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Theorem dvfsumge 26150

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

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

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 dvfsumge**
Step	Hyp	Ref	Expression
1		dvfsumge.m	. . . 4 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
2		df-neg 11444	. . . . . 6 ⊢ -𝐴 = (0 − 𝐴)
3	2	mpteq2i 5211	. . . . 5 ⊢ (𝑥 ∈ (𝑀[,]𝑁) ↦ -𝐴) = (𝑥 ∈ (𝑀[,]𝑁) ↦ (0 − 𝐴))
4		eqid 2769	. . . . . 6 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
5	4	subcn 24993	. . . . . 6 ⊢ − ∈ (((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld)) Cn (TopOpen‘ℂ_fld))
6		0red 11211	. . . . . . 7 ⊢ (𝜑 → 0 ∈ ℝ)
7		eluzel2 12867	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑀 ∈ ℤ)
8	1, 7	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℤ)
9	8	zred 12700	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℝ)
10		eluzelz 12872	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑁 ∈ ℤ)
11	1, 10	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℤ)
12	11	zred 12700	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℝ)
13		iccssre 13456	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝑀[,]𝑁) ⊆ ℝ)
14	9, 12, 13	syl2anc 595	. . . . . . . 8 ⊢ (𝜑 → (𝑀[,]𝑁) ⊆ ℝ)
15		ax-resscn 11157	. . . . . . . 8 ⊢ ℝ ⊆ ℂ
16	14, 15	sstrdi 3957	. . . . . . 7 ⊢ (𝜑 → (𝑀[,]𝑁) ⊆ ℂ)
17	15	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℝ ⊆ ℂ)
18		cncfmptc 25040	. . . . . . 7 ⊢ ((0 ∈ ℝ ∧ (𝑀[,]𝑁) ⊆ ℂ ∧ ℝ ⊆ ℂ) → (𝑥 ∈ (𝑀[,]𝑁) ↦ 0) ∈ ((𝑀[,]𝑁)–cn→ℝ))
19	6, 16, 17, 18	syl3anc 1396	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (𝑀[,]𝑁) ↦ 0) ∈ ((𝑀[,]𝑁)–cn→ℝ))
20		dvfsumge.a	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴) ∈ ((𝑀[,]𝑁)–cn→ℝ))
21		resubcl 11522	. . . . . 6 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (0 − 𝐴) ∈ ℝ)
22	4, 5, 19, 20, 15, 21	cncfmpt2ss 25044	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (𝑀[,]𝑁) ↦ (0 − 𝐴)) ∈ ((𝑀[,]𝑁)–cn→ℝ))
23	3, 22	eqeltrid 2873	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (𝑀[,]𝑁) ↦ -𝐴) ∈ ((𝑀[,]𝑁)–cn→ℝ))
24		negex 11455	. . . . 5 ⊢ -𝐵 ∈ V
25	24	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀(,)𝑁)) → -𝐵 ∈ V)
26		reelprrecn 11192	. . . . . 6 ⊢ ℝ ∈ {ℝ, ℂ}
27	26	a1i 11	. . . . 5 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
28		ioossicc 13460	. . . . . . . 8 ⊢ (𝑀(,)𝑁) ⊆ (𝑀[,]𝑁)
29	28	sseli 3941	. . . . . . 7 ⊢ (𝑥 ∈ (𝑀(,)𝑁) → 𝑥 ∈ (𝑀[,]𝑁))
30		cncff 25021	. . . . . . . . 9 ⊢ ((𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴) ∈ ((𝑀[,]𝑁)–cn→ℝ) → (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴):(𝑀[,]𝑁)⟶ℝ)
31	20, 30	syl 18	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴):(𝑀[,]𝑁)⟶ℝ)
32	31	fvmptelcdm 7109	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → 𝐴 ∈ ℝ)
33	29, 32	sylan2 604	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 𝐴 ∈ ℝ)
34	33	recnd 11237	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 𝐴 ∈ ℂ)
35		dvfsumge.v	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 𝐵 ∈ 𝑉)
36		dvfsumge.b	. . . . 5 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)) = (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵))
37	27, 34, 35, 36	dvmptneg 26094	. . . 4 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ -𝐴)) = (𝑥 ∈ (𝑀(,)𝑁) ↦ -𝐵))
38		dvfsumge.c	. . . . 5 ⊢ (𝑥 = 𝑀 → 𝐴 = 𝐶)
39	38	negeqd 11451	. . . 4 ⊢ (𝑥 = 𝑀 → -𝐴 = -𝐶)
40		dvfsumge.d	. . . . 5 ⊢ (𝑥 = 𝑁 → 𝐴 = 𝐷)
41	40	negeqd 11451	. . . 4 ⊢ (𝑥 = 𝑁 → -𝐴 = -𝐷)
