Theorem dchrvmasum2if 27854

Description: Combine the results of dchrvmasumlem1 27852 and dchrvmasum2lem 27853 inside a conditional. (Contributed by Mario Carneiro, 4-May-2016.)

Hypotheses

Ref	Expression
rpvmasum.z	⊢ 𝑍 = (ℤ/nℤ‘𝑁)
rpvmasum.l	⊢ 𝐿 = (ℤRHom‘𝑍)
rpvmasum.a	⊢ (𝜑 → 𝑁 ∈ ℕ)
rpvmasum.g	⊢ 𝐺 = (DChr‘𝑁)
rpvmasum.d	⊢ 𝐷 = (Base‘𝐺)
rpvmasum.1	⊢ 1 = (0g‘𝐺)
dchrisum.b	⊢ (𝜑 → 𝑋 ∈ 𝐷)
dchrisum.n1	⊢ (𝜑 → 𝑋 ≠ 1 )
dchrvmasum.a	⊢ (𝜑 → 𝐴 ∈ ℝ+)
dchrvmasum2.2	⊢ (𝜑 → 1 ≤ 𝐴)

Assertion

Ref	Expression
dchrvmasum2if	⊢ (𝜑 → (Σ𝑛 ∈ (1...(⌊‘𝐴))((𝑋‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) + if(𝜓, (log‘𝐴), 0)) = Σ𝑑 ∈ (1...(⌊‘𝐴))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))((𝑋‘(𝐿‘𝑚)) · ((log‘if(𝜓, (𝐴 / 𝑑), 𝑚)) / 𝑚))))

Distinct variable groups:   𝑚,𝑛, 1   𝑚,𝑑,𝑛,𝐴   𝑚,𝑁,𝑛   𝜑,𝑑,𝑚,𝑛   𝜓,𝑑,𝑚   𝑚,𝑍,𝑛   𝐷,𝑚,𝑛   𝐿,𝑑,𝑚,𝑛   𝑋,𝑑,𝑚,𝑛   𝐴,𝑛

Allowed substitution hints:   𝜓(𝑛)   𝐷(𝑑)   1 (𝑑)   𝐺(𝑚, 𝑛, 𝑑)   𝑁(𝑑)   𝑍(𝑑)

Proof of Theorem dchrvmasum2if

Step	Hyp	Ref	Expression
1		fzfid 14143	. . . . . 6 ⊢ (𝜑 → (1...(⌊‘𝐴)) ∈ Fin)
2		rpvmasum.g	. . . . . . . . 9 ⊢ 𝐺 = (DChr‘𝑁)
3		rpvmasum.z	. . . . . . . . 9 ⊢ 𝑍 = (ℤ/nℤ‘𝑁)
4		rpvmasum.d	. . . . . . . . 9 ⊢ 𝐷 = (Base‘𝐺)
5		rpvmasum.l	. . . . . . . . 9 ⊢ 𝐿 = (ℤRHom‘𝑍)
6		dchrisum.b	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ 𝐷)
7	6	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → 𝑋 ∈ 𝐷)
8		elfzelz 13683	. . . . . . . . . 10 ⊢ (𝑑 ∈ (1...(⌊‘𝐴)) → 𝑑 ∈ ℤ)
9	8	adantl 487	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → 𝑑 ∈ ℤ)
10	2, 3, 4, 5, 7, 9	dchrzrhcl 27602	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (𝑋‘(𝐿‘𝑑)) ∈ ℂ)
11		elfznn 13714	. . . . . . . . . . 11 ⊢ (𝑑 ∈ (1...(⌊‘𝐴)) → 𝑑 ∈ ℕ)
12	11	adantl 487	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → 𝑑 ∈ ℕ)
13		mucl 27498	. . . . . . . . . . . 12 ⊢ (𝑑 ∈ ℕ → (μ‘𝑑) ∈ ℤ)
14	13	zred 12830	. . . . . . . . . . 11 ⊢ (𝑑 ∈ ℕ → (μ‘𝑑) ∈ ℝ)
15		nndivre 12406	. . . . . . . . . . 11 ⊢ (((μ‘𝑑) ∈ ℝ ∧ 𝑑 ∈ ℕ) → ((μ‘𝑑) / 𝑑) ∈ ℝ)
16	14, 15	mpancom 701	. . . . . . . . . 10 ⊢ (𝑑 ∈ ℕ → ((μ‘𝑑) / 𝑑) ∈ ℝ)
17	12, 16	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → ((μ‘𝑑) / 𝑑) ∈ ℝ)
18	17	recnd 11364	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → ((μ‘𝑑) / 𝑑) ∈ ℂ)
19	10, 18	mulcld 11356	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → ((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) ∈ ℂ)
20		fzfid 14143	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (1...(⌊‘(𝐴 / 𝑑))) ∈ Fin)
21	7	adantr 486	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → 𝑋 ∈ 𝐷)
22		elfzelz 13683	. . . . . . . . . . 11 ⊢ (𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑))) → 𝑚 ∈ ℤ)
23	22	adantl 487	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → 𝑚 ∈ ℤ)
