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Theorem dvfsumle 24618

Description: Compare a finite sum to an integral (the integral here is given as a function with a known derivative). (Contributed by Mario Carneiro, 14-May-2016.)

**Hypotheses**
Ref	Expression
dvfsumle.m	⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
dvfsumle.a	⊢ (𝜑 → (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴) ∈ ((𝑀[,]𝑁)–cn→ℝ))
dvfsumle.v	⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 𝐵 ∈ 𝑉)
dvfsumle.b	⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)) = (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵))
dvfsumle.c	⊢ (𝑥 = 𝑀 → 𝐴 = 𝐶)
dvfsumle.d	⊢ (𝑥 = 𝑁 → 𝐴 = 𝐷)
dvfsumle.x	⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑋 ∈ ℝ)
dvfsumle.l	⊢ ((𝜑 ∧ (𝑘 ∈ (𝑀..^𝑁) ∧ 𝑥 ∈ (𝑘(,)(𝑘 + 1)))) → 𝑋 ≤ 𝐵)

**Assertion**
Ref	Expression
dvfsumle	⊢ (𝜑 → Σ𝑘 ∈ (𝑀..^𝑁)𝑋 ≤ (𝐷 − 𝐶))

Distinct variable groups: 𝐴,𝑘 𝑥,𝑘,𝑀 𝑘,𝑁,𝑥 𝜑,𝑘,𝑥 𝑥,𝑋 𝑥,𝐶 𝑥,𝐷 𝑥,𝑉 Allowed substitution hints: 𝐴(𝑥) 𝐵(𝑥,𝑘) 𝐶(𝑘) 𝐷(𝑘) 𝑉(𝑘) 𝑋(𝑘)

Proof of Theorem dvfsumle Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fzofi 13343	. . . 4 ⊢ (𝑀..^𝑁) ∈ Fin
2	1	a1i 11	. . 3 ⊢ (𝜑 → (𝑀..^𝑁) ∈ Fin)
3		dvfsumle.x	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑋 ∈ ℝ)
4		dvfsumle.m	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
5		eluzel2 12249	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑀 ∈ ℤ)
6	4, 5	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℤ)
7		eluzelz 12254	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑁 ∈ ℤ)
8	4, 7	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℤ)
9		fzval2 12896	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀...𝑁) = ((𝑀[,]𝑁) ∩ ℤ))
10	6, 8, 9	syl2anc 586	. . . . . . . . 9 ⊢ (𝜑 → (𝑀...𝑁) = ((𝑀[,]𝑁) ∩ ℤ))
11		inss1 4205	. . . . . . . . 9 ⊢ ((𝑀[,]𝑁) ∩ ℤ) ⊆ (𝑀[,]𝑁)
12	10, 11	eqsstrdi 4021	. . . . . . . 8 ⊢ (𝜑 → (𝑀...𝑁) ⊆ (𝑀[,]𝑁))
13	12	sselda 3967	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑀...𝑁)) → 𝑦 ∈ (𝑀[,]𝑁))
14		dvfsumle.a	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴) ∈ ((𝑀[,]𝑁)–cn→ℝ))
15		cncff 23501	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴) ∈ ((𝑀[,]𝑁)–cn→ℝ) → (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴):(𝑀[,]𝑁)⟶ℝ)
16	14, 15	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴):(𝑀[,]𝑁)⟶ℝ)
17		eqid 2821	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴) = (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴)
18	17	fmpt 6874	. . . . . . . . 9 ⊢ (∀𝑥 ∈ (𝑀[,]𝑁)𝐴 ∈ ℝ ↔ (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴):(𝑀[,]𝑁)⟶ℝ)
19	16, 18	sylibr 236	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ (𝑀[,]𝑁)𝐴 ∈ ℝ)
