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Theorem dchrisumlema 25177

Description: Lemma for dchrisum 25181. Lemma 9.4.1 of [Shapiro], p. 377. (Contributed by Mario Carneiro, 2-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 = (0_g‘𝐺)
dchrisum.b	⊢ (𝜑 → 𝑋 ∈ 𝐷)
dchrisum.n1	⊢ (𝜑 → 𝑋 ≠ 1 )
dchrisum.2	⊢ (𝑛 = 𝑥 → 𝐴 = 𝐵)
dchrisum.3	⊢ (𝜑 → 𝑀 ∈ ℕ)
dchrisum.4	⊢ ((𝜑 ∧ 𝑛 ∈ ℝ⁺) → 𝐴 ∈ ℝ)
dchrisum.5	⊢ ((𝜑 ∧ (𝑛 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑀 ≤ 𝑛 ∧ 𝑛 ≤ 𝑥)) → 𝐵 ≤ 𝐴)
dchrisum.6	⊢ (𝜑 → (𝑛 ∈ ℝ⁺ ↦ 𝐴) ⇝_𝑟 0)
dchrisum.7	⊢ 𝐹 = (𝑛 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑛)) · 𝐴))

**Assertion**
Ref	Expression
dchrisumlema	⊢ (𝜑 → ((𝐼 ∈ ℝ⁺ → ⦋𝐼 / 𝑛⦌𝐴 ∈ ℝ) ∧ (𝐼 ∈ (𝑀[,)+∞) → 0 ≤ ⦋𝐼 / 𝑛⦌𝐴)))

Distinct variable groups: 𝑥,𝑛, 1 𝑛,𝐹,𝑥 𝑛,𝐼,𝑥 𝑥,𝐴 𝑛,𝑁,𝑥 𝜑,𝑛,𝑥 𝐵,𝑛 𝑛,𝑍,𝑥 𝐷,𝑛,𝑥 𝑛,𝐿,𝑥 𝑛,𝑀,𝑥 𝑛,𝑋,𝑥 Allowed substitution hints: 𝐴(𝑛) 𝐵(𝑥) 𝐺(𝑥,𝑛)

Proof of Theorem dchrisumlema Dummy variable 𝑖 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dchrisum.4	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℝ⁺) → 𝐴 ∈ ℝ)
2	1	ralrimiva 2966	. . 3 ⊢ (𝜑 → ∀𝑛 ∈ ℝ⁺ 𝐴 ∈ ℝ)
3		nfcsb1v 3549	. . . . 5 ⊢ Ⅎ𝑛⦋𝐼 / 𝑛⦌𝐴
4	3	nfel1 2779	. . . 4 ⊢ Ⅎ𝑛⦋𝐼 / 𝑛⦌𝐴 ∈ ℝ
5		csbeq1a 3542	. . . . 5 ⊢ (𝑛 = 𝐼 → 𝐴 = ⦋𝐼 / 𝑛⦌𝐴)
6	5	eleq1d 2686	. . . 4 ⊢ (𝑛 = 𝐼 → (𝐴 ∈ ℝ ↔ ⦋𝐼 / 𝑛⦌𝐴 ∈ ℝ))
7	4, 6	rspc 3303	. . 3 ⊢ (𝐼 ∈ ℝ⁺ → (∀𝑛 ∈ ℝ⁺ 𝐴 ∈ ℝ → ⦋𝐼 / 𝑛⦌𝐴 ∈ ℝ))
8	2, 7	syl5com 31	. 2 ⊢ (𝜑 → (𝐼 ∈ ℝ⁺ → ⦋𝐼 / 𝑛⦌𝐴 ∈ ℝ))
9		eqid 2622	. . . 4 ⊢ (ℤ_≥‘((⌊‘𝐼) + 1)) = (ℤ_≥‘((⌊‘𝐼) + 1))
10		dchrisum.3	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℕ)
11	10	nnred 11035	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℝ)
