Theorem dchrvmasum2if 27441

Description: Combine the results of dchrvmasumlem1 27439 and dchrvmasum2lem 27440 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 13886	. . . . . 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 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → 𝑋 ∈ 𝐷)
8		elfzelz 13430	. . . . . . . . . 10 ⊢ (𝑑 ∈ (1...(⌊‘𝐴)) → 𝑑 ∈ ℤ)
9	8	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → 𝑑 ∈ ℤ)
10	2, 3, 4, 5, 7, 9	dchrzrhcl 27189	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (𝑋‘(𝐿‘𝑑)) ∈ ℂ)
11		elfznn 13459	. . . . . . . . . . 11 ⊢ (𝑑 ∈ (1...(⌊‘𝐴)) → 𝑑 ∈ ℕ)
12	11	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → 𝑑 ∈ ℕ)
13		mucl 27084	. . . . . . . . . . . 12 ⊢ (𝑑 ∈ ℕ → (μ‘𝑑) ∈ ℤ)
14	13	zred 12583	. . . . . . . . . . 11 ⊢ (𝑑 ∈ ℕ → (μ‘𝑑) ∈ ℝ)
15		nndivre 12172	. . . . . . . . . . 11 ⊢ (((μ‘𝑑) ∈ ℝ ∧ 𝑑 ∈ ℕ) → ((μ‘𝑑) / 𝑑) ∈ ℝ)
16	14, 15	mpancom 688	. . . . . . . . . 10 ⊢ (𝑑 ∈ ℕ → ((μ‘𝑑) / 𝑑) ∈ ℝ)
17	12, 16	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → ((μ‘𝑑) / 𝑑) ∈ ℝ)
18	17	recnd 11146	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → ((μ‘𝑑) / 𝑑) ∈ ℂ)
19	10, 18	mulcld 11138	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → ((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) ∈ ℂ)
20		fzfid 13886	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (1...(⌊‘(𝐴 / 𝑑))) ∈ Fin)
21	7	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → 𝑋 ∈ 𝐷)
22		elfzelz 13430	. . . . . . . . . . 11 ⊢ (𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑))) → 𝑚 ∈ ℤ)
23	22	adantl 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → 𝑚 ∈ ℤ)
24	2, 3, 4, 5, 21, 23	dchrzrhcl 27189	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → (𝑋‘(𝐿‘𝑚)) ∈ ℂ)
25		elfznn 13459	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑))) → 𝑚 ∈ ℕ)
26	25	adantl 481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → 𝑚 ∈ ℕ)
27	26	nnrpd 12938	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → 𝑚 ∈ ℝ+)
28	27	relogcld 26565	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → (log‘𝑚) ∈ ℝ)
29	28, 26	nndivred 12185	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → ((log‘𝑚) / 𝑚) ∈ ℝ)
30	29	recnd 11146	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → ((log‘𝑚) / 𝑚) ∈ ℂ)
31	24, 30	mulcld 11138	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → ((𝑋‘(𝐿‘𝑚)) · ((log‘𝑚) / 𝑚)) ∈ ℂ)
32	20, 31	fsumcl 15646	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))((𝑋‘(𝐿‘𝑚)) · ((log‘𝑚) / 𝑚)) ∈ ℂ)
33	19, 32	mulcld 11138	. . . . . 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 12938	. . . . . . . . . . . . . . 15 ⊢ (𝑑 ∈ (1...(⌊‘𝐴)) → 𝑑 ∈ ℝ+)
36		rpdivcl 12923	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+) → (𝐴 / 𝑑) ∈ ℝ+)
37	34, 35, 36	syl2an 596	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (𝐴 / 𝑑) ∈ ℝ+)
38	37	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → (𝐴 / 𝑑) ∈ ℝ+)
39	38, 27	rpdivcld 12957	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → ((𝐴 / 𝑑) / 𝑚) ∈ ℝ+)
40	39	relogcld 26565	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → (log‘((𝐴 / 𝑑) / 𝑚)) ∈ ℝ)
41	40, 26	nndivred 12185	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → ((log‘((𝐴 / 𝑑) / 𝑚)) / 𝑚) ∈ ℝ)
42	41	recnd 11146	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → ((log‘((𝐴 / 𝑑) / 𝑚)) / 𝑚) ∈ ℂ)
43	24, 42	mulcld 11138	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → ((𝑋‘(𝐿‘𝑚)) · ((log‘((𝐴 / 𝑑) / 𝑚)) / 𝑚)) ∈ ℂ)
