Theorem phisum 12812

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 4091	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ∥ 𝑁 ↔ 𝑦 ∥ 𝑁))
2	1	elrab 2962	. . . . 5 ⊢ (𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} ↔ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝑁))
3		hashgcdeq 12811	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (♯‘{𝑧 ∈ (0..^𝑁) ∣ (𝑧 gcd 𝑁) = 𝑦}) = if(𝑦 ∥ 𝑁, (ϕ‘(𝑁 / 𝑦)), 0))
4	3	adantrr 479	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝑁)) → (♯‘{𝑧 ∈ (0..^𝑁) ∣ (𝑧 gcd 𝑁) = 𝑦}) = if(𝑦 ∥ 𝑁, (ϕ‘(𝑁 / 𝑦)), 0))
5		iftrue 3610	. . . . . . 7 ⊢ (𝑦 ∥ 𝑁 → if(𝑦 ∥ 𝑁, (ϕ‘(𝑁 / 𝑦)), 0) = (ϕ‘(𝑁 / 𝑦)))
6	5	ad2antll 491	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝑁)) → if(𝑦 ∥ 𝑁, (ϕ‘(𝑁 / 𝑦)), 0) = (ϕ‘(𝑁 / 𝑦)))
7	4, 6	eqtrd 2264	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝑁)) → (♯‘{𝑧 ∈ (0..^𝑁) ∣ (𝑧 gcd 𝑁) = 𝑦}) = (ϕ‘(𝑁 / 𝑦)))
8	2, 7	sylan2b 287	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) → (♯‘{𝑧 ∈ (0..^𝑁) ∣ (𝑧 gcd 𝑁) = 𝑦}) = (ϕ‘(𝑁 / 𝑦)))
9	8	sumeq2dv 11928	. . 3 ⊢ (𝑁 ∈ ℕ → Σ𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} (♯‘{𝑧 ∈ (0..^𝑁) ∣ (𝑧 gcd 𝑁) = 𝑦}) = Σ𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} (ϕ‘(𝑁 / 𝑦)))
10		dvdsfi 12810	. . . 4 ⊢ (𝑁 ∈ ℕ → {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} ∈ Fin)
11		0z 9489	. . . . . . 7 ⊢ 0 ∈ ℤ
12		nnz 9497	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
13		fzofig 10693	. . . . . . 7 ⊢ ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (0..^𝑁) ∈ Fin)
14	11, 12, 13	sylancr 414	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (0..^𝑁) ∈ Fin)
15	14	adantr 276	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) → (0..^𝑁) ∈ Fin)
16		ssrab2 3312	. . . . . 6 ⊢ {𝑧 ∈ (0..^𝑁) ∣ (𝑧 gcd 𝑁) = 𝑦} ⊆ (0..^𝑁)
17	16	a1i 9	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) → {𝑧 ∈ (0..^𝑁) ∣ (𝑧 gcd 𝑁) = 𝑦} ⊆ (0..^𝑁))
18		elfzoelz 10381	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (0..^𝑁) → 𝑗 ∈ ℤ)
19	18	adantl 277	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) ∧ 𝑗 ∈ (0..^𝑁)) → 𝑗 ∈ ℤ)
20	12	ad2antrr 488	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) ∧ 𝑗 ∈ (0..^𝑁)) → 𝑁 ∈ ℤ)
21	19, 20	gcdcld 12538	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) ∧ 𝑗 ∈ (0..^𝑁)) → (𝑗 gcd 𝑁) ∈ ℕ0)
22	21	nn0zd 9599	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) ∧ 𝑗 ∈ (0..^𝑁)) → (𝑗 gcd 𝑁) ∈ ℤ)
23		elrabi 2959	. . . . . . . . . 10 ⊢ (𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} → 𝑦 ∈ ℕ)
24	23	ad2antlr 489	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) ∧ 𝑗 ∈ (0..^𝑁)) → 𝑦 ∈ ℕ)
25	24	nnzd 9600	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) ∧ 𝑗 ∈ (0..^𝑁)) → 𝑦 ∈ ℤ)
26		zdceq 9554	. . . . . . . 8 ⊢ (((𝑗 gcd 𝑁) ∈ ℤ ∧ 𝑦 ∈ ℤ) → DECID (𝑗 gcd 𝑁) = 𝑦)