12		elicopnf 12269	. . . . . . . 8 ⊢ (𝑀 ∈ ℝ → (𝐼 ∈ (𝑀[,)+∞) ↔ (𝐼 ∈ ℝ ∧ 𝑀 ≤ 𝐼)))
13	11, 12	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝐼 ∈ (𝑀[,)+∞) ↔ (𝐼 ∈ ℝ ∧ 𝑀 ≤ 𝐼)))
14	13	simprbda 653	. . . . . 6 ⊢ ((𝜑 ∧ 𝐼 ∈ (𝑀[,)+∞)) → 𝐼 ∈ ℝ)
15	14	flcld 12599	. . . . 5 ⊢ ((𝜑 ∧ 𝐼 ∈ (𝑀[,)+∞)) → (⌊‘𝐼) ∈ ℤ)
16	15	peano2zd 11485	. . . 4 ⊢ ((𝜑 ∧ 𝐼 ∈ (𝑀[,)+∞)) → ((⌊‘𝐼) + 1) ∈ ℤ)
17		nnuz 11723	. . . . . 6 ⊢ ℕ = (ℤ_≥‘1)
18		1zzd 11408	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℤ)
19		dchrisum.6	. . . . . 6 ⊢ (𝜑 → (𝑛 ∈ ℝ⁺ ↦ 𝐴) ⇝_𝑟 0)
20		nnrp 11842	. . . . . . . 8 ⊢ (𝑖 ∈ ℕ → 𝑖 ∈ ℝ⁺)
21	20	ssriv 3607	. . . . . . 7 ⊢ ℕ ⊆ ℝ⁺
22		eqid 2622	. . . . . . . 8 ⊢ (𝑛 ∈ ℝ⁺ ↦ 𝐴) = (𝑛 ∈ ℝ⁺ ↦ 𝐴)
23	22, 1	dmmptd 6024	. . . . . . 7 ⊢ (𝜑 → dom (𝑛 ∈ ℝ⁺ ↦ 𝐴) = ℝ⁺)
24	21, 23	syl5sseqr 3654	. . . . . 6 ⊢ (𝜑 → ℕ ⊆ dom (𝑛 ∈ ℝ⁺ ↦ 𝐴))
25	17, 18, 19, 24	rlimclim1 14276	. . . . 5 ⊢ (𝜑 → (𝑛 ∈ ℝ⁺ ↦ 𝐴) ⇝ 0)
26	25	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝐼 ∈ (𝑀[,)+∞)) → (𝑛 ∈ ℝ⁺ ↦ 𝐴) ⇝ 0)
27		0red 10041	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐼 ∈ (𝑀[,)+∞)) → 0 ∈ ℝ)
28	11	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐼 ∈ (𝑀[,)+∞)) → 𝑀 ∈ ℝ)
29	10	nngt0d 11064	. . . . . . . . . 10 ⊢ (𝜑 → 0 < 𝑀)
30	29	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐼 ∈ (𝑀[,)+∞)) → 0 < 𝑀)
31	13	simplbda 654	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐼 ∈ (𝑀[,)+∞)) → 𝑀 ≤ 𝐼)
32	27, 28, 14, 30, 31	ltletrd 10197	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐼 ∈ (𝑀[,)+∞)) → 0 < 𝐼)
33	14, 32	elrpd 11869	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐼 ∈ (𝑀[,)+∞)) → 𝐼 ∈ ℝ⁺)
