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Theorem dvfsumle 23588

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 12635	. . . 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 11568	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑀 ∈ ℤ)
6	4, 5	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℤ)
7		eluzelz 11573	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑁 ∈ ℤ)
8	4, 7	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℤ)
9		fzval2 12200	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀...𝑁) = ((𝑀[,]𝑁) ∩ ℤ))
10	6, 8, 9	syl2anc 691	. . . . . . . . 9 ⊢ (𝜑 → (𝑀...𝑁) = ((𝑀[,]𝑁) ∩ ℤ))
11		inss1 3795	. . . . . . . . 9 ⊢ ((𝑀[,]𝑁) ∩ ℤ) ⊆ (𝑀[,]𝑁)
12	10, 11	syl6eqss 3618	. . . . . . . 8 ⊢ (𝜑 → (𝑀...𝑁) ⊆ (𝑀[,]𝑁))
13	12	sselda 3568	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑀...𝑁)) → 𝑦 ∈ (𝑀[,]𝑁))
14		dvfsumle.a	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴) ∈ ((𝑀[,]𝑁)–cn→ℝ))
15		cncff 22504	. . . . . . . . . 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 2610	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴) = (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴)
18	17	fmpt 6289	. . . . . . . . 9 ⊢ (∀𝑥 ∈ (𝑀[,]𝑁)𝐴 ∈ ℝ ↔ (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴):(𝑀[,]𝑁)⟶ℝ)
19	16, 18	sylibr 223	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ (𝑀[,]𝑁)𝐴 ∈ ℝ)
20		nfcsb1v 3515	. . . . . . . . . 10 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐴
21	20	nfel1 2765	. . . . . . . . 9 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐴 ∈ ℝ
22		csbeq1a 3508	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → 𝐴 = ⦋𝑦 / 𝑥⦌𝐴)
23	22	eleq1d 2672	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝐴 ∈ ℝ ↔ ⦋𝑦 / 𝑥⦌𝐴 ∈ ℝ))
24	21, 23	rspc 3276	. . . . . . . 8 ⊢ (𝑦 ∈ (𝑀[,]𝑁) → (∀𝑥 ∈ (𝑀[,]𝑁)𝐴 ∈ ℝ → ⦋𝑦 / 𝑥⦌𝐴 ∈ ℝ))
25	19, 24	mpan9 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑀[,]𝑁)) → ⦋𝑦 / 𝑥⦌𝐴 ∈ ℝ)
26	13, 25	syldan 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑀...𝑁)) → ⦋𝑦 / 𝑥⦌𝐴 ∈ ℝ)
27	26	ralrimiva 2949	. . . . 5 ⊢ (𝜑 → ∀𝑦 ∈ (𝑀...𝑁)⦋𝑦 / 𝑥⦌𝐴 ∈ ℝ)
28		fzofzp1 12431	. . . . 5 ⊢ (𝑘 ∈ (𝑀..^𝑁) → (𝑘 + 1) ∈ (𝑀...𝑁))
29		csbeq1 3502	. . . . . . 7 ⊢ (𝑦 = (𝑘 + 1) → ⦋𝑦 / 𝑥⦌𝐴 = ⦋(𝑘 + 1) / 𝑥⦌𝐴)
30	29	eleq1d 2672	. . . . . 6 ⊢ (𝑦 = (𝑘 + 1) → (⦋𝑦 / 𝑥⦌𝐴 ∈ ℝ ↔ ⦋(𝑘 + 1) / 𝑥⦌𝐴 ∈ ℝ))
31	30	rspccva 3281	. . . . 5 ⊢ ((∀𝑦 ∈ (𝑀...𝑁)⦋𝑦 / 𝑥⦌𝐴 ∈ ℝ ∧ (𝑘 + 1) ∈ (𝑀...𝑁)) → ⦋(𝑘 + 1) / 𝑥⦌𝐴 ∈ ℝ)
32	27, 28, 31	syl2an 493	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ⦋(𝑘 + 1) / 𝑥⦌𝐴 ∈ ℝ)
33		elfzofz 12354	. . . . 5 ⊢ (𝑘 ∈ (𝑀..^𝑁) → 𝑘 ∈ (𝑀...𝑁))
34		csbeq1 3502	. . . . . . 7 ⊢ (𝑦 = 𝑘 → ⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑘 / 𝑥⦌𝐴)
35	34	eleq1d 2672	. . . . . 6 ⊢ (𝑦 = 𝑘 → (⦋𝑦 / 𝑥⦌𝐴 ∈ ℝ ↔ ⦋𝑘 / 𝑥⦌𝐴 ∈ ℝ))
