Theorem dvdsflsumcom 27114

Description: A sum commutation from Σ𝑛 ≤ 𝐴, Σ𝑑 ∥ 𝑛, 𝐵(𝑛, 𝑑) to Σ𝑑 ≤ 𝐴, Σ𝑚 ≤ 𝐴 / 𝑑, 𝐵(𝑛, 𝑑𝑚). (Contributed by Mario Carneiro, 4-May-2016.)

Hypotheses

Ref	Expression
dvdsflsumcom.1	⊢ (𝑛 = (𝑑 · 𝑚) → 𝐵 = 𝐶)
dvdsflsumcom.2	⊢ (𝜑 → 𝐴 ∈ ℝ)
dvdsflsumcom.3	⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛})) → 𝐵 ∈ ℂ)

Assertion

Ref	Expression
dvdsflsumcom	⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝐴))Σ𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛}𝐵 = Σ𝑑 ∈ (1...(⌊‘𝐴))Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))𝐶)

Distinct variable groups:   𝑚,𝑑,𝑛,𝑥,𝐴   𝐵,𝑚   𝐶,𝑛   𝜑,𝑑,𝑚,𝑛

Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥,𝑛,𝑑)   𝐶(𝑥,𝑚,𝑑)

Proof of Theorem dvdsflsumcom

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fzfid 13898	. . 3 ⊢ (𝜑 → (1...(⌊‘𝐴)) ∈ Fin)
2		fzfid 13898	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (1...𝑛) ∈ Fin)
3		elfznn 13474	. . . . . 6 ⊢ (𝑛 ∈ (1...(⌊‘𝐴)) → 𝑛 ∈ ℕ)
4	3	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℕ)
5		dvdsssfz1 16247	. . . . 5 ⊢ (𝑛 ∈ ℕ → {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛} ⊆ (1...𝑛))
6	4, 5	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛} ⊆ (1...𝑛))
7	2, 6	ssfid 9170	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛} ∈ Fin)
8		nnre 12153	. . . . . . . . . . . 12 ⊢ (𝑑 ∈ ℕ → 𝑑 ∈ ℝ)
9	8	ad2antrl 728	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) ∧ (𝑑 ∈ ℕ ∧ 𝑑 ∥ 𝑛)) → 𝑑 ∈ ℝ)
10	4	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) ∧ (𝑑 ∈ ℕ ∧ 𝑑 ∥ 𝑛)) → 𝑛 ∈ ℕ)
11	10	nnred 12161	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) ∧ (𝑑 ∈ ℕ ∧ 𝑑 ∥ 𝑛)) → 𝑛 ∈ ℝ)
12		dvdsflsumcom.2	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ ℝ)
13	12	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) ∧ (𝑑 ∈ ℕ ∧ 𝑑 ∥ 𝑛)) → 𝐴 ∈ ℝ)
14		nnz 12510	. . . . . . . . . . . . 13 ⊢ (𝑑 ∈ ℕ → 𝑑 ∈ ℤ)
15		dvdsle 16239	. . . . . . . . . . . . 13 ⊢ ((𝑑 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑑 ∥ 𝑛 → 𝑑 ≤ 𝑛))
16	14, 4, 15	syl2anr 597	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) ∧ 𝑑 ∈ ℕ) → (𝑑 ∥ 𝑛 → 𝑑 ≤ 𝑛))
17	16	impr 454	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) ∧ (𝑑 ∈ ℕ ∧ 𝑑 ∥ 𝑛)) → 𝑑 ≤ 𝑛)
18		fznnfl 13784	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℝ → (𝑛 ∈ (1...(⌊‘𝐴)) ↔ (𝑛 ∈ ℕ ∧ 𝑛 ≤ 𝐴)))
19	12, 18	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑛 ∈ (1...(⌊‘𝐴)) ↔ (𝑛 ∈ ℕ ∧ 𝑛 ≤ 𝐴)))
20	19	simplbda 499	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ≤ 𝐴)
21	20	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) ∧ (𝑑 ∈ ℕ ∧ 𝑑 ∥ 𝑛)) → 𝑛 ≤ 𝐴)
22	9, 11, 13, 17, 21	letrd 11291	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) ∧ (𝑑 ∈ ℕ ∧ 𝑑 ∥ 𝑛)) → 𝑑 ≤ 𝐴)
23	22	ex 412	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → ((𝑑 ∈ ℕ ∧ 𝑑 ∥ 𝑛) → 𝑑 ≤ 𝐴))
24	23	pm4.71rd 562	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → ((𝑑 ∈ ℕ ∧ 𝑑 ∥ 𝑛) ↔ (𝑑 ≤ 𝐴 ∧ (𝑑 ∈ ℕ ∧ 𝑑 ∥ 𝑛))))
25		ancom 460	. . . . . . . . 9 ⊢ ((𝑑 ≤ 𝐴 ∧ (𝑑 ∈ ℕ ∧ 𝑑 ∥ 𝑛)) ↔ ((𝑑 ∈ ℕ ∧ 𝑑 ∥ 𝑛) ∧ 𝑑 ≤ 𝐴))
26		an32 646	. . . . . . . . 9 ⊢ (((𝑑 ∈ ℕ ∧ 𝑑 ∥ 𝑛) ∧ 𝑑 ≤ 𝐴) ↔ ((𝑑 ∈ ℕ ∧ 𝑑 ≤ 𝐴) ∧ 𝑑 ∥ 𝑛))
