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Theorem dvfsumle 25385

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 13879	. . . 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 12768	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑀 ∈ ℤ)
6	4, 5	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℤ)
7		eluzelz 12773	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑁 ∈ ℤ)
8	4, 7	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℤ)
9		fzval2 13427	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀...𝑁) = ((𝑀[,]𝑁) ∩ ℤ))
10	6, 8, 9	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → (𝑀...𝑁) = ((𝑀[,]𝑁) ∩ ℤ))
11		inss1 4188	. . . . . . . . 9 ⊢ ((𝑀[,]𝑁) ∩ ℤ) ⊆ (𝑀[,]𝑁)
12	10, 11	eqsstrdi 3998	. . . . . . . 8 ⊢ (𝜑 → (𝑀...𝑁) ⊆ (𝑀[,]𝑁))
13	12	sselda 3944	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑀...𝑁)) → 𝑦 ∈ (𝑀[,]𝑁))
14		dvfsumle.a	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴) ∈ ((𝑀[,]𝑁)–cn→ℝ))
15		cncff 24256	. . . . . . . . . 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 2736	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴) = (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴)
18	17	fmpt 7058	. . . . . . . . 9 ⊢ (∀𝑥 ∈ (𝑀[,]𝑁)𝐴 ∈ ℝ ↔ (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴):(𝑀[,]𝑁)⟶ℝ)
19	16, 18	sylibr 233	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ (𝑀[,]𝑁)𝐴 ∈ ℝ)
20		nfcsb1v 3880	. . . . . . . . . 10 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐴
21	20	nfel1 2923	. . . . . . . . 9 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐴 ∈ ℝ
22		csbeq1a 3869	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → 𝐴 = ⦋𝑦 / 𝑥⦌𝐴)
23	22	eleq1d 2822	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝐴 ∈ ℝ ↔ ⦋𝑦 / 𝑥⦌𝐴 ∈ ℝ))
24	21, 23	rspc 3569	. . . . . . . 8 ⊢ (𝑦 ∈ (𝑀[,]𝑁) → (∀𝑥 ∈ (𝑀[,]𝑁)𝐴 ∈ ℝ → ⦋𝑦 / 𝑥⦌𝐴 ∈ ℝ))
25	19, 24	mpan9 507	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑀[,]𝑁)) → ⦋𝑦 / 𝑥⦌𝐴 ∈ ℝ)
26	13, 25	syldan 591	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑀...𝑁)) → ⦋𝑦 / 𝑥⦌𝐴 ∈ ℝ)
27	26	ralrimiva 3143	. . . . 5 ⊢ (𝜑 → ∀𝑦 ∈ (𝑀...𝑁)⦋𝑦 / 𝑥⦌𝐴 ∈ ℝ)
28		fzofzp1 13669	. . . . 5 ⊢ (𝑘 ∈ (𝑀..^𝑁) → (𝑘 + 1) ∈ (𝑀...𝑁))
29		csbeq1 3858	. . . . . . 7 ⊢ (𝑦 = (𝑘 + 1) → ⦋𝑦 / 𝑥⦌𝐴 = ⦋(𝑘 + 1) / 𝑥⦌𝐴)
30	29	eleq1d 2822	. . . . . 6 ⊢ (𝑦 = (𝑘 + 1) → (⦋𝑦 / 𝑥⦌𝐴 ∈ ℝ ↔ ⦋(𝑘 + 1) / 𝑥⦌𝐴 ∈ ℝ))
31	30	rspccva 3580	. . . . 5 ⊢ ((∀𝑦 ∈ (𝑀...𝑁)⦋𝑦 / 𝑥⦌𝐴 ∈ ℝ ∧ (𝑘 + 1) ∈ (𝑀...𝑁)) → ⦋(𝑘 + 1) / 𝑥⦌𝐴 ∈ ℝ)
32	27, 28, 31	syl2an 596	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ⦋(𝑘 + 1) / 𝑥⦌𝐴 ∈ ℝ)
33		elfzofz 13588	. . . . 5 ⊢ (𝑘 ∈ (𝑀..^𝑁) → 𝑘 ∈ (𝑀...𝑁))
34		csbeq1 3858	. . . . . . 7 ⊢ (𝑦 = 𝑘 → ⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑘 / 𝑥⦌𝐴)
35	34	eleq1d 2822	. . . . . 6 ⊢ (𝑦 = 𝑘 → (⦋𝑦 / 𝑥⦌𝐴 ∈ ℝ ↔ ⦋𝑘 / 𝑥⦌𝐴 ∈ ℝ))
36	35	rspccva 3580	. . . . 5 ⊢ ((∀𝑦 ∈ (𝑀...𝑁)⦋𝑦 / 𝑥⦌𝐴 ∈ ℝ ∧ 𝑘 ∈ (𝑀...𝑁)) → ⦋𝑘 / 𝑥⦌𝐴 ∈ ℝ)
37	27, 33, 36	syl2an 596	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ⦋𝑘 / 𝑥⦌𝐴 ∈ ℝ)
38	32, 37	resubcld 11583	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (⦋(𝑘 + 1) / 𝑥⦌𝐴 − ⦋𝑘 / 𝑥⦌𝐴) ∈ ℝ)
