dfphi2 - Intuitionistic Logic Explorer

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Theorem dfphi2 12750

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 9810	. 2 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 = 1 ∨ 𝑁 ∈ (ℤ_≥‘2)))
2		phi1 12749	. . . . 5 ⊢ (ϕ‘1) = 1
3		0z 9465	. . . . . 6 ⊢ 0 ∈ ℤ
4		hashsng 11028	. . . . . 6 ⊢ (0 ∈ ℤ → (♯‘{0}) = 1)
5	3, 4	ax-mp 5	. . . . 5 ⊢ (♯‘{0}) = 1
6		rabid2 2708	. . . . . . 7 ⊢ ({0} = {𝑥 ∈ {0} ∣ (𝑥 gcd 1) = 1} ↔ ∀𝑥 ∈ {0} (𝑥 gcd 1) = 1)
7		elsni 3684	. . . . . . . . 9 ⊢ (𝑥 ∈ {0} → 𝑥 = 0)
8	7	oveq1d 6022	. . . . . . . 8 ⊢ (𝑥 ∈ {0} → (𝑥 gcd 1) = (0 gcd 1))
9		gcd1 12516	. . . . . . . . 9 ⊢ (0 ∈ ℤ → (0 gcd 1) = 1)
10	3, 9	ax-mp 5	. . . . . . . 8 ⊢ (0 gcd 1) = 1
11	8, 10	eqtrdi 2278	. . . . . . 7 ⊢ (𝑥 ∈ {0} → (𝑥 gcd 1) = 1)
12	6, 11	mprgbir 2588	. . . . . 6 ⊢ {0} = {𝑥 ∈ {0} ∣ (𝑥 gcd 1) = 1}
13	12	fveq2i 5632	. . . . 5 ⊢ (♯‘{0}) = (♯‘{𝑥 ∈ {0} ∣ (𝑥 gcd 1) = 1})
14	2, 5, 13	3eqtr2i 2256	. . . 4 ⊢ (ϕ‘1) = (♯‘{𝑥 ∈ {0} ∣ (𝑥 gcd 1) = 1})
15		fveq2 5629	. . . 4 ⊢ (𝑁 = 1 → (ϕ‘𝑁) = (ϕ‘1))
16		oveq2 6015	. . . . . . 7 ⊢ (𝑁 = 1 → (0..^𝑁) = (0..^1))
17		fzo01 10430	. . . . . . 7 ⊢ (0..^1) = {0}
18	16, 17	eqtrdi 2278	. . . . . 6 ⊢ (𝑁 = 1 → (0..^𝑁) = {0})
19		oveq2 6015	. . . . . . 7 ⊢ (𝑁 = 1 → (𝑥 gcd 𝑁) = (𝑥 gcd 1))
20	19	eqeq1d 2238	. . . . . 6 ⊢ (𝑁 = 1 → ((𝑥 gcd 𝑁) = 1 ↔ (𝑥 gcd 1) = 1))
21	18, 20	rabeqbidv 2794	. . . . 5 ⊢ (𝑁 = 1 → {𝑥 ∈ (0..^𝑁) ∣ (𝑥 gcd 𝑁) = 1} = {𝑥 ∈ {0} ∣ (𝑥 gcd 1) = 1})
22	21	fveq2d 5633	. . . 4 ⊢ (𝑁 = 1 → (♯‘{𝑥 ∈ (0..^𝑁) ∣ (𝑥 gcd 𝑁) = 1}) = (♯‘{𝑥 ∈ {0} ∣ (𝑥 gcd 1) = 1}))
23	14, 15, 22	3eqtr4a 2288	. . 3 ⊢ (𝑁 = 1 → (ϕ‘𝑁) = (♯‘{𝑥 ∈ (0..^𝑁) ∣ (𝑥 gcd 𝑁) = 1}))
24		eluz2nn 9769	. . . . 5 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 𝑁 ∈ ℕ)
25		phival 12743	. . . . 5 ⊢ (𝑁 ∈ ℕ → (ϕ‘𝑁) = (♯‘{𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1}))
26	24, 25	syl 14	. . . 4 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (ϕ‘𝑁) = (♯‘{𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1}))
27		fzossfz 10370	. . . . . . . . . . 11 ⊢ (1..^𝑁) ⊆ (1...𝑁)
28	27	a1i 9	. . . . . . . . . 10 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (1..^𝑁) ⊆ (1...𝑁))
29		sseqin2 3423	. . . . . . . . . 10 ⊢ ((1..^𝑁) ⊆ (1...𝑁) ↔ ((1...𝑁) ∩ (1..^𝑁)) = (1..^𝑁))
30	28, 29	sylib 122	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ_≥‘2) → ((1...𝑁) ∩ (1..^𝑁)) = (1..^𝑁))
31		fzo0ss1 10380	. . . . . . . . . 10 ⊢ (1..^𝑁) ⊆ (0..^𝑁)