27	22, 25, 26	syl2anc 411	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) ∧ 𝑗 ∈ (0..^𝑁)) → DECID (𝑗 gcd 𝑁) = 𝑦)
28		oveq1 6024	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑗 → (𝑧 gcd 𝑁) = (𝑗 gcd 𝑁))
29	28	eqeq1d 2240	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑗 → ((𝑧 gcd 𝑁) = 𝑦 ↔ (𝑗 gcd 𝑁) = 𝑦))
30	29	elrab 2962	. . . . . . . . . 10 ⊢ (𝑗 ∈ {𝑧 ∈ (0..^𝑁) ∣ (𝑧 gcd 𝑁) = 𝑦} ↔ (𝑗 ∈ (0..^𝑁) ∧ (𝑗 gcd 𝑁) = 𝑦))
31	30	baibr 927	. . . . . . . . 9 ⊢ (𝑗 ∈ (0..^𝑁) → ((𝑗 gcd 𝑁) = 𝑦 ↔ 𝑗 ∈ {𝑧 ∈ (0..^𝑁) ∣ (𝑧 gcd 𝑁) = 𝑦}))
32	31	dcbid 845	. . . . . . . 8 ⊢ (𝑗 ∈ (0..^𝑁) → (DECID (𝑗 gcd 𝑁) = 𝑦 ↔ DECID 𝑗 ∈ {𝑧 ∈ (0..^𝑁) ∣ (𝑧 gcd 𝑁) = 𝑦}))
33	32	adantl 277	. . . . . . 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}))
34	27, 33	mpbid 147	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) ∧ 𝑗 ∈ (0..^𝑁)) → DECID 𝑗 ∈ {𝑧 ∈ (0..^𝑁) ∣ (𝑧 gcd 𝑁) = 𝑦})
35	34	ralrimiva 2605	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) → ∀𝑗 ∈ (0..^𝑁)DECID 𝑗 ∈ {𝑧 ∈ (0..^𝑁) ∣ (𝑧 gcd 𝑁) = 𝑦})
36		ssfidc 7129	. . . . 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)
37	15, 17, 35, 36	syl3anc 1273	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) → {𝑧 ∈ (0..^𝑁) ∣ (𝑧 gcd 𝑁) = 𝑦} ∈ Fin)
38		oveq1 6024	. . . . . . . . . 10 ⊢ (𝑧 = 𝑤 → (𝑧 gcd 𝑁) = (𝑤 gcd 𝑁))
39	38	eqeq1d 2240	. . . . . . . . 9 ⊢ (𝑧 = 𝑤 → ((𝑧 gcd 𝑁) = 𝑦 ↔ (𝑤 gcd 𝑁) = 𝑦))
40	39	elrab 2962	. . . . . . . 8 ⊢ (𝑤 ∈ {𝑧 ∈ (0..^𝑁) ∣ (𝑧 gcd 𝑁) = 𝑦} ↔ (𝑤 ∈ (0..^𝑁) ∧ (𝑤 gcd 𝑁) = 𝑦))
41	40	simprbi 275	. . . . . . 7 ⊢ (𝑤 ∈ {𝑧 ∈ (0..^𝑁) ∣ (𝑧 gcd 𝑁) = 𝑦} → (𝑤 gcd 𝑁) = 𝑦)
42	41	rgen 2585	. . . . . 6 ⊢ ∀𝑤 ∈ {𝑧 ∈ (0..^𝑁) ∣ (𝑧 gcd 𝑁) = 𝑦} (𝑤 gcd 𝑁) = 𝑦
43	42	rgenw 2587	. . . . 5 ⊢ ∀𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}∀𝑤 ∈ {𝑧 ∈ (0..^𝑁) ∣ (𝑧 gcd 𝑁) = 𝑦} (𝑤 gcd 𝑁) = 𝑦
44		invdisj 4081	. . . . 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})
45	43, 44	mp1i 10	. . . 4 ⊢ (𝑁 ∈ ℕ → Disj 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} {𝑧 ∈ (0..^𝑁) ∣ (𝑧 gcd 𝑁) = 𝑦})
46	10, 37, 45	hashiun 12038	. . 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}))
47		fveq2 5639	. . . 4 ⊢ (𝑑 = (𝑁 / 𝑦) → (ϕ‘𝑑) = (ϕ‘(𝑁 / 𝑦)))
48		eqid 2231	. . . . 5 ⊢ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} = {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}
49		eqid 2231	. . . . 5 ⊢ (𝑧 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} ↦ (𝑁 / 𝑧)) = (𝑧 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} ↦ (𝑁 / 𝑧))
50	48, 49	dvdsflip 12411	. . . 4 ⊢ (𝑁 ∈ ℕ → (𝑧 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} ↦ (𝑁 / 𝑧)):{𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}–1-1-onto→{𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁})
51		oveq2 6025	. . . . 5 ⊢ (𝑧 = 𝑦 → (𝑁 / 𝑧) = (𝑁 / 𝑦))
52		simpr 110	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) → 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁})
