Theorem dvdsflsumcom 27165

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 13926	. . 3 ⊢ (𝜑 → (1...(⌊‘𝐴)) ∈ Fin)
2		fzfid 13926	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (1...𝑛) ∈ Fin)
3		elfznn 13498	. . . . . 6 ⊢ (𝑛 ∈ (1...(⌊‘𝐴)) → 𝑛 ∈ ℕ)
4	3	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℕ)
5		dvdsssfz1 16278	. . . . 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 9172	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛} ∈ Fin)
8		nnre 12172	. . . . . . . . . . . 12 ⊢ (𝑑 ∈ ℕ → 𝑑 ∈ ℝ)
9	8	ad2antrl 729	. . . . . . . . . . 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 12180	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) ∧ (𝑑 ∈ ℕ ∧ 𝑑 ∥ 𝑛)) → 𝑛 ∈ ℝ)
12		dvdsflsumcom.2	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ ℝ)
13	12	ad2antrr 727	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) ∧ (𝑑 ∈ ℕ ∧ 𝑑 ∥ 𝑛)) → 𝐴 ∈ ℝ)
14		nnz 12536	. . . . . . . . . . . . 13 ⊢ (𝑑 ∈ ℕ → 𝑑 ∈ ℤ)
15		dvdsle 16270	. . . . . . . . . . . . 13 ⊢ ((𝑑 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑑 ∥ 𝑛 → 𝑑 ≤ 𝑛))
16	14, 4, 15	syl2anr 598	. . . . . . . . . . . 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 13812	. . . . . . . . . . . . . 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 11294	. . . . . . . . . 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 647	. . . . . . . . 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 13812	. . . . . . . . . 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 632	. . . . . . 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 646	. . . . 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 5089	. . . . . 6 ⊢ (𝑥 = 𝑑 → (𝑥 ∥ 𝑛 ↔ 𝑑 ∥ 𝑛))
38	37	elrab 3635	. . . . 5 ⊢ (𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛} ↔ (𝑑 ∈ ℕ ∧ 𝑑 ∥ 𝑛))
39	38	anbi2i 624	. . . 4 ⊢ ((𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛}) ↔ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ (𝑑 ∈ ℕ ∧ 𝑑 ∥ 𝑛)))
40		breq2 5090	. . . . . 6 ⊢ (𝑥 = 𝑛 → (𝑑 ∥ 𝑥 ↔ 𝑑 ∥ 𝑛))
41	40	elrab 3635	. . . . 5 ⊢ (𝑛 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑑 ∥ 𝑥} ↔ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑑 ∥ 𝑛))
42	41	anbi2i 624	. . . 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 15727	. 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 13926	. . . 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 2737	. . . . 5 ⊢ (𝑦 ∈ (1...(⌊‘(𝐴 / 𝑑))) ↦ (𝑑 · 𝑦)) = (𝑦 ∈ (1...(⌊‘(𝐴 / 𝑑))) ↦ (𝑑 · 𝑦))
51	48, 49, 50	dvdsflf1o 27164	. . . 4 ⊢ ((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (𝑦 ∈ (1...(⌊‘(𝐴 / 𝑑))) ↦ (𝑑 · 𝑦)):(1...(⌊‘(𝐴 / 𝑑)))–1-1-onto→{𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑑 ∥ 𝑥})
52		oveq2 7368	. . . . . 6 ⊢ (𝑦 = 𝑚 → (𝑑 · 𝑦) = (𝑑 · 𝑚))
53		ovex 7393	. . . . . 6 ⊢ (𝑑 · 𝑚) ∈ V
54	52, 50, 53	fvmpt 6941	. . . . 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 592	. . . . 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 15676	. . 3 ⊢ ((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → Σ𝑛 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑑 ∥ 𝑥}𝐵 = Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))𝐶)
60	59	sumeq2dv 15655	. 2 ⊢ (𝜑 → Σ𝑑 ∈ (1...(⌊‘𝐴))Σ𝑛 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑑 ∥ 𝑥}𝐵 = Σ𝑑 ∈ (1...(⌊‘𝐴))Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))𝐶)
61	45, 60	eqtrd 2772	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 1542   ∈ wcel 2114  {crab 3390   ⊆ wss 3890   class class class wbr 5086   ↦ cmpt 5167  ‘cfv 6492  (class class class)co 7360  ℂcc 11027  ℝcr 11028  1c1 11030   · cmul 11034   ≤ cle 11171   / cdiv 11798  ℕcn 12165  ℤcz 12515  ...cfz 13452  ⌊cfl 13740  Σcsu 15639   ∥ cdvds 16212

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-inf2 9553  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106  ax-pre-sup 11107

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-1o 8398  df-er 8636  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-sup 9348  df-inf 9349  df-oi 9418  df-card 9854  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-div 11799  df-nn 12166  df-2 12235  df-3 12236  df-n0 12429  df-z 12516  df-uz 12780  df-rp 12934  df-fz 13453  df-fzo 13600  df-fl 13742  df-seq 13955  df-exp 14015  df-hash 14284  df-cj 15052  df-re 15053  df-im 15054  df-sqrt 15188  df-abs 15189  df-clim 15441  df-sum 15640  df-dvds 16213

This theorem is referenced by:  dchrmusum2  27471  dchrvmasumlem1  27472  dchrvmasum2lem  27473  dchrisum0  27497  mudivsum  27507  mulogsum  27509  mulog2sumlem2  27512  vmalogdivsum2  27515  selberglem3  27524  selberg  27525  selberg34r  27548  pntsval2  27553