20		nfcsb1v 3907	. . . . . . . . . 10 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐴
21	20	nfel1 2994	. . . . . . . . 9 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐴 ∈ ℝ
22		csbeq1a 3897	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → 𝐴 = ⦋𝑦 / 𝑥⦌𝐴)
23	22	eleq1d 2897	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝐴 ∈ ℝ ↔ ⦋𝑦 / 𝑥⦌𝐴 ∈ ℝ))
24	21, 23	rspc 3611	. . . . . . . 8 ⊢ (𝑦 ∈ (𝑀[,]𝑁) → (∀𝑥 ∈ (𝑀[,]𝑁)𝐴 ∈ ℝ → ⦋𝑦 / 𝑥⦌𝐴 ∈ ℝ))
25	19, 24	mpan9 509	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑀[,]𝑁)) → ⦋𝑦 / 𝑥⦌𝐴 ∈ ℝ)
26	13, 25	syldan 593	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑀...𝑁)) → ⦋𝑦 / 𝑥⦌𝐴 ∈ ℝ)
27	26	ralrimiva 3182	. . . . 5 ⊢ (𝜑 → ∀𝑦 ∈ (𝑀...𝑁)⦋𝑦 / 𝑥⦌𝐴 ∈ ℝ)
28		fzofzp1 13135	. . . . 5 ⊢ (𝑘 ∈ (𝑀..^𝑁) → (𝑘 + 1) ∈ (𝑀...𝑁))
29		csbeq1 3886	. . . . . . 7 ⊢ (𝑦 = (𝑘 + 1) → ⦋𝑦 / 𝑥⦌𝐴 = ⦋(𝑘 + 1) / 𝑥⦌𝐴)
30	29	eleq1d 2897	. . . . . 6 ⊢ (𝑦 = (𝑘 + 1) → (⦋𝑦 / 𝑥⦌𝐴 ∈ ℝ ↔ ⦋(𝑘 + 1) / 𝑥⦌𝐴 ∈ ℝ))
31	30	rspccva 3622	. . . . 5 ⊢ ((∀𝑦 ∈ (𝑀...𝑁)⦋𝑦 / 𝑥⦌𝐴 ∈ ℝ ∧ (𝑘 + 1) ∈ (𝑀...𝑁)) → ⦋(𝑘 + 1) / 𝑥⦌𝐴 ∈ ℝ)
32	27, 28, 31	syl2an 597	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ⦋(𝑘 + 1) / 𝑥⦌𝐴 ∈ ℝ)
33		elfzofz 13054	. . . . 5 ⊢ (𝑘 ∈ (𝑀..^𝑁) → 𝑘 ∈ (𝑀...𝑁))
34		csbeq1 3886	. . . . . . 7 ⊢ (𝑦 = 𝑘 → ⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑘 / 𝑥⦌𝐴)
35	34	eleq1d 2897	. . . . . 6 ⊢ (𝑦 = 𝑘 → (⦋𝑦 / 𝑥⦌𝐴 ∈ ℝ ↔ ⦋𝑘 / 𝑥⦌𝐴 ∈ ℝ))
36	35	rspccva 3622	. . . . 5 ⊢ ((∀𝑦 ∈ (𝑀...𝑁)⦋𝑦 / 𝑥⦌𝐴 ∈ ℝ ∧ 𝑘 ∈ (𝑀...𝑁)) → ⦋𝑘 / 𝑥⦌𝐴 ∈ ℝ)
37	27, 33, 36	syl2an 597	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ⦋𝑘 / 𝑥⦌𝐴 ∈ ℝ)
38	32, 37	resubcld 11068	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (⦋(𝑘 + 1) / 𝑥⦌𝐴 − ⦋𝑘 / 𝑥⦌𝐴) ∈ ℝ)
39		elfzoelz 13039	. . . . . . . . . 10 ⊢ (𝑘 ∈ (𝑀..^𝑁) → 𝑘 ∈ ℤ)
40	39	adantl 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑘 ∈ ℤ)
41	40	zred 12088	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑘 ∈ ℝ)
42	41	recnd 10669	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑘 ∈ ℂ)
43		ax-1cn 10595	. . . . . . 7 ⊢ 1 ∈ ℂ
44		pncan2 10893	. . . . . . 7 ⊢ ((𝑘 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑘 + 1) − 𝑘) = 1)
45	42, 43, 44	sylancl 588	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ((𝑘 + 1) − 𝑘) = 1)
46	45	oveq2d 7172	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑋 · ((𝑘 + 1) − 𝑘)) = (𝑋 · 1))
47	3	recnd 10669	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑋 ∈ ℂ)
48		peano2re 10813	. . . . . . . 8 ⊢ (𝑘 ∈ ℝ → (𝑘 + 1) ∈ ℝ)
49	41, 48	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘 + 1) ∈ ℝ)
50	49	recnd 10669	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘 + 1) ∈ ℂ)
51	47, 50, 42	subdid 11096	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑋 · ((𝑘 + 1) − 𝑘)) = ((𝑋 · (𝑘 + 1)) − (𝑋 · 𝑘)))
