Theorem phisum 12119

Description: The divisor sum identity of the totient function. Theorem 2.2 in [ApostolNT] p. 26. (Contributed by Stefan O'Rear, 12-Sep-2015.)

Assertion

Ref	Expression
phisum	⊢ (𝑁 ∈ ℕ → Σ𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} (ϕ‘𝑑) = 𝑁)

Distinct variable group:   𝑥,𝑁,𝑑

Proof of Theorem phisum

Dummy variables 𝑧 𝑦 𝑤 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq1 3969	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ∥ 𝑁 ↔ 𝑦 ∥ 𝑁))
2	1	elrab 2868	. . . . 5 ⊢ (𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} ↔ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝑁))
3		hashgcdeq 12118	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (♯‘{𝑧 ∈ (0..^𝑁) ∣ (𝑧 gcd 𝑁) = 𝑦}) = if(𝑦 ∥ 𝑁, (ϕ‘(𝑁 / 𝑦)), 0))
4	3	adantrr 471	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝑁)) → (♯‘{𝑧 ∈ (0..^𝑁) ∣ (𝑧 gcd 𝑁) = 𝑦}) = if(𝑦 ∥ 𝑁, (ϕ‘(𝑁 / 𝑦)), 0))
5		iftrue 3510	. . . . . . 7 ⊢ (𝑦 ∥ 𝑁 → if(𝑦 ∥ 𝑁, (ϕ‘(𝑁 / 𝑦)), 0) = (ϕ‘(𝑁 / 𝑦)))
6	5	ad2antll 483	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝑁)) → if(𝑦 ∥ 𝑁, (ϕ‘(𝑁 / 𝑦)), 0) = (ϕ‘(𝑁 / 𝑦)))
7	4, 6	eqtrd 2190	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝑁)) → (♯‘{𝑧 ∈ (0..^𝑁) ∣ (𝑧 gcd 𝑁) = 𝑦}) = (ϕ‘(𝑁 / 𝑦)))
8	2, 7	sylan2b 285	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) → (♯‘{𝑧 ∈ (0..^𝑁) ∣ (𝑧 gcd 𝑁) = 𝑦}) = (ϕ‘(𝑁 / 𝑦)))
9	8	sumeq2dv 11269	. . 3 ⊢ (𝑁 ∈ ℕ → Σ𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} (♯‘{𝑧 ∈ (0..^𝑁) ∣ (𝑧 gcd 𝑁) = 𝑦}) = Σ𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} (ϕ‘(𝑁 / 𝑦)))
10		1zzd 9195	. . . . . 6 ⊢ (𝑁 ∈ ℕ → 1 ∈ ℤ)
11		nnz 9187	. . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
12	10, 11	fzfigd 10334	. . . . 5 ⊢ (𝑁 ∈ ℕ → (1...𝑁) ∈ Fin)
13		dvdsssfz1 11748	. . . . 5 ⊢ (𝑁 ∈ ℕ → {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} ⊆ (1...𝑁))
14		elfznn 9957	. . . . . . . 8 ⊢ (𝑗 ∈ (1...𝑁) → 𝑗 ∈ ℕ)
15		dvdsdc 11698	. . . . . . . 8 ⊢ ((𝑗 ∈ ℕ ∧ 𝑁 ∈ ℤ) → DECID 𝑗 ∥ 𝑁)
16	14, 11, 15	syl2anr 288	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (1...𝑁)) → DECID 𝑗 ∥ 𝑁)
17		ibar 299	. . . . . . . . . . 11 ⊢ (𝑗 ∈ ℕ → (𝑗 ∥ 𝑁 ↔ (𝑗 ∈ ℕ ∧ 𝑗 ∥ 𝑁)))
18	14, 17	syl 14	. . . . . . . . . 10 ⊢ (𝑗 ∈ (1...𝑁) → (𝑗 ∥ 𝑁 ↔ (𝑗 ∈ ℕ ∧ 𝑗 ∥ 𝑁)))
19		breq1 3969	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑗 → (𝑥 ∥ 𝑁 ↔ 𝑗 ∥ 𝑁))
20	19	elrab 2868	. . . . . . . . . 10 ⊢ (𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} ↔ (𝑗 ∈ ℕ ∧ 𝑗 ∥ 𝑁))
21	18, 20	bitr4di 197	. . . . . . . . 9 ⊢ (𝑗 ∈ (1...𝑁) → (𝑗 ∥ 𝑁 ↔ 𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}))
