dfphi2 - Metamath Proof Explorer

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Theorem dfphi2 16703

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 12905	. 2 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 = 1 ∨ 𝑁 ∈ (ℤ_≥‘2)))
2		phi1 16702	. . . . 5 ⊢ (ϕ‘1) = 1
3		0z 12565	. . . . . 6 ⊢ 0 ∈ ℤ
4		hashsng 14325	. . . . . 6 ⊢ (0 ∈ ℤ → (♯‘{0}) = 1)
5	3, 4	ax-mp 5	. . . . 5 ⊢ (♯‘{0}) = 1
6		rabid2 3464	. . . . . . 7 ⊢ ({0} = {𝑥 ∈ {0} ∣ (𝑥 gcd 1) = 1} ↔ ∀𝑥 ∈ {0} (𝑥 gcd 1) = 1)
7		elsni 4644	. . . . . . . . 9 ⊢ (𝑥 ∈ {0} → 𝑥 = 0)
8	7	oveq1d 7420	. . . . . . . 8 ⊢ (𝑥 ∈ {0} → (𝑥 gcd 1) = (0 gcd 1))
9		gcd1 16465	. . . . . . . . 9 ⊢ (0 ∈ ℤ → (0 gcd 1) = 1)
10	3, 9	ax-mp 5	. . . . . . . 8 ⊢ (0 gcd 1) = 1
11	8, 10	eqtrdi 2788	. . . . . . 7 ⊢ (𝑥 ∈ {0} → (𝑥 gcd 1) = 1)
12	6, 11	mprgbir 3068	. . . . . 6 ⊢ {0} = {𝑥 ∈ {0} ∣ (𝑥 gcd 1) = 1}
13	12	fveq2i 6891	. . . . 5 ⊢ (♯‘{0}) = (♯‘{𝑥 ∈ {0} ∣ (𝑥 gcd 1) = 1})
14	2, 5, 13	3eqtr2i 2766	. . . 4 ⊢ (ϕ‘1) = (♯‘{𝑥 ∈ {0} ∣ (𝑥 gcd 1) = 1})
15		fveq2 6888	. . . 4 ⊢ (𝑁 = 1 → (ϕ‘𝑁) = (ϕ‘1))
16		oveq2 7413	. . . . . . 7 ⊢ (𝑁 = 1 → (0..^𝑁) = (0..^1))
17		fzo01 13710	. . . . . . 7 ⊢ (0..^1) = {0}
18	16, 17	eqtrdi 2788	. . . . . 6 ⊢ (𝑁 = 1 → (0..^𝑁) = {0})
19		oveq2 7413	. . . . . . 7 ⊢ (𝑁 = 1 → (𝑥 gcd 𝑁) = (𝑥 gcd 1))
20	19	eqeq1d 2734	. . . . . 6 ⊢ (𝑁 = 1 → ((𝑥 gcd 𝑁) = 1 ↔ (𝑥 gcd 1) = 1))
21	18, 20	rabeqbidv 3449	. . . . 5 ⊢ (𝑁 = 1 → {𝑥 ∈ (0..^𝑁) ∣ (𝑥 gcd 𝑁) = 1} = {𝑥 ∈ {0} ∣ (𝑥 gcd 1) = 1})
22	21	fveq2d 6892	. . . 4 ⊢ (𝑁 = 1 → (♯‘{𝑥 ∈ (0..^𝑁) ∣ (𝑥 gcd 𝑁) = 1}) = (♯‘{𝑥 ∈ {0} ∣ (𝑥 gcd 1) = 1}))
23	14, 15, 22	3eqtr4a 2798	. . 3 ⊢ (𝑁 = 1 → (ϕ‘𝑁) = (♯‘{𝑥 ∈ (0..^𝑁) ∣ (𝑥 gcd 𝑁) = 1}))
24		eluz2nn 12864	. . . . 5 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 𝑁 ∈ ℕ)
25		phival 16696	. . . . 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 13647	. . . . . . . . . . 11 ⊢ (1..^𝑁) ⊆ (1...𝑁)
28	27	a1i 11	. . . . . . . . . 10 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (1..^𝑁) ⊆ (1...𝑁))
29		sseqin2 4214	. . . . . . . . . 10 ⊢ ((1..^𝑁) ⊆ (1...𝑁) ↔ ((1...𝑁) ∩ (1..^𝑁)) = (1..^𝑁))
30	28, 29	sylib 217	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ_≥‘2) → ((1...𝑁) ∩ (1..^𝑁)) = (1..^𝑁))
31		fzo0ss1 13658	. . . . . . . . . 10 ⊢ (1..^𝑁) ⊆ (0..^𝑁)
32		sseqin2 4214	. . . . . . . . . 10 ⊢ ((1..^𝑁) ⊆ (0..^𝑁) ↔ ((0..^𝑁) ∩ (1..^𝑁)) = (1..^𝑁))
33	31, 32	mpbi 229	. . . . . . . . 9 ⊢ ((0..^𝑁) ∩ (1..^𝑁)) = (1..^𝑁)
34	30, 33	eqtr4di 2790	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘2) → ((1...𝑁) ∩ (1..^𝑁)) = ((0..^𝑁) ∩ (1..^𝑁)))