27	25, 26	bitri 275	. . . . . . . 8 ⊢ ((𝑑 ≤ 𝐴 ∧ (𝑑 ∈ ℕ ∧ 𝑑 ∥ 𝑛)) ↔ ((𝑑 ∈ ℕ ∧ 𝑑 ≤ 𝐴) ∧ 𝑑 ∥ 𝑛))
28	24, 27	bitrdi 287	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → ((𝑑 ∈ ℕ ∧ 𝑑 ∥ 𝑛) ↔ ((𝑑 ∈ ℕ ∧ 𝑑 ≤ 𝐴) ∧ 𝑑 ∥ 𝑛)))
29		fznnfl 13784	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ → (𝑑 ∈ (1...(⌊‘𝐴)) ↔ (𝑑 ∈ ℕ ∧ 𝑑 ≤ 𝐴)))
30	12, 29	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝑑 ∈ (1...(⌊‘𝐴)) ↔ (𝑑 ∈ ℕ ∧ 𝑑 ≤ 𝐴)))
31	30	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (𝑑 ∈ (1...(⌊‘𝐴)) ↔ (𝑑 ∈ ℕ ∧ 𝑑 ≤ 𝐴)))
32	31	anbi1d 631	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → ((𝑑 ∈ (1...(⌊‘𝐴)) ∧ 𝑑 ∥ 𝑛) ↔ ((𝑑 ∈ ℕ ∧ 𝑑 ≤ 𝐴) ∧ 𝑑 ∥ 𝑛)))
33	28, 32	bitr4d 282	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → ((𝑑 ∈ ℕ ∧ 𝑑 ∥ 𝑛) ↔ (𝑑 ∈ (1...(⌊‘𝐴)) ∧ 𝑑 ∥ 𝑛)))
34	33	pm5.32da 579	. . . . 5 ⊢ (𝜑 → ((𝑛 ∈ (1...(⌊‘𝐴)) ∧ (𝑑 ∈ ℕ ∧ 𝑑 ∥ 𝑛)) ↔ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ (𝑑 ∈ (1...(⌊‘𝐴)) ∧ 𝑑 ∥ 𝑛))))
35		an12 645	. . . . 5 ⊢ ((𝑛 ∈ (1...(⌊‘𝐴)) ∧ (𝑑 ∈ (1...(⌊‘𝐴)) ∧ 𝑑 ∥ 𝑛)) ↔ (𝑑 ∈ (1...(⌊‘𝐴)) ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑑 ∥ 𝑛)))
36	34, 35	bitrdi 287	. . . 4 ⊢ (𝜑 → ((𝑛 ∈ (1...(⌊‘𝐴)) ∧ (𝑑 ∈ ℕ ∧ 𝑑 ∥ 𝑛)) ↔ (𝑑 ∈ (1...(⌊‘𝐴)) ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑑 ∥ 𝑛))))
37		breq1 5098	. . . . . 6 ⊢ (𝑥 = 𝑑 → (𝑥 ∥ 𝑛 ↔ 𝑑 ∥ 𝑛))
38	37	elrab 3650	. . . . 5 ⊢ (𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛} ↔ (𝑑 ∈ ℕ ∧ 𝑑 ∥ 𝑛))
39	38	anbi2i 623	. . . 4 ⊢ ((𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛}) ↔ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ (𝑑 ∈ ℕ ∧ 𝑑 ∥ 𝑛)))
40		breq2 5099	. . . . . 6 ⊢ (𝑥 = 𝑛 → (𝑑 ∥ 𝑥 ↔ 𝑑 ∥ 𝑛))
41	40	elrab 3650	. . . . 5 ⊢ (𝑛 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑑 ∥ 𝑥} ↔ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑑 ∥ 𝑛))
42	41	anbi2i 623	. . . 4 ⊢ ((𝑑 ∈ (1...(⌊‘𝐴)) ∧ 𝑛 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑑 ∥ 𝑥}) ↔ (𝑑 ∈ (1...(⌊‘𝐴)) ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑑 ∥ 𝑛)))
43	36, 39, 42	3bitr4g 314	. . 3 ⊢ (𝜑 → ((𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛}) ↔ (𝑑 ∈ (1...(⌊‘𝐴)) ∧ 𝑛 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑑 ∥ 𝑥})))
44		dvdsflsumcom.3	. . 3 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛})) → 𝐵 ∈ ℂ)
45	1, 1, 7, 43, 44	fsumcom2 15699	. 2 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝐴))Σ𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛}𝐵 = Σ𝑑 ∈ (1...(⌊‘𝐴))Σ𝑛 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑑 ∥ 𝑥}𝐵)
46		dvdsflsumcom.1	. . . 4 ⊢ (𝑛 = (𝑑 · 𝑚) → 𝐵 = 𝐶)
47		fzfid 13898	. . . 4 ⊢ ((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (1...(⌊‘(𝐴 / 𝑑))) ∈ Fin)
48	12	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → 𝐴 ∈ ℝ)
49	30	simprbda 498	. . . . 5 ⊢ ((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → 𝑑 ∈ ℕ)
50		eqid 2729	. . . . 5 ⊢ (𝑦 ∈ (1...(⌊‘(𝐴 / 𝑑))) ↦ (𝑑 · 𝑦)) = (𝑦 ∈ (1...(⌊‘(𝐴 / 𝑑))) ↦ (𝑑 · 𝑦))
51	48, 49, 50	dvdsflf1o 27113	. . . 4 ⊢ ((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (𝑦 ∈ (1...(⌊‘(𝐴 / 𝑑))) ↦ (𝑑 · 𝑦)):(1...(⌊‘(𝐴 / 𝑑)))–1-1-onto→{𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑑 ∥ 𝑥})