22	21	dcbid 824	. . . . . . . 8 ⊢ (𝑗 ∈ (1...𝑁) → (DECID 𝑗 ∥ 𝑁 ↔ DECID 𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}))
23	22	adantl 275	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (1...𝑁)) → (DECID 𝑗 ∥ 𝑁 ↔ DECID 𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}))
24	16, 23	mpbid 146	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (1...𝑁)) → DECID 𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁})
25	24	ralrimiva 2530	. . . . 5 ⊢ (𝑁 ∈ ℕ → ∀𝑗 ∈ (1...𝑁)DECID 𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁})
26		ssfidc 6880	. . . . 5 ⊢ (((1...𝑁) ∈ Fin ∧ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} ⊆ (1...𝑁) ∧ ∀𝑗 ∈ (1...𝑁)DECID 𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) → {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} ∈ Fin)
27	12, 13, 25, 26	syl3anc 1220	. . . 4 ⊢ (𝑁 ∈ ℕ → {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} ∈ Fin)
28		0z 9179	. . . . . . 7 ⊢ 0 ∈ ℤ
29		fzofig 10335	. . . . . . 7 ⊢ ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (0..^𝑁) ∈ Fin)
30	28, 11, 29	sylancr 411	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (0..^𝑁) ∈ Fin)
31	30	adantr 274	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) → (0..^𝑁) ∈ Fin)
32		ssrab2 3213	. . . . . 6 ⊢ {𝑧 ∈ (0..^𝑁) ∣ (𝑧 gcd 𝑁) = 𝑦} ⊆ (0..^𝑁)
33	32	a1i 9	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) → {𝑧 ∈ (0..^𝑁) ∣ (𝑧 gcd 𝑁) = 𝑦} ⊆ (0..^𝑁))
34		elfzoelz 10050	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (0..^𝑁) → 𝑗 ∈ ℤ)
35	34	adantl 275	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) ∧ 𝑗 ∈ (0..^𝑁)) → 𝑗 ∈ ℤ)
36	11	ad2antrr 480	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) ∧ 𝑗 ∈ (0..^𝑁)) → 𝑁 ∈ ℤ)
37	35, 36	gcdcld 11856	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) ∧ 𝑗 ∈ (0..^𝑁)) → (𝑗 gcd 𝑁) ∈ ℕ0)
38	37	nn0zd 9285	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) ∧ 𝑗 ∈ (0..^𝑁)) → (𝑗 gcd 𝑁) ∈ ℤ)
39		elrabi 2865	. . . . . . . . . 10 ⊢ (𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} → 𝑦 ∈ ℕ)
40	39	ad2antlr 481	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) ∧ 𝑗 ∈ (0..^𝑁)) → 𝑦 ∈ ℕ)
41	40	nnzd 9286	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) ∧ 𝑗 ∈ (0..^𝑁)) → 𝑦 ∈ ℤ)
42		zdceq 9240	. . . . . . . 8 ⊢ (((𝑗 gcd 𝑁) ∈ ℤ ∧ 𝑦 ∈ ℤ) → DECID (𝑗 gcd 𝑁) = 𝑦)
43	38, 41, 42	syl2anc 409	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) ∧ 𝑗 ∈ (0..^𝑁)) → DECID (𝑗 gcd 𝑁) = 𝑦)