32		sseqin2 3423	. . . . . . . . . 10 ⊢ ((1..^𝑁) ⊆ (0..^𝑁) ↔ ((0..^𝑁) ∩ (1..^𝑁)) = (1..^𝑁))
33	31, 32	mpbi 145	. . . . . . . . 9 ⊢ ((0..^𝑁) ∩ (1..^𝑁)) = (1..^𝑁)
34	30, 33	eqtr4di 2280	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘2) → ((1...𝑁) ∩ (1..^𝑁)) = ((0..^𝑁) ∩ (1..^𝑁)))
35	34	rabeqdv 2793	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘2) → {𝑥 ∈ ((1...𝑁) ∩ (1..^𝑁)) ∣ (𝑥 gcd 𝑁) = 1} = {𝑥 ∈ ((0..^𝑁) ∩ (1..^𝑁)) ∣ (𝑥 gcd 𝑁) = 1})
36		inrab2 3477	. . . . . . 7 ⊢ ({𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1} ∩ (1..^𝑁)) = {𝑥 ∈ ((1...𝑁) ∩ (1..^𝑁)) ∣ (𝑥 gcd 𝑁) = 1}
37		inrab2 3477	. . . . . . 7 ⊢ ({𝑥 ∈ (0..^𝑁) ∣ (𝑥 gcd 𝑁) = 1} ∩ (1..^𝑁)) = {𝑥 ∈ ((0..^𝑁) ∩ (1..^𝑁)) ∣ (𝑥 gcd 𝑁) = 1}
38	35, 36, 37	3eqtr4g 2287	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘2) → ({𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1} ∩ (1..^𝑁)) = ({𝑥 ∈ (0..^𝑁) ∣ (𝑥 gcd 𝑁) = 1} ∩ (1..^𝑁)))
39		phibndlem 12746	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘2) → {𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1} ⊆ (1...(𝑁 − 1)))
40		eluzelz 9739	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 𝑁 ∈ ℤ)
41		fzoval 10352	. . . . . . . . 9 ⊢ (𝑁 ∈ ℤ → (1..^𝑁) = (1...(𝑁 − 1)))
42	40, 41	syl 14	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (1..^𝑁) = (1...(𝑁 − 1)))
43	39, 42	sseqtrrd 3263	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘2) → {𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1} ⊆ (1..^𝑁))
44		df-ss 3210	. . . . . . 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 122	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘2) → ({𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1} ∩ (1..^𝑁)) = {𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1})
46		gcd0id 12508	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℤ → (0 gcd 𝑁) = (abs‘𝑁))
47	40, 46	syl 14	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (0 gcd 𝑁) = (abs‘𝑁))
48		eluzelre 9740	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 𝑁 ∈ ℝ)
49		eluzge2nn0 9772	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 𝑁 ∈ ℕ₀)
50	49	nn0ge0d 9433	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 0 ≤ 𝑁)
51	48, 50	absidd 11686	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (abs‘𝑁) = 𝑁)
52	47, 51	eqtrd 2262	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (0 gcd 𝑁) = 𝑁)
53		eluz2b3 9807	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ (ℤ_≥‘2) ↔ (𝑁 ∈ ℕ ∧ 𝑁 ≠ 1))