42		dvfsumge.x	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑋 ∈ ℝ)
43	42	renegcld 11641	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → -𝑋 ∈ ℝ)
44		dvfsumge.l	. . . . 5 ⊢ ((𝜑 ∧ (𝑘 ∈ (𝑀..^𝑁) ∧ 𝑥 ∈ (𝑘(,)(𝑘 + 1)))) → 𝐵 ≤ 𝑋)
45	9	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑀 ∈ ℝ)
46	45	rexrd 11259	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑀 ∈ ℝ^*)
47		elfzole1 13696	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (𝑀..^𝑁) → 𝑀 ≤ 𝑘)
48	47	adantl 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑀 ≤ 𝑘)
49		iooss1 13407	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℝ^* ∧ 𝑀 ≤ 𝑘) → (𝑘(,)(𝑘 + 1)) ⊆ (𝑀(,)(𝑘 + 1)))
50	46, 48, 49	syl2anc 595	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘(,)(𝑘 + 1)) ⊆ (𝑀(,)(𝑘 + 1)))
51	12	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑁 ∈ ℝ)
52	51	rexrd 11259	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑁 ∈ ℝ^*)
53		fzofzp1 13793	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (𝑀..^𝑁) → (𝑘 + 1) ∈ (𝑀...𝑁))
54	53	adantl 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘 + 1) ∈ (𝑀...𝑁))
55		elfzle2 13556	. . . . . . . . . . . 12 ⊢ ((𝑘 + 1) ∈ (𝑀...𝑁) → (𝑘 + 1) ≤ 𝑁)
56	54, 55	syl 18	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘 + 1) ≤ 𝑁)
57		iooss2 13408	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℝ^* ∧ (𝑘 + 1) ≤ 𝑁) → (𝑀(,)(𝑘 + 1)) ⊆ (𝑀(,)𝑁))
58	52, 56, 57	syl2anc 595	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑀(,)(𝑘 + 1)) ⊆ (𝑀(,)𝑁))
59	50, 58	sstrd 3955	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘(,)(𝑘 + 1)) ⊆ (𝑀(,)𝑁))
60	59	sselda 3945	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑘(,)(𝑘 + 1))) → 𝑥 ∈ (𝑀(,)𝑁))
61	32	adantlr 727	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀[,]𝑁)) → 𝐴 ∈ ℝ)
62	29, 61	sylan2 604	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 𝐴 ∈ ℝ)
63	62	fmpttd 7111	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴):(𝑀(,)𝑁)⟶ℝ)
64		ioossre 13434	. . . . . . . . . . 11 ⊢ (𝑀(,)𝑁) ⊆ ℝ
65		dvfre 26079	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴):(𝑀(,)𝑁)⟶ℝ ∧ (𝑀(,)𝑁) ⊆ ℝ) → (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)):dom (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴))⟶ℝ)
66	63, 64, 65	sylancl 597	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)):dom (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴))⟶ℝ)
67	36	adantr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)) = (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵))
68	67	dmeqd 5896	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → dom (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)) = dom (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵))
69	35	adantlr 727	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 𝐵 ∈ 𝑉)
70	69	ralrimiva 3163	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ∀𝑥 ∈ (𝑀(,)𝑁)𝐵 ∈ 𝑉)
71		dmmptg 6244	. . . . . . . . . . . . 13 ⊢ (∀𝑥 ∈ (𝑀(,)𝑁)𝐵 ∈ 𝑉 → dom (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵) = (𝑀(,)𝑁))
72	70, 71	syl 18	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → dom (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵) = (𝑀(,)𝑁))
73	68, 72	eqtrd 2804	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → dom (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)) = (𝑀(,)𝑁))
74	67, 73	feq12d 6694	. . . . . . . . . 10 ⊢ ((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))
75	66, 74	mpbid 235	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵):(𝑀(,)𝑁)⟶ℝ)