36	35	rspccva 3281	. . . . 5 ⊢ ((∀𝑦 ∈ (𝑀...𝑁)⦋𝑦 / 𝑥⦌𝐴 ∈ ℝ ∧ 𝑘 ∈ (𝑀...𝑁)) → ⦋𝑘 / 𝑥⦌𝐴 ∈ ℝ)
37	27, 33, 36	syl2an 493	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ⦋𝑘 / 𝑥⦌𝐴 ∈ ℝ)
38	32, 37	resubcld 10337	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (⦋(𝑘 + 1) / 𝑥⦌𝐴 − ⦋𝑘 / 𝑥⦌𝐴) ∈ ℝ)
39		elfzoelz 12339	. . . . . . . . . 10 ⊢ (𝑘 ∈ (𝑀..^𝑁) → 𝑘 ∈ ℤ)
40	39	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑘 ∈ ℤ)
41	40	zred 11358	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑘 ∈ ℝ)
42	41	recnd 9947	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑘 ∈ ℂ)
43		ax-1cn 9873	. . . . . . 7 ⊢ 1 ∈ ℂ
44		pncan2 10167	. . . . . . 7 ⊢ ((𝑘 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑘 + 1) − 𝑘) = 1)
45	42, 43, 44	sylancl 693	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ((𝑘 + 1) − 𝑘) = 1)
46	45	oveq2d 6565	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑋 · ((𝑘 + 1) − 𝑘)) = (𝑋 · 1))
47	3	recnd 9947	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑋 ∈ ℂ)
48		peano2re 10088	. . . . . . . 8 ⊢ (𝑘 ∈ ℝ → (𝑘 + 1) ∈ ℝ)
49	41, 48	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘 + 1) ∈ ℝ)
50	49	recnd 9947	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘 + 1) ∈ ℂ)
51	47, 50, 42	subdid 10365	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑋 · ((𝑘 + 1) − 𝑘)) = ((𝑋 · (𝑘 + 1)) − (𝑋 · 𝑘)))
52	47	mulid1d 9936	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑋 · 1) = 𝑋)
53	46, 51, 52	3eqtr3d 2652	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ((𝑋 · (𝑘 + 1)) − (𝑋 · 𝑘)) = 𝑋)
54		eqid 2610	. . . . . 6 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
55	54	mulcn 22478	. . . . . 6 ⊢ · ∈ (((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld)) Cn (TopOpen‘ℂ_fld))
56	6	zred 11358	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℝ)
57	56	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑀 ∈ ℝ)
58	8	zred 11358	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℝ)
59	58	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑁 ∈ ℝ)
60		elfzole1 12347	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (𝑀..^𝑁) → 𝑀 ≤ 𝑘)
61	60	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑀 ≤ 𝑘)
62	28	adantl 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘 + 1) ∈ (𝑀...𝑁))
63		elfzle2 12216	. . . . . . . . . . 11 ⊢ ((𝑘 + 1) ∈ (𝑀...𝑁) → (𝑘 + 1) ≤ 𝑁)
64	62, 63	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘 + 1) ≤ 𝑁)
65		iccss 12112	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) ∧ (𝑀 ≤ 𝑘 ∧ (𝑘 + 1) ≤ 𝑁)) → (𝑘[,](𝑘 + 1)) ⊆ (𝑀[,]𝑁))
66	57, 59, 61, 64, 65	syl22anc 1319	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘[,](𝑘 + 1)) ⊆ (𝑀[,]𝑁))
67		iccssre 12126	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝑀[,]𝑁) ⊆ ℝ)
