Theorem dvdsflsumcom 26242

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 13621	. . 3 ⊢ (𝜑 → (1...(⌊‘𝐴)) ∈ Fin)
2		fzfid 13621	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (1...𝑛) ∈ Fin)
3		elfznn 13214	. . . . . 6 ⊢ (𝑛 ∈ (1...(⌊‘𝐴)) → 𝑛 ∈ ℕ)
4	3	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℕ)
5		dvdsssfz1 15955	. . . . 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 8971	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛} ∈ Fin)
8		nnre 11910	. . . . . . . . . . . 12 ⊢ (𝑑 ∈ ℕ → 𝑑 ∈ ℝ)
9	8	ad2antrl 724	. . . . . . . . . . 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 11918	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) ∧ (𝑑 ∈ ℕ ∧ 𝑑 ∥ 𝑛)) → 𝑛 ∈ ℝ)
12		dvdsflsumcom.2	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ ℝ)
13	12	ad2antrr 722	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) ∧ (𝑑 ∈ ℕ ∧ 𝑑 ∥ 𝑛)) → 𝐴 ∈ ℝ)
14		nnz 12272	. . . . . . . . . . . . 13 ⊢ (𝑑 ∈ ℕ → 𝑑 ∈ ℤ)
15		dvdsle 15947	. . . . . . . . . . . . 13 ⊢ ((𝑑 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑑 ∥ 𝑛 → 𝑑 ≤ 𝑛))
16	14, 4, 15	syl2anr 596	. . . . . . . . . . . 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 13510	. . . . . . . . . . . . . 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 11062	. . . . . . . . . 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 642	. . . . . . . . 9 ⊢ (((𝑑 ∈ ℕ ∧ 𝑑 ∥ 𝑛) ∧ 𝑑 ≤ 𝐴) ↔ ((𝑑 ∈ ℕ ∧ 𝑑 ≤ 𝐴) ∧ 𝑑 ∥ 𝑛))
27	25, 26	bitri 274	. . . . . . . 8 ⊢ ((𝑑 ≤ 𝐴 ∧ (𝑑 ∈ ℕ ∧ 𝑑 ∥ 𝑛)) ↔ ((𝑑 ∈ ℕ ∧ 𝑑 ≤ 𝐴) ∧ 𝑑 ∥ 𝑛))
28	24, 27	bitrdi 286	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → ((𝑑 ∈ ℕ ∧ 𝑑 ∥ 𝑛) ↔ ((𝑑 ∈ ℕ ∧ 𝑑 ≤ 𝐴) ∧ 𝑑 ∥ 𝑛)))
29		fznnfl 13510	. . . . . . . . . 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 629	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → ((𝑑 ∈ (1...(⌊‘𝐴)) ∧ 𝑑 ∥ 𝑛) ↔ ((𝑑 ∈ ℕ ∧ 𝑑 ≤ 𝐴) ∧ 𝑑 ∥ 𝑛)))
33	28, 32	bitr4d 281	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → ((𝑑 ∈ ℕ ∧ 𝑑 ∥ 𝑛) ↔ (𝑑 ∈ (1...(⌊‘𝐴)) ∧ 𝑑 ∥ 𝑛)))
34	33	pm5.32da 578	. . . . 5 ⊢ (𝜑 → ((𝑛 ∈ (1...(⌊‘𝐴)) ∧ (𝑑 ∈ ℕ ∧ 𝑑 ∥ 𝑛)) ↔ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ (𝑑 ∈ (1...(⌊‘𝐴)) ∧ 𝑑 ∥ 𝑛))))
35		an12 641	. . . . 5 ⊢ ((𝑛 ∈ (1...(⌊‘𝐴)) ∧ (𝑑 ∈ (1...(⌊‘𝐴)) ∧ 𝑑 ∥ 𝑛)) ↔ (𝑑 ∈ (1...(⌊‘𝐴)) ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑑 ∥ 𝑛)))
36	34, 35	bitrdi 286	. . . 4 ⊢ (𝜑 → ((𝑛 ∈ (1...(⌊‘𝐴)) ∧ (𝑑 ∈ ℕ ∧ 𝑑 ∥ 𝑛)) ↔ (𝑑 ∈ (1...(⌊‘𝐴)) ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑑 ∥ 𝑛))))
37		breq1 5073	. . . . . 6 ⊢ (𝑥 = 𝑑 → (𝑥 ∥ 𝑛 ↔ 𝑑 ∥ 𝑛))
38	37	elrab 3617	. . . . 5 ⊢ (𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛} ↔ (𝑑 ∈ ℕ ∧ 𝑑 ∥ 𝑛))
39	38	anbi2i 622	. . . 4 ⊢ ((𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛}) ↔ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ (𝑑 ∈ ℕ ∧ 𝑑 ∥ 𝑛)))
40		breq2 5074	. . . . . 6 ⊢ (𝑥 = 𝑛 → (𝑑 ∥ 𝑥 ↔ 𝑑 ∥ 𝑛))
41	40	elrab 3617	. . . . 5 ⊢ (𝑛 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑑 ∥ 𝑥} ↔ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑑 ∥ 𝑛))