34	2	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐼 ∈ (𝑀[,)+∞)) → ∀𝑛 ∈ ℝ⁺ 𝐴 ∈ ℝ)
35	33, 34, 7	sylc 65	. . . . . 6 ⊢ ((𝜑 ∧ 𝐼 ∈ (𝑀[,)+∞)) → ⦋𝐼 / 𝑛⦌𝐴 ∈ ℝ)
36	35	recnd 10068	. . . . 5 ⊢ ((𝜑 ∧ 𝐼 ∈ (𝑀[,)+∞)) → ⦋𝐼 / 𝑛⦌𝐴 ∈ ℂ)
37		ssid 3624	. . . . . 6 ⊢ (ℤ_≥‘((⌊‘𝐼) + 1)) ⊆ (ℤ_≥‘((⌊‘𝐼) + 1))
38		fvex 6201	. . . . . 6 ⊢ (ℤ_≥‘((⌊‘𝐼) + 1)) ∈ V
39	37, 38	climconst2 14279	. . . . 5 ⊢ ((⦋𝐼 / 𝑛⦌𝐴 ∈ ℂ ∧ ((⌊‘𝐼) + 1) ∈ ℤ) → ((ℤ_≥‘((⌊‘𝐼) + 1)) × {⦋𝐼 / 𝑛⦌𝐴}) ⇝ ⦋𝐼 / 𝑛⦌𝐴)
40	36, 16, 39	syl2anc 693	. . . 4 ⊢ ((𝜑 ∧ 𝐼 ∈ (𝑀[,)+∞)) → ((ℤ_≥‘((⌊‘𝐼) + 1)) × {⦋𝐼 / 𝑛⦌𝐴}) ⇝ ⦋𝐼 / 𝑛⦌𝐴)
41	33	rpge0d 11876	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐼 ∈ (𝑀[,)+∞)) → 0 ≤ 𝐼)
42		flge0nn0 12621	. . . . . . . . . 10 ⊢ ((𝐼 ∈ ℝ ∧ 0 ≤ 𝐼) → (⌊‘𝐼) ∈ ℕ₀)
43	14, 41, 42	syl2anc 693	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐼 ∈ (𝑀[,)+∞)) → (⌊‘𝐼) ∈ ℕ₀)
44		nn0p1nn 11332	. . . . . . . . 9 ⊢ ((⌊‘𝐼) ∈ ℕ₀ → ((⌊‘𝐼) + 1) ∈ ℕ)
45	43, 44	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐼 ∈ (𝑀[,)+∞)) → ((⌊‘𝐼) + 1) ∈ ℕ)
46		eluznn 11758	. . . . . . . 8 ⊢ ((((⌊‘𝐼) + 1) ∈ ℕ ∧ 𝑖 ∈ (ℤ_≥‘((⌊‘𝐼) + 1))) → 𝑖 ∈ ℕ)
47	45, 46	sylan 488	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐼 ∈ (𝑀[,)+∞)) ∧ 𝑖 ∈ (ℤ_≥‘((⌊‘𝐼) + 1))) → 𝑖 ∈ ℕ)
48	47	nnrpd 11870	. . . . . 6 ⊢ (((𝜑 ∧ 𝐼 ∈ (𝑀[,)+∞)) ∧ 𝑖 ∈ (ℤ_≥‘((⌊‘𝐼) + 1))) → 𝑖 ∈ ℝ⁺)
49	2	ad2antrr 762	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐼 ∈ (𝑀[,)+∞)) ∧ 𝑖 ∈ (ℤ_≥‘((⌊‘𝐼) + 1))) → ∀𝑛 ∈ ℝ⁺ 𝐴 ∈ ℝ)
50		nfcsb1v 3549	. . . . . . . . 9 ⊢ Ⅎ𝑛⦋𝑖 / 𝑛⦌𝐴
51	50	nfel1 2779	. . . . . . . 8 ⊢ Ⅎ𝑛⦋𝑖 / 𝑛⦌𝐴 ∈ ℝ
52		csbeq1a 3542	. . . . . . . . 9 ⊢ (𝑛 = 𝑖 → 𝐴 = ⦋𝑖 / 𝑛⦌𝐴)
