^𝑁) ∩ (1..^𝑁)) ∣ (𝑥 gcd 𝑁) = 1})
36		inrab2 4282	. . . . . . 7 ⊢ ({𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1} ∩ (1..^𝑁)) = {𝑥 ∈ ((1...𝑁) ∩ (1..^𝑁)) ∣ (𝑥 gcd 𝑁) = 1}
37		inrab2 4282	. . . . . . 7 ⊢ ({𝑥 ∈ (0..^𝑁) ∣ (𝑥 gcd 𝑁) = 1} ∩ (1..^𝑁)) = {𝑥 ∈ ((0..^𝑁) ∩ (1..^𝑁)) ∣ (𝑥 gcd 𝑁) = 1}
38	35, 36, 37	3eqtr4g 2790	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘2) → ({𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1} ∩ (1..^𝑁)) = ({𝑥 ∈ (0..^𝑁) ∣ (𝑥 gcd 𝑁) = 1} ∩ (1..^𝑁)))
39		phibndlem 16746	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘2) → {𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1} ⊆ (1...(𝑁 − 1)))
40		eluzelz 12809	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 𝑁 ∈ ℤ)
41		fzoval 13627	. . . . . . . . 9 ⊢ (𝑁 ∈ ℤ → (1..^𝑁) = (1...(𝑁 − 1)))
42	40, 41	syl 17	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (1..^𝑁) = (1...(𝑁 − 1)))
43	39, 42	sseqtrrd 3986	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘2) → {𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1} ⊆ (1..^𝑁))
44		dfss2 3934	. . . . . . 7 ⊢ ({𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1} ⊆ (1..^𝑁) ↔ ({𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1} ∩ (1..^𝑁)) = {𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1})
45	43, 44	sylib 218	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘2) → ({𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1} ∩ (1..^𝑁)) = {𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1})
46		gcd0id 16495	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℤ → (0 gcd 𝑁) = (abs‘𝑁))
47	40, 46	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (0 gcd 𝑁) = (abs‘𝑁))
48		eluzelre 12810	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 𝑁 ∈ ℝ)
49		eluzge2nn0 12857	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 𝑁 ∈ ℕ₀)
50	49	nn0ge0d 12512	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 0 ≤ 𝑁)
51	48, 50	absidd 15395	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (abs‘𝑁) = 𝑁)
52	47, 51	eqtrd 2765	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (0 gcd 𝑁) = 𝑁)
53		eluz2b3 12887	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ (ℤ_≥‘2) ↔ (𝑁 ∈ ℕ ∧ 𝑁 ≠ 1))
54	53	simprbi 496	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 𝑁 ≠ 1)
55	52, 54	eqnetrd 2993	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (0 gcd 𝑁) ≠ 1)
56	55	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ (ℤ_≥‘2) ∧ 𝑥 ∈ (0..^𝑁)) → (0 gcd 𝑁) ≠ 1)
57	7	oveq1d 7404	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ {0} → (𝑥 gcd 𝑁) = (0 gcd 𝑁))
58	57, 17	eleq2s 2847	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (0..^1) → (𝑥 gcd 𝑁) = (0 gcd 𝑁))
59	58	neeq1d 2985	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (0..^1) → ((𝑥 gcd 𝑁) ≠ 1 ↔ (0 gcd 𝑁) ≠ 1))
60	56, 59	syl5ibrcom 247	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ (ℤ_≥‘2) ∧ 𝑥 ∈ (0..^𝑁)) → (𝑥 ∈ (0..^1) → (𝑥 gcd 𝑁) ≠ 1))
61	60	necon2bd 2942	. . . . . . . . . 10 ⊢ ((𝑁 ∈ (ℤ_≥‘2) ∧ 𝑥 ∈ (0..^𝑁)) → ((𝑥 gcd 𝑁) = 1 → ¬ 𝑥 ∈ (0..^1)))
62		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ (ℤ_≥‘2) ∧ 𝑥 ∈ (0..^𝑁)) → 𝑥 ∈ (0..^𝑁))
63		1z 12569	. . . . . . . . . . . 12 ⊢ 1 ∈ ℤ
64		fzospliti 13658	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (0..^𝑁) ∧ 1 ∈ ℤ) → (𝑥 ∈ (0..^1) ∨ 𝑥 ∈ (1..^𝑁)))
65	62, 63, 64	sylancl 586	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ (ℤ_≥‘2) ∧ 𝑥 ∈ (0..^𝑁)) → (𝑥 ∈ (0..^1) ∨ 𝑥 ∈ (1..^𝑁)))
66	65	ord 864	. . . . . . . . . 10 ⊢ ((𝑁 ∈ (ℤ_≥‘2) ∧ 𝑥 ∈ (0..^𝑁)) → (¬ 𝑥 ∈ (0..^1) → 𝑥 ∈ (1..^𝑁)))
67	61, 66	syld 47	. . . . . . . . 9 ⊢ ((𝑁 ∈ (ℤ_≥‘2) ∧ 𝑥 ∈ (0..^𝑁)) → ((𝑥 gcd 𝑁) = 1 → 𝑥 ∈ (1..^𝑁)))
68	67	ralrimiva 3126	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘2) → ∀𝑥 ∈ (0..^𝑁)((𝑥 gcd 𝑁) = 1 → 𝑥 ∈ (1..^𝑁)))
69		rabss 4037	. . . . . . . 8 ⊢ ({𝑥 ∈ (0..^𝑁) ∣ (𝑥 gcd 𝑁) = 1} ⊆ (1..^𝑁) ↔ ∀𝑥 ∈ (0..^𝑁)((𝑥 gcd 𝑁) = 1 → 𝑥 ∈ (1..^𝑁)))
70	68, 69	sylibr 234	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘2) → {𝑥 ∈ (0..^𝑁) ∣ (𝑥 gcd 𝑁) = 1} ⊆ (1..^𝑁))
71		dfss2 3934	. . . . . . 7 ⊢ ({𝑥 ∈ (0..^𝑁) ∣ (𝑥 gcd 𝑁) = 1} ⊆ (1..^𝑁) ↔ ({𝑥 ∈ (0..^𝑁) ∣ (𝑥 gcd 𝑁) = 1} ∩ (1..^𝑁)) = {𝑥 ∈ (0..^𝑁) ∣ (𝑥 gcd 𝑁) = 1})
72	70, 71	sylib 218	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘2) → ({𝑥 ∈ (0..^𝑁) ∣ (𝑥 gcd 𝑁) = 1} ∩ (1..^𝑁)) = {𝑥 ∈ (0..^𝑁) ∣ (𝑥 gcd 𝑁) = 1})
73	38, 45, 72	3eqtr3d 2773	. . . . 5 ⊢ (𝑁 ∈ (ℤ_≥‘2) → {𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1} = {𝑥 ∈ (0..^𝑁) ∣ (𝑥 gcd 𝑁) = 1})
74	73	fveq2d 6864	. . . 4 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (♯‘{𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1}) = (♯‘{𝑥 ∈ (0..^𝑁) ∣ (𝑥 gcd 𝑁) = 1}))
75	26, 74	eqtrd 2765	. . 3 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (ϕ‘𝑁) = (♯‘{𝑥 ∈ (0..^𝑁) ∣ (𝑥 gcd 𝑁) = 1}))
76	23, 75	jaoi 857	. 2 ⊢ ((𝑁 = 1 ∨ 𝑁 ∈ (ℤ_≥‘2)) → (ϕ‘𝑁) = (♯‘{𝑥 ∈ (0..^𝑁) ∣ (𝑥 gcd 𝑁) = 1}))
77	1, 76	sylbi 217	1 ⊢ (𝑁 ∈ ℕ → (ϕ‘𝑁) = (♯‘{𝑥 ∈ (0..^𝑁) ∣ (𝑥 gcd 𝑁) = 1}))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∨ wo 847 = wceq 1540 ∈ wcel 2109 ≠ wne 2926 ∀wral 3045 {crab 3408 ∩ cin 3915 ⊆ wss 3916 {csn 4591 ‘cfv 6513 (class class class)co 7389 0cc0 11074 1c1 11075 − cmin 11411 ℕcn 12187 2c2 12242 ℤcz 12535 ℤ_≥cuz 12799 ...cfz 13474 ..^cfzo 13621 ♯chash 14301 abscabs 15206 gcd cgcd 16470 ϕcphi 16740