54	53	simprbi 275	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 𝑁 ≠ 1)
55	52, 54	eqnetrd 2424	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (0 gcd 𝑁) ≠ 1)
56	55	adantr 276	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ (ℤ_≥‘2) ∧ 𝑥 ∈ (0..^𝑁)) → (0 gcd 𝑁) ≠ 1)
57	7	oveq1d 6022	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ {0} → (𝑥 gcd 𝑁) = (0 gcd 𝑁))
58	57, 17	eleq2s 2324	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (0..^1) → (𝑥 gcd 𝑁) = (0 gcd 𝑁))
59	58	neeq1d 2418	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (0..^1) → ((𝑥 gcd 𝑁) ≠ 1 ↔ (0 gcd 𝑁) ≠ 1))
60	56, 59	syl5ibrcom 157	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ (ℤ_≥‘2) ∧ 𝑥 ∈ (0..^𝑁)) → (𝑥 ∈ (0..^1) → (𝑥 gcd 𝑁) ≠ 1))
61	60	necon2bd 2458	. . . . . . . . . 10 ⊢ ((𝑁 ∈ (ℤ_≥‘2) ∧ 𝑥 ∈ (0..^𝑁)) → ((𝑥 gcd 𝑁) = 1 → ¬ 𝑥 ∈ (0..^1)))
62		simpr 110	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ (ℤ_≥‘2) ∧ 𝑥 ∈ (0..^𝑁)) → 𝑥 ∈ (0..^𝑁))
63		1z 9480	. . . . . . . . . . . 12 ⊢ 1 ∈ ℤ
64		fzospliti 10382	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (0..^𝑁) ∧ 1 ∈ ℤ) → (𝑥 ∈ (0..^1) ∨ 𝑥 ∈ (1..^𝑁)))
65	62, 63, 64	sylancl 413	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ (ℤ_≥‘2) ∧ 𝑥 ∈ (0..^𝑁)) → (𝑥 ∈ (0..^1) ∨ 𝑥 ∈ (1..^𝑁)))
66	65	ord 729	. . . . . . . . . 10 ⊢ ((𝑁 ∈ (ℤ_≥‘2) ∧ 𝑥 ∈ (0..^𝑁)) → (¬ 𝑥 ∈ (0..^1) → 𝑥 ∈ (1..^𝑁)))
67	61, 66	syld 45	. . . . . . . . 9 ⊢ ((𝑁 ∈ (ℤ_≥‘2) ∧ 𝑥 ∈ (0..^𝑁)) → ((𝑥 gcd 𝑁) = 1 → 𝑥 ∈ (1..^𝑁)))
68	67	ralrimiva 2603	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘2) → ∀𝑥 ∈ (0..^𝑁)((𝑥 gcd 𝑁) = 1 → 𝑥 ∈ (1..^𝑁)))
69		rabss 3301	. . . . . . . 8 ⊢ ({𝑥 ∈ (0..^𝑁) ∣ (𝑥 gcd 𝑁) = 1} ⊆ (1..^𝑁) ↔ ∀𝑥 ∈ (0..^𝑁)((𝑥 gcd 𝑁) = 1 → 𝑥 ∈ (1..^𝑁)))
70	68, 69	sylibr 134	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘2) → {𝑥 ∈ (0..^𝑁) ∣ (𝑥 gcd 𝑁) = 1} ⊆ (1..^𝑁))
71		df-ss 3210	. . . . . . 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 122	. . . . . 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 2270	. . . . 5 ⊢ (𝑁 ∈ (ℤ_≥‘2) → {𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1} = {𝑥 ∈ (0..^𝑁) ∣ (𝑥 gcd 𝑁) = 1})
74	73	fveq2d 5633	. . . 4 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (♯‘{𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1}) = (♯‘{𝑥 ∈ (0..^𝑁) ∣ (𝑥 gcd 𝑁) = 1}))
75	26, 74	eqtrd 2262	. . 3 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (ϕ‘𝑁) = (♯‘{𝑥 ∈ (0..^𝑁) ∣ (𝑥 gcd 𝑁) = 1}))