24	2, 3, 4, 5, 21, 23	dchrzrhcl 27602	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → (𝑋‘(𝐿‘𝑚)) ∈ ℂ)
25		elfznn 13714	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑))) → 𝑚 ∈ ℕ)
26	25	adantl 487	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → 𝑚 ∈ ℕ)
27	26	nnrpd 13189	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → 𝑚 ∈ ℝ+)
28	27	relogcld 26981	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → (log‘𝑚) ∈ ℝ)
29	28, 26	nndivred 12419	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → ((log‘𝑚) / 𝑚) ∈ ℝ)
30	29	recnd 11364	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → ((log‘𝑚) / 𝑚) ∈ ℂ)
31	24, 30	mulcld 11356	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → ((𝑋‘(𝐿‘𝑚)) · ((log‘𝑚) / 𝑚)) ∈ ℂ)
32	20, 31	fsumcl 15928	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))((𝑋‘(𝐿‘𝑚)) · ((log‘𝑚) / 𝑚)) ∈ ℂ)
33	19, 32	mulcld 11356	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))((𝑋‘(𝐿‘𝑚)) · ((log‘𝑚) / 𝑚))) ∈ ℂ)
34		dchrvmasum.a	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐴 ∈ ℝ+)
35	11	nnrpd 13189	. . . . . . . . . . . . . . 15 ⊢ (𝑑 ∈ (1...(⌊‘𝐴)) → 𝑑 ∈ ℝ+)
36		rpdivcl 13174	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+) → (𝐴 / 𝑑) ∈ ℝ+)
37	34, 35, 36	syl2an 608	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (𝐴 / 𝑑) ∈ ℝ+)
38	37	adantr 486	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → (𝐴 / 𝑑) ∈ ℝ+)
39	38, 27	rpdivcld 13208	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → ((𝐴 / 𝑑) / 𝑚) ∈ ℝ+)
40	39	relogcld 26981	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → (log‘((𝐴 / 𝑑) / 𝑚)) ∈ ℝ)
41	40, 26	nndivred 12419	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → ((log‘((𝐴 / 𝑑) / 𝑚)) / 𝑚) ∈ ℝ)
42	41	recnd 11364	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → ((log‘((𝐴 / 𝑑) / 𝑚)) / 𝑚) ∈ ℂ)
43	24, 42	mulcld 11356	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → ((𝑋‘(𝐿‘𝑚)) · ((log‘((𝐴 / 𝑑) / 𝑚)) / 𝑚)) ∈ ℂ)
44	20, 43	fsumcl 15928	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))((𝑋‘(𝐿‘𝑚)) · ((log‘((𝐴 / 𝑑) / 𝑚)) / 𝑚)) ∈ ℂ)
45	19, 44	mulcld 11356	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))((𝑋‘(𝐿‘𝑚)) · ((log‘((𝐴 / 𝑑) / 𝑚)) / 𝑚))) ∈ ℂ)
46	1, 33, 45	fsumadd 15935	. . . . 5 ⊢ (𝜑 → Σ𝑑 ∈ (1...(⌊‘𝐴))((((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))((𝑋‘(𝐿‘𝑚)) · ((log‘𝑚) / 𝑚))) + (((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))((𝑋‘(𝐿‘𝑚)) · ((log‘((𝐴 / 𝑑) / 𝑚)) / 𝑚)))) = (Σ𝑑 ∈ (1...(⌊‘𝐴))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))((𝑋‘(𝐿‘𝑚)) · ((log‘𝑚) / 𝑚))) + Σ𝑑 ∈ (1...(⌊‘𝐴))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))((𝑋‘(𝐿‘𝑚)) · ((log‘((𝐴 / 𝑑) / 𝑚)) / 𝑚)))))