76	75	fvmptelcdm 7109	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 𝐵 ∈ ℝ)
77	60, 76	syldan 602	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑘(,)(𝑘 + 1))) → 𝐵 ∈ ℝ)
78	77	anasss 471	. . . . . 6 ⊢ ((𝜑 ∧ (𝑘 ∈ (𝑀..^𝑁) ∧ 𝑥 ∈ (𝑘(,)(𝑘 + 1)))) → 𝐵 ∈ ℝ)
79	42	adantrr 729	. . . . . 6 ⊢ ((𝜑 ∧ (𝑘 ∈ (𝑀..^𝑁) ∧ 𝑥 ∈ (𝑘(,)(𝑘 + 1)))) → 𝑋 ∈ ℝ)
80	78, 79	lenegd 11793	. . . . 5 ⊢ ((𝜑 ∧ (𝑘 ∈ (𝑀..^𝑁) ∧ 𝑥 ∈ (𝑘(,)(𝑘 + 1)))) → (𝐵 ≤ 𝑋 ↔ -𝑋 ≤ -𝐵))
81	44, 80	mpbid 235	. . . 4 ⊢ ((𝜑 ∧ (𝑘 ∈ (𝑀..^𝑁) ∧ 𝑥 ∈ (𝑘(,)(𝑘 + 1)))) → -𝑋 ≤ -𝐵)
82	1, 23, 25, 37, 39, 41, 43, 81	dvfsumle 26149	. . 3 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀..^𝑁)-𝑋 ≤ (-𝐷 − -𝐶))
83		fzofi 14010	. . . . 5 ⊢ (𝑀..^𝑁) ∈ Fin
84	83	a1i 11	. . . 4 ⊢ (𝜑 → (𝑀..^𝑁) ∈ Fin)
85	42	recnd 11237	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑋 ∈ ℂ)
86	84, 85	fsumneg 15838	. . 3 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀..^𝑁)-𝑋 = -Σ𝑘 ∈ (𝑀..^𝑁)𝑋)
87	40	eleq1d 2854	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (𝐴 ∈ ℝ ↔ 𝐷 ∈ ℝ))
88		eqid 2769	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴) = (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴)
89	88	fmpt 7106	. . . . . . . 8 ⊢ (∀𝑥 ∈ (𝑀[,]𝑁)𝐴 ∈ ℝ ↔ (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴):(𝑀[,]𝑁)⟶ℝ)
90	31, 89	sylibr 237	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ (𝑀[,]𝑁)𝐴 ∈ ℝ)
91	9	rexrd 11259	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℝ^*)
92	12	rexrd 11259	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℝ^*)
93		eluzle 12875	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑀 ≤ 𝑁)
94	1, 93	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ≤ 𝑁)
95		ubicc2 13492	. . . . . . . 8 ⊢ ((𝑀 ∈ ℝ^* ∧ 𝑁 ∈ ℝ^* ∧ 𝑀 ≤ 𝑁) → 𝑁 ∈ (𝑀[,]𝑁))
96	91, 92, 94, 95	syl3anc 1396	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ (𝑀[,]𝑁))
97	87, 90, 96	rspcdva 3591	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ ℝ)
98	97	recnd 11237	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ ℂ)
99	38	eleq1d 2854	. . . . . . 7 ⊢ (𝑥 = 𝑀 → (𝐴 ∈ ℝ ↔ 𝐶 ∈ ℝ))
100		lbicc2 13491	. . . . . . . 8 ⊢ ((𝑀 ∈ ℝ^* ∧ 𝑁 ∈ ℝ^* ∧ 𝑀 ≤ 𝑁) → 𝑀 ∈ (𝑀[,]𝑁))
101	91, 92, 94, 100	syl3anc 1396	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ (𝑀[,]𝑁))
102	99, 90, 101	rspcdva 3591	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ ℝ)
103	102	recnd 11237	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℂ)
104	98, 103	neg2subd 11586	. . . 4 ⊢ (𝜑 → (-𝐷 − -𝐶) = (𝐶 − 𝐷))
105	98, 103	negsubdi2d 11585	. . . 4 ⊢ (𝜑 → -(𝐷 − 𝐶) = (𝐶 − 𝐷))
106	104, 105	eqtr4d 2807	. . 3 ⊢ (𝜑 → (-𝐷 − -𝐶) = -(𝐷 − 𝐶))
107	82, 86, 106	3brtr3d 5146	. 2 ⊢ (𝜑 → -Σ𝑘 ∈ (𝑀..^𝑁)𝑋 ≤ -(𝐷 − 𝐶))
108	97, 102	resubcld 11642	. . 3 ⊢ (𝜑 → (𝐷 − 𝐶) ∈ ℝ)
109	84, 42	fsumrecl 15785	. . 3 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀..^𝑁)𝑋 ∈ ℝ)
110	108, 109	lenegd 11793	. 2 ⊢ (𝜑 → ((𝐷 − 𝐶) ≤ Σ𝑘 ∈ (𝑀..^𝑁)𝑋 ↔ -Σ𝑘 ∈ (𝑀..^𝑁)𝑋 ≤ -(𝐷 − 𝐶)))
111	107, 110	mpbird 260	1 ⊢ (𝜑 → (𝐷 − 𝐶) ≤ Σ𝑘 ∈ (𝑀..^𝑁)𝑋)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 = wceq 1567 ∈ wcel 2149 ∀wral 3085 Vcvv 3463 ⊆ wss 3913 {cpr 4596 class class class wbr 5113 ↦ cmpt 5196 dom cdm 5662 ⟶wf 6533 ‘cfv 6537 (class class class)co 7411 Fincfn 8943 ℂcc 11098 ℝcr 11099 0cc0 11100 1c1 11101 + caddc 11103 ℝ^*cxr 11242 ≤ cle 11244 − cmin 11441 -cneg 11442 ℤcz 12591 ℤ_≥cuz 12862 (,)cioo 13372 [,]cicc 13375 ...cfz 13535 ..^cfzo 13682 Σcsu 15737 TopOpenctopn 17474 ℂ_fldccnfld 21491 –cn→ccncf 25004 D cdv 25991

