dfphi2 - Metamath Proof Explorer

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Theorem dfphi2 16956

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 13055	. 2 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 = 1 ∨ 𝑁 ∈ (ℤ_≥‘2)))
2		phi1 16955	. . . . 5 ⊢ (ϕ‘1) = 1
3		0z 12707	. . . . . 6 ⊢ 0 ∈ ℤ
4		hashsng 14516	. . . . . 6 ⊢ (0 ∈ ℤ → (♯‘{0}) = 1)
5	3, 4	ax-mp 5	. . . . 5 ⊢ (♯‘{0}) = 1
6		rabid2 3444	. . . . . . 7 ⊢ ({0} = {𝑥 ∈ {0} ∣ (𝑥 gcd 1) = 1} ↔ ∀𝑥 ∈ {0} (𝑥 gcd 1) = 1)
7		elsni 4597	. . . . . . . . 9 ⊢ (𝑥 ∈ {0} → 𝑥 = 0)
8	7	oveq1d 7435	. . . . . . . 8 ⊢ (𝑥 ∈ {0} → (𝑥 gcd 1) = (0 gcd 1))
9		gcd1 16706	. . . . . . . . 9 ⊢ (0 ∈ ℤ → (0 gcd 1) = 1)
10	3, 9	ax-mp 5	. . . . . . . 8 ⊢ (0 gcd 1) = 1
11	8, 10	eqtrdi 2811	. . . . . . 7 ⊢ (𝑥 ∈ {0} → (𝑥 gcd 1) = 1)
12	6, 11	mprgbir 3083	. . . . . 6 ⊢ {0} = {𝑥 ∈ {0} ∣ (𝑥 gcd 1) = 1}
13	12	fveq2i 6888	. . . . 5 ⊢ (♯‘{0}) = (♯‘{𝑥 ∈ {0} ∣ (𝑥 gcd 1) = 1})
14	2, 5, 13	3eqtr2i 2789	. . . 4 ⊢ (ϕ‘1) = (♯‘{𝑥 ∈ {0} ∣ (𝑥 gcd 1) = 1})
15		fveq2 6885	. . . 4 ⊢ (𝑁 = 1 → (ϕ‘𝑁) = (ϕ‘1))
16		oveq2 7428	. . . . . . 7 ⊢ (𝑁 = 1 → (0..^𝑁) = (0..^1))
17		fzo01 13885	. . . . . . 7 ⊢ (0..^1) = {0}
18	16, 17	eqtrdi 2811	. . . . . 6 ⊢ (𝑁 = 1 → (0..^𝑁) = {0})
19		oveq2 7428	. . . . . . 7 ⊢ (𝑁 = 1 → (𝑥 gcd 𝑁) = (𝑥 gcd 1))
20	19	eqeq1d 2762	. . . . . 6 ⊢ (𝑁 = 1 → ((𝑥 gcd 𝑁) = 1 ↔ (𝑥 gcd 1) = 1))
21	18, 20	rabeqbidv 3429	. . . . 5 ⊢ (𝑁 = 1 → {𝑥 ∈ (0..^𝑁) ∣ (𝑥 gcd 𝑁) = 1} = {𝑥 ∈ {0} ∣ (𝑥 gcd 1) = 1})
22	21	fveq2d 6889	. . . 4 ⊢ (𝑁 = 1 → (♯‘{𝑥 ∈ (0..^𝑁) ∣ (𝑥 gcd 𝑁) = 1}) = (♯‘{𝑥 ∈ {0} ∣ (𝑥 gcd 1) = 1}))
23	14, 15, 22	3eqtr4a 2821	. . 3 ⊢ (𝑁 = 1 → (ϕ‘𝑁) = (♯‘{𝑥 ∈ (0..^𝑁) ∣ (𝑥 gcd 𝑁) = 1}))
24		eluz2nn 13018	. . . . 5 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 𝑁 ∈ ℕ)
25		phival 16949	. . . . 5 ⊢ (𝑁 ∈ ℕ → (ϕ‘𝑁) = (♯‘{𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1}))
26	24, 25	syl 18	. . . 4 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (ϕ‘𝑁) = (♯‘{𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1}))
27		fzossfz 13816	. . . . . . . . . . 11 ⊢ (1..^𝑁) ⊆ (1...𝑁)
28	27	a1i 11	. . . . . . . . . 10 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (1..^𝑁) ⊆ (1...𝑁))
29		sseqin2 4120	. . . . . . . . . 10 ⊢ ((1..^𝑁) ⊆ (1...𝑁) ↔ ((1...𝑁) ∩ (1..^𝑁)) = (1..^𝑁))
30	28, 29	sylib 221	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ_≥‘2) → ((1...𝑁) ∩ (1..^𝑁)) = (1..^𝑁))
31		fzo0ss1 13827	. . . . . . . . . 10 ⊢ (1..^𝑁) ⊆ (0..^𝑁)
32		sseqin2 4120	. . . . . . . . . 10 ⊢ ((1..^𝑁) ⊆ (0..^𝑁) ↔ ((0..^𝑁) ∩ (1..^𝑁)) = (1..^𝑁))
33	31, 32	mpbi 233	. . . . . . . . 9 ⊢ ((0..^𝑁) ∩ (1..^𝑁)) = (1..^𝑁)
34	30, 33	eqtr4di 2813	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘2) → ((1...𝑁) ∩ (1..^𝑁)) = ((0..^𝑁) ∩ (1..^𝑁)))
35	34	rabeqdv 3427	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘2) → {𝑥 ∈ ((1...𝑁) ∩ (1..^𝑁)) ∣ (𝑥 gcd 𝑁) = 1} = {𝑥 ∈ ((0..^𝑁) ∩ (1..^𝑁)) ∣ (𝑥 gcd 𝑁) = 1})
36		inrab2 4262	. . . . . . 7 ⊢ ({𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1} ∩ (1..^𝑁)) = {𝑥 ∈ ((1...𝑁) ∩ (1..^𝑁)) ∣ (𝑥 gcd 𝑁) = 1}
37		inrab2 4262	. . . . . . 7 ⊢ ({𝑥 ∈ (0..^𝑁) ∣ (𝑥 gcd 𝑁) = 1} ∩ (1..^𝑁)) = {𝑥 ∈ ((0..^𝑁) ∩ (1..^𝑁)) ∣ (𝑥 gcd 𝑁) = 1}
38	35, 36, 37	3eqtr4g 2820	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘2) → ({𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1} ∩ (1..^𝑁)) = ({𝑥 ∈ (0..^𝑁) ∣ (𝑥 gcd 𝑁) = 1} ∩ (1..^𝑁)))
39		phibndlem 16952	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘2) → {𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1} ⊆ (1...(𝑁 − 1)))
40		eluzelz 12978	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 𝑁 ∈ ℤ)
41		fzoval 13797	. . . . . . . . 9 ⊢ (𝑁 ∈ ℤ → (1..^𝑁) = (1...(𝑁 − 1)))
42	40, 41	syl 18	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (1..^𝑁) = (1...(𝑁 − 1)))
43	39, 42	sseqtrrd 3945	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘2) → {𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1} ⊆ (1..^𝑁))