47	38, 27	relogdivd 26984	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → (log‘((𝐴 / 𝑑) / 𝑚)) = ((log‘(𝐴 / 𝑑)) − (log‘𝑚)))
48	47	oveq2d 7459	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → ((log‘𝑚) + (log‘((𝐴 / 𝑑) / 𝑚))) = ((log‘𝑚) + ((log‘(𝐴 / 𝑑)) − (log‘𝑚))))
49	28	recnd 11364	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → (log‘𝑚) ∈ ℂ)
50	37	relogcld 26981	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (log‘(𝐴 / 𝑑)) ∈ ℝ)
51	50	recnd 11364	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (log‘(𝐴 / 𝑑)) ∈ ℂ)
52	51	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → (log‘(𝐴 / 𝑑)) ∈ ℂ)
53	49, 52	pncan3d 11699	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → ((log‘𝑚) + ((log‘(𝐴 / 𝑑)) − (log‘𝑚))) = (log‘(𝐴 / 𝑑)))
54	48, 53	eqtr2d 2799	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → (log‘(𝐴 / 𝑑)) = ((log‘𝑚) + (log‘((𝐴 / 𝑑) / 𝑚))))
55	54	oveq1d 7458	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → ((log‘(𝐴 / 𝑑)) / 𝑚) = (((log‘𝑚) + (log‘((𝐴 / 𝑑) / 𝑚))) / 𝑚))
56	40	recnd 11364	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → (log‘((𝐴 / 𝑑) / 𝑚)) ∈ ℂ)
57	26	nncnd 12378	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → 𝑚 ∈ ℂ)
58	26	nnne0d 12415	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → 𝑚 ≠ 0)
59	49, 56, 57, 58	divdird 12158	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → (((log‘𝑚) + (log‘((𝐴 / 𝑑) / 𝑚))) / 𝑚) = (((log‘𝑚) / 𝑚) + ((log‘((𝐴 / 𝑑) / 𝑚)) / 𝑚)))
60	55, 59	eqtrd 2798	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → ((log‘(𝐴 / 𝑑)) / 𝑚) = (((log‘𝑚) / 𝑚) + ((log‘((𝐴 / 𝑑) / 𝑚)) / 𝑚)))
61	60	oveq2d 7459	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → ((𝑋‘(𝐿‘𝑚)) · ((log‘(𝐴 / 𝑑)) / 𝑚)) = ((𝑋‘(𝐿‘𝑚)) · (((log‘𝑚) / 𝑚) + ((log‘((𝐴 / 𝑑) / 𝑚)) / 𝑚))))
62	24, 30, 42	adddid 11360	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → ((𝑋‘(𝐿‘𝑚)) · (((log‘𝑚) / 𝑚) + ((log‘((𝐴 / 𝑑) / 𝑚)) / 𝑚))) = (((𝑋‘(𝐿‘𝑚)) · ((log‘𝑚) / 𝑚)) + ((𝑋‘(𝐿‘𝑚)) · ((log‘((𝐴 / 𝑑) / 𝑚)) / 𝑚))))
63	61, 62	eqtrd 2798	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → ((𝑋‘(𝐿‘𝑚)) · ((log‘(𝐴 / 𝑑)) / 𝑚)) = (((𝑋‘(𝐿‘𝑚)) · ((log‘𝑚) / 𝑚)) + ((𝑋‘(𝐿‘𝑚)) · ((log‘((𝐴 / 𝑑) / 𝑚)) / 𝑚))))
64	63	sumeq2dv 15898	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))((𝑋‘(𝐿‘𝑚)) · ((log‘(𝐴 / 𝑑)) / 𝑚)) = Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))(((𝑋‘(𝐿‘𝑚)) · ((log‘𝑚) / 𝑚)) + ((𝑋‘(𝐿‘𝑚)) · ((log‘((𝐴 / 𝑑) / 𝑚)) / 𝑚))))
65	20, 31, 43	fsumadd 15935	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))(((𝑋‘(𝐿‘𝑚)) · ((log‘𝑚) / 𝑚)) + ((𝑋‘(𝐿‘𝑚)) · ((log‘((𝐴 / 𝑑) / 𝑚)) / 𝑚))) = (Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))((𝑋‘(𝐿‘𝑚)) · ((log‘𝑚) / 𝑚)) + Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))((𝑋‘(𝐿‘𝑚)) · ((log‘((𝐴 / 𝑑) / 𝑚)) / 𝑚))))