44	20, 43	fsumcl 15646	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))((𝑋‘(𝐿‘𝑚)) · ((log‘((𝐴 / 𝑑) / 𝑚)) / 𝑚)) ∈ ℂ)
45	19, 44	mulcld 11138	. . . . . 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 15653	. . . . 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 26568	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → (log‘((𝐴 / 𝑑) / 𝑚)) = ((log‘(𝐴 / 𝑑)) − (log‘𝑚)))
48	47	oveq2d 7368	. . . . . . . . . . . . . . 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 11146	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → (log‘𝑚) ∈ ℂ)
50	37	relogcld 26565	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (log‘(𝐴 / 𝑑)) ∈ ℝ)
51	50	recnd 11146	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (log‘(𝐴 / 𝑑)) ∈ ℂ)
52	51	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → (log‘(𝐴 / 𝑑)) ∈ ℂ)
53	49, 52	pncan3d 11481	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → ((log‘𝑚) + ((log‘(𝐴 / 𝑑)) − (log‘𝑚))) = (log‘(𝐴 / 𝑑)))
54	48, 53	eqtr2d 2767	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → (log‘(𝐴 / 𝑑)) = ((log‘𝑚) + (log‘((𝐴 / 𝑑) / 𝑚))))
55	54	oveq1d 7367	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → ((log‘(𝐴 / 𝑑)) / 𝑚) = (((log‘𝑚) + (log‘((𝐴 / 𝑑) / 𝑚))) / 𝑚))
56	40	recnd 11146	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → (log‘((𝐴 / 𝑑) / 𝑚)) ∈ ℂ)
57	26	nncnd 12147	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → 𝑚 ∈ ℂ)
58	26	nnne0d 12181	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → 𝑚 ≠ 0)
59	49, 56, 57, 58	divdird 11941	. . . . . . . . . . . . 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 2766	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → ((log‘(𝐴 / 𝑑)) / 𝑚) = (((log‘𝑚) / 𝑚) + ((log‘((𝐴 / 𝑑) / 𝑚)) / 𝑚)))
61	60	oveq2d 7368	. . . . . . . . . . 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 11142	. . . . . . . . . . 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 2766	. . . . . . . . . 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 15615	. . . . . . . . 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 15653	. . . . . . . . 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 2766	. . . . . . . 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 7368	. . . . . . 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 11142	. . . . . . 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 2766	. . . . . 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 15615	. . . . 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 27439	. . . . . 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 27440	. . . . . 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 7370	. . . . 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 2777	. . . 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 480	. . 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 4480	. . . . 5 ⊢ (𝜓 → if(𝜓, (log‘𝐴), 0) = (log‘𝐴))
81	80	oveq2d 7368	. . . 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 481	. . 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 4480	. . . . . . . . . 10 ⊢ (𝜓 → if(𝜓, (𝐴 / 𝑑), 𝑚) = (𝐴 / 𝑑))
84	83	fveq2d 6832	. . . . . . . . 9 ⊢ (𝜓 → (log‘if(𝜓, (𝐴 / 𝑑), 𝑚)) = (log‘(𝐴 / 𝑑)))
85	84	oveq1d 7367	. . . . . . . 8 ⊢ (𝜓 → ((log‘if(𝜓, (𝐴 / 𝑑), 𝑚)) / 𝑚) = ((log‘(𝐴 / 𝑑)) / 𝑚))
86	85	oveq2d 7368	. . . . . . 7 ⊢ (𝜓 → ((𝑋‘(𝐿‘𝑚)) · ((log‘if(𝜓, (𝐴 / 𝑑), 𝑚)) / 𝑚)) = ((𝑋‘(𝐿‘𝑚)) · ((log‘(𝐴 / 𝑑)) / 𝑚)))