53	52	eleq1d 2686	. . . . . . . 8 ⊢ (𝑛 = 𝑖 → (𝐴 ∈ ℝ ↔ ⦋𝑖 / 𝑛⦌𝐴 ∈ ℝ))
54	51, 53	rspc 3303	. . . . . . 7 ⊢ (𝑖 ∈ ℝ⁺ → (∀𝑛 ∈ ℝ⁺ 𝐴 ∈ ℝ → ⦋𝑖 / 𝑛⦌𝐴 ∈ ℝ))
55	48, 49, 54	sylc 65	. . . . . 6 ⊢ (((𝜑 ∧ 𝐼 ∈ (𝑀[,)+∞)) ∧ 𝑖 ∈ (ℤ_≥‘((⌊‘𝐼) + 1))) → ⦋𝑖 / 𝑛⦌𝐴 ∈ ℝ)
56	22	fvmpts 6285	. . . . . 6 ⊢ ((𝑖 ∈ ℝ⁺ ∧ ⦋𝑖 / 𝑛⦌𝐴 ∈ ℝ) → ((𝑛 ∈ ℝ⁺ ↦ 𝐴)‘𝑖) = ⦋𝑖 / 𝑛⦌𝐴)
57	48, 55, 56	syl2anc 693	. . . . 5 ⊢ (((𝜑 ∧ 𝐼 ∈ (𝑀[,)+∞)) ∧ 𝑖 ∈ (ℤ_≥‘((⌊‘𝐼) + 1))) → ((𝑛 ∈ ℝ⁺ ↦ 𝐴)‘𝑖) = ⦋𝑖 / 𝑛⦌𝐴)
58	57, 55	eqeltrd 2701	. . . 4 ⊢ (((𝜑 ∧ 𝐼 ∈ (𝑀[,)+∞)) ∧ 𝑖 ∈ (ℤ_≥‘((⌊‘𝐼) + 1))) → ((𝑛 ∈ ℝ⁺ ↦ 𝐴)‘𝑖) ∈ ℝ)
59		fvconst2g 6467	. . . . . 6 ⊢ ((⦋𝐼 / 𝑛⦌𝐴 ∈ ℝ ∧ 𝑖 ∈ (ℤ_≥‘((⌊‘𝐼) + 1))) → (((ℤ_≥‘((⌊‘𝐼) + 1)) × {⦋𝐼 / 𝑛⦌𝐴})‘𝑖) = ⦋𝐼 / 𝑛⦌𝐴)
60	35, 59	sylan 488	. . . . 5 ⊢ (((𝜑 ∧ 𝐼 ∈ (𝑀[,)+∞)) ∧ 𝑖 ∈ (ℤ_≥‘((⌊‘𝐼) + 1))) → (((ℤ_≥‘((⌊‘𝐼) + 1)) × {⦋𝐼 / 𝑛⦌𝐴})‘𝑖) = ⦋𝐼 / 𝑛⦌𝐴)
61	35	adantr 481	. . . . 5 ⊢ (((𝜑 ∧ 𝐼 ∈ (𝑀[,)+∞)) ∧ 𝑖 ∈ (ℤ_≥‘((⌊‘𝐼) + 1))) → ⦋𝐼 / 𝑛⦌𝐴 ∈ ℝ)
62	60, 61	eqeltrd 2701	. . . 4 ⊢ (((𝜑 ∧ 𝐼 ∈ (𝑀[,)+∞)) ∧ 𝑖 ∈ (ℤ_≥‘((⌊‘𝐼) + 1))) → (((ℤ_≥‘((⌊‘𝐼) + 1)) × {⦋𝐼 / 𝑛⦌𝐴})‘𝑖) ∈ ℝ)
63	33	adantr 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐼 ∈ (𝑀[,)+∞)) ∧ 𝑖 ∈ (ℤ_≥‘((⌊‘𝐼) + 1))) → 𝐼 ∈ ℝ⁺)
64		dchrisum.5	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑀 ≤ 𝑛 ∧ 𝑛 ≤ 𝑥)) → 𝐵 ≤ 𝐴)
65	64	3expia 1267	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺)) → ((𝑀 ≤ 𝑛 ∧ 𝑛 ≤ 𝑥) → 𝐵 ≤ 𝐴))
66	65	ralrimivva 2971	. . . . . . . 8 ⊢ (𝜑 → ∀𝑛 ∈ ℝ⁺ ∀𝑥 ∈ ℝ⁺ ((𝑀 ≤ 𝑛 ∧ 𝑛 ≤ 𝑥) → 𝐵 ≤ 𝐴))
67	66	ad2antrr 762	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐼 ∈ (𝑀[,)+∞)) ∧ 𝑖 ∈ (ℤ_≥‘((⌊‘𝐼) + 1))) → ∀𝑛 ∈ ℝ⁺ ∀𝑥 ∈ ℝ⁺ ((𝑀 ≤ 𝑛 ∧ 𝑛 ≤ 𝑥) → 𝐵 ≤ 𝐴))
68		nfcv 2764	. . . . . . . . 9 ⊢ Ⅎ𝑛ℝ⁺