66	64, 65	eqtrd 2798	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))((𝑋‘(𝐿‘𝑚)) · ((log‘(𝐴 / 𝑑)) / 𝑚)) = (Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))((𝑋‘(𝐿‘𝑚)) · ((log‘𝑚) / 𝑚)) + Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))((𝑋‘(𝐿‘𝑚)) · ((log‘((𝐴 / 𝑑) / 𝑚)) / 𝑚))))
67	66	oveq2d 7459	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))((𝑋‘(𝐿‘𝑚)) · ((log‘(𝐴 / 𝑑)) / 𝑚))) = (((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · (Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))((𝑋‘(𝐿‘𝑚)) · ((log‘𝑚) / 𝑚)) + Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))((𝑋‘(𝐿‘𝑚)) · ((log‘((𝐴 / 𝑑) / 𝑚)) / 𝑚)))))
68	19, 32, 44	adddid 11360	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · (Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))((𝑋‘(𝐿‘𝑚)) · ((log‘𝑚) / 𝑚)) + Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))((𝑋‘(𝐿‘𝑚)) · ((log‘((𝐴 / 𝑑) / 𝑚)) / 𝑚)))) = ((((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))((𝑋‘(𝐿‘𝑚)) · ((log‘𝑚) / 𝑚))) + (((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))((𝑋‘(𝐿‘𝑚)) · ((log‘((𝐴 / 𝑑) / 𝑚)) / 𝑚)))))
69	67, 68	eqtrd 2798	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))((𝑋‘(𝐿‘𝑚)) · ((log‘(𝐴 / 𝑑)) / 𝑚))) = ((((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))((𝑋‘(𝐿‘𝑚)) · ((log‘𝑚) / 𝑚))) + (((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))((𝑋‘(𝐿‘𝑚)) · ((log‘((𝐴 / 𝑑) / 𝑚)) / 𝑚)))))
70	69	sumeq2dv 15898	. . . . 5 ⊢ (𝜑 → Σ𝑑 ∈ (1...(⌊‘𝐴))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))((𝑋‘(𝐿‘𝑚)) · ((log‘(𝐴 / 𝑑)) / 𝑚))) = Σ𝑑 ∈ (1...(⌊‘𝐴))((((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))((𝑋‘(𝐿‘𝑚)) · ((log‘𝑚) / 𝑚))) + (((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))((𝑋‘(𝐿‘𝑚)) · ((log‘((𝐴 / 𝑑) / 𝑚)) / 𝑚)))))
71		rpvmasum.a	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ)
72		rpvmasum.1	. . . . . . 7 ⊢ 1 = (0g‘𝐺)
73		dchrisum.n1	. . . . . . 7 ⊢ (𝜑 → 𝑋 ≠ 1 )
74	3, 5, 71, 2, 4, 72, 6, 73, 34	dchrvmasumlem1 27852	. . . . . 6 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝐴))((𝑋‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) = Σ𝑑 ∈ (1...(⌊‘𝐴))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))((𝑋‘(𝐿‘𝑚)) · ((log‘𝑚) / 𝑚))))
75		dchrvmasum2.2	. . . . . . 7 ⊢ (𝜑 → 1 ≤ 𝐴)
76	3, 5, 71, 2, 4, 72, 6, 73, 34, 75	dchrvmasum2lem 27853	. . . . . 6 ⊢ (𝜑 → (log‘𝐴) = Σ𝑑 ∈ (1...(⌊‘𝐴))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))((𝑋‘(𝐿‘𝑚)) · ((log‘((𝐴 / 𝑑) / 𝑚)) / 𝑚))))
77	74, 76	oveq12d 7461	. . . . 5 ⊢ (𝜑 → (Σ𝑛 ∈ (1...(⌊‘𝐴))((𝑋‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) + (log‘𝐴)) = (Σ𝑑 ∈ (1...(⌊‘𝐴))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))((𝑋‘(𝐿‘𝑚)) · ((log‘𝑚) / 𝑚))) + Σ𝑑 ∈ (1...(⌊‘𝐴))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))((𝑋‘(𝐿‘𝑚)) · ((log‘((𝐴 / 𝑑) / 𝑚)) / 𝑚)))))