87	86	sumeq2sdv 15616	. . . . . 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 7368	. . . . 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 15616	. . . 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 481	. . 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 2776	. 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 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝑋 ∈ 𝐷)
93		elfzelz 13430	. . . . . . . . 9 ⊢ (𝑛 ∈ (1...(⌊‘𝐴)) → 𝑛 ∈ ℤ)
94	93	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℤ)
95	2, 3, 4, 5, 92, 94	dchrzrhcl 27189	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (𝑋‘(𝐿‘𝑛)) ∈ ℂ)
96		elfznn 13459	. . . . . . . . 9 ⊢ (𝑛 ∈ (1...(⌊‘𝐴)) → 𝑛 ∈ ℕ)
97	96	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℕ)
98		vmacl 27061	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (Λ‘𝑛) ∈ ℝ)
99		nndivre 12172	. . . . . . . . . 10 ⊢ (((Λ‘𝑛) ∈ ℝ ∧ 𝑛 ∈ ℕ) → ((Λ‘𝑛) / 𝑛) ∈ ℝ)
100	98, 99	mpancom 688	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → ((Λ‘𝑛) / 𝑛) ∈ ℝ)
101	100	recnd 11146	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → ((Λ‘𝑛) / 𝑛) ∈ ℂ)
102	97, 101	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → ((Λ‘𝑛) / 𝑛) ∈ ℂ)
103	95, 102	mulcld 11138	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → ((𝑋‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) ∈ ℂ)
104	1, 103	fsumcl 15646	. . . . 5 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝐴))((𝑋‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) ∈ ℂ)
105	104	adantr 480	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝜓) → Σ𝑛 ∈ (1...(⌊‘𝐴))((𝑋‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) ∈ ℂ)
106	105	addridd 11319	. . 3 ⊢ ((𝜑 ∧ ¬ 𝜓) → (Σ𝑛 ∈ (1...(⌊‘𝐴))((𝑋‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) + 0) = Σ𝑛 ∈ (1...(⌊‘𝐴))((𝑋‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)))
107		iffalse 4483	. . . . 5 ⊢ (¬ 𝜓 → if(𝜓, (log‘𝐴), 0) = 0)
108	107	adantl 481	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝜓) → if(𝜓, (log‘𝐴), 0) = 0)
109	108	oveq2d 7368	. . 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 4483	. . . . . . . . . 10 ⊢ (¬ 𝜓 → if(𝜓, (𝐴 / 𝑑), 𝑚) = 𝑚)
111	110	fveq2d 6832	. . . . . . . . 9 ⊢ (¬ 𝜓 → (log‘if(𝜓, (𝐴 / 𝑑), 𝑚)) = (log‘𝑚))
112	111	oveq1d 7367	. . . . . . . 8 ⊢ (¬ 𝜓 → ((log‘if(𝜓, (𝐴 / 𝑑), 𝑚)) / 𝑚) = ((log‘𝑚) / 𝑚))
113	112	oveq2d 7368	. . . . . . 7 ⊢ (¬ 𝜓 → ((𝑋‘(𝐿‘𝑚)) · ((log‘if(𝜓, (𝐴 / 𝑑), 𝑚)) / 𝑚)) = ((𝑋‘(𝐿‘𝑚)) · ((log‘𝑚) / 𝑚)))
114	113	sumeq2sdv 15616	. . . . . 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 7368	. . . . 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 15616	. . . 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 2737	. . . 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 2788	. . 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 2776	. 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 812	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))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111   ≠ wne 2928  ifcif 4474   class class class wbr 5093  ‘cfv 6487  (class class class)co 7352  ℂcc 11010  ℝcr 11011  0cc0 11012  1c1 11013   + caddc 11015   · cmul 11017   ≤ cle 11153   − cmin 11350   / cdiv 11780  ℕcn 12131  ℤcz 12474  ℝ⁺crp 12896  ...cfz 13413  ⌊cfl 13700  Σcsu 15599  Basecbs 17126  0_gc0g 17349  ℤRHomczrh 21442  ℤ/nℤczn 21445  logclog 26496  Λcvma 27035  μcmu 27038  DChrcdchr 27176