69		nfv 1843	. . . . . . . . . 10 ⊢ Ⅎ𝑛(𝑀 ≤ 𝐼 ∧ 𝐼 ≤ 𝑥)
70		nfcv 2764	. . . . . . . . . . 11 ⊢ Ⅎ𝑛𝐵
71		nfcv 2764	. . . . . . . . . . 11 ⊢ Ⅎ𝑛 ≤
72	70, 71, 3	nfbr 4699	. . . . . . . . . 10 ⊢ Ⅎ𝑛 𝐵 ≤ ⦋𝐼 / 𝑛⦌𝐴
73	69, 72	nfim 1825	. . . . . . . . 9 ⊢ Ⅎ𝑛((𝑀 ≤ 𝐼 ∧ 𝐼 ≤ 𝑥) → 𝐵 ≤ ⦋𝐼 / 𝑛⦌𝐴)
74	68, 73	nfral 2945	. . . . . . . 8 ⊢ Ⅎ𝑛∀𝑥 ∈ ℝ⁺ ((𝑀 ≤ 𝐼 ∧ 𝐼 ≤ 𝑥) → 𝐵 ≤ ⦋𝐼 / 𝑛⦌𝐴)
75		breq2 4657	. . . . . . . . . . 11 ⊢ (𝑛 = 𝐼 → (𝑀 ≤ 𝑛 ↔ 𝑀 ≤ 𝐼))
76		breq1 4656	. . . . . . . . . . 11 ⊢ (𝑛 = 𝐼 → (𝑛 ≤ 𝑥 ↔ 𝐼 ≤ 𝑥))
77	75, 76	anbi12d 747	. . . . . . . . . 10 ⊢ (𝑛 = 𝐼 → ((𝑀 ≤ 𝑛 ∧ 𝑛 ≤ 𝑥) ↔ (𝑀 ≤ 𝐼 ∧ 𝐼 ≤ 𝑥)))
78	5	breq2d 4665	. . . . . . . . . 10 ⊢ (𝑛 = 𝐼 → (𝐵 ≤ 𝐴 ↔ 𝐵 ≤ ⦋𝐼 / 𝑛⦌𝐴))
79	77, 78	imbi12d 334	. . . . . . . . 9 ⊢ (𝑛 = 𝐼 → (((𝑀 ≤ 𝑛 ∧ 𝑛 ≤ 𝑥) → 𝐵 ≤ 𝐴) ↔ ((𝑀 ≤ 𝐼 ∧ 𝐼 ≤ 𝑥) → 𝐵 ≤ ⦋𝐼 / 𝑛⦌𝐴)))
80	79	ralbidv 2986	. . . . . . . 8 ⊢ (𝑛 = 𝐼 → (∀𝑥 ∈ ℝ⁺ ((𝑀 ≤ 𝑛 ∧ 𝑛 ≤ 𝑥) → 𝐵 ≤ 𝐴) ↔ ∀𝑥 ∈ ℝ⁺ ((𝑀 ≤ 𝐼 ∧ 𝐼 ≤ 𝑥) → 𝐵 ≤ ⦋𝐼 / 𝑛⦌𝐴)))
81	74, 80	rspc 3303	. . . . . . 7 ⊢ (𝐼 ∈ ℝ⁺ → (∀𝑛 ∈ ℝ⁺ ∀𝑥 ∈ ℝ⁺ ((𝑀 ≤ 𝑛 ∧ 𝑛 ≤ 𝑥) → 𝐵 ≤ 𝐴) → ∀𝑥 ∈ ℝ⁺ ((𝑀 ≤ 𝐼 ∧ 𝐼 ≤ 𝑥) → 𝐵 ≤ ⦋𝐼 / 𝑛⦌𝐴)))
82	63, 67, 81	sylc 65	. . . . . 6 ⊢ (((𝜑 ∧ 𝐼 ∈ (𝑀[,)+∞)) ∧ 𝑖 ∈ (ℤ_≥‘((⌊‘𝐼) + 1))) → ∀𝑥 ∈ ℝ⁺ ((𝑀 ≤ 𝐼 ∧ 𝐼 ≤ 𝑥) → 𝐵 ≤ ⦋𝐼 / 𝑛⦌𝐴))
83	31	adantr 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐼 ∈ (𝑀[,)+∞)) ∧ 𝑖 ∈ (ℤ_≥‘((⌊‘𝐼) + 1))) → 𝑀 ≤ 𝐼)
84	14	adantr 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐼 ∈ (𝑀[,)+∞)) ∧ 𝑖 ∈ (ℤ_≥‘((⌊‘𝐼) + 1))) → 𝐼 ∈ ℝ)
85		reflcl 12597	. . . . . . . . 9 ⊢ (𝐼 ∈ ℝ → (⌊‘𝐼) ∈ ℝ)
86		peano2re 10209	. . . . . . . . 9 ⊢ ((⌊‘𝐼) ∈ ℝ → ((⌊‘𝐼) + 1) ∈ ℝ)
87	84, 85, 86	3syl 18	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐼 ∈ (𝑀[,)+∞)) ∧ 𝑖 ∈ (ℤ_≥‘((⌊‘𝐼) + 1))) → ((⌊‘𝐼) + 1) ∈ ℝ)