68	56, 58, 67	syl2anc 691	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀[,]𝑁) ⊆ ℝ)
69	68	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑀[,]𝑁) ⊆ ℝ)
70	66, 69	sstrd 3578	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘[,](𝑘 + 1)) ⊆ ℝ)
71		ax-resscn 9872	. . . . . . . 8 ⊢ ℝ ⊆ ℂ
72	70, 71	syl6ss 3580	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘[,](𝑘 + 1)) ⊆ ℂ)
73	71	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ℝ ⊆ ℂ)
74		cncfmptc 22522	. . . . . . 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 1318	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑦 ∈ (𝑘[,](𝑘 + 1)) ↦ 𝑋) ∈ ((𝑘[,](𝑘 + 1))–cn→ℝ))
76		cncfmptid 22523	. . . . . . 7 ⊢ (((𝑘[,](𝑘 + 1)) ⊆ ℝ ∧ ℝ ⊆ ℂ) → (𝑦 ∈ (𝑘[,](𝑘 + 1)) ↦ 𝑦) ∈ ((𝑘[,](𝑘 + 1))–cn→ℝ))
77	70, 71, 76	sylancl 693	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑦 ∈ (𝑘[,](𝑘 + 1)) ↦ 𝑦) ∈ ((𝑘[,](𝑘 + 1))–cn→ℝ))
78		remulcl 9900	. . . . . 6 ⊢ ((𝑋 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑋 · 𝑦) ∈ ℝ)
79	54, 55, 75, 77, 71, 78	cncfmpt2ss 22526	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑦 ∈ (𝑘[,](𝑘 + 1)) ↦ (𝑋 · 𝑦)) ∈ ((𝑘[,](𝑘 + 1))–cn→ℝ))
80		reelprrecn 9907	. . . . . . . 8 ⊢ ℝ ∈ {ℝ, ℂ}
81	80	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ℝ ∈ {ℝ, ℂ})
82	57	rexrd 9968	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑀 ∈ ℝ^*)
83		iooss1 12081	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℝ^* ∧ 𝑀 ≤ 𝑘) → (𝑘(,)(𝑘 + 1)) ⊆ (𝑀(,)(𝑘 + 1)))
84	82, 61, 83	syl2anc 691	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘(,)(𝑘 + 1)) ⊆ (𝑀(,)(𝑘 + 1)))
85	59	rexrd 9968	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑁 ∈ ℝ^*)
86		iooss2 12082	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℝ^* ∧ (𝑘 + 1) ≤ 𝑁) → (𝑀(,)(𝑘 + 1)) ⊆ (𝑀(,)𝑁))
87	85, 64, 86	syl2anc 691	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑀(,)(𝑘 + 1)) ⊆ (𝑀(,)𝑁))
88	84, 87	sstrd 3578	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘(,)(𝑘 + 1)) ⊆ (𝑀(,)𝑁))
89		ioossicc 12130	. . . . . . . . . 10 ⊢ (𝑀(,)𝑁) ⊆ (𝑀[,]𝑁)
90	69, 71	syl6ss 3580	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑀[,]𝑁) ⊆ ℂ)
91	89, 90	syl5ss 3579	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑀(,)𝑁) ⊆ ℂ)
92	88, 91	sstrd 3578	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘(,)(𝑘 + 1)) ⊆ ℂ)
93	92	sselda 3568	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑦 ∈ (𝑘(,)(𝑘 + 1))) → 𝑦 ∈ ℂ)
94		1cnd 9935	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑦 ∈ (𝑘(,)(𝑘 + 1))) → 1 ∈ ℂ)
95	73	sselda 3568	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℂ)
96		1cnd 9935	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑦 ∈ ℝ) → 1 ∈ ℂ)