44		dfss2 4094	. . . . . . 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 221	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘2) → ({𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1} ∩ (1..^𝑁)) = {𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1})
46		gcd0id 16696	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℤ → (0 gcd 𝑁) = (abs‘𝑁))
47	40, 46	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (0 gcd 𝑁) = (abs‘𝑁))
48		eluzelre 12979	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 𝑁 ∈ ℝ)
49		eluzge2nn0 13022	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 𝑁 ∈ ℕ₀)
50	49	nn0ge0d 12673	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 0 ≤ 𝑁)
51	48, 50	absidd 15595	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (abs‘𝑁) = 𝑁)
52	47, 51	eqtrd 2795	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (0 gcd 𝑁) = 𝑁)
53		eluz2b3 13052	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ (ℤ_≥‘2) ↔ (𝑁 ∈ ℕ ∧ 𝑁 ≠ 1))
54	53	simprbi 503	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 𝑁 ≠ 1)
55	52, 54	eqnetrd 3022	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (0 gcd 𝑁) ≠ 1)
56	55	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ (ℤ_≥‘2) ∧ 𝑥 ∈ (0..^𝑁)) → (0 gcd 𝑁) ≠ 1)
57	7	oveq1d 7435	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ {0} → (𝑥 gcd 𝑁) = (0 gcd 𝑁))
58	57, 17	eleq2s 2878	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (0..^1) → (𝑥 gcd 𝑁) = (0 gcd 𝑁))
59	58	neeq1d 3014	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (0..^1) → ((𝑥 gcd 𝑁) ≠ 1 ↔ (0 gcd 𝑁) ≠ 1))
60	56, 59	syl5ibrcom 250	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ (ℤ_≥‘2) ∧ 𝑥 ∈ (0..^𝑁)) → (𝑥 ∈ (0..^1) → (𝑥 gcd 𝑁) ≠ 1))
61	60	necon2bd 2971	. . . . . . . . . 10 ⊢ ((𝑁 ∈ (ℤ_≥‘2) ∧ 𝑥 ∈ (0..^𝑁)) → ((𝑥 gcd 𝑁) = 1 → ¬ 𝑥 ∈ (0..^1)))
62		simpr 490	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ (ℤ_≥‘2) ∧ 𝑥 ∈ (0..^𝑁)) → 𝑥 ∈ (0..^𝑁))
63		1z 12729	. . . . . . . . . . . 12 ⊢ 1 ∈ ℤ
64		fzospliti 13829	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (0..^𝑁) ∧ 1 ∈ ℤ) → (𝑥 ∈ (0..^1) ∨ 𝑥 ∈ (1..^𝑁)))
65	62, 63, 64	sylancl 598	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ (ℤ_≥‘2) ∧ 𝑥 ∈ (0..^𝑁)) → (𝑥 ∈ (0..^1) ∨ 𝑥 ∈ (1..^𝑁)))
66	65	ord 878	. . . . . . . . . 10 ⊢ ((𝑁 ∈ (ℤ_≥‘2) ∧ 𝑥 ∈ (0..^𝑁)) → (¬ 𝑥 ∈ (0..^1) → 𝑥 ∈ (1..^𝑁)))
67	61, 66	syld 48	. . . . . . . . 9 ⊢ ((𝑁 ∈ (ℤ_≥‘2) ∧ 𝑥 ∈ (0..^𝑁)) → ((𝑥 gcd 𝑁) = 1 → 𝑥 ∈ (1..^𝑁)))
68	67	ralrimiva 3154	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘2) → ∀𝑥 ∈ (0..^𝑁)((𝑥 gcd 𝑁) = 1 → 𝑥 ∈ (1..^𝑁)))
69		rabss 3995	. . . . . . . 8 ⊢ ({𝑥 ∈ (0..^𝑁) ∣ (𝑥 gcd 𝑁) = 1} ⊆ (1..^𝑁) ↔ ∀𝑥 ∈ (0..^𝑁)((𝑥 gcd 𝑁) = 1 → 𝑥 ∈ (1..^𝑁)))
70	68, 69	sylibr 237	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘2) → {𝑥 ∈ (0..^𝑁) ∣ (𝑥 gcd 𝑁) = 1} ⊆ (1..^𝑁))
71		dfss2 4094	. . . . . . 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 221	. . . . . 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 2803	. . . . 5 ⊢ (𝑁 ∈ (ℤ_≥‘2) → {𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1} = {𝑥 ∈ (0..^𝑁) ∣ (𝑥 gcd 𝑁) = 1})
74	73	fveq2d 6889	. . . 4 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (♯‘{𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1}) = (♯‘{𝑥 ∈ (0..^𝑁) ∣ (𝑥 gcd 𝑁) = 1}))
75	26, 74	eqtrd 2795	. . 3 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (ϕ‘𝑁) = (♯‘{𝑥 ∈ (0..^𝑁) ∣ (𝑥 gcd 𝑁) = 1}))
76	23, 75	jaoi 871	. 2 ⊢ ((𝑁 = 1 ∨ 𝑁 ∈ (ℤ_≥‘2)) → (ϕ‘𝑁) = (♯‘{𝑥 ∈ (0..^𝑁) ∣ (𝑥 gcd 𝑁) = 1}))
77	1, 76	sylbi 220	1 ⊢ (𝑁 ∈ ℕ → (ϕ‘𝑁) = (♯‘{𝑥 ∈ (0..^𝑁) ∣ (𝑥 gcd 𝑁) = 1}))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 ∨ wo 861 = wceq 1569 ∈ wcel 2144 ≠ wne 2955 ∀wral 3076 {crab 3412 ⊆ wss 3895 ∩ cin 4089 {csn 4584 ‘cfv 6538 (class class class)co 7420 0cc0 11203 1c1 11204 − cmin 11544 ℕcn 12338 2c2 12400 ℤcz 12696 ℤ_≥cuz 12968 ...cfz 13642 ..^cfzo 13791 ♯chash 14477 abscabs 15406 gcd cgcd 16669 ϕcphi 16946