39		elfzoelz 13572	. . . . . . . . . 10 ⊢ (𝑘 ∈ (𝑀..^𝑁) → 𝑘 ∈ ℤ)
40	39	adantl 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑘 ∈ ℤ)
41	40	zred 12607	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑘 ∈ ℝ)
42	41	recnd 11183	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑘 ∈ ℂ)
43		ax-1cn 11109	. . . . . . 7 ⊢ 1 ∈ ℂ
44		pncan2 11408	. . . . . . 7 ⊢ ((𝑘 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑘 + 1) − 𝑘) = 1)
45	42, 43, 44	sylancl 586	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ((𝑘 + 1) − 𝑘) = 1)
46	45	oveq2d 7373	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑋 · ((𝑘 + 1) − 𝑘)) = (𝑋 · 1))
47	3	recnd 11183	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑋 ∈ ℂ)
48		peano2re 11328	. . . . . . . 8 ⊢ (𝑘 ∈ ℝ → (𝑘 + 1) ∈ ℝ)
49	41, 48	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘 + 1) ∈ ℝ)
50	49	recnd 11183	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘 + 1) ∈ ℂ)
51	47, 50, 42	subdid 11611	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑋 · ((𝑘 + 1) − 𝑘)) = ((𝑋 · (𝑘 + 1)) − (𝑋 · 𝑘)))
52	47	mulid1d 11172	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑋 · 1) = 𝑋)
53	46, 51, 52	3eqtr3d 2784	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ((𝑋 · (𝑘 + 1)) − (𝑋 · 𝑘)) = 𝑋)
54		eqid 2736	. . . . . 6 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
55	54	mulcn 24230	. . . . . 6 ⊢ · ∈ (((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld)) Cn (TopOpen‘ℂ_fld))
56	6	zred 12607	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℝ)
57	56	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑀 ∈ ℝ)
58	8	zred 12607	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℝ)
59	58	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑁 ∈ ℝ)
60		elfzole1 13580	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (𝑀..^𝑁) → 𝑀 ≤ 𝑘)
61	60	adantl 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑀 ≤ 𝑘)
62	28	adantl 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘 + 1) ∈ (𝑀...𝑁))
63		elfzle2 13445	. . . . . . . . . . 11 ⊢ ((𝑘 + 1) ∈ (𝑀...𝑁) → (𝑘 + 1) ≤ 𝑁)
64	62, 63	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘 + 1) ≤ 𝑁)
65		iccss 13332	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) ∧ (𝑀 ≤ 𝑘 ∧ (𝑘 + 1) ≤ 𝑁)) → (𝑘[,](𝑘 + 1)) ⊆ (𝑀[,]𝑁))
66	57, 59, 61, 64, 65	syl22anc 837	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘[,](𝑘 + 1)) ⊆ (𝑀[,]𝑁))
67		iccssre 13346	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝑀[,]𝑁) ⊆ ℝ)
68	56, 58, 67	syl2anc 584	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀[,]𝑁) ⊆ ℝ)
69	68	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑀[,]𝑁) ⊆ ℝ)
70	66, 69	sstrd 3954	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘[,](𝑘 + 1)) ⊆ ℝ)
71		ax-resscn 11108	. . . . . . . 8 ⊢ ℝ ⊆ ℂ
72	70, 71	sstrdi 3956	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘[,](𝑘 + 1)) ⊆ ℂ)
73	71	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ℝ ⊆ ℂ)