97	81	dvmptid 23526	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (ℝ D (𝑦 ∈ ℝ ↦ 𝑦)) = (𝑦 ∈ ℝ ↦ 1))
98		ioossre 12106	. . . . . . . . 9 ⊢ (𝑘(,)(𝑘 + 1)) ⊆ ℝ
99	98	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘(,)(𝑘 + 1)) ⊆ ℝ)
100	54	tgioo2 22414	. . . . . . . 8 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
101		iooretop 22379	. . . . . . . . 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 23532	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (ℝ D (𝑦 ∈ (𝑘(,)(𝑘 + 1)) ↦ 𝑦)) = (𝑦 ∈ (𝑘(,)(𝑘 + 1)) ↦ 1))
104	81, 93, 94, 103, 47	dvmptcmul 23533	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (ℝ D (𝑦 ∈ (𝑘(,)(𝑘 + 1)) ↦ (𝑋 · 𝑦))) = (𝑦 ∈ (𝑘(,)(𝑘 + 1)) ↦ (𝑋 · 1)))
105	52	mpteq2dv 4673	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑦 ∈ (𝑘(,)(𝑘 + 1)) ↦ (𝑋 · 1)) = (𝑦 ∈ (𝑘(,)(𝑘 + 1)) ↦ 𝑋))
106	104, 105	eqtrd 2644	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (ℝ D (𝑦 ∈ (𝑘(,)(𝑘 + 1)) ↦ (𝑋 · 𝑦))) = (𝑦 ∈ (𝑘(,)(𝑘 + 1)) ↦ 𝑋))
107		nfcv 2751	. . . . . . 7 ⊢ Ⅎ𝑦𝐴
108	107, 20, 22	cbvmpt 4677	. . . . . 6 ⊢ (𝑥 ∈ (𝑘[,](𝑘 + 1)) ↦ 𝐴) = (𝑦 ∈ (𝑘[,](𝑘 + 1)) ↦ ⦋𝑦 / 𝑥⦌𝐴)
109	66	resmptd 5371	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ((𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴) ↾ (𝑘[,](𝑘 + 1))) = (𝑥 ∈ (𝑘[,](𝑘 + 1)) ↦ 𝐴))
110	14	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴) ∈ ((𝑀[,]𝑁)–cn→ℝ))
111		rescncf 22508	. . . . . . . 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 63	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ((𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴) ↾ (𝑘[,](𝑘 + 1))) ∈ ((𝑘[,](𝑘 + 1))–cn→ℝ))
113	109, 112	eqeltrrd 2689	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑥 ∈ (𝑘[,](𝑘 + 1)) ↦ 𝐴) ∈ ((𝑘[,](𝑘 + 1))–cn→ℝ))
114	108, 113	syl5eqelr 2693	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑦 ∈ (𝑘[,](𝑘 + 1)) ↦ ⦋𝑦 / 𝑥⦌𝐴) ∈ ((𝑘[,](𝑘 + 1))–cn→ℝ))
115	16	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴):(𝑀[,]𝑁)⟶ℝ)
116	115, 18	sylibr 223	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ∀𝑥 ∈ (𝑀[,]𝑁)𝐴 ∈ ℝ)
117	89	sseli 3564	. . . . . . . 8 ⊢ (𝑦 ∈ (𝑀(,)𝑁) → 𝑦 ∈ (𝑀[,]𝑁))
118	24	impcom 445	. . . . . . . 8 ⊢ ((∀𝑥 ∈ (𝑀[,]𝑁)𝐴 ∈ ℝ ∧ 𝑦 ∈ (𝑀[,]𝑁)) → ⦋𝑦 / 𝑥⦌𝐴 ∈ ℝ)
119	116, 117, 118	syl2an 493	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑦 ∈ (𝑀(,)𝑁)) → ⦋𝑦 / 𝑥⦌𝐴 ∈ ℝ)
120	119	recnd 9947	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑦 ∈ (𝑀(,)𝑁)) → ⦋𝑦 / 𝑥⦌𝐴 ∈ ℂ)
121	89	sseli 3564	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝑀(,)𝑁) → 𝑥 ∈ (𝑀[,]𝑁))
122	19	r19.21bi 2916	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → 𝐴 ∈ ℝ)
123	122	adantlr 747	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀[,]𝑁)) → 𝐴 ∈ ℝ)
124	121, 123	sylan2 490	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 𝐴 ∈ ℝ)