52	47	mulid1d 10658	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑋 · 1) = 𝑋)
53	46, 51, 52	3eqtr3d 2864	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ((𝑋 · (𝑘 + 1)) − (𝑋 · 𝑘)) = 𝑋)
54		eqid 2821	. . . . . 6 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
55	54	mulcn 23475	. . . . . 6 ⊢ · ∈ (((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld)) Cn (TopOpen‘ℂ_fld))
56	6	zred 12088	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℝ)
57	56	adantr 483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑀 ∈ ℝ)
58	8	zred 12088	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℝ)
59	58	adantr 483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑁 ∈ ℝ)
60		elfzole1 13047	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (𝑀..^𝑁) → 𝑀 ≤ 𝑘)
61	60	adantl 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑀 ≤ 𝑘)
62	28	adantl 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘 + 1) ∈ (𝑀...𝑁))
63		elfzle2 12912	. . . . . . . . . . 11 ⊢ ((𝑘 + 1) ∈ (𝑀...𝑁) → (𝑘 + 1) ≤ 𝑁)
64	62, 63	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘 + 1) ≤ 𝑁)
65		iccss 12805	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) ∧ (𝑀 ≤ 𝑘 ∧ (𝑘 + 1) ≤ 𝑁)) → (𝑘[,](𝑘 + 1)) ⊆ (𝑀[,]𝑁))
66	57, 59, 61, 64, 65	syl22anc 836	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘[,](𝑘 + 1)) ⊆ (𝑀[,]𝑁))
67		iccssre 12819	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝑀[,]𝑁) ⊆ ℝ)
68	56, 58, 67	syl2anc 586	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀[,]𝑁) ⊆ ℝ)
69	68	adantr 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑀[,]𝑁) ⊆ ℝ)
70	66, 69	sstrd 3977	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘[,](𝑘 + 1)) ⊆ ℝ)
71		ax-resscn 10594	. . . . . . . 8 ⊢ ℝ ⊆ ℂ
72	70, 71	sstrdi 3979	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘[,](𝑘 + 1)) ⊆ ℂ)
73	71	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ℝ ⊆ ℂ)
74		cncfmptc 23519	. . . . . . 7 ⊢ ((𝑋 ∈ ℝ ∧ (𝑘[,](𝑘 + 1)) ⊆ ℂ ∧ ℝ ⊆ ℂ) → (𝑦 ∈ (𝑘[,](𝑘 + 1)) ↦ 𝑋) ∈ ((𝑘[,](𝑘 + 1))–cn→ℝ))
75	3, 72, 73, 74	syl3anc 1367	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑦 ∈ (𝑘[,](𝑘 + 1)) ↦ 𝑋) ∈ ((𝑘[,](𝑘 + 1))–cn→ℝ))
76		cncfmptid 23520	. . . . . . 7 ⊢ (((𝑘[,](𝑘 + 1)) ⊆ ℝ ∧ ℝ ⊆ ℂ) → (𝑦 ∈ (𝑘[,](𝑘 + 1)) ↦ 𝑦) ∈ ((𝑘[,](𝑘 + 1))–cn→ℝ))
77	70, 71, 76	sylancl 588	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑦 ∈ (𝑘[,](𝑘 + 1)) ↦ 𝑦) ∈ ((𝑘[,](𝑘 + 1))–cn→ℝ))
78		remulcl 10622	. . . . . 6 ⊢ ((𝑋 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑋 · 𝑦) ∈ ℝ)
79	54, 55, 75, 77, 71, 78	cncfmpt2ss 23523	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑦 ∈ (𝑘[,](𝑘 + 1)) ↦ (𝑋 · 𝑦)) ∈ ((𝑘[,](𝑘 + 1))–cn→ℝ))
80		reelprrecn 10629	. . . . . . . 8 ⊢ ℝ ∈ {ℝ, ℂ}