78	46, 70, 77	3eqtr4rd 2809	. . . 4 ⊢ (𝜑 → (Σ𝑛 ∈ (1...(⌊‘𝐴))((𝑋‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) + (log‘𝐴)) = Σ𝑑 ∈ (1...(⌊‘𝐴))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))((𝑋‘(𝐿‘𝑚)) · ((log‘(𝐴 / 𝑑)) / 𝑚))))
79	78	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝜓) → (Σ𝑛 ∈ (1...(⌊‘𝐴))((𝑋‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) + (log‘𝐴)) = Σ𝑑 ∈ (1...(⌊‘𝐴))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))((𝑋‘(𝐿‘𝑚)) · ((log‘(𝐴 / 𝑑)) / 𝑚))))
80		iftrue 4491	. . . . 5 ⊢ (𝜓 → if(𝜓, (log‘𝐴), 0) = (log‘𝐴))
81	80	oveq2d 7459	. . . 4 ⊢ (𝜓 → (Σ𝑛 ∈ (1...(⌊‘𝐴))((𝑋‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) + if(𝜓, (log‘𝐴), 0)) = (Σ𝑛 ∈ (1...(⌊‘𝐴))((𝑋‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) + (log‘𝐴)))
82	81	adantl 487	. . 3 ⊢ ((𝜑 ∧ 𝜓) → (Σ𝑛 ∈ (1...(⌊‘𝐴))((𝑋‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) + if(𝜓, (log‘𝐴), 0)) = (Σ𝑛 ∈ (1...(⌊‘𝐴))((𝑋‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) + (log‘𝐴)))
83		iftrue 4491	. . . . . . . . . 10 ⊢ (𝜓 → if(𝜓, (𝐴 / 𝑑), 𝑚) = (𝐴 / 𝑑))
84	83	fveq2d 6912	. . . . . . . . 9 ⊢ (𝜓 → (log‘if(𝜓, (𝐴 / 𝑑), 𝑚)) = (log‘(𝐴 / 𝑑)))
85	84	oveq1d 7458	. . . . . . . 8 ⊢ (𝜓 → ((log‘if(𝜓, (𝐴 / 𝑑), 𝑚)) / 𝑚) = ((log‘(𝐴 / 𝑑)) / 𝑚))
86	85	oveq2d 7459	. . . . . . 7 ⊢ (𝜓 → ((𝑋‘(𝐿‘𝑚)) · ((log‘if(𝜓, (𝐴 / 𝑑), 𝑚)) / 𝑚)) = ((𝑋‘(𝐿‘𝑚)) · ((log‘(𝐴 / 𝑑)) / 𝑚)))
87	86	sumeq2sdv 15899	. . . . . 6 ⊢ (𝜓 → Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))((𝑋‘(𝐿‘𝑚)) · ((log‘if(𝜓, (𝐴 / 𝑑), 𝑚)) / 𝑚)) = Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))((𝑋‘(𝐿‘𝑚)) · ((log‘(𝐴 / 𝑑)) / 𝑚)))
88	87	oveq2d 7459	. . . . 5 ⊢ (𝜓 → (((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))((𝑋‘(𝐿‘𝑚)) · ((log‘if(𝜓, (𝐴 / 𝑑), 𝑚)) / 𝑚))) = (((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))((𝑋‘(𝐿‘𝑚)) · ((log‘(𝐴 / 𝑑)) / 𝑚))))
89	88	sumeq2sdv 15899	. . . 4 ⊢ (𝜓 → Σ𝑑 ∈ (1...(⌊‘𝐴))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))((𝑋‘(𝐿‘𝑚)) · ((log‘if(𝜓, (𝐴 / 𝑑), 𝑚)) / 𝑚))) = Σ𝑑 ∈ (1...(⌊‘𝐴))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))((𝑋‘(𝐿‘𝑚)) · ((log‘(𝐴 / 𝑑)) / 𝑚))))
90	89	adantl 487	. . 3 ⊢ ((𝜑 ∧ 𝜓) → Σ𝑑 ∈ (1...(⌊‘𝐴))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))((𝑋‘(𝐿‘𝑚)) · ((log‘if(𝜓, (𝐴 / 𝑑), 𝑚)) / 𝑚))) = Σ𝑑 ∈ (1...(⌊‘𝐴))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))((𝑋‘(𝐿‘𝑚)) · ((log‘(𝐴 / 𝑑)) / 𝑚))))
91	79, 82, 90	3eqtr4d 2808	. 2 ⊢ ((𝜑 ∧ 𝜓) → (Σ𝑛 ∈ (1...(⌊‘𝐴))((𝑋‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) + if(𝜓, (log‘𝐴), 0)) = Σ𝑑 ∈ (1...(⌊‘𝐴))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))((𝑋‘(𝐿‘𝑚)) · ((log‘if(𝜓, (𝐴 / 𝑑), 𝑚)) / 𝑚))))