35	34	rabeqdv 3447	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘2) → {𝑥 ∈ ((1...𝑁) ∩ (1..^𝑁)) ∣ (𝑥 gcd 𝑁) = 1} = {𝑥 ∈ ((0..^𝑁) ∩ (1..^𝑁)) ∣ (𝑥 gcd 𝑁) = 1})
36		inrab2 4306	. . . . . . 7 ⊢ ({𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1} ∩ (1..^𝑁)) = {𝑥 ∈ ((1...𝑁) ∩ (1..^𝑁)) ∣ (𝑥 gcd 𝑁) = 1}
37		inrab2 4306	. . . . . . 7 ⊢ ({𝑥 ∈ (0..^𝑁) ∣ (𝑥 gcd 𝑁) = 1} ∩ (1..^𝑁)) = {𝑥 ∈ ((0..^𝑁) ∩ (1..^𝑁)) ∣ (𝑥 gcd 𝑁) = 1}
38	35, 36, 37	3eqtr4g 2797	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘2) → ({𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1} ∩ (1..^𝑁)) = ({𝑥 ∈ (0..^𝑁) ∣ (𝑥 gcd 𝑁) = 1} ∩ (1..^𝑁)))
39		phibndlem 16699	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘2) → {𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1} ⊆ (1...(𝑁 − 1)))
40		eluzelz 12828	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 𝑁 ∈ ℤ)
41		fzoval 13629	. . . . . . . . 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 4022	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘2) → {𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1} ⊆ (1..^𝑁))
44		df-ss 3964	. . . . . . 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 217	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘2) → ({𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1} ∩ (1..^𝑁)) = {𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1})
46		gcd0id 16456	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℤ → (0 gcd 𝑁) = (abs‘𝑁))
47	40, 46	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (0 gcd 𝑁) = (abs‘𝑁))
48		eluzelre 12829	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 𝑁 ∈ ℝ)
49		eluzge2nn0 12867	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 𝑁 ∈ ℕ₀)
50	49	nn0ge0d 12531	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 0 ≤ 𝑁)
51	48, 50	absidd 15365	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (abs‘𝑁) = 𝑁)
52	47, 51	eqtrd 2772	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (0 gcd 𝑁) = 𝑁)
53		eluz2b3 12902	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ (ℤ_≥‘2) ↔ (𝑁 ∈ ℕ ∧ 𝑁 ≠ 1))
54	53	simprbi 497	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 𝑁 ≠ 1)
55	52, 54	eqnetrd 3008	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (0 gcd 𝑁) ≠ 1)
56	55	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ (ℤ_≥‘2) ∧ 𝑥 ∈ (0..^𝑁)) → (0 gcd 𝑁) ≠ 1)
57	7	oveq1d 7420	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ {0} → (𝑥 gcd 𝑁) = (0 gcd 𝑁))
58	57, 17	eleq2s 2851	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (0..^1) → (𝑥 gcd 𝑁) = (0 gcd 𝑁))
59	58	neeq1d 3000	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (0..^1) → ((𝑥 gcd 𝑁) ≠ 1 ↔ (0 gcd 𝑁) ≠ 1))
