dfphi2 - Metamath Proof Explorer

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Theorem dfphi2 16713

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 12915	. 2 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 = 1 ∨ 𝑁 ∈ (ℤ_≥‘2)))
2		phi1 16712	. . . . 5 ⊢ (ϕ‘1) = 1
3		0z 12575	. . . . . 6 ⊢ 0 ∈ ℤ
4		hashsng 14335	. . . . . 6 ⊢ (0 ∈ ℤ → (♯‘{0}) = 1)
5	3, 4	ax-mp 5	. . . . 5 ⊢ (♯‘{0}) = 1
6		rabid2 3462	. . . . . . 7 ⊢ ({0} = {𝑥 ∈ {0} ∣ (𝑥 gcd 1) = 1} ↔ ∀𝑥 ∈ {0} (𝑥 gcd 1) = 1)
7		elsni 4646	. . . . . . . . 9 ⊢ (𝑥 ∈ {0} → 𝑥 = 0)
8	7	oveq1d 7428	. . . . . . . 8 ⊢ (𝑥 ∈ {0} → (𝑥 gcd 1) = (0 gcd 1))
9		gcd1 16475	. . . . . . . . 9 ⊢ (0 ∈ ℤ → (0 gcd 1) = 1)
10	3, 9	ax-mp 5	. . . . . . . 8 ⊢ (0 gcd 1) = 1
11	8, 10	eqtrdi 2786	. . . . . . 7 ⊢ (𝑥 ∈ {0} → (𝑥 gcd 1) = 1)
12	6, 11	mprgbir 3066	. . . . . 6 ⊢ {0} = {𝑥 ∈ {0} ∣ (𝑥 gcd 1) = 1}
13	12	fveq2i 6895	. . . . 5 ⊢ (♯‘{0}) = (♯‘{𝑥 ∈ {0} ∣ (𝑥 gcd 1) = 1})
14	2, 5, 13	3eqtr2i 2764	. . . 4 ⊢ (ϕ‘1) = (♯‘{𝑥 ∈ {0} ∣ (𝑥 gcd 1) = 1})
15		fveq2 6892	. . . 4 ⊢ (𝑁 = 1 → (ϕ‘𝑁) = (ϕ‘1))
16		oveq2 7421	. . . . . . 7 ⊢ (𝑁 = 1 → (0..^𝑁) = (0..^1))
17		fzo01 13720	. . . . . . 7 ⊢ (0..^1) = {0}
18	16, 17	eqtrdi 2786	. . . . . 6 ⊢ (𝑁 = 1 → (0..^𝑁) = {0})
19		oveq2 7421	. . . . . . 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 3447	. . . . 5 ⊢ (𝑁 = 1 → {𝑥 ∈ (0..^𝑁) ∣ (𝑥 gcd 𝑁) = 1} = {𝑥 ∈ {0} ∣ (𝑥 gcd 1) = 1})
22	21	fveq2d 6896	. . . 4 ⊢ (𝑁 = 1 → (♯‘{𝑥 ∈ (0..^𝑁) ∣ (𝑥 gcd 𝑁) = 1}) = (♯‘{𝑥 ∈ {0} ∣ (𝑥 gcd 1) = 1}))
23	14, 15, 22	3eqtr4a 2796	. . 3 ⊢ (𝑁 = 1 → (ϕ‘𝑁) = (♯‘{𝑥 ∈ (0..^𝑁) ∣ (𝑥 gcd 𝑁) = 1}))
24		eluz2nn 12874	. . . . 5 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 𝑁 ∈ ℕ)
25		phival 16706	. . . . 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 13657	. . . . . . . . . . 11 ⊢ (1..^𝑁) ⊆ (1...𝑁)
28	27	a1i 11	. . . . . . . . . 10 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (1..^𝑁) ⊆ (1...𝑁))
29		sseqin2 4216	. . . . . . . . . 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 13668	. . . . . . . . . 10 ⊢ (1..^𝑁) ⊆ (0..^𝑁)
32		sseqin2 4216	. . . . . . . . . 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 2788	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘2) → ((1...𝑁) ∩ (1..^𝑁)) = ((0..^𝑁) ∩ (1..^𝑁)))
35	34	rabeqdv 3445	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘2) → {𝑥 ∈ ((1...𝑁) ∩ (1..^𝑁)) ∣ (𝑥 gcd 𝑁) = 1} = {𝑥 ∈ ((0..^𝑁) ∩ (1..^𝑁)) ∣ (𝑥 gcd 𝑁) = 1})
36		inrab2 4308	. . . . . . 7 ⊢ ({𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1} ∩ (1..^𝑁)) = {𝑥 ∈ ((1...𝑁) ∩ (1..^𝑁)) ∣ (𝑥 gcd 𝑁) = 1}
37		inrab2 4308	. . . . . . 7 ⊢ ({𝑥 ∈ (0..^𝑁) ∣ (𝑥 gcd 𝑁) = 1} ∩ (1..^𝑁)) = {𝑥 ∈ ((0..^𝑁) ∩ (1..^𝑁)) ∣ (𝑥 gcd 𝑁) = 1}
38	35, 36, 37	3eqtr4g 2795	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘2) → ({𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1} ∩ (1..^𝑁)) = ({𝑥 ∈ (0..^𝑁) ∣ (𝑥 gcd 𝑁) = 1} ∩ (1..^𝑁)))
39		phibndlem 16709	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘2) → {𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1} ⊆ (1...(𝑁 − 1)))
40		eluzelz 12838	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 𝑁 ∈ ℤ)
41		fzoval 13639	. . . . . . . . 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 4024	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘2) → {𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1} ⊆ (1..^𝑁))
44		df-ss 3966	. . . . . . 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 16466	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℤ → (0 gcd 𝑁) = (abs‘𝑁))
47	40, 46	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (0 gcd 𝑁) = (abs‘𝑁))