92	6	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝑋 ∈ 𝐷)
93		elfzelz 13683	. . . . . . . . 9 ⊢ (𝑛 ∈ (1...(⌊‘𝐴)) → 𝑛 ∈ ℤ)
94	93	adantl 487	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℤ)
95	2, 3, 4, 5, 92, 94	dchrzrhcl 27602	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (𝑋‘(𝐿‘𝑛)) ∈ ℂ)
96		elfznn 13714	. . . . . . . . 9 ⊢ (𝑛 ∈ (1...(⌊‘𝐴)) → 𝑛 ∈ ℕ)
97	96	adantl 487	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℕ)
98		vmacl 27475	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (Λ‘𝑛) ∈ ℝ)
99		nndivre 12406	. . . . . . . . . 10 ⊢ (((Λ‘𝑛) ∈ ℝ ∧ 𝑛 ∈ ℕ) → ((Λ‘𝑛) / 𝑛) ∈ ℝ)
100	98, 99	mpancom 701	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → ((Λ‘𝑛) / 𝑛) ∈ ℝ)
101	100	recnd 11364	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → ((Λ‘𝑛) / 𝑛) ∈ ℂ)
102	97, 101	syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → ((Λ‘𝑛) / 𝑛) ∈ ℂ)
103	95, 102	mulcld 11356	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → ((𝑋‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) ∈ ℂ)
104	1, 103	fsumcl 15928	. . . . 5 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝐴))((𝑋‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) ∈ ℂ)
105	104	adantr 486	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝜓) → Σ𝑛 ∈ (1...(⌊‘𝐴))((𝑋‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) ∈ ℂ)
106	105	addridd 11537	. . 3 ⊢ ((𝜑 ∧ ¬ 𝜓) → (Σ𝑛 ∈ (1...(⌊‘𝐴))((𝑋‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) + 0) = Σ𝑛 ∈ (1...(⌊‘𝐴))((𝑋‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)))
107		iffalse 4494	. . . . 5 ⊢ (¬ 𝜓 → if(𝜓, (log‘𝐴), 0) = 0)
108	107	adantl 487	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝜓) → if(𝜓, (log‘𝐴), 0) = 0)
109	108	oveq2d 7459	. . 3 ⊢ ((𝜑 ∧ ¬ 𝜓) → (Σ𝑛 ∈ (1...(⌊‘𝐴))((𝑋‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) + if(𝜓, (log‘𝐴), 0)) = (Σ𝑛 ∈ (1...(⌊‘𝐴))((𝑋‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) + 0))
110		iffalse 4494	. . . . . . . . . 10 ⊢ (¬ 𝜓 → if(𝜓, (𝐴 / 𝑑), 𝑚) = 𝑚)
111	110	fveq2d 6912	. . . . . . . . 9 ⊢ (¬ 𝜓 → (log‘if(𝜓, (𝐴 / 𝑑), 𝑚)) = (log‘𝑚))
112	111	oveq1d 7458	. . . . . . . 8 ⊢ (¬ 𝜓 → ((log‘if(𝜓, (𝐴 / 𝑑), 𝑚)) / 𝑚) = ((log‘𝑚) / 𝑚))
113	112	oveq2d 7459	. . . . . . 7 ⊢ (¬ 𝜓 → ((𝑋‘(𝐿‘𝑚)) · ((log‘if(𝜓, (𝐴 / 𝑑), 𝑚)) / 𝑚)) = ((𝑋‘(𝐿‘𝑚)) · ((log‘𝑚) / 𝑚)))
114	113	sumeq2sdv 15899	. . . . . 6 ⊢ (¬ 𝜓 → Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))((𝑋‘(𝐿‘𝑚)) · ((log‘if(𝜓, (𝐴 / 𝑑), 𝑚)) / 𝑚)) = Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))((𝑋‘(𝐿‘𝑚)) · ((log‘𝑚) / 𝑚)))