44		ibar 299	. . . . . . . . . 10 ⊢ (𝑗 ∈ (0..^𝑁) → ((𝑗 gcd 𝑁) = 𝑦 ↔ (𝑗 ∈ (0..^𝑁) ∧ (𝑗 gcd 𝑁) = 𝑦)))
45		oveq1 5832	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑗 → (𝑧 gcd 𝑁) = (𝑗 gcd 𝑁))
46	45	eqeq1d 2166	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑗 → ((𝑧 gcd 𝑁) = 𝑦 ↔ (𝑗 gcd 𝑁) = 𝑦))
47	46	elrab 2868	. . . . . . . . . 10 ⊢ (𝑗 ∈ {𝑧 ∈ (0..^𝑁) ∣ (𝑧 gcd 𝑁) = 𝑦} ↔ (𝑗 ∈ (0..^𝑁) ∧ (𝑗 gcd 𝑁) = 𝑦))
48	44, 47	bitr4di 197	. . . . . . . . 9 ⊢ (𝑗 ∈ (0..^𝑁) → ((𝑗 gcd 𝑁) = 𝑦 ↔ 𝑗 ∈ {𝑧 ∈ (0..^𝑁) ∣ (𝑧 gcd 𝑁) = 𝑦}))
49	48	dcbid 824	. . . . . . . 8 ⊢ (𝑗 ∈ (0..^𝑁) → (DECID (𝑗 gcd 𝑁) = 𝑦 ↔ DECID 𝑗 ∈ {𝑧 ∈ (0..^𝑁) ∣ (𝑧 gcd 𝑁) = 𝑦}))
50	49	adantl 275	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) ∧ 𝑗 ∈ (0..^𝑁)) → (DECID (𝑗 gcd 𝑁) = 𝑦 ↔ DECID 𝑗 ∈ {𝑧 ∈ (0..^𝑁) ∣ (𝑧 gcd 𝑁) = 𝑦}))
51	43, 50	mpbid 146	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) ∧ 𝑗 ∈ (0..^𝑁)) → DECID 𝑗 ∈ {𝑧 ∈ (0..^𝑁) ∣ (𝑧 gcd 𝑁) = 𝑦})
52	51	ralrimiva 2530	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) → ∀𝑗 ∈ (0..^𝑁)DECID 𝑗 ∈ {𝑧 ∈ (0..^𝑁) ∣ (𝑧 gcd 𝑁) = 𝑦})
53		ssfidc 6880	. . . . 5 ⊢ (((0..^𝑁) ∈ Fin ∧ {𝑧 ∈ (0..^𝑁) ∣ (𝑧 gcd 𝑁) = 𝑦} ⊆ (0..^𝑁) ∧ ∀𝑗 ∈ (0..^𝑁)DECID 𝑗 ∈ {𝑧 ∈ (0..^𝑁) ∣ (𝑧 gcd 𝑁) = 𝑦}) → {𝑧 ∈ (0..^𝑁) ∣ (𝑧 gcd 𝑁) = 𝑦} ∈ Fin)
54	31, 33, 52, 53	syl3anc 1220	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) → {𝑧 ∈ (0..^𝑁) ∣ (𝑧 gcd 𝑁) = 𝑦} ∈ Fin)
55		oveq1 5832	. . . . . . . . . 10 ⊢ (𝑧 = 𝑤 → (𝑧 gcd 𝑁) = (𝑤 gcd 𝑁))
56	55	eqeq1d 2166	. . . . . . . . 9 ⊢ (𝑧 = 𝑤 → ((𝑧 gcd 𝑁) = 𝑦 ↔ (𝑤 gcd 𝑁) = 𝑦))
57	56	elrab 2868	. . . . . . . 8 ⊢ (𝑤 ∈ {𝑧 ∈ (0..^𝑁) ∣ (𝑧 gcd 𝑁) = 𝑦} ↔ (𝑤 ∈ (0..^𝑁) ∧ (𝑤 gcd 𝑁) = 𝑦))
58	57	simprbi 273	. . . . . . 7 ⊢ (𝑤 ∈ {𝑧 ∈ (0..^𝑁) ∣ (𝑧 gcd 𝑁) = 𝑦} → (𝑤 gcd 𝑁) = 𝑦)
59	58	rgen 2510	. . . . . 6 ⊢ ∀𝑤 ∈ {𝑧 ∈ (0..^𝑁) ∣ (𝑧 gcd 𝑁) = 𝑦} (𝑤 gcd 𝑁) = 𝑦
60	59	rgenw 2512	. . . . 5 ⊢ ∀𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}∀𝑤 ∈ {𝑧 ∈ (0..^𝑁) ∣ (𝑧 gcd 𝑁) = 𝑦} (𝑤 gcd 𝑁) = 𝑦
61		invdisj 3960	. . . . 5 ⊢ (∀𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}∀𝑤 ∈ {𝑧 ∈ (0..^𝑁) ∣ (𝑧 gcd 𝑁) = 𝑦} (𝑤 gcd 𝑁) = 𝑦 → Disj 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} {𝑧 ∈ (0..^𝑁) ∣ (𝑧 gcd 𝑁) = 𝑦})
62	60, 61	mp1i 10	. . . 4 ⊢ (𝑁 ∈ ℕ → Disj 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} {𝑧 ∈ (0..^𝑁) ∣ (𝑧 gcd 𝑁) = 𝑦})
63	27, 54, 62	hashiun 11379	. . 3 ⊢ (𝑁 ∈ ℕ → (♯‘∪ 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} {𝑧 ∈ (0..^𝑁) ∣ (𝑧 gcd 𝑁) = 𝑦}) = Σ𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} (♯‘{𝑧 ∈ (0..^𝑁) ∣ (𝑧 gcd 𝑁) = 𝑦}))
64		fveq2 5469	. . . 4 ⊢ (𝑑 = (𝑁 / 𝑦) → (ϕ‘𝑑) = (ϕ‘(𝑁 / 𝑦)))