88	47	nnred 11035	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐼 ∈ (𝑀[,)+∞)) ∧ 𝑖 ∈ (ℤ_≥‘((⌊‘𝐼) + 1))) → 𝑖 ∈ ℝ)
89		fllep1 12602	. . . . . . . . . 10 ⊢ (𝐼 ∈ ℝ → 𝐼 ≤ ((⌊‘𝐼) + 1))
90	14, 89	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐼 ∈ (𝑀[,)+∞)) → 𝐼 ≤ ((⌊‘𝐼) + 1))
91	90	adantr 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐼 ∈ (𝑀[,)+∞)) ∧ 𝑖 ∈ (ℤ_≥‘((⌊‘𝐼) + 1))) → 𝐼 ≤ ((⌊‘𝐼) + 1))
92		eluzle 11700	. . . . . . . . 9 ⊢ (𝑖 ∈ (ℤ_≥‘((⌊‘𝐼) + 1)) → ((⌊‘𝐼) + 1) ≤ 𝑖)
93	92	adantl 482	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐼 ∈ (𝑀[,)+∞)) ∧ 𝑖 ∈ (ℤ_≥‘((⌊‘𝐼) + 1))) → ((⌊‘𝐼) + 1) ≤ 𝑖)
94	84, 87, 88, 91, 93	letrd 10194	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐼 ∈ (𝑀[,)+∞)) ∧ 𝑖 ∈ (ℤ_≥‘((⌊‘𝐼) + 1))) → 𝐼 ≤ 𝑖)
95	83, 94	jca 554	. . . . . 6 ⊢ (((𝜑 ∧ 𝐼 ∈ (𝑀[,)+∞)) ∧ 𝑖 ∈ (ℤ_≥‘((⌊‘𝐼) + 1))) → (𝑀 ≤ 𝐼 ∧ 𝐼 ≤ 𝑖))
96		breq2 4657	. . . . . . . . 9 ⊢ (𝑥 = 𝑖 → (𝐼 ≤ 𝑥 ↔ 𝐼 ≤ 𝑖))
97	96	anbi2d 740	. . . . . . . 8 ⊢ (𝑥 = 𝑖 → ((𝑀 ≤ 𝐼 ∧ 𝐼 ≤ 𝑥) ↔ (𝑀 ≤ 𝐼 ∧ 𝐼 ≤ 𝑖)))
98		eqvisset 3211	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑖 → 𝑖 ∈ V)
99		equtr2 1954	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑖 ∧ 𝑛 = 𝑖) → 𝑥 = 𝑛)
100		dchrisum.2	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑥 → 𝐴 = 𝐵)
101	100	equcoms 1947	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑛 → 𝐴 = 𝐵)
102	99, 101	syl 17	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑖 ∧ 𝑛 = 𝑖) → 𝐴 = 𝐵)
103	98, 102	csbied 3560	. . . . . . . . . 10 ⊢ (𝑥 = 𝑖 → ⦋𝑖 / 𝑛⦌𝐴 = 𝐵)
104	103	eqcomd 2628	. . . . . . . . 9 ⊢ (𝑥 = 𝑖 → 𝐵 = ⦋𝑖 / 𝑛⦌𝐴)