76	23, 75	jaoi 721	. 2 ⊢ ((𝑁 = 1 ∨ 𝑁 ∈ (ℤ_≥‘2)) → (ϕ‘𝑁) = (♯‘{𝑥 ∈ (0..^𝑁) ∣ (𝑥 gcd 𝑁) = 1}))
77	1, 76	sylbi 121	1 ⊢ (𝑁 ∈ ℕ → (ϕ‘𝑁) = (♯‘{𝑥 ∈ (0..^𝑁) ∣ (𝑥 gcd 𝑁) = 1}))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ∨ wo 713 = wceq 1395 ∈ wcel 2200 ≠ wne 2400 ∀wral 2508 {crab 2512 ∩ cin 3196 ⊆ wss 3197 {csn 3666 ‘cfv 5318 (class class class)co 6007 0cc0 8007 1c1 8008 − cmin 8325 ℕcn 9118 2c2 9169 ℤcz 9454 ℤ_≥cuz 9730 ...cfz 10212 ..^cfzo 10346 ♯chash 11005 abscabs 11516 gcd cgcd 12482 ϕcphi 12739

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4199 ax-sep 4202 ax-nul 4210 ax-pow 4258 ax-pr 4293 ax-un 4524 ax-setind 4629 ax-iinf 4680 ax-cnex 8098 ax-resscn 8099 ax-1cn 8100 ax-1re 8101 ax-icn 8102 ax-addcl 8103 ax-addrcl 8104 ax-mulcl 8105 ax-mulrcl 8106 ax-addcom 8107 ax-mulcom 8108 ax-addass 8109 ax-mulass 8110 ax-distr 8111 ax-i2m1 8112 ax-0lt1 8113 ax-1rid 8114 ax-0id 8115 ax-rnegex 8116 ax-precex 8117 ax-cnre 8118 ax-pre-ltirr 8119 ax-pre-ltwlin 8120 ax-pre-lttrn 8121 ax-pre-apti 8122 ax-pre-ltadd 8123 ax-pre-mulgt0 8124 ax-pre-mulext 8125 ax-arch 8126 ax-caucvg 8127

This theorem depends on definitions: df-bi 117 df-stab 836 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rmo 2516 df-rab 2517 df-v 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-nul 3492 df-if 3603 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3889 df-int 3924 df-iun 3967 df-br 4084 df-opab 4146 df-mpt 4147 df-tr 4183 df-id 4384 df-po 4387 df-iso 4388 df-iord 4457 df-on 4459 df-ilim 4460 df-suc 4462 df-iom 4683 df-xp 4725 df-rel 4726 df-cnv 4727 df-co 4728 df-dm 4729 df-rn 4730 df-res 4731 df-ima 4732 df-iota 5278 df-fun 5320 df-fn 5321 df-f 5322 df-f1 5323 df-fo 5324 df-f1o 5325 df-fv 5326 df-riota 5960 df-ov 6010 df-oprab 6011 df-mpo 6012 df-1st 6292 df-2nd 6293 df-recs 6457 df-frec 6543 df-1o 6568 df-er 6688 df-en 6896 df-dom 6897 df-fin 6898 df-sup 7159 df-pnf 8191 df-mnf 8192 df-xr 8193 df-ltxr 8194 df-le 8195 df-sub 8327 df-neg 8328 df-reap 8730 df-ap 8737 df-div 8828 df-inn 9119 df-2 9177 df-3 9178 df-4 9179 df-n0 9378 df-z 9455 df-uz 9731 df-q 9823 df-rp 9858 df-fz 10213 df-fzo 10347 df-fl 10498 df-mod 10553 df-seqfrec 10678 df-exp 10769 df-ihash 11006 df-cj 11361 df-re 11362 df-im 11363 df-rsqrt 11517 df-abs 11518 df-dvds 12307 df-gcd 12483 df-phi 12741

This theorem is referenced by: phimullem 12755 eulerth 12763 hashgcdeq 12770

W3C validator