42	41	anbi2i 622	. . . 4 ⊢ ((𝑑 ∈ (1...(⌊‘𝐴)) ∧ 𝑛 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑑 ∥ 𝑥}) ↔ (𝑑 ∈ (1...(⌊‘𝐴)) ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑑 ∥ 𝑛)))
43	36, 39, 42	3bitr4g 313	. . 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 15414	. 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 13621	. . . 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 2738	. . . . 5 ⊢ (𝑦 ∈ (1...(⌊‘(𝐴 / 𝑑))) ↦ (𝑑 · 𝑦)) = (𝑦 ∈ (1...(⌊‘(𝐴 / 𝑑))) ↦ (𝑑 · 𝑦))
51	48, 49, 50	dvdsflf1o 26241	. . . 4 ⊢ ((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (𝑦 ∈ (1...(⌊‘(𝐴 / 𝑑))) ↦ (𝑑 · 𝑦)):(1...(⌊‘(𝐴 / 𝑑)))–1-1-onto→{𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑑 ∥ 𝑥})
52		oveq2 7263	. . . . . 6 ⊢ (𝑦 = 𝑚 → (𝑑 · 𝑦) = (𝑑 · 𝑚))
53		ovex 7288	. . . . . 6 ⊢ (𝑑 · 𝑚) ∈ V
54	52, 50, 53	fvmpt 6857	. . . . 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 590	. . . . 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 15363	. . 3 ⊢ ((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → Σ𝑛 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑑 ∥ 𝑥}𝐵 = Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))𝐶)
60	59	sumeq2dv 15343	. 2 ⊢ (𝜑 → Σ𝑑 ∈ (1...(⌊‘𝐴))Σ𝑛 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑑 ∥ 𝑥}𝐵 = Σ𝑑 ∈ (1...(⌊‘𝐴))Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))𝐶)
61	45, 60	eqtrd 2778	1 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝐴))Σ𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛}𝐵 = Σ𝑑 ∈ (1...(⌊‘𝐴))Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))𝐶)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  {crab 3067   ⊆ wss 3883   class class class wbr 5070   ↦ cmpt 5153  ‘cfv 6418  (class class class)co 7255  ℂcc 10800  ℝcr 10801  1c1 10803   · cmul 10807   ≤ cle 10941   / cdiv 11562  ℕcn 11903  ℤcz 12249  ...cfz 13168  ⌊cfl 13438  Σcsu 15325   ∥ cdvds 15891

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-inf2 9329  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879  ax-pre-sup 10880

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-se 5536  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-isom 6427  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-er 8456  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-sup 9131  df-inf 9132  df-oi 9199  df-card 9628  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-div 11563  df-nn 11904  df-2 11966  df-3 11967  df-n0 12164  df-z 12250  df-uz 12512  df-rp 12660  df-fz 13169  df-fzo 13312  df-fl 13440  df-seq 13650  df-exp 13711  df-hash 13973  df-cj 14738  df-re 14739  df-im 14740  df-sqrt 14874  df-abs 14875  df-clim 15125  df-sum 15326  df-dvds 15892

This theorem is referenced by:  dchrmusum2  26547  dchrvmasumlem1  26548  dchrvmasum2lem  26549  dchrisum0  26573  mudivsum  26583  mulogsum  26585  mulog2sumlem2  26588  vmalogdivsum2  26591  selberglem3  26600  selberg  26601  selberg34r  26624  pntsval2  26629