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5242 ax-sep 5261 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733 ax-inf2 9610 ax-cnex 11156 ax-resscn 11157 ax-1cn 11158 ax-icn 11159 ax-addcl 11160 ax-addrcl 11161 ax-mulcl 11162 ax-mulrcl 11163 ax-mulcom 11164 ax-addass 11165 ax-mulass 11166 ax-distr 11167 ax-i2m1 11168 ax-1ne0 11169 ax-1rid 11170 ax-rnegex 11171 ax-rrecex 11172 ax-cnre 11173 ax-pre-lttri 11174 ax-pre-lttrn 11175 ax-pre-ltadd 11176 ax-pre-mulgt0 11177 ax-pre-sup 11178 ax-addf 11179

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-tp 4599 df-op 4601 df-uni 4877 df-int 4917 df-iun 4962 df-iin 4963 df-br 5114 df-opab 5178 df-mpt 5197 df-tr 5223 df-id 5557 df-eprel 5562 df-po 5570 df-so 5571 df-fr 5615 df-se 5616 df-we 5617 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7368 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7675 df-om 7863 df-1st 7986 df-2nd 7987 df-supp 8157 df-frecs 8278 df-wrecs 8309 df-recs 8358 df-rdg 8397 df-1o 8453 df-2o 8454 df-er 8694 df-map 8826 df-pm 8827 df-ixp 8896 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-fsupp 9322 df-fi 9371 df-sup 9402 df-inf 9403 df-oi 9472 df-card 9925 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11443 df-neg 11444 df-div 11872 df-nn 12234 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12505 df-z 12592 df-dec 12712 df-uz 12863 df-q 12973 df-rp 13017 df-xneg 13137 df-xadd 13138 df-xmul 13139 df-ioo 13376 df-ico 13378 df-icc 13379 df-fz 13536 df-fzo 13683 df-seq 14038 df-exp 14098 df-hash 14367 df-cj 15150 df-re 15151 df-im 15152 df-sqrt 15286 df-abs 15287 df-clim 15539 df-sum 15738 df-struct 17207 df-sets 17224 df-slot 17242 df-ndx 17254 df-base 17270 df-ress 17291 df-plusg 17323 df-mulr 17324 df-starv 17325 df-sca 17326 df-vsca 17327 df-ip 17328 df-tset 17329 df-ple 17330 df-ds 17332 df-unif 17333 df-hom 17334 df-cco 17335 df-rest 17475 df-topn 17476 df-0g 17494 df-gsum 17495 df-topgen 17496 df-pt 17497 df-prds 17500 df-xrs 17556 df-qtop 17561 df-imas 17562 df-xps 17564 df-mre 17638 df-mrc 17639 df-acs 17641 df-mgm 18698 df-sgrp 18777 df-mnd 18793 df-submnd 18842 df-mulg 19134 df-cntz 19387 df-cmn 19852 df-psmet 21483 df-xmet 21484 df-met 21485 df-bl 21486 df-mopn 21487 df-fbas 21488 df-fg 21489 df-cnfld 21492 df-top 23020 df-topon 23037 df-topsp 23059 df-bases 23072 df-cld 23145 df-ntr 23146 df-cls 23147 df-nei 23224 df-lp 23262 df-perf 23263 df-cn 23353 df-cnp 23354 df-haus 23441 df-cmp 23513 df-tx 23688 df-hmeo 23881 df-fil 23972 df-fm 24064 df-flim 24065 df-flf 24066 df-xms 24446 df-ms 24447 df-tms 24448 df-cncf 25006 df-limc 25994 df-dv 25995

This theorem is referenced by: (None)

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