65		eqid 2157	. . . . 5 ⊢ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} = {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}
66		eqid 2157	. . . . 5 ⊢ (𝑧 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} ↦ (𝑁 / 𝑧)) = (𝑧 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} ↦ (𝑁 / 𝑧))
67	65, 66	dvdsflip 11747	. . . 4 ⊢ (𝑁 ∈ ℕ → (𝑧 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} ↦ (𝑁 / 𝑧)):{𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}–1-1-onto→{𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁})
68		oveq2 5833	. . . . 5 ⊢ (𝑧 = 𝑦 → (𝑁 / 𝑧) = (𝑁 / 𝑦))
69		simpr 109	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) → 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁})
70	11	adantr 274	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) → 𝑁 ∈ ℤ)
71	39	adantl 275	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) → 𝑦 ∈ ℕ)
72		znq 9534	. . . . . 6 ⊢ ((𝑁 ∈ ℤ ∧ 𝑦 ∈ ℕ) → (𝑁 / 𝑦) ∈ ℚ)
73	70, 71, 72	syl2anc 409	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) → (𝑁 / 𝑦) ∈ ℚ)
74	66, 68, 69, 73	fvmptd3 5562	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) → ((𝑧 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} ↦ (𝑁 / 𝑧))‘𝑦) = (𝑁 / 𝑦))
75		elrabi 2865	. . . . . . 7 ⊢ (𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} → 𝑑 ∈ ℕ)
76	75	adantl 275	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) → 𝑑 ∈ ℕ)
77	76	phicld 12097	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) → (ϕ‘𝑑) ∈ ℕ)
78	77	nncnd 8848	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) → (ϕ‘𝑑) ∈ ℂ)
79	64, 27, 67, 74, 78	fsumf1o 11291	. . 3 ⊢ (𝑁 ∈ ℕ → Σ𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} (ϕ‘𝑑) = Σ𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} (ϕ‘(𝑁 / 𝑦)))
80	9, 63, 79	3eqtr4rd 2201	. 2 ⊢ (𝑁 ∈ ℕ → Σ𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} (ϕ‘𝑑) = (♯‘∪ 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} {𝑧 ∈ (0..^𝑁) ∣ (𝑧 gcd 𝑁) = 𝑦}))
81		iunrab 3897	. . . . 5 ⊢ ∪ 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} {𝑧 ∈ (0..^𝑁) ∣ (𝑧 gcd 𝑁) = 𝑦} = {𝑧 ∈ (0..^𝑁) ∣ ∃𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} (𝑧 gcd 𝑁) = 𝑦}
82		breq1 3969	. . . . . . . . 9 ⊢ (𝑥 = (𝑧 gcd 𝑁) → (𝑥 ∥ 𝑁 ↔ (𝑧 gcd 𝑁) ∥ 𝑁))
83		elfzoelz 10050	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (0..^𝑁) → 𝑧 ∈ ℤ)
84	83	adantl 275	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ (0..^𝑁)) → 𝑧 ∈ ℤ)
85	11	adantr 274	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ (0..^𝑁)) → 𝑁 ∈ ℤ)