115	114	oveq2d 7459	. . . . 5 ⊢ (¬ 𝜓 → (((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))((𝑋‘(𝐿‘𝑚)) · ((log‘if(𝜓, (𝐴 / 𝑑), 𝑚)) / 𝑚))) = (((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))((𝑋‘(𝐿‘𝑚)) · ((log‘𝑚) / 𝑚))))
116	115	sumeq2sdv 15899	. . . 4 ⊢ (¬ 𝜓 → Σ𝑑 ∈ (1...(⌊‘𝐴))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))((𝑋‘(𝐿‘𝑚)) · ((log‘if(𝜓, (𝐴 / 𝑑), 𝑚)) / 𝑚))) = Σ𝑑 ∈ (1...(⌊‘𝐴))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))((𝑋‘(𝐿‘𝑚)) · ((log‘𝑚) / 𝑚))))
117	74	eqcomd 2769	. . . 4 ⊢ (𝜑 → Σ𝑑 ∈ (1...(⌊‘𝐴))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))((𝑋‘(𝐿‘𝑚)) · ((log‘𝑚) / 𝑚))) = Σ𝑛 ∈ (1...(⌊‘𝐴))((𝑋‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)))
118	116, 117	sylan9eqr 2820	. . 3 ⊢ ((𝜑 ∧ ¬ 𝜓) → Σ𝑑 ∈ (1...(⌊‘𝐴))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))((𝑋‘(𝐿‘𝑚)) · ((log‘if(𝜓, (𝐴 / 𝑑), 𝑚)) / 𝑚))) = Σ𝑛 ∈ (1...(⌊‘𝐴))((𝑋‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)))
119	106, 109, 118	3eqtr4d 2808	. 2 ⊢ ((𝜑 ∧ ¬ 𝜓) → (Σ𝑛 ∈ (1...(⌊‘𝐴))((𝑋‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) + if(𝜓, (log‘𝐴), 0)) = Σ𝑑 ∈ (1...(⌊‘𝐴))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))((𝑋‘(𝐿‘𝑚)) · ((log‘if(𝜓, (𝐴 / 𝑑), 𝑚)) / 𝑚))))
120	91, 119	pm2.61dan 825	1 ⊢ (𝜑 → (Σ𝑛 ∈ (1...(⌊‘𝐴))((𝑋‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) + if(𝜓, (log‘𝐴), 0)) = Σ𝑑 ∈ (1...(⌊‘𝐴))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))((𝑋‘(𝐿‘𝑚)) · ((log‘if(𝜓, (𝐴 / 𝑑), 𝑚)) / 𝑚))))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  ifcif 4485   class class class wbr 5106  ‘cfv 6558  (class class class)co 7443  ℂcc 11225  ℝcr 11226  0cc0 11227  1c1 11228   + caddc 11230   · cmul 11232   ≤ cle 11371   − cmin 11568   / cdiv 12000  ℕcn 12362  ℤcz 12720  ℝ⁺crp 13147  ...cfz 13666  ⌊cfl 13957  Σcsu 15882  Basecbs 17416  0_gc0g 17639  ℤRHomczrh 21835  ℤ/nℤczn 21838  logclog 26912  Λcvma 27449  μcmu 27452  DChrcdchr 27589

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775  ax-inf2 9669  ax-cnex 11283  ax-resscn 11284  ax-1cn 11285  ax-icn 11286  ax-addcl 11287  ax-addrcl 11288  ax-mulcl 11289  ax-mulrcl 11290  ax-mulcom 11291  ax-addass 11292  ax-mulass 11293  ax-distr 11294  ax-i2m1 11295  ax-1ne0 11296  ax-1rid 11297  ax-rnegex 11298  ax-rrecex 11299  ax-cnre 11300  ax-pre-lttri 11301  ax-pre-lttrn 11302  ax-pre-ltadd 11303  ax-pre-mulgt0 11304  ax-pre-sup 11305  ax-addf 11306  ax-mulf 11307