53	12	adantr 276	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) → 𝑁 ∈ ℤ)
54	23	adantl 277	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) → 𝑦 ∈ ℕ)
55		znq 9857	. . . . . 6 ⊢ ((𝑁 ∈ ℤ ∧ 𝑦 ∈ ℕ) → (𝑁 / 𝑦) ∈ ℚ)
56	53, 54, 55	syl2anc 411	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) → (𝑁 / 𝑦) ∈ ℚ)
57	49, 51, 52, 56	fvmptd3 5740	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) → ((𝑧 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} ↦ (𝑁 / 𝑧))‘𝑦) = (𝑁 / 𝑦))
58		elrabi 2959	. . . . . . 7 ⊢ (𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} → 𝑑 ∈ ℕ)
59	58	adantl 277	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) → 𝑑 ∈ ℕ)
60	59	phicld 12789	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) → (ϕ‘𝑑) ∈ ℕ)
61	60	nncnd 9156	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) → (ϕ‘𝑑) ∈ ℂ)
62	47, 10, 50, 57, 61	fsumf1o 11950	. . 3 ⊢ (𝑁 ∈ ℕ → Σ𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} (ϕ‘𝑑) = Σ𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} (ϕ‘(𝑁 / 𝑦)))
63	9, 46, 62	3eqtr4rd 2275	. 2 ⊢ (𝑁 ∈ ℕ → Σ𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} (ϕ‘𝑑) = (♯‘∪ 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} {𝑧 ∈ (0..^𝑁) ∣ (𝑧 gcd 𝑁) = 𝑦}))
64		iunrab 4018	. . . . 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}
65		breq1 4091	. . . . . . . . 9 ⊢ (𝑥 = (𝑧 gcd 𝑁) → (𝑥 ∥ 𝑁 ↔ (𝑧 gcd 𝑁) ∥ 𝑁))
66		elfzoelz 10381	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (0..^𝑁) → 𝑧 ∈ ℤ)
67	66	adantl 277	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ (0..^𝑁)) → 𝑧 ∈ ℤ)
68	12	adantr 276	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ (0..^𝑁)) → 𝑁 ∈ ℤ)
69		nnne0 9170	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ → 𝑁 ≠ 0)
70	69	neneqd 2423	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → ¬ 𝑁 = 0)
71	70	intnand 938	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → ¬ (𝑧 = 0 ∧ 𝑁 = 0))
72	71	adantr 276	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ (0..^𝑁)) → ¬ (𝑧 = 0 ∧ 𝑁 = 0))
73		gcdn0cl 12532	. . . . . . . . . 10 ⊢ (((𝑧 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑧 = 0 ∧ 𝑁 = 0)) → (𝑧 gcd 𝑁) ∈ ℕ)
74	67, 68, 72, 73	syl21anc 1272	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ (0..^𝑁)) → (𝑧 gcd 𝑁) ∈ ℕ)
75		gcddvds 12533	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑧 gcd 𝑁) ∥ 𝑧 ∧ (𝑧 gcd 𝑁) ∥ 𝑁))
76	67, 68, 75	syl2anc 411	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ (0..^𝑁)) → ((𝑧 gcd 𝑁) ∥ 𝑧 ∧ (𝑧 gcd 𝑁) ∥ 𝑁))
77	76	simprd 114	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ (0..^𝑁)) → (𝑧 gcd 𝑁) ∥ 𝑁)
78	65, 74, 77	elrabd 2964	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ (0..^𝑁)) → (𝑧 gcd 𝑁) ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁})