105	104	breq1d 4663	. . . . . . . 8 ⊢ (𝑥 = 𝑖 → (𝐵 ≤ ⦋𝐼 / 𝑛⦌𝐴 ↔ ⦋𝑖 / 𝑛⦌𝐴 ≤ ⦋𝐼 / 𝑛⦌𝐴))
106	97, 105	imbi12d 334	. . . . . . 7 ⊢ (𝑥 = 𝑖 → (((𝑀 ≤ 𝐼 ∧ 𝐼 ≤ 𝑥) → 𝐵 ≤ ⦋𝐼 / 𝑛⦌𝐴) ↔ ((𝑀 ≤ 𝐼 ∧ 𝐼 ≤ 𝑖) → ⦋𝑖 / 𝑛⦌𝐴 ≤ ⦋𝐼 / 𝑛⦌𝐴)))
107	106	rspcv 3305	. . . . . 6 ⊢ (𝑖 ∈ ℝ⁺ → (∀𝑥 ∈ ℝ⁺ ((𝑀 ≤ 𝐼 ∧ 𝐼 ≤ 𝑥) → 𝐵 ≤ ⦋𝐼 / 𝑛⦌𝐴) → ((𝑀 ≤ 𝐼 ∧ 𝐼 ≤ 𝑖) → ⦋𝑖 / 𝑛⦌𝐴 ≤ ⦋𝐼 / 𝑛⦌𝐴)))
108	48, 82, 95, 107	syl3c 66	. . . . 5 ⊢ (((𝜑 ∧ 𝐼 ∈ (𝑀[,)+∞)) ∧ 𝑖 ∈ (ℤ_≥‘((⌊‘𝐼) + 1))) → ⦋𝑖 / 𝑛⦌𝐴 ≤ ⦋𝐼 / 𝑛⦌𝐴)
109	108, 57, 60	3brtr4d 4685	. . . 4 ⊢ (((𝜑 ∧ 𝐼 ∈ (𝑀[,)+∞)) ∧ 𝑖 ∈ (ℤ_≥‘((⌊‘𝐼) + 1))) → ((𝑛 ∈ ℝ⁺ ↦ 𝐴)‘𝑖) ≤ (((ℤ_≥‘((⌊‘𝐼) + 1)) × {⦋𝐼 / 𝑛⦌𝐴})‘𝑖))
110	9, 16, 26, 40, 58, 62, 109	climle 14370	. . 3 ⊢ ((𝜑 ∧ 𝐼 ∈ (𝑀[,)+∞)) → 0 ≤ ⦋𝐼 / 𝑛⦌𝐴)
111	110	ex 450	. 2 ⊢ (𝜑 → (𝐼 ∈ (𝑀[,)+∞) → 0 ≤ ⦋𝐼 / 𝑛⦌𝐴))
112	8, 111	jca 554	1 ⊢ (𝜑 → ((𝐼 ∈ ℝ⁺ → ⦋𝐼 / 𝑛⦌𝐴 ∈ ℝ) ∧ (𝐼 ∈ (𝑀[,)+∞) → 0 ≤ ⦋𝐼 / 𝑛⦌𝐴)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 Vcvv 3200 ⦋csb 3533 {csn 4177 class class class wbr 4653 ↦ cmpt 4729 × cxp 5112 dom cdm 5114 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 ℝcr 9935 0cc0 9936 1c1 9937 + caddc 9939 · cmul 9941 +∞cpnf 10071 < clt 10074 ≤ cle 10075 ℕcn 11020 ℕ₀cn0 11292 ℤcz 11377 ℤ_≥cuz 11687 ℝ⁺crp 11832 [,)cico 12177 ⌊cfl 12591 ⇝ cli 14215 ⇝_𝑟 crli 14216 Basecbs 15857 0_gc0g 16100 ℤRHomczrh 19848 ℤ/nℤczn 19851 DChrcdchr 24957

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-inf 8349 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-ico 12181 df-fl 12593 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-rlim 14220

This theorem is referenced by: dchrisumlem2 25179 dchrisumlem3 25180

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