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-iin 4957  df-disj 5074  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-isom 6567  df-riota 7400  df-ov 7446  df-oprab 7447  df-mpo 7448  df-of 7717  df-om 7902  df-1st 8025  df-2nd 8026  df-supp 8205  df-tpos 8271  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-rdg 8446  df-1o 8504  df-2o 8505  df-oadd 8508  df-er 8745  df-ec 8747  df-qs 8751  df-map 8877  df-pm 8878  df-ixp 8954  df-en 9002  df-dom 9003  df-sdom 9004  df-fin 9005  df-fsupp 9382  df-fi 9430  df-sup 9461  df-inf 9462  df-oi 9531  df-dju 10009  df-card 10047  df-pnf 11372  df-mnf 11373  df-xr 11374  df-ltxr 11375  df-le 11376  df-sub 11570  df-neg 11571  df-div 12001  df-nn 12363  df-2 12432  df-3 12433  df-4 12434  df-5 12435  df-6 12436  df-7 12437  df-8 12438  df-9 12439  df-n0 12634  df-xnn0 12707  df-z 12721  df-dec 12842  df-uz 12993  df-q 13103  df-rp 13148  df-xneg 13268  df-xadd 13269  df-xmul 13270  df-ioo 13507  df-ioc 13508  df-ico 13509  df-icc 13510  df-fz 13667  df-fzo 13816  df-fl 13959  df-mod 14037  df-seq 14172  df-exp 14232  df-fac 14445  df-bc 14474  df-hash 14502  df-shft 15249  df-cj 15295  df-re 15296  df-im 15297  df-sqrt 15431  df-abs 15432  df-limsup 15667  df-clim 15684  df-rlim 15685  df-sum 15883  df-ef 16262  df-sin 16264  df-cos 16265  df-pi 16267  df-dvds 16452  df-gcd 16694  df-prm 16876  df-pc 17044  df-struct 17354  df-sets 17371  df-slot 17389  df-ndx 17401  df-base 17417  df-ress 17438  df-plusg 17470  df-mulr 17471  df-starv 17472  df-sca 17473  df-vsca 17474  df-ip 17475  df-tset 17476  df-ple 17477  df-ds 17479  df-unif 17480  df-hom 17481  df-cco 17482  df-rest 17622  df-topn 17623  df-0g 17641  df-gsum 17642  df-topgen 17643  df-pt 17644  df-prds 17647  df-xrs 17703  df-qtop 17708  df-imas 17709  df-qus 17710  df-xps 17711  df-mre 17785  df-mrc 17786  df-acs 17788  df-mgm 18845  df-sgrp 18937  df-mnd 18953  df-mhm 19007  df-submnd 19008  df-grp 19176  df-minusg 19177  df-sbg 19178  df-mulg 19307  df-subg 19362  df-nsg 19363  df-eqg 19364  df-ghm 19457  df-cntz 19561  df-cmn 20026  df-abl 20027  df-mgp 20391  df-rng 20405  df-ur 20438  df-ring 20491  df-cring 20492  df-oppr 20597  df-dvdsr 20617  df-unit 20618  df-rhm 20732  df-subrng 20828  df-subrg 20852  df-lmod 21167  df-lss 21237  df-lsp 21277  df-sra 21478  df-rgmod 21479  df-lidl 21516  df-rsp 21517  df-2idl 21573  df-psmet 21700  df-xmet 21701  df-met 21702  df-bl 21703  df-mopn 21704  df-fbas 21705  df-fg 21706  df-cnfld 21709  df-zring 21783  df-zrh 21839  df-zn 21842  df-top 23242  df-topon 23259  df-topsp 23281  df-bases 23294  df-cld 23367  df-ntr 23368  df-cls 23369  df-nei 23446  df-lp 23484  df-perf 23485  df-cn 23575  df-cnp 23576  df-haus 23663  df-tx 23911  df-hmeo 24104  df-fil 24195  df-fm 24287  df-flim 24288  df-flf 24289  df-xms 24669  df-ms 24670  df-tms 24671  df-cncf 25229  df-limc 26216  df-dv 26217  df-log 26914  df-vma 27455  df-mu 27458  df-dchr 27590

This theorem is used by:  dchrvmasumiflem2  27859