60	56, 59	syl5ibrcom 246	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ (ℤ_≥‘2) ∧ 𝑥 ∈ (0..^𝑁)) → (𝑥 ∈ (0..^1) → (𝑥 gcd 𝑁) ≠ 1))
61	60	necon2bd 2956	. . . . . . . . . 10 ⊢ ((𝑁 ∈ (ℤ_≥‘2) ∧ 𝑥 ∈ (0..^𝑁)) → ((𝑥 gcd 𝑁) = 1 → ¬ 𝑥 ∈ (0..^1)))
62		simpr 485	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ (ℤ_≥‘2) ∧ 𝑥 ∈ (0..^𝑁)) → 𝑥 ∈ (0..^𝑁))
63		1z 12588	. . . . . . . . . . . 12 ⊢ 1 ∈ ℤ
64		fzospliti 13660	. . . . . . . . . . . 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 862	. . . . . . . . . 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 3146	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘2) → ∀𝑥 ∈ (0..^𝑁)((𝑥 gcd 𝑁) = 1 → 𝑥 ∈ (1..^𝑁)))
69		rabss 4068	. . . . . . . 8 ⊢ ({𝑥 ∈ (0..^𝑁) ∣ (𝑥 gcd 𝑁) = 1} ⊆ (1..^𝑁) ↔ ∀𝑥 ∈ (0..^𝑁)((𝑥 gcd 𝑁) = 1 → 𝑥 ∈ (1..^𝑁)))
70	68, 69	sylibr 233	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘2) → {𝑥 ∈ (0..^𝑁) ∣ (𝑥 gcd 𝑁) = 1} ⊆ (1..^𝑁))
71		df-ss 3964	. . . . . . 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 217	. . . . . 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 2780	. . . . 5 ⊢ (𝑁 ∈ (ℤ_≥‘2) → {𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1} = {𝑥 ∈ (0..^𝑁) ∣ (𝑥 gcd 𝑁) = 1})
74	73	fveq2d 6892	. . . 4 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (♯‘{𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1}) = (♯‘{𝑥 ∈ (0..^𝑁) ∣ (𝑥 gcd 𝑁) = 1}))
75	26, 74	eqtrd 2772	. . 3 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (ϕ‘𝑁) = (♯‘{𝑥 ∈ (0..^𝑁) ∣ (𝑥 gcd 𝑁) = 1}))
76	23, 75	jaoi 855	. 2 ⊢ ((𝑁 = 1 ∨ 𝑁 ∈ (ℤ_≥‘2)) → (ϕ‘𝑁) = (♯‘{𝑥 ∈ (0..^𝑁) ∣ (𝑥 gcd 𝑁) = 1}))
77	1, 76	sylbi 216	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 396 ∨ wo 845 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ∀wral 3061 {crab 3432 ∩ cin 3946 ⊆ wss 3947 {csn 4627 ‘cfv 6540 (class class class)co 7405 0cc0 11106 1c1 11107 − cmin 11440 ℕcn 12208 2c2 12263 ℤcz 12554 ℤ_≥cuz 12818 ...cfz 13480 ..^cfzo 13623 ♯chash 14286 abscabs 15177 gcd cgcd 16431 ϕcphi 16693

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-oadd 8466 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-sup 9433 df-inf 9434 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-n0 12469 df-xnn0 12541 df-z 12555 df-uz 12819 df-rp 12971 df-fz 13481 df-fzo 13624 df-seq 13963 df-exp 14024 df-hash 14287 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-dvds 16194 df-gcd 16432 df-phi 16695

This theorem is referenced by: phimullem 16708 eulerth 16712 hashgcdeq 16718 odngen 19439 znunithash 21111

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