125		eqid 2610	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴) = (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)
126	124, 125	fmptd 6292	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴):(𝑀(,)𝑁)⟶ℝ)
127		ioossre 12106	. . . . . . . . . 10 ⊢ (𝑀(,)𝑁) ⊆ ℝ
128		dvfre 23520	. . . . . . . . . 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)
129	126, 127, 128	sylancl 693	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)):dom (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴))⟶ℝ)
130		dvfsumle.b	. . . . . . . . . . 11 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)) = (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵))
131	130	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)) = (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵))
132	131	dmeqd 5248	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → dom (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)) = dom (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵))
133		dvfsumle.v	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 𝐵 ∈ 𝑉)
134	133	adantlr 747	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 𝐵 ∈ 𝑉)
135	134	ralrimiva 2949	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ∀𝑥 ∈ (𝑀(,)𝑁)𝐵 ∈ 𝑉)
136		dmmptg 5549	. . . . . . . . . . . 12 ⊢ (∀𝑥 ∈ (𝑀(,)𝑁)𝐵 ∈ 𝑉 → dom (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵) = (𝑀(,)𝑁))
137	135, 136	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → dom (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵) = (𝑀(,)𝑁))
138	132, 137	eqtrd 2644	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → dom (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)) = (𝑀(,)𝑁))
139	131, 138	feq12d 5946	. . . . . . . . 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))
140	129, 139	mpbid 221	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵):(𝑀(,)𝑁)⟶ℝ)
141		eqid 2610	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵) = (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵)
142	141	fmpt 6289	. . . . . . . 8 ⊢ (∀𝑥 ∈ (𝑀(,)𝑁)𝐵 ∈ ℝ ↔ (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵):(𝑀(,)𝑁)⟶ℝ)
143	140, 142	sylibr 223	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ∀𝑥 ∈ (𝑀(,)𝑁)𝐵 ∈ ℝ)
144		nfcsb1v 3515	. . . . . . . . 9 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵
145	144	nfel1 2765	. . . . . . . 8 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵 ∈ ℝ
146		csbeq1a 3508	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)
147	146	eleq1d 2672	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝐵 ∈ ℝ ↔ ⦋𝑦 / 𝑥⦌𝐵 ∈ ℝ))
148	145, 147	rspc 3276	. . . . . . 7 ⊢ (𝑦 ∈ (𝑀(,)𝑁) → (∀𝑥 ∈ (𝑀(,)𝑁)𝐵 ∈ ℝ → ⦋𝑦 / 𝑥⦌𝐵 ∈ ℝ))