81	80	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ℝ ∈ {ℝ, ℂ})
82	57	rexrd 10691	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑀 ∈ ℝ^*)
83		iooss1 12774	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℝ^* ∧ 𝑀 ≤ 𝑘) → (𝑘(,)(𝑘 + 1)) ⊆ (𝑀(,)(𝑘 + 1)))
84	82, 61, 83	syl2anc 586	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘(,)(𝑘 + 1)) ⊆ (𝑀(,)(𝑘 + 1)))
85	59	rexrd 10691	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑁 ∈ ℝ^*)
86		iooss2 12775	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℝ^* ∧ (𝑘 + 1) ≤ 𝑁) → (𝑀(,)(𝑘 + 1)) ⊆ (𝑀(,)𝑁))
87	85, 64, 86	syl2anc 586	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑀(,)(𝑘 + 1)) ⊆ (𝑀(,)𝑁))
88	84, 87	sstrd 3977	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘(,)(𝑘 + 1)) ⊆ (𝑀(,)𝑁))
89		ioossicc 12823	. . . . . . . . . 10 ⊢ (𝑀(,)𝑁) ⊆ (𝑀[,]𝑁)
90	69, 71	sstrdi 3979	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑀[,]𝑁) ⊆ ℂ)
91	89, 90	sstrid 3978	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑀(,)𝑁) ⊆ ℂ)
92	88, 91	sstrd 3977	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘(,)(𝑘 + 1)) ⊆ ℂ)
93	92	sselda 3967	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑦 ∈ (𝑘(,)(𝑘 + 1))) → 𝑦 ∈ ℂ)
94		1cnd 10636	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑦 ∈ (𝑘(,)(𝑘 + 1))) → 1 ∈ ℂ)
95	73	sselda 3967	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℂ)
96		1cnd 10636	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑦 ∈ ℝ) → 1 ∈ ℂ)
97	81	dvmptid 24554	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (ℝ D (𝑦 ∈ ℝ ↦ 𝑦)) = (𝑦 ∈ ℝ ↦ 1))
98		ioossre 12799	. . . . . . . . 9 ⊢ (𝑘(,)(𝑘 + 1)) ⊆ ℝ
99	98	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘(,)(𝑘 + 1)) ⊆ ℝ)
100	54	tgioo2 23411	. . . . . . . 8 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
101		iooretop 23374	. . . . . . . . 9 ⊢ (𝑘(,)(𝑘 + 1)) ∈ (topGen‘ran (,))
102	101	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘(,)(𝑘 + 1)) ∈ (topGen‘ran (,)))
103	81, 95, 96, 97, 99, 100, 54, 102	dvmptres 24560	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (ℝ D (𝑦 ∈ (𝑘(,)(𝑘 + 1)) ↦ 𝑦)) = (𝑦 ∈ (𝑘(,)(𝑘 + 1)) ↦ 1))
104	81, 93, 94, 103, 47	dvmptcmul 24561	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (ℝ D (𝑦 ∈ (𝑘(,)(𝑘 + 1)) ↦ (𝑋 · 𝑦))) = (𝑦 ∈ (𝑘(,)(𝑘 + 1)) ↦ (𝑋 · 1)))
105	52	mpteq2dv 5162	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑦 ∈ (𝑘(,)(𝑘 + 1)) ↦ (𝑋 · 1)) = (𝑦 ∈ (𝑘(,)(𝑘 + 1)) ↦ 𝑋))
106	104, 105	eqtrd 2856	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (ℝ D (𝑦 ∈ (𝑘(,)(𝑘 + 1)) ↦ (𝑋 · 𝑦))) = (𝑦 ∈ (𝑘(,)(𝑘 + 1)) ↦ 𝑋))
107		nfcv 2977	. . . . . . 7 ⊢ Ⅎ𝑦𝐴
108	107, 20, 22	cbvmpt 5167	. . . . . 6 ⊢ (𝑥 ∈ (𝑘[,](𝑘 + 1)) ↦ 𝐴) = (𝑦 ∈ (𝑘[,](𝑘 + 1)) ↦ ⦋𝑦 / 𝑥⦌𝐴)