86		nnne0 8862	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ → 𝑁 ≠ 0)
87	86	neneqd 2348	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → ¬ 𝑁 = 0)
88	87	intnand 917	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → ¬ (𝑧 = 0 ∧ 𝑁 = 0))
89	88	adantr 274	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ (0..^𝑁)) → ¬ (𝑧 = 0 ∧ 𝑁 = 0))
90		gcdn0cl 11850	. . . . . . . . . 10 ⊢ (((𝑧 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑧 = 0 ∧ 𝑁 = 0)) → (𝑧 gcd 𝑁) ∈ ℕ)
91	84, 85, 89, 90	syl21anc 1219	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ (0..^𝑁)) → (𝑧 gcd 𝑁) ∈ ℕ)
92		gcddvds 11851	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑧 gcd 𝑁) ∥ 𝑧 ∧ (𝑧 gcd 𝑁) ∥ 𝑁))
93	84, 85, 92	syl2anc 409	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ (0..^𝑁)) → ((𝑧 gcd 𝑁) ∥ 𝑧 ∧ (𝑧 gcd 𝑁) ∥ 𝑁))
94	93	simprd 113	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ (0..^𝑁)) → (𝑧 gcd 𝑁) ∥ 𝑁)
95	82, 91, 94	elrabd 2870	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ (0..^𝑁)) → (𝑧 gcd 𝑁) ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁})
96		clel5 2849	. . . . . . . 8 ⊢ ((𝑧 gcd 𝑁) ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} ↔ ∃𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} (𝑧 gcd 𝑁) = 𝑦)
97	95, 96	sylib 121	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ (0..^𝑁)) → ∃𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} (𝑧 gcd 𝑁) = 𝑦)
98	97	ralrimiva 2530	. . . . . 6 ⊢ (𝑁 ∈ ℕ → ∀𝑧 ∈ (0..^𝑁)∃𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} (𝑧 gcd 𝑁) = 𝑦)
99		rabid2 2633	. . . . . 6 ⊢ ((0..^𝑁) = {𝑧 ∈ (0..^𝑁) ∣ ∃𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} (𝑧 gcd 𝑁) = 𝑦} ↔ ∀𝑧 ∈ (0..^𝑁)∃𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} (𝑧 gcd 𝑁) = 𝑦)
100	98, 99	sylibr 133	. . . . 5 ⊢ (𝑁 ∈ ℕ → (0..^𝑁) = {𝑧 ∈ (0..^𝑁) ∣ ∃𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} (𝑧 gcd 𝑁) = 𝑦})
101	81, 100	eqtr4id 2209	. . . 4 ⊢ (𝑁 ∈ ℕ → ∪ 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} {𝑧 ∈ (0..^𝑁) ∣ (𝑧 gcd 𝑁) = 𝑦} = (0..^𝑁))
102	101	fveq2d 5473	. . 3 ⊢ (𝑁 ∈ ℕ → (♯‘∪ 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} {𝑧 ∈ (0..^𝑁) ∣ (𝑧 gcd 𝑁) = 𝑦}) = (♯‘(0..^𝑁)))
103		nnnn0 9098	. . . 4 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0)
104		hashfzo0 10701	. . . 4 ⊢ (𝑁 ∈ ℕ0 → (♯‘(0..^𝑁)) = 𝑁)
105	103, 104	syl 14	. . 3 ⊢ (𝑁 ∈ ℕ → (♯‘(0..^𝑁)) = 𝑁)
106	102, 105	eqtrd 2190	. 2 ⊢ (𝑁 ∈ ℕ → (♯‘∪ 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} {𝑧 ∈ (0..^𝑁) ∣ (𝑧 gcd 𝑁) = 𝑦}) = 𝑁)