149	143, 148	mpan9 485	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑦 ∈ (𝑀(,)𝑁)) → ⦋𝑦 / 𝑥⦌𝐵 ∈ ℝ)
150	107, 20, 22	cbvmpt 4677	. . . . . . . 8 ⊢ (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴) = (𝑦 ∈ (𝑀(,)𝑁) ↦ ⦋𝑦 / 𝑥⦌𝐴)
151	150	oveq2i 6560	. . . . . . 7 ⊢ (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)) = (ℝ D (𝑦 ∈ (𝑀(,)𝑁) ↦ ⦋𝑦 / 𝑥⦌𝐴))
152		nfcv 2751	. . . . . . . 8 ⊢ Ⅎ𝑦𝐵
153	152, 144, 146	cbvmpt 4677	. . . . . . 7 ⊢ (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵) = (𝑦 ∈ (𝑀(,)𝑁) ↦ ⦋𝑦 / 𝑥⦌𝐵)
154	131, 151, 153	3eqtr3g 2667	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (ℝ D (𝑦 ∈ (𝑀(,)𝑁) ↦ ⦋𝑦 / 𝑥⦌𝐴)) = (𝑦 ∈ (𝑀(,)𝑁) ↦ ⦋𝑦 / 𝑥⦌𝐵))
155	81, 120, 149, 154, 88, 100, 54, 102	dvmptres 23532	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (ℝ D (𝑦 ∈ (𝑘(,)(𝑘 + 1)) ↦ ⦋𝑦 / 𝑥⦌𝐴)) = (𝑦 ∈ (𝑘(,)(𝑘 + 1)) ↦ ⦋𝑦 / 𝑥⦌𝐵))
156		dvfsumle.l	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ (𝑀..^𝑁) ∧ 𝑥 ∈ (𝑘(,)(𝑘 + 1)))) → 𝑋 ≤ 𝐵)
157	156	anassrs 678	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑘(,)(𝑘 + 1))) → 𝑋 ≤ 𝐵)
158	157	ralrimiva 2949	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ∀𝑥 ∈ (𝑘(,)(𝑘 + 1))𝑋 ≤ 𝐵)
159		nfcv 2751	. . . . . . . 8 ⊢ Ⅎ𝑥𝑋
160		nfcv 2751	. . . . . . . 8 ⊢ Ⅎ𝑥 ≤
161	159, 160, 144	nfbr 4629	. . . . . . 7 ⊢ Ⅎ𝑥 𝑋 ≤ ⦋𝑦 / 𝑥⦌𝐵
162	146	breq2d 4595	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑋 ≤ 𝐵 ↔ 𝑋 ≤ ⦋𝑦 / 𝑥⦌𝐵))
163	161, 162	rspc 3276	. . . . . 6 ⊢ (𝑦 ∈ (𝑘(,)(𝑘 + 1)) → (∀𝑥 ∈ (𝑘(,)(𝑘 + 1))𝑋 ≤ 𝐵 → 𝑋 ≤ ⦋𝑦 / 𝑥⦌𝐵))
164	158, 163	mpan9 485	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑦 ∈ (𝑘(,)(𝑘 + 1))) → 𝑋 ≤ ⦋𝑦 / 𝑥⦌𝐵)
165	41	rexrd 9968	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑘 ∈ ℝ^*)
166	49	rexrd 9968	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘 + 1) ∈ ℝ^*)
167	41	lep1d 10834	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑘 ≤ (𝑘 + 1))
168		lbicc2 12159	. . . . . 6 ⊢ ((𝑘 ∈ ℝ^* ∧ (𝑘 + 1) ∈ ℝ^* ∧ 𝑘 ≤ (𝑘 + 1)) → 𝑘 ∈ (𝑘[,](𝑘 + 1)))
169	165, 166, 167, 168	syl3anc 1318	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑘 ∈ (𝑘[,](𝑘 + 1)))
170		ubicc2 12160	. . . . . 6 ⊢ ((𝑘 ∈ ℝ^* ∧ (𝑘 + 1) ∈ ℝ^* ∧ 𝑘 ≤ (𝑘 + 1)) → (𝑘 + 1) ∈ (𝑘[,](𝑘 + 1)))
171	165, 166, 167, 170	syl3anc 1318	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘 + 1) ∈ (𝑘[,](𝑘 + 1)))
172		oveq2 6557	. . . . 5 ⊢ (𝑦 = 𝑘 → (𝑋 · 𝑦) = (𝑋 · 𝑘))
173		oveq2 6557	. . . . 5 ⊢ (𝑦 = (𝑘 + 1) → (𝑋 · 𝑦) = (𝑋 · (𝑘 + 1)))
174	41, 49, 79, 106, 114, 155, 164, 169, 171, 167, 172, 34, 173, 29	dvle 23574	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ((𝑋 · (𝑘 + 1)) − (𝑋 · 𝑘)) ≤ (⦋(𝑘 + 1) / 𝑥⦌𝐴 − ⦋𝑘 / 𝑥⦌𝐴))
175	53, 174	eqbrtrrd 4607	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑋 ≤ (⦋(𝑘 + 1) / 𝑥⦌𝐴 − ⦋𝑘 / 𝑥⦌𝐴))