74		cncfmptc 24275	. . . . . . 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 1371	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑦 ∈ (𝑘[,](𝑘 + 1)) ↦ 𝑋) ∈ ((𝑘[,](𝑘 + 1))–cn→ℝ))
76		cncfmptid 24276	. . . . . . 7 ⊢ (((𝑘[,](𝑘 + 1)) ⊆ ℝ ∧ ℝ ⊆ ℂ) → (𝑦 ∈ (𝑘[,](𝑘 + 1)) ↦ 𝑦) ∈ ((𝑘[,](𝑘 + 1))–cn→ℝ))
77	70, 71, 76	sylancl 586	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑦 ∈ (𝑘[,](𝑘 + 1)) ↦ 𝑦) ∈ ((𝑘[,](𝑘 + 1))–cn→ℝ))
78		remulcl 11136	. . . . . 6 ⊢ ((𝑋 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑋 · 𝑦) ∈ ℝ)
79	54, 55, 75, 77, 71, 78	cncfmpt2ss 24279	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑦 ∈ (𝑘[,](𝑘 + 1)) ↦ (𝑋 · 𝑦)) ∈ ((𝑘[,](𝑘 + 1))–cn→ℝ))
80		reelprrecn 11143	. . . . . . . 8 ⊢ ℝ ∈ {ℝ, ℂ}
81	80	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ℝ ∈ {ℝ, ℂ})
82	57	rexrd 11205	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑀 ∈ ℝ^*)
83		iooss1 13299	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℝ^* ∧ 𝑀 ≤ 𝑘) → (𝑘(,)(𝑘 + 1)) ⊆ (𝑀(,)(𝑘 + 1)))
84	82, 61, 83	syl2anc 584	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘(,)(𝑘 + 1)) ⊆ (𝑀(,)(𝑘 + 1)))
85	59	rexrd 11205	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑁 ∈ ℝ^*)
86		iooss2 13300	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℝ^* ∧ (𝑘 + 1) ≤ 𝑁) → (𝑀(,)(𝑘 + 1)) ⊆ (𝑀(,)𝑁))
87	85, 64, 86	syl2anc 584	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑀(,)(𝑘 + 1)) ⊆ (𝑀(,)𝑁))
88	84, 87	sstrd 3954	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘(,)(𝑘 + 1)) ⊆ (𝑀(,)𝑁))
89		ioossicc 13350	. . . . . . . . . 10 ⊢ (𝑀(,)𝑁) ⊆ (𝑀[,]𝑁)
90	69, 71	sstrdi 3956	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑀[,]𝑁) ⊆ ℂ)
91	89, 90	sstrid 3955	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑀(,)𝑁) ⊆ ℂ)
92	88, 91	sstrd 3954	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘(,)(𝑘 + 1)) ⊆ ℂ)
93	92	sselda 3944	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑦 ∈ (𝑘(,)(𝑘 + 1))) → 𝑦 ∈ ℂ)
94		1cnd 11150	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑦 ∈ (𝑘(,)(𝑘 + 1))) → 1 ∈ ℂ)
95	73	sselda 3944	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℂ)
96		1cnd 11150	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑦 ∈ ℝ) → 1 ∈ ℂ)
97	81	dvmptid 25321	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (ℝ D (𝑦 ∈ ℝ ↦ 𝑦)) = (𝑦 ∈ ℝ ↦ 1))
98		ioossre 13325	. . . . . . . . 9 ⊢ (𝑘(,)(𝑘 + 1)) ⊆ ℝ
99	98	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘(,)(𝑘 + 1)) ⊆ ℝ)
100	54	tgioo2 24166	. . . . . . . 8 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
101		iooretop 24129	. . . . . . . . 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 25327	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (ℝ D (𝑦 ∈ (𝑘(,)(𝑘 + 1)) ↦ 𝑦)) = (𝑦 ∈ (𝑘(,)(𝑘 + 1)) ↦ 1))
104	81, 93, 94, 103, 47	dvmptcmul 25328	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (ℝ D (𝑦 ∈ (𝑘(,)(𝑘 + 1)) ↦ (𝑋 · 𝑦))) = (𝑦 ∈ (𝑘(,)(𝑘 + 1)) ↦ (𝑋 · 1)))
105	52	mpteq2dv 5207	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑦 ∈ (𝑘(,)(𝑘 + 1)) ↦ (𝑋 · 1)) = (𝑦 ∈ (𝑘(,)(𝑘 + 1)) ↦ 𝑋))