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-sep 5253 ax-nul 5263 ax-pow 5322 ax-pr 5389 ax-un 7713 ax-cnex 11130 ax-resscn 11131 ax-1cn 11132 ax-icn 11133 ax-addcl 11134 ax-addrcl 11135 ax-mulcl 11136 ax-mulrcl 11137 ax-mulcom 11138 ax-addass 11139 ax-mulass 11140 ax-distr 11141 ax-i2m1 11142 ax-1ne0 11143 ax-1rid 11144 ax-rnegex 11145 ax-rrecex 11146 ax-cnre 11147 ax-pre-lttri 11148 ax-pre-lttrn 11149 ax-pre-ltadd 11150 ax-pre-mulgt0 11151 ax-pre-sup 11152

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3756 df-csb 3865 df-dif 3919 df-un 3921 df-in 3923 df-ss 3933 df-pss 3936 df-nul 4299 df-if 4491 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4874 df-int 4913 df-iun 4959 df-br 5110 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5535 df-eprel 5540 df-po 5548 df-so 5549 df-fr 5593 df-we 5595 df-xp 5646 df-rel 5647 df-cnv 5648 df-co 5649 df-dm 5650 df-rn 5651 df-res 5652 df-ima 5653 df-pred 6276 df-ord 6337 df-on 6338 df-lim 6339 df-suc 6340 df-iota 6466 df-fun 6515 df-fn 6516 df-f 6517 df-f1 6518 df-fo 6519 df-f1o 6520 df-fv 6521 df-riota 7346 df-ov 7392 df-oprab 7393 df-mpo 7394 df-om 7845 df-1st 7970 df-2nd 7971 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8380 df-1o 8436 df-oadd 8440 df-er 8673 df-en 8921 df-dom 8922 df-sdom 8923 df-fin 8924 df-sup 9399 df-inf 9400 df-card 9898 df-pnf 11216 df-mnf 11217 df-xr 11218 df-ltxr 11219 df-le 11220 df-sub 11413 df-neg 11414 df-div 11842 df-nn 12188 df-2 12250 df-3 12251 df-n0 12449 df-xnn0 12522 df-z 12536 df-uz 12800 df-rp 12958 df-fz 13475 df-fzo 13622 df-seq 13973 df-exp 14033 df-hash 14302 df-cj 15071 df-re 15072 df-im 15073 df-sqrt 15207 df-abs 15208 df-dvds 16229 df-gcd 16471 df-phi 16742

This theorem is referenced by: phimullem 16755 eulerth 16759 hashgcdeq 16766 odngen 19513 znunithash 21480

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