109	66	resmptd 5908	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ((𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴) ↾ (𝑘[,](𝑘 + 1))) = (𝑥 ∈ (𝑘[,](𝑘 + 1)) ↦ 𝐴))
110	14	adantr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴) ∈ ((𝑀[,]𝑁)–cn→ℝ))
111		rescncf 23505	. . . . . . . 8 ⊢ ((𝑘[,](𝑘 + 1)) ⊆ (𝑀[,]𝑁) → ((𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴) ∈ ((𝑀[,]𝑁)–cn→ℝ) → ((𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴) ↾ (𝑘[,](𝑘 + 1))) ∈ ((𝑘[,](𝑘 + 1))–cn→ℝ)))
112	66, 110, 111	sylc 65	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ((𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴) ↾ (𝑘[,](𝑘 + 1))) ∈ ((𝑘[,](𝑘 + 1))–cn→ℝ))
113	109, 112	eqeltrrd 2914	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑥 ∈ (𝑘[,](𝑘 + 1)) ↦ 𝐴) ∈ ((𝑘[,](𝑘 + 1))–cn→ℝ))
114	108, 113	eqeltrrid 2918	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑦 ∈ (𝑘[,](𝑘 + 1)) ↦ ⦋𝑦 / 𝑥⦌𝐴) ∈ ((𝑘[,](𝑘 + 1))–cn→ℝ))
115	16	adantr 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴):(𝑀[,]𝑁)⟶ℝ)
116	115, 18	sylibr 236	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ∀𝑥 ∈ (𝑀[,]𝑁)𝐴 ∈ ℝ)
117	89	sseli 3963	. . . . . . . 8 ⊢ (𝑦 ∈ (𝑀(,)𝑁) → 𝑦 ∈ (𝑀[,]𝑁))
118	24	impcom 410	. . . . . . . 8 ⊢ ((∀𝑥 ∈ (𝑀[,]𝑁)𝐴 ∈ ℝ ∧ 𝑦 ∈ (𝑀[,]𝑁)) → ⦋𝑦 / 𝑥⦌𝐴 ∈ ℝ)
119	116, 117, 118	syl2an 597	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑦 ∈ (𝑀(,)𝑁)) → ⦋𝑦 / 𝑥⦌𝐴 ∈ ℝ)
120	119	recnd 10669	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑦 ∈ (𝑀(,)𝑁)) → ⦋𝑦 / 𝑥⦌𝐴 ∈ ℂ)
121	89	sseli 3963	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝑀(,)𝑁) → 𝑥 ∈ (𝑀[,]𝑁))
122	16	fvmptelrn 6877	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → 𝐴 ∈ ℝ)
123	122	adantlr 713	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀[,]𝑁)) → 𝐴 ∈ ℝ)
124	121, 123	sylan2 594	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 𝐴 ∈ ℝ)
125	124	fmpttd 6879	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴):(𝑀(,)𝑁)⟶ℝ)
126		ioossre 12799	. . . . . . . . . 10 ⊢ (𝑀(,)𝑁) ⊆ ℝ
127		dvfre 24548	. . . . . . . . . 10 ⊢ (((𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴):(𝑀(,)𝑁)⟶ℝ ∧ (𝑀(,)𝑁) ⊆ ℝ) → (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)):dom (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴))⟶ℝ)
128	125, 126, 127	sylancl 588	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)):dom (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴))⟶ℝ)
129		dvfsumle.b	. . . . . . . . . . 11 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)) = (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵))
130	129	adantr 483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)) = (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵))