106	104, 105	eqtrd 2776	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (ℝ D (𝑦 ∈ (𝑘(,)(𝑘 + 1)) ↦ (𝑋 · 𝑦))) = (𝑦 ∈ (𝑘(,)(𝑘 + 1)) ↦ 𝑋))
107		nfcv 2907	. . . . . . 7 ⊢ Ⅎ𝑦𝐴
108	107, 20, 22	cbvmpt 5216	. . . . . 6 ⊢ (𝑥 ∈ (𝑘[,](𝑘 + 1)) ↦ 𝐴) = (𝑦 ∈ (𝑘[,](𝑘 + 1)) ↦ ⦋𝑦 / 𝑥⦌𝐴)
109	66	resmptd 5994	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ((𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴) ↾ (𝑘[,](𝑘 + 1))) = (𝑥 ∈ (𝑘[,](𝑘 + 1)) ↦ 𝐴))
110	14	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴) ∈ ((𝑀[,]𝑁)–cn→ℝ))
111		rescncf 24260	. . . . . . . 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 2839	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑥 ∈ (𝑘[,](𝑘 + 1)) ↦ 𝐴) ∈ ((𝑘[,](𝑘 + 1))–cn→ℝ))
114	108, 113	eqeltrrid 2843	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑦 ∈ (𝑘[,](𝑘 + 1)) ↦ ⦋𝑦 / 𝑥⦌𝐴) ∈ ((𝑘[,](𝑘 + 1))–cn→ℝ))
115	16	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴):(𝑀[,]𝑁)⟶ℝ)
116	115, 18	sylibr 233	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ∀𝑥 ∈ (𝑀[,]𝑁)𝐴 ∈ ℝ)
117	89	sseli 3940	. . . . . . . 8 ⊢ (𝑦 ∈ (𝑀(,)𝑁) → 𝑦 ∈ (𝑀[,]𝑁))
118	24	impcom 408	. . . . . . . 8 ⊢ ((∀𝑥 ∈ (𝑀[,]𝑁)𝐴 ∈ ℝ ∧ 𝑦 ∈ (𝑀[,]𝑁)) → ⦋𝑦 / 𝑥⦌𝐴 ∈ ℝ)
119	116, 117, 118	syl2an 596	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑦 ∈ (𝑀(,)𝑁)) → ⦋𝑦 / 𝑥⦌𝐴 ∈ ℝ)
120	119	recnd 11183	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑦 ∈ (𝑀(,)𝑁)) → ⦋𝑦 / 𝑥⦌𝐴 ∈ ℂ)
121	89	sseli 3940	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝑀(,)𝑁) → 𝑥 ∈ (𝑀[,]𝑁))
122	16	fvmptelcdm 7061	. . . . . . . . . . . . 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 593	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 𝐴 ∈ ℝ)
125	124	fmpttd 7063	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴):(𝑀(,)𝑁)⟶ℝ)
126		ioossre 13325	. . . . . . . . . 10 ⊢ (𝑀(,)𝑁) ⊆ ℝ
127		dvfre 25315	. . . . . . . . . 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 586	. . . . . . . . 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 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)) = (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵))
131	130	dmeqd 5861	. . . . . . . . . . 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 3143	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ∀𝑥 ∈ (𝑀(,)𝑁)𝐵 ∈ 𝑉)
135		dmmptg 6194	. . . . . . . . . . . 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 2776	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → dom (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)) = (𝑀(,)𝑁))
138	130, 137	feq12d 6656	. . . . . . . . 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 231	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵):(𝑀(,)𝑁)⟶ℝ)
140		eqid 2736	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵) = (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵)
141	140	fmpt 7058	. . . . . . . 8 ⊢ (∀𝑥 ∈ (𝑀(,)𝑁)𝐵 ∈ ℝ ↔ (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵):(𝑀(,)𝑁)⟶ℝ)
142	139, 141	sylibr 233	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ∀𝑥 ∈ (𝑀(,)𝑁)𝐵 ∈ ℝ)
143		nfcsb1v 3880	. . . . . . . . 9 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵