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674  ax-inf2 9537  ax-cnex 11068  ax-resscn 11069  ax-1cn 11070  ax-icn 11071  ax-addcl 11072  ax-addrcl 11073  ax-mulcl 11074  ax-mulrcl 11075  ax-mulcom 11076  ax-addass 11077  ax-mulass 11078  ax-distr 11079  ax-i2m1 11080  ax-1ne0 11081  ax-1rid 11082  ax-rnegex 11083  ax-rrecex 11084  ax-cnre 11085  ax-pre-lttri 11086  ax-pre-lttrn 11087  ax-pre-ltadd 11088  ax-pre-mulgt0 11089  ax-pre-sup 11090  ax-addf 11091  ax-mulf 11092

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-tp 4580  df-op 4582  df-uni 4859  df-int 4898  df-iun 4943  df-iin 4944  df-disj 5061  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6254  df-ord 6315  df-on 6316  df-lim 6317  df-suc 6318  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-fv 6495  df-isom 6496  df-riota 7309  df-ov 7355  df-oprab 7356  df-mpo 7357  df-of 7616  df-om 7803  df-1st 7927  df-2nd 7928  df-supp 8097  df-tpos 8162  df-frecs 8217  df-wrecs 8248  df-recs 8297  df-rdg 8335  df-1o 8391  df-2o 8392  df-oadd 8395  df-er 8628  df-ec 8630  df-qs 8634  df-map 8758  df-pm 8759  df-ixp 8828  df-en 8876  df-dom 8877  df-sdom 8878  df-fin 8879  df-fsupp 9252  df-fi 9301  df-sup 9332  df-inf 9333  df-oi 9402  df-dju 9800  df-card 9838  df-pnf 11154  df-mnf 11155  df-xr 11156  df-ltxr 11157  df-le 11158  df-sub 11352  df-neg 11353  df-div 11781  df-nn 12132  df-2 12194  df-3 12195  df-4 12196  df-5 12197  df-6 12198  df-7 12199  df-8 12200  df-9 12201  df-n0 12388  df-xnn0 12461  df-z 12475  df-dec 12595  df-uz 12739  df-q 12853  df-rp 12897  df-xneg 13017  df-xadd 13018  df-xmul 13019  df-ioo 13255  df-ioc 13256  df-ico 13257  df-icc 13258  df-fz 13414  df-fzo 13561  df-fl 13702  df-mod 13780  df-seq 13915  df-exp 13975  df-fac 14187  df-bc 14216  df-hash 14244  df-shft 14980  df-cj 15012  df-re 15013  df-im 15014  df-sqrt 15148  df-abs 15149  df-limsup 15384  df-clim 15401  df-rlim 15402  df-sum 15600  df-ef 15980  df-sin 15982  df-cos 15983  df-pi 15985  df-dvds 16170  df-gcd 16412  df-prm 16589  df-pc 16755  df-struct 17064  df-sets 17081  df-slot 17099  df-ndx 17111  df-base 17127  df-ress 17148  df-plusg 17180  df-mulr 17181  df-starv 17182  df-sca 17183  df-vsca 17184  df-ip 17185  df-tset 17186  df-ple 17187  df-ds 17189  df-unif 17190  df-hom 17191  df-cco 17192  df-rest 17332  df-topn 17333  df-0g 17351  df-gsum 17352  df-topgen 17353  df-pt 17354  df-prds 17357  df-xrs 17412  df-qtop 17417  df-imas 17418  df-qus 17419  df-xps 17420  df-mre 17494  df-mrc 17495  df-acs 17497  df-mgm 18554  df-sgrp 18633  df-mnd 18649  df-mhm 18697  df-submnd 18698  df-grp 18855  df-minusg 18856  df-sbg 18857  df-mulg 18987  df-subg 19042  df-nsg 19043  df-eqg 19044  df-ghm 19131  df-cntz 19235  df-cmn 19700  df-abl 19701  df-mgp 20065  df-rng 20077  df-ur 20106  df-ring 20159  df-cring 20160  df-oppr 20261  df-dvdsr 20281  df-unit 20282  df-rhm 20396  df-subrng 20467  df-subrg 20491  df-lmod 20801  df-lss 20871  df-lsp 20911  df-sra 21113  df-rgmod 21114  df-lidl 21151  df-rsp 21152  df-2idl 21193  df-psmet 21289  df-xmet 21290  df-met 21291  df-bl 21292  df-mopn 21293  df-fbas 21294  df-fg 21295  df-cnfld 21298  df-zring 21390  df-zrh 21446  df-zn 21449  df-top 22815  df-topon 22832  df-topsp 22854  df-bases 22867  df-cld 22940  df-ntr 22941  df-cls 22942  df-nei 23019  df-lp 23057  df-perf 23058  df-cn 23148  df-cnp 23149  df-haus 23236  df-tx 23483  df-hmeo 23676  df-fil 23767  df-fm 23859  df-flim 23860  df-flf 23861  df-xms 24241  df-ms 24242  df-tms 24243  df-cncf 24804  df-limc 25800  df-dv 25801  df-log 26498  df-vma 27041  df-mu 27044  df-dchr 27177

This theorem is referenced by:  dchrvmasumiflem2  27446