131	130	dmeqd 5774	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → dom (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)) = dom (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵))
132		dvfsumle.v	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 𝐵 ∈ 𝑉)
133	132	adantlr 713	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 𝐵 ∈ 𝑉)
134	133	ralrimiva 3182	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ∀𝑥 ∈ (𝑀(,)𝑁)𝐵 ∈ 𝑉)
135		dmmptg 6096	. . . . . . . . . . . 12 ⊢ (∀𝑥 ∈ (𝑀(,)𝑁)𝐵 ∈ 𝑉 → dom (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵) = (𝑀(,)𝑁))
136	134, 135	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → dom (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵) = (𝑀(,)𝑁))
137	131, 136	eqtrd 2856	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → dom (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)) = (𝑀(,)𝑁))
138	130, 137	feq12d 6502	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ((ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)):dom (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴))⟶ℝ ↔ (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵):(𝑀(,)𝑁)⟶ℝ))
139	128, 138	mpbid 234	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵):(𝑀(,)𝑁)⟶ℝ)
140		eqid 2821	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵) = (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵)
141	140	fmpt 6874	. . . . . . . 8 ⊢ (∀𝑥 ∈ (𝑀(,)𝑁)𝐵 ∈ ℝ ↔ (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵):(𝑀(,)𝑁)⟶ℝ)
142	139, 141	sylibr 236	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ∀𝑥 ∈ (𝑀(,)𝑁)𝐵 ∈ ℝ)
143		nfcsb1v 3907	. . . . . . . . 9 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵
144	143	nfel1 2994	. . . . . . . 8 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵 ∈ ℝ
145		csbeq1a 3897	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)
146	145	eleq1d 2897	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝐵 ∈ ℝ ↔ ⦋𝑦 / 𝑥⦌𝐵 ∈ ℝ))
147	144, 146	rspc 3611	. . . . . . 7 ⊢ (𝑦 ∈ (𝑀(,)𝑁) → (∀𝑥 ∈ (𝑀(,)𝑁)𝐵 ∈ ℝ → ⦋𝑦 / 𝑥⦌𝐵 ∈ ℝ))
148	142, 147	mpan9 509	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑦 ∈ (𝑀(,)𝑁)) → ⦋𝑦 / 𝑥⦌𝐵 ∈ ℝ)
149	107, 20, 22	cbvmpt 5167	. . . . . . . 8 ⊢ (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴) = (𝑦 ∈ (𝑀(,)𝑁) ↦ ⦋𝑦 / 𝑥⦌𝐴)
150	149	oveq2i 7167	. . . . . . 7 ⊢ (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)) = (ℝ D (𝑦 ∈ (𝑀(,)𝑁) ↦ ⦋𝑦 / 𝑥⦌𝐴))
151		nfcv 2977	. . . . . . . 8 ⊢ Ⅎ𝑦𝐵
152	151, 143, 145	cbvmpt 5167	. . . . . . 7 ⊢ (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵) = (𝑦 ∈ (𝑀(,)𝑁) ↦ ⦋𝑦 / 𝑥⦌𝐵)
153	130, 150, 152	3eqtr3g 2879	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (ℝ D (𝑦 ∈ (𝑀(,)𝑁) ↦ ⦋𝑦 / 𝑥⦌𝐴)) = (𝑦 ∈ (𝑀(,)𝑁) ↦ ⦋𝑦 / 𝑥⦌𝐵))