79		clel5 2943	. . . . . . . 8 ⊢ ((𝑧 gcd 𝑁) ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} ↔ ∃𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} (𝑧 gcd 𝑁) = 𝑦)
80	78, 79	sylib 122	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ (0..^𝑁)) → ∃𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} (𝑧 gcd 𝑁) = 𝑦)
81	80	ralrimiva 2605	. . . . . 6 ⊢ (𝑁 ∈ ℕ → ∀𝑧 ∈ (0..^𝑁)∃𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} (𝑧 gcd 𝑁) = 𝑦)
82		rabid2 2710	. . . . . 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)
83	81, 82	sylibr 134	. . . . 5 ⊢ (𝑁 ∈ ℕ → (0..^𝑁) = {𝑧 ∈ (0..^𝑁) ∣ ∃𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} (𝑧 gcd 𝑁) = 𝑦})
84	64, 83	eqtr4id 2283	. . . 4 ⊢ (𝑁 ∈ ℕ → ∪ 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} {𝑧 ∈ (0..^𝑁) ∣ (𝑧 gcd 𝑁) = 𝑦} = (0..^𝑁))
85	84	fveq2d 5643	. . 3 ⊢ (𝑁 ∈ ℕ → (♯‘∪ 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} {𝑧 ∈ (0..^𝑁) ∣ (𝑧 gcd 𝑁) = 𝑦}) = (♯‘(0..^𝑁)))
86		nnnn0 9408	. . . 4 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0)
87		hashfzo0 11086	. . . 4 ⊢ (𝑁 ∈ ℕ0 → (♯‘(0..^𝑁)) = 𝑁)
88	86, 87	syl 14	. . 3 ⊢ (𝑁 ∈ ℕ → (♯‘(0..^𝑁)) = 𝑁)
89	85, 88	eqtrd 2264	. 2 ⊢ (𝑁 ∈ ℕ → (♯‘∪ 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} {𝑧 ∈ (0..^𝑁) ∣ (𝑧 gcd 𝑁) = 𝑦}) = 𝑁)
90	63, 89	eqtrd 2264	1 ⊢ (𝑁 ∈ ℕ → Σ𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} (ϕ‘𝑑) = 𝑁)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105  DECID wdc 841   = wceq 1397   ∈ wcel 2202  ∀wral 2510  ∃wrex 2511  {crab 2514   ⊆ wss 3200  ifcif 3605  ∪ ciun 3970  Disj wdisj 4064   class class class wbr 4088   ↦ cmpt 4150  ‘cfv 5326  (class class class)co 6017  Fincfn 6908  0cc0 8031   / cdiv 8851  ℕcn 9142  ℕ₀cn0 9401  ℤcz 9478  ℚcq 9852  ..^cfzo 10376  ♯chash 11036  Σcsu 11913   ∥ cdvds 12347   gcd cgcd 12523  ϕcphi 12780

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-precex 8141  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147  ax-pre-mulgt0 8148  ax-pre-mulext 8149  ax-arch 8150  ax-caucvg 8151

This theorem depends on definitions:  df-bi 117  df-stab 838  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-disj 4065  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-isom 5335  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-irdg 6535  df-frec 6556  df-1o 6581  df-oadd 6585  df-er 6701  df-en 6909  df-dom 6910  df-fin 6911  df-sup 7182  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-reap 8754  df-ap 8761  df-div 8852  df-inn 9143  df-2 9201  df-3 9202  df-4 9203  df-n0 9402  df-z 9479  df-uz 9755  df-q 9853  df-rp 9888  df-fz 10243  df-fzo 10377  df-fl 10529  df-mod 10584  df-seqfrec 10709  df-exp 10800  df-ihash 11037  df-cj 11402  df-re 11403  df-im 11404  df-rsqrt 11558  df-abs 11559  df-clim 11839  df-sumdc 11914  df-dvds 12348  df-gcd 12524  df-phi 12782

This theorem is referenced by: (None)