176	2, 3, 38, 175	fsumle 14372	. 2 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀..^𝑁)𝑋 ≤ Σ𝑘 ∈ (𝑀..^𝑁)(⦋(𝑘 + 1) / 𝑥⦌𝐴 − ⦋𝑘 / 𝑥⦌𝐴))
177		vex 3176	. . . . 5 ⊢ 𝑦 ∈ V
178	177	a1i 11	. . . 4 ⊢ (𝑦 = 𝑀 → 𝑦 ∈ V)
179		eqeq2 2621	. . . . . 6 ⊢ (𝑦 = 𝑀 → (𝑥 = 𝑦 ↔ 𝑥 = 𝑀))
180	179	biimpa 500	. . . . 5 ⊢ ((𝑦 = 𝑀 ∧ 𝑥 = 𝑦) → 𝑥 = 𝑀)
181		dvfsumle.c	. . . . 5 ⊢ (𝑥 = 𝑀 → 𝐴 = 𝐶)
182	180, 181	syl 17	. . . 4 ⊢ ((𝑦 = 𝑀 ∧ 𝑥 = 𝑦) → 𝐴 = 𝐶)
183	178, 182	csbied 3526	. . 3 ⊢ (𝑦 = 𝑀 → ⦋𝑦 / 𝑥⦌𝐴 = 𝐶)
184	177	a1i 11	. . . 4 ⊢ (𝑦 = 𝑁 → 𝑦 ∈ V)
185		eqeq2 2621	. . . . . 6 ⊢ (𝑦 = 𝑁 → (𝑥 = 𝑦 ↔ 𝑥 = 𝑁))
186	185	biimpa 500	. . . . 5 ⊢ ((𝑦 = 𝑁 ∧ 𝑥 = 𝑦) → 𝑥 = 𝑁)
187		dvfsumle.d	. . . . 5 ⊢ (𝑥 = 𝑁 → 𝐴 = 𝐷)
188	186, 187	syl 17	. . . 4 ⊢ ((𝑦 = 𝑁 ∧ 𝑥 = 𝑦) → 𝐴 = 𝐷)
189	184, 188	csbied 3526	. . 3 ⊢ (𝑦 = 𝑁 → ⦋𝑦 / 𝑥⦌𝐴 = 𝐷)
190	26	recnd 9947	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑀...𝑁)) → ⦋𝑦 / 𝑥⦌𝐴 ∈ ℂ)
191	34, 29, 183, 189, 4, 190	telfsumo2 14376	. 2 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀..^𝑁)(⦋(𝑘 + 1) / 𝑥⦌𝐴 − ⦋𝑘 / 𝑥⦌𝐴) = (𝐷 − 𝐶))
192	176, 191	breqtrd 4609	1 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀..^𝑁)𝑋 ≤ (𝐷 − 𝐶))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 Vcvv 3173 ⦋csb 3499 ∩ cin 3539 ⊆ wss 3540 {cpr 4127 class class class wbr 4583 ↦ cmpt 4643 dom cdm 5038 ran crn 5039 ↾ cres 5040 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 Fincfn 7841 ℂcc 9813 ℝcr 9814 1c1 9816 + caddc 9818 · cmul 9820 ℝ^*cxr 9952 ≤ cle 9954 − cmin 10145 ℤcz 11254 ℤ_≥cuz 11563 (,)cioo 12046 [,]cicc 12049 ...cfz 12197 ..^cfzo 12334 Σcsu 14264 TopOpenctopn 15905 topGenctg 15921 ℂ_fldccnfld 19567 –cn→ccncf 22487 D cdv 23433

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-fi 8200 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-ioo 12050 df-ico 12052 df-icc 12053 df-fz 12198 df-fzo 12335 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-sum 14265 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-hom 15793 df-cco 15794 df-rest 15906 df-topn 15907 df-0g 15925 df-gsum 15926 df-topgen 15927 df-pt 15928 df-prds 15931 df-xrs 15985 df-qtop 15990 df-imas 15991 df-xps 15993 df-mre 16069 df-mrc 16070 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-mulg 17364 df-cntz 17573 df-cmn 18018 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-fbas 19564 df-fg 19565 df-cnfld 19568 df-top 20521 df-bases 20522 df-topon 20523 df-topsp 20524 df-cld 20633 df-ntr 20634 df-cls 20635 df-nei 20712 df-lp 20750 df-perf 20751 df-cn 20841 df-cnp 20842 df-haus 20929 df-cmp 21000 df-tx 21175 df-hmeo 21368 df-fil 21460 df-fm 21552 df-flim 21553 df-flf 21554 df-xms 21935 df-ms 21936 df-tms 21937 df-cncf 22489 df-limc 23436 df-dv 23437

This theorem is referenced by: dvfsumge 23589

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