52		oveq2 7361	. . . . . 6 ⊢ (𝑦 = 𝑚 → (𝑑 · 𝑦) = (𝑑 · 𝑚))
53		ovex 7386	. . . . . 6 ⊢ (𝑑 · 𝑚) ∈ V
54	52, 50, 53	fvmpt 6934	. . . . 5 ⊢ (𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑))) → ((𝑦 ∈ (1...(⌊‘(𝐴 / 𝑑))) ↦ (𝑑 · 𝑦))‘𝑚) = (𝑑 · 𝑚))
55	54	adantl 481	. . . 4 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → ((𝑦 ∈ (1...(⌊‘(𝐴 / 𝑑))) ↦ (𝑑 · 𝑦))‘𝑚) = (𝑑 · 𝑚))
56	43	biimpar 477	. . . . . 6 ⊢ ((𝜑 ∧ (𝑑 ∈ (1...(⌊‘𝐴)) ∧ 𝑛 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑑 ∥ 𝑥})) → (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛}))
57	56, 44	syldan 591	. . . . 5 ⊢ ((𝜑 ∧ (𝑑 ∈ (1...(⌊‘𝐴)) ∧ 𝑛 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑑 ∥ 𝑥})) → 𝐵 ∈ ℂ)
58	57	anassrs 467	. . . 4 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑛 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑑 ∥ 𝑥}) → 𝐵 ∈ ℂ)
59	46, 47, 51, 55, 58	fsumf1o 15648	. . 3 ⊢ ((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → Σ𝑛 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑑 ∥ 𝑥}𝐵 = Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))𝐶)
60	59	sumeq2dv 15627	. 2 ⊢ (𝜑 → Σ𝑑 ∈ (1...(⌊‘𝐴))Σ𝑛 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑑 ∥ 𝑥}𝐵 = Σ𝑑 ∈ (1...(⌊‘𝐴))Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))𝐶)
61	45, 60	eqtrd 2764	1 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝐴))Σ𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛}𝐵 = Σ𝑑 ∈ (1...(⌊‘𝐴))Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))𝐶)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {crab 3396   ⊆ wss 3905   class class class wbr 5095   ↦ cmpt 5176  ‘cfv 6486  (class class class)co 7353  ℂcc 11026  ℝcr 11027  1c1 11029   · cmul 11033   ≤ cle 11169   / cdiv 11795  ℕcn 12146  ℤcz 12489  ...cfz 13428  ⌊cfl 13712  Σcsu 15611   ∥ cdvds 16181

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675  ax-inf2 9556  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105  ax-pre-sup 11106

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3345  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-int 4900  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-se 5577  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-isom 6495  df-riota 7310  df-ov 7356  df-oprab 7357  df-mpo 7358  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-er 8632  df-en 8880  df-dom 8881  df-sdom 8882  df-fin 8883  df-sup 9351  df-inf 9352  df-oi 9421  df-card 9854  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11367  df-neg 11368  df-div 11796  df-nn 12147  df-2 12209  df-3 12210  df-n0 12403  df-z 12490  df-uz 12754  df-rp 12912  df-fz 13429  df-fzo 13576  df-fl 13714  df-seq 13927  df-exp 13987  df-hash 14256  df-cj 15024  df-re 15025  df-im 15026  df-sqrt 15160  df-abs 15161  df-clim 15413  df-sum 15612  df-dvds 16182

This theorem is referenced by:  dchrmusum2  27421  dchrvmasumlem1  27422  dchrvmasum2lem  27423  dchrisum0  27447  mudivsum  27457  mulogsum  27459  mulog2sumlem2  27462  vmalogdivsum2  27465  selberglem3  27474  selberg  27475  selberg34r  27498  pntsval2  27503