154	81, 120, 148, 153, 88, 100, 54, 102	dvmptres 24560	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (ℝ D (𝑦 ∈ (𝑘(,)(𝑘 + 1)) ↦ ⦋𝑦 / 𝑥⦌𝐴)) = (𝑦 ∈ (𝑘(,)(𝑘 + 1)) ↦ ⦋𝑦 / 𝑥⦌𝐵))
155		dvfsumle.l	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ (𝑀..^𝑁) ∧ 𝑥 ∈ (𝑘(,)(𝑘 + 1)))) → 𝑋 ≤ 𝐵)
156	155	anassrs 470	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑘(,)(𝑘 + 1))) → 𝑋 ≤ 𝐵)
157	156	ralrimiva 3182	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ∀𝑥 ∈ (𝑘(,)(𝑘 + 1))𝑋 ≤ 𝐵)
158		nfcv 2977	. . . . . . . 8 ⊢ Ⅎ𝑥𝑋
159		nfcv 2977	. . . . . . . 8 ⊢ Ⅎ𝑥 ≤
160	158, 159, 143	nfbr 5113	. . . . . . 7 ⊢ Ⅎ𝑥 𝑋 ≤ ⦋𝑦 / 𝑥⦌𝐵
161	145	breq2d 5078	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑋 ≤ 𝐵 ↔ 𝑋 ≤ ⦋𝑦 / 𝑥⦌𝐵))
162	160, 161	rspc 3611	. . . . . 6 ⊢ (𝑦 ∈ (𝑘(,)(𝑘 + 1)) → (∀𝑥 ∈ (𝑘(,)(𝑘 + 1))𝑋 ≤ 𝐵 → 𝑋 ≤ ⦋𝑦 / 𝑥⦌𝐵))
163	157, 162	mpan9 509	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑦 ∈ (𝑘(,)(𝑘 + 1))) → 𝑋 ≤ ⦋𝑦 / 𝑥⦌𝐵)
164	41	rexrd 10691	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑘 ∈ ℝ^*)
165	49	rexrd 10691	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘 + 1) ∈ ℝ^*)
166	41	lep1d 11571	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑘 ≤ (𝑘 + 1))
167		lbicc2 12853	. . . . . 6 ⊢ ((𝑘 ∈ ℝ^* ∧ (𝑘 + 1) ∈ ℝ^* ∧ 𝑘 ≤ (𝑘 + 1)) → 𝑘 ∈ (𝑘[,](𝑘 + 1)))
168	164, 165, 166, 167	syl3anc 1367	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑘 ∈ (𝑘[,](𝑘 + 1)))
169		ubicc2 12854	. . . . . 6 ⊢ ((𝑘 ∈ ℝ^* ∧ (𝑘 + 1) ∈ ℝ^* ∧ 𝑘 ≤ (𝑘 + 1)) → (𝑘 + 1) ∈ (𝑘[,](𝑘 + 1)))
170	164, 165, 166, 169	syl3anc 1367	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘 + 1) ∈ (𝑘[,](𝑘 + 1)))
171		oveq2 7164	. . . . 5 ⊢ (𝑦 = 𝑘 → (𝑋 · 𝑦) = (𝑋 · 𝑘))
172		oveq2 7164	. . . . 5 ⊢ (𝑦 = (𝑘 + 1) → (𝑋 · 𝑦) = (𝑋 · (𝑘 + 1)))
173	41, 49, 79, 106, 114, 154, 163, 168, 170, 166, 171, 34, 172, 29	dvle 24604	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ((𝑋 · (𝑘 + 1)) − (𝑋 · 𝑘)) ≤ (⦋(𝑘 + 1) / 𝑥⦌𝐴 − ⦋𝑘 / 𝑥⦌𝐴))
174	53, 173	eqbrtrrd 5090	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑋 ≤ (⦋(𝑘 + 1) / 𝑥⦌𝐴 − ⦋𝑘 / 𝑥⦌𝐴))
175	2, 3, 38, 174	fsumle 15154	. 2 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀..^𝑁)𝑋 ≤ Σ𝑘 ∈ (𝑀..^𝑁)(⦋(𝑘 + 1) / 𝑥⦌𝐴 − ⦋𝑘 / 𝑥⦌𝐴))
176		vex 3497	. . . . 5 ⊢ 𝑦 ∈ V
177	176	a1i 11	. . . 4 ⊢ (𝑦 = 𝑀 → 𝑦 ∈ V)
178		eqeq2 2833	. . . . . 6 ⊢ (𝑦 = 𝑀 → (𝑥 = 𝑦 ↔ 𝑥 = 𝑀))
179	178	biimpa 479	. . . . 5 ⊢ ((𝑦 = 𝑀 ∧ 𝑥 = 𝑦) → 𝑥 = 𝑀)
180		dvfsumle.c	. . . . 5 ⊢ (𝑥 = 𝑀 → 𝐴 = 𝐶)
181	179, 180	syl 17	. . . 4 ⊢ ((𝑦 = 𝑀 ∧ 𝑥 = 𝑦) → 𝐴 = 𝐶)
182	177, 181	csbied 3919	. . 3 ⊢ (𝑦 = 𝑀 → ⦋𝑦 / 𝑥⦌𝐴 = 𝐶)
183	176	a1i 11	. . . 4 ⊢ (𝑦 = 𝑁 → 𝑦 ∈ V)
184		eqeq2 2833	. . . . . 6 ⊢ (𝑦 = 𝑁 → (𝑥 = 𝑦 ↔ 𝑥 = 𝑁))
185	184	biimpa 479	. . . . 5 ⊢ ((𝑦 = 𝑁 ∧ 𝑥 = 𝑦) → 𝑥 = 𝑁)