107	80, 106	eqtrd 2190	1 ⊢ (𝑁 ∈ ℕ → Σ𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} (ϕ‘𝑑) = 𝑁)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104  DECID wdc 820   = wceq 1335   ∈ wcel 2128  ∀wral 2435  ∃wrex 2436  {crab 2439   ⊆ wss 3102  ifcif 3505  ∪ ciun 3850  Disj wdisj 3943   class class class wbr 3966   ↦ cmpt 4026  ‘cfv 5171  (class class class)co 5825  Fincfn 6686  0cc0 7733  1c1 7734   / cdiv 8546  ℕcn 8834  ℕ₀cn0 9091  ℤcz 9168  ℚcq 9529  ...cfz 9913  ..^cfzo 10045  ♯chash 10653  Σcsu 11254   ∥ cdvds 11687   gcd cgcd 11833  ϕcphi 12088

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4080  ax-sep 4083  ax-nul 4091  ax-pow 4136  ax-pr 4170  ax-un 4394  ax-setind 4497  ax-iinf 4548  ax-cnex 7824  ax-resscn 7825  ax-1cn 7826  ax-1re 7827  ax-icn 7828  ax-addcl 7829  ax-addrcl 7830  ax-mulcl 7831  ax-mulrcl 7832  ax-addcom 7833  ax-mulcom 7834  ax-addass 7835  ax-mulass 7836  ax-distr 7837  ax-i2m1 7838  ax-0lt1 7839  ax-1rid 7840  ax-0id 7841  ax-rnegex 7842  ax-precex 7843  ax-cnre 7844  ax-pre-ltirr 7845  ax-pre-ltwlin 7846  ax-pre-lttrn 7847  ax-pre-apti 7848  ax-pre-ltadd 7849  ax-pre-mulgt0 7850  ax-pre-mulext 7851  ax-arch 7852  ax-caucvg 7853

This theorem depends on definitions:  df-bi 116  df-stab 817  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-if 3506  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3774  df-int 3809  df-iun 3852  df-disj 3944  df-br 3967  df-opab 4027  df-mpt 4028  df-tr 4064  df-id 4254  df-po 4257  df-iso 4258  df-iord 4327  df-on 4329  df-ilim 4330  df-suc 4332  df-iom 4551  df-xp 4593  df-rel 4594  df-cnv 4595  df-co 4596  df-dm 4597  df-rn 4598  df-res 4599  df-ima 4600  df-iota 5136  df-fun 5173  df-fn 5174  df-f 5175  df-f1 5176  df-fo 5177  df-f1o 5178  df-fv 5179  df-isom 5180  df-riota 5781  df-ov 5828  df-oprab 5829  df-mpo 5830  df-1st 6089  df-2nd 6090  df-recs 6253  df-irdg 6318  df-frec 6339  df-1o 6364  df-oadd 6368  df-er 6481  df-en 6687  df-dom 6688  df-fin 6689  df-sup 6929  df-pnf 7915  df-mnf 7916  df-xr 7917  df-ltxr 7918  df-le 7919  df-sub 8049  df-neg 8050  df-reap 8451  df-ap 8458  df-div 8547  df-inn 8835  df-2 8893  df-3 8894  df-4 8895  df-n0 9092  df-z 9169  df-uz 9441  df-q 9530  df-rp 9562  df-fz 9914  df-fzo 10046  df-fl 10173  df-mod 10226  df-seqfrec 10349  df-exp 10423  df-ihash 10654  df-cj 10746  df-re 10747  df-im 10748  df-rsqrt 10902  df-abs 10903  df-clim 11180  df-sumdc 11255  df-dvds 11688  df-gcd 11834  df-phi 12090

This theorem is referenced by: (None)