144	143	nfel1 2923	. . . . . . . 8 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵 ∈ ℝ
145		csbeq1a 3869	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)
146	145	eleq1d 2822	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝐵 ∈ ℝ ↔ ⦋𝑦 / 𝑥⦌𝐵 ∈ ℝ))
147	144, 146	rspc 3569	. . . . . . 7 ⊢ (𝑦 ∈ (𝑀(,)𝑁) → (∀𝑥 ∈ (𝑀(,)𝑁)𝐵 ∈ ℝ → ⦋𝑦 / 𝑥⦌𝐵 ∈ ℝ))
148	142, 147	mpan9 507	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑦 ∈ (𝑀(,)𝑁)) → ⦋𝑦 / 𝑥⦌𝐵 ∈ ℝ)
149	107, 20, 22	cbvmpt 5216	. . . . . . . 8 ⊢ (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴) = (𝑦 ∈ (𝑀(,)𝑁) ↦ ⦋𝑦 / 𝑥⦌𝐴)
150	149	oveq2i 7368	. . . . . . 7 ⊢ (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)) = (ℝ D (𝑦 ∈ (𝑀(,)𝑁) ↦ ⦋𝑦 / 𝑥⦌𝐴))
151		nfcv 2907	. . . . . . . 8 ⊢ Ⅎ𝑦𝐵
152	151, 143, 145	cbvmpt 5216	. . . . . . 7 ⊢ (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵) = (𝑦 ∈ (𝑀(,)𝑁) ↦ ⦋𝑦 / 𝑥⦌𝐵)
153	130, 150, 152	3eqtr3g 2799	. . . . . 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 25327	. . . . 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 468	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑘(,)(𝑘 + 1))) → 𝑋 ≤ 𝐵)
157	156	ralrimiva 3143	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ∀𝑥 ∈ (𝑘(,)(𝑘 + 1))𝑋 ≤ 𝐵)
158		nfcv 2907	. . . . . . . 8 ⊢ Ⅎ𝑥𝑋
159		nfcv 2907	. . . . . . . 8 ⊢ Ⅎ𝑥 ≤
160	158, 159, 143	nfbr 5152	. . . . . . 7 ⊢ Ⅎ𝑥 𝑋 ≤ ⦋𝑦 / 𝑥⦌𝐵
161	145	breq2d 5117	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑋 ≤ 𝐵 ↔ 𝑋 ≤ ⦋𝑦 / 𝑥⦌𝐵))
162	160, 161	rspc 3569	. . . . . 6 ⊢ (𝑦 ∈ (𝑘(,)(𝑘 + 1)) → (∀𝑥 ∈ (𝑘(,)(𝑘 + 1))𝑋 ≤ 𝐵 → 𝑋 ≤ ⦋𝑦 / 𝑥⦌𝐵))
163	157, 162	mpan9 507	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑦 ∈ (𝑘(,)(𝑘 + 1))) → 𝑋 ≤ ⦋𝑦 / 𝑥⦌𝐵)
164	41	rexrd 11205	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑘 ∈ ℝ^*)
165	49	rexrd 11205	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘 + 1) ∈ ℝ^*)
166	41	lep1d 12086	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑘 ≤ (𝑘 + 1))
167		lbicc2 13381	. . . . . 6 ⊢ ((𝑘 ∈ ℝ^* ∧ (𝑘 + 1) ∈ ℝ^* ∧ 𝑘 ≤ (𝑘 + 1)) → 𝑘 ∈ (𝑘[,](𝑘 + 1)))
168	164, 165, 166, 167	syl3anc 1371	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑘 ∈ (𝑘[,](𝑘 + 1)))
169		ubicc2 13382	. . . . . 6 ⊢ ((𝑘 ∈ ℝ^* ∧ (𝑘 + 1) ∈ ℝ^* ∧ 𝑘 ≤ (𝑘 + 1)) → (𝑘 + 1) ∈ (𝑘[,](𝑘 + 1)))
170	164, 165, 166, 169	syl3anc 1371	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘 + 1) ∈ (𝑘[,](𝑘 + 1)))
171		oveq2 7365	. . . . 5 ⊢ (𝑦 = 𝑘 → (𝑋 · 𝑦) = (𝑋 · 𝑘))
172		oveq2 7365	. . . . 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 25371	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ((𝑋 · (𝑘 + 1)) − (𝑋 · 𝑘)) ≤ (⦋(𝑘 + 1) / 𝑥⦌𝐴 − ⦋𝑘 / 𝑥⦌𝐴))
174	53, 173	eqbrtrrd 5129	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑋 ≤ (⦋(𝑘 + 1) / 𝑥⦌𝐴 − ⦋𝑘 / 𝑥⦌𝐴))
175	2, 3, 38, 174	fsumle 15684	. 2 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀..^𝑁)𝑋 ≤ Σ𝑘 ∈ (𝑀..^𝑁)(⦋(𝑘 + 1) / 𝑥⦌𝐴 − ⦋𝑘 / 𝑥⦌𝐴))
176		vex 3449	. . . . 5 ⊢ 𝑦 ∈ V
177	176	a1i 11	. . . 4 ⊢ (𝑦 = 𝑀 → 𝑦 ∈ V)
178		eqeq2 2748	. . . . . 6 ⊢ (𝑦 = 𝑀 → (𝑥 = 𝑦 ↔ 𝑥 = 𝑀))