186		dvfsumle.d	. . . . 5 ⊢ (𝑥 = 𝑁 → 𝐴 = 𝐷)
187	185, 186	syl 17	. . . 4 ⊢ ((𝑦 = 𝑁 ∧ 𝑥 = 𝑦) → 𝐴 = 𝐷)
188	183, 187	csbied 3919	. . 3 ⊢ (𝑦 = 𝑁 → ⦋𝑦 / 𝑥⦌𝐴 = 𝐷)
189	26	recnd 10669	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑀...𝑁)) → ⦋𝑦 / 𝑥⦌𝐴 ∈ ℂ)
190	34, 29, 182, 188, 4, 189	telfsumo2 15158	. 2 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀..^𝑁)(⦋(𝑘 + 1) / 𝑥⦌𝐴 − ⦋𝑘 / 𝑥⦌𝐴) = (𝐷 − 𝐶))
191	175, 190	breqtrd 5092	1 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀..^𝑁)𝑋 ≤ (𝐷 − 𝐶))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3138 Vcvv 3494 ⦋csb 3883 ∩ cin 3935 ⊆ wss 3936 {cpr 4569 class class class wbr 5066 ↦ cmpt 5146 dom cdm 5555 ran crn 5556 ↾ cres 5557 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 Fincfn 8509 ℂcc 10535 ℝcr 10536 1c1 10538 + caddc 10540 · cmul 10542 ℝ^*cxr 10674 ≤ cle 10676 − cmin 10870 ℤcz 11982 ℤ_≥cuz 12244 (,)cioo 12739 [,]cicc 12742 ...cfz 12893 ..^cfzo 13034 Σcsu 15042 TopOpenctopn 16695 topGenctg 16711 ℂ_fldccnfld 20545 –cn→ccncf 23484 D cdv 24461

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 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 ax-addf 10616 ax-mulf 10617

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-er 8289 df-map 8408 df-pm 8409 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-fi 8875 df-sup 8906 df-inf 8907 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-q 12350 df-rp 12391 df-xneg 12508 df-xadd 12509 df-xmul 12510 df-ioo 12743 df-ico 12745 df-icc 12746 df-fz 12894 df-fzo 13035 df-seq 13371 df-exp 13431 df-hash 13692 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-clim 14845 df-sum 15043 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-starv 16580 df-sca 16581 df-vsca 16582 df-ip 16583 df-tset 16584 df-ple 16585 df-ds 16587 df-unif 16588 df-hom 16589 df-cco 16590 df-rest 16696 df-topn 16697 df-0g 16715 df-gsum 16716 df-topgen 16717 df-pt 16718 df-prds 16721 df-xrs 16775 df-qtop 16780 df-imas 16781 df-xps 16783 df-mre 16857 df-mrc 16858 df-acs 16860 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-submnd 17957 df-mulg 18225 df-cntz 18447 df-cmn 18908 df-psmet 20537 df-xmet 20538 df-met 20539 df-bl 20540 df-mopn 20541 df-fbas 20542 df-fg 20543 df-cnfld 20546 df-top 21502 df-topon 21519 df-topsp 21541 df-bases 21554 df-cld 21627 df-ntr 21628 df-cls 21629 df-nei 21706 df-lp 21744 df-perf 21745 df-cn 21835 df-cnp 21836 df-haus 21923 df-cmp 21995 df-tx 22170 df-hmeo 22363 df-fil 22454 df-fm 22546 df-flim 22547 df-flf 22548 df-xms 22930 df-ms 22931 df-tms 22932 df-cncf 23486 df-limc 24464 df-dv 24465

This theorem is referenced by: dvfsumge 24619

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