48		eluzelre 12839	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 𝑁 ∈ ℝ)
49		eluzge2nn0 12877	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 𝑁 ∈ ℕ₀)
50	49	nn0ge0d 12541	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 0 ≤ 𝑁)
51	48, 50	absidd 15375	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (abs‘𝑁) = 𝑁)
52	47, 51	eqtrd 2770	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (0 gcd 𝑁) = 𝑁)
53		eluz2b3 12912	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ (ℤ_≥‘2) ↔ (𝑁 ∈ ℕ ∧ 𝑁 ≠ 1))
54	53	simprbi 495	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 𝑁 ≠ 1)
55	52, 54	eqnetrd 3006	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (0 gcd 𝑁) ≠ 1)
56	55	adantr 479	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ (ℤ_≥‘2) ∧ 𝑥 ∈ (0..^𝑁)) → (0 gcd 𝑁) ≠ 1)
57	7	oveq1d 7428	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ {0} → (𝑥 gcd 𝑁) = (0 gcd 𝑁))
58	57, 17	eleq2s 2849	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (0..^1) → (𝑥 gcd 𝑁) = (0 gcd 𝑁))
59	58	neeq1d 2998	. . . . . . . . . . . 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 2954	. . . . . . . . . 10 ⊢ ((𝑁 ∈ (ℤ_≥‘2) ∧ 𝑥 ∈ (0..^𝑁)) → ((𝑥 gcd 𝑁) = 1 → ¬ 𝑥 ∈ (0..^1)))
62		simpr 483	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ (ℤ_≥‘2) ∧ 𝑥 ∈ (0..^𝑁)) → 𝑥 ∈ (0..^𝑁))
63		1z 12598	. . . . . . . . . . . 12 ⊢ 1 ∈ ℤ
64		fzospliti 13670	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (0..^𝑁) ∧ 1 ∈ ℤ) → (𝑥 ∈ (0..^1) ∨ 𝑥 ∈ (1..^𝑁)))
65	62, 63, 64	sylancl 584	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ (ℤ_≥‘2) ∧ 𝑥 ∈ (0..^𝑁)) → (𝑥 ∈ (0..^1) ∨ 𝑥 ∈ (1..^𝑁)))
66	65	ord 860	. . . . . . . . . 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 3144	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘2) → ∀𝑥 ∈ (0..^𝑁)((𝑥 gcd 𝑁) = 1 → 𝑥 ∈ (1..^𝑁)))
69		rabss 4070	. . . . . . . 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 3966	. . . . . . 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 2778	. . . . 5 ⊢ (𝑁 ∈ (ℤ_≥‘2) → {𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1} = {𝑥 ∈ (0..^𝑁) ∣ (𝑥 gcd 𝑁) = 1})
74	73	fveq2d 6896	. . . 4 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (♯‘{𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1}) = (♯‘{𝑥 ∈ (0..^𝑁) ∣ (𝑥 gcd 𝑁) = 1}))
75	26, 74	eqtrd 2770	. . 3 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (ϕ‘𝑁) = (♯‘{𝑥 ∈ (0..^𝑁) ∣ (𝑥 gcd 𝑁) = 1}))
76	23, 75	jaoi 853	. 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 394 ∨ wo 843 = wceq 1539 ∈ wcel 2104 ≠ wne 2938 ∀wral 3059 {crab 3430 ∩ cin 3948 ⊆ wss 3949 {csn 4629 ‘cfv 6544 (class class class)co 7413 0cc0 11114 1c1 11115 − cmin 11450 ℕcn 12218 2c2 12273 ℤcz 12564 ℤ_≥cuz 12828 ...cfz 13490 ..^cfzo 13633 ♯chash 14296 abscabs 15187 gcd cgcd 16441 ϕcphi 16703

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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7729 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191 ax-pre-sup 11192

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-1o 8470 df-oadd 8474 df-er 8707 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-sup 9441 df-inf 9442 df-card 9938 df-pnf 11256 df-mnf 11257 df-xr 11258 df-ltxr 11259 df-le 11260 df-sub 11452 df-neg 11453 df-div 11878 df-nn 12219 df-2 12281 df-3 12282 df-n0 12479 df-xnn0 12551 df-z 12565 df-uz 12829 df-rp 12981 df-fz 13491 df-fzo 13634 df-seq 13973 df-exp 14034 df-hash 14297 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-dvds 16204 df-gcd 16442 df-phi 16705

This theorem is referenced by: phimullem 16718 eulerth 16722 hashgcdeq 16728 odngen 19488 znunithash 21341

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