179	178	biimpa 477	. . . . 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 3893	. . 3 ⊢ (𝑦 = 𝑀 → ⦋𝑦 / 𝑥⦌𝐴 = 𝐶)
183	176	a1i 11	. . . 4 ⊢ (𝑦 = 𝑁 → 𝑦 ∈ V)
184		eqeq2 2748	. . . . . 6 ⊢ (𝑦 = 𝑁 → (𝑥 = 𝑦 ↔ 𝑥 = 𝑁))
185	184	biimpa 477	. . . . 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 3893	. . 3 ⊢ (𝑦 = 𝑁 → ⦋𝑦 / 𝑥⦌𝐴 = 𝐷)
189	26	recnd 11183	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑀...𝑁)) → ⦋𝑦 / 𝑥⦌𝐴 ∈ ℂ)
190	34, 29, 182, 188, 4, 189	telfsumo2 15688	. 2 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀..^𝑁)(⦋(𝑘 + 1) / 𝑥⦌𝐴 − ⦋𝑘 / 𝑥⦌𝐴) = (𝐷 − 𝐶))
191	175, 190	breqtrd 5131	1 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀..^𝑁)𝑋 ≤ (𝐷 − 𝐶))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3064 Vcvv 3445 ⦋csb 3855 ∩ cin 3909 ⊆ wss 3910 {cpr 4588 class class class wbr 5105 ↦ cmpt 5188 dom cdm 5633 ran crn 5634 ↾ cres 5635 ⟶wf 6492 ‘cfv 6496 (class class class)co 7357 Fincfn 8883 ℂcc 11049 ℝcr 11050 1c1 11052 + caddc 11054 · cmul 11056 ℝ^*cxr 11188 ≤ cle 11190 − cmin 11385 ℤcz 12499 ℤ_≥cuz 12763 (,)cioo 13264 [,]cicc 13267 ...cfz 13424 ..^cfzo 13567 Σcsu 15570 TopOpenctopn 17303 topGenctg 17319 ℂ_fldccnfld 20796 –cn→ccncf 24239 D cdv 25227

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-inf2 9577 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129 ax-addf 11130 ax-mulf 11131

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-om 7803 df-1st 7921 df-2nd 7922 df-supp 8093 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-2o 8413 df-er 8648 df-map 8767 df-pm 8768 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9306 df-fi 9347 df-sup 9378 df-inf 9379 df-oi 9446 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-z 12500 df-dec 12619 df-uz 12764 df-q 12874 df-rp 12916 df-xneg 13033 df-xadd 13034 df-xmul 13035 df-ioo 13268 df-ico 13270 df-icc 13271 df-fz 13425 df-fzo 13568 df-seq 13907 df-exp 13968 df-hash 14231 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-clim 15370 df-sum 15571 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-starv 17148 df-sca 17149 df-vsca 17150 df-ip 17151 df-tset 17152 df-ple 17153 df-ds 17155 df-unif 17156 df-hom 17157 df-cco 17158 df-rest 17304 df-topn 17305 df-0g 17323 df-gsum 17324 df-topgen 17325 df-pt 17326 df-prds 17329 df-xrs 17384 df-qtop 17389 df-imas 17390 df-xps 17392 df-mre 17466 df-mrc 17467 df-acs 17469 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-submnd 18602 df-mulg 18873 df-cntz 19097 df-cmn 19564 df-psmet 20788 df-xmet 20789 df-met 20790 df-bl 20791 df-mopn 20792 df-fbas 20793 df-fg 20794 df-cnfld 20797 df-top 22243 df-topon 22260 df-topsp 22282 df-bases 22296 df-cld 22370 df-ntr 22371 df-cls 22372 df-nei 22449 df-lp 22487 df-perf 22488 df-cn 22578 df-cnp 22579 df-haus 22666 df-cmp 22738 df-tx 22913 df-hmeo 23106 df-fil 23197 df-fm 23289 df-flim 23290 df-flf 23291 df-xms 23673 df-ms 23674 df-tms 23675 df-cncf 24241 df-limc 25230 df-dv 25231

This theorem is referenced by: dvfsumge 25386

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