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7751 ax-cnex 11259 ax-resscn 11260 ax-1cn 11261 ax-icn 11262 ax-addcl 11263 ax-addrcl 11264 ax-mulcl 11265 ax-mulrcl 11266 ax-mulcom 11267 ax-addass 11268 ax-mulass 11269 ax-distr 11270 ax-i2m1 11271 ax-1ne0 11272 ax-1rid 11273 ax-rnegex 11274 ax-rrecex 11275 ax-cnre 11276 ax-pre-lttri 11277 ax-pre-lttrn 11278 ax-pre-ltadd 11279 ax-pre-mulgt0 11280 ax-pre-sup 11281

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-op 4835 df-uni 4867 df-int 4907 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7878 df-1st 8001 df-2nd 8002 df-frecs 8302 df-wrecs 8333 df-recs 8382 df-rdg 8421 df-1o 8479 df-oadd 8483 df-er 8720 df-en 8977 df-dom 8978 df-sdom 8979 df-fin 8980 df-sup 9437 df-inf 9438 df-card 10023 df-pnf 11348 df-mnf 11349 df-xr 11350 df-ltxr 11351 df-le 11352 df-sub 11546 df-neg 11547 df-div 11977 df-nn 12339 df-2 12408 df-3 12409 df-n0 12610 df-xnn0 12683 df-z 12697 df-uz 12969 df-rp 13124 df-fz 13643 df-fzo 13792 df-seq 14148 df-exp 14208 df-hash 14478 df-cj 15271 df-re 15272 df-im 15273 df-sqrt 15407 df-abs 15408 df-dvds 16428 df-gcd 16670 df-phi 16948

This theorem is used by: phimullem 16961 eulerth 16965 hashgcdeq 16972 odngen 19796 znunithash 21875

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