dvfsumge - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > dvfsumge		Structured version Visualization version GIF version

Theorem dvfsumge 24625

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)
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		dvfsumle.m	. . . 4 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
2		df-neg 10862	. . . . . 6 ⊢ -𝐴 = (0 − 𝐴)
3	2	mpteq2i 5122	. . . . 5 ⊢ (𝑥 ∈ (𝑀[,]𝑁) ↦ -𝐴) = (𝑥 ∈ (𝑀[,]𝑁) ↦ (0 − 𝐴))
4		eqid 2798	. . . . . 6 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
5	4	subcn 23471	. . . . . 6 ⊢ − ∈ (((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld)) Cn (TopOpen‘ℂ_fld))
6		0red 10633	. . . . . . 7 ⊢ (𝜑 → 0 ∈ ℝ)
7		eluzel2 12236	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑀 ∈ ℤ)
8	1, 7	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℤ)
9	8	zred 12075	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℝ)
10		eluzelz 12241	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑁 ∈ ℤ)
11	1, 10	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℤ)
12	11	zred 12075	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℝ)
13		iccssre 12807	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝑀[,]𝑁) ⊆ ℝ)
14	9, 12, 13	syl2anc 587	. . . . . . . 8 ⊢ (𝜑 → (𝑀[,]𝑁) ⊆ ℝ)
15		ax-resscn 10583	. . . . . . . 8 ⊢ ℝ ⊆ ℂ
16	14, 15	sstrdi 3927	. . . . . . 7 ⊢ (𝜑 → (𝑀[,]𝑁) ⊆ ℂ)
17	15	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℝ ⊆ ℂ)
18		cncfmptc 23517	. . . . . . 7 ⊢ ((0 ∈ ℝ ∧ (𝑀[,]𝑁) ⊆ ℂ ∧ ℝ ⊆ ℂ) → (𝑥 ∈ (𝑀[,]𝑁) ↦ 0) ∈ ((𝑀[,]𝑁)–cn→ℝ))
19	6, 16, 17, 18	syl3anc 1368	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (𝑀[,]𝑁) ↦ 0) ∈ ((𝑀[,]𝑁)–cn→ℝ))
20		dvfsumle.a	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴) ∈ ((𝑀[,]𝑁)–cn→ℝ))
21		resubcl 10939	. . . . . 6 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (0 − 𝐴) ∈ ℝ)
22	4, 5, 19, 20, 15, 21	cncfmpt2ss 23521	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (𝑀[,]𝑁) ↦ (0 − 𝐴)) ∈ ((𝑀[,]𝑁)–cn→ℝ))
23	3, 22	eqeltrid 2894	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (𝑀[,]𝑁) ↦ -𝐴) ∈ ((𝑀[,]𝑁)–cn→ℝ))
24		negex 10873	. . . . 5 ⊢ -𝐵 ∈ V
25	24	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀(,)𝑁)) → -𝐵 ∈ V)
26		reelprrecn 10618	. . . . . 6 ⊢ ℝ ∈ {ℝ, ℂ}
27	26	a1i 11	. . . . 5 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
28		ioossicc 12811	. . . . . . . 8 ⊢ (𝑀(,)𝑁) ⊆ (𝑀[,]𝑁)
29	28	sseli 3911	. . . . . . 7 ⊢ (𝑥 ∈ (𝑀(,)𝑁) → 𝑥 ∈ (𝑀[,]𝑁))
30		cncff 23498	. . . . . . . . 9 ⊢ ((𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴) ∈ ((𝑀[,]𝑁)–cn→ℝ) → (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴):(𝑀[,]𝑁)⟶ℝ)
31	20, 30	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴):(𝑀[,]𝑁)⟶ℝ)
32	31	fvmptelrn 6854	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → 𝐴 ∈ ℝ)
33	29, 32	sylan2 595	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 𝐴 ∈ ℝ)
34	33	recnd 10658	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 𝐴 ∈ ℂ)
35		dvfsumle.v	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 𝐵 ∈ 𝑉)
36		dvfsumle.b	. . . . 5 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)) = (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵))
37	27, 34, 35, 36	dvmptneg 24569	. . . 4 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ -𝐴)) = (𝑥 ∈ (𝑀(,)𝑁) ↦ -𝐵))
38		dvfsumle.c	. . . . 5 ⊢ (𝑥 = 𝑀 → 𝐴 = 𝐶)
39	38	negeqd 10869	. . . 4 ⊢ (𝑥 = 𝑀 → -𝐴 = -𝐶)
40		dvfsumle.d	. . . . 5 ⊢ (𝑥 = 𝑁 → 𝐴 = 𝐷)
41	40	negeqd 10869	. . . 4 ⊢ (𝑥 = 𝑁 → -𝐴 = -𝐷)
42		dvfsumle.x	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑋 ∈ ℝ)
43	42	renegcld 11056	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → -𝑋 ∈ ℝ)
44		dvfsumge.l	. . . . 5 ⊢ ((𝜑 ∧ (𝑘 ∈ (𝑀..^𝑁) ∧ 𝑥 ∈ (𝑘(,)(𝑘 + 1)))) → 𝐵 ≤ 𝑋)
45	9	adantr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑀 ∈ ℝ)
46	45	rexrd 10680	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑀 ∈ ℝ^*)
47		elfzole1 13041	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (𝑀..^𝑁) → 𝑀 ≤ 𝑘)
48	47	adantl 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑀 ≤ 𝑘)
49		iooss1 12761	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℝ^* ∧ 𝑀 ≤ 𝑘) → (𝑘(,)(𝑘 + 1)) ⊆ (𝑀(,)(𝑘 + 1)))
50	46, 48, 49	syl2anc 587	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘(,)(𝑘 + 1)) ⊆ (𝑀(,)(𝑘 + 1)))
51	12	adantr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑁 ∈ ℝ)
52	51	rexrd 10680	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑁 ∈ ℝ^*)
53		fzofzp1 13129	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (𝑀..^𝑁) → (𝑘 + 1) ∈ (𝑀...𝑁))
54	53	adantl 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘 + 1) ∈ (𝑀...𝑁))
55		elfzle2 12906	. . . . . . . . . . . 12 ⊢ ((𝑘 + 1) ∈ (𝑀...𝑁) → (𝑘 + 1) ≤ 𝑁)
56	54, 55	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘 + 1) ≤ 𝑁)
57		iooss2 12762	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℝ^* ∧ (𝑘 + 1) ≤ 𝑁) → (𝑀(,)(𝑘 + 1)) ⊆ (𝑀(,)𝑁))
58	52, 56, 57	syl2anc 587	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑀(,)(𝑘 + 1)) ⊆ (𝑀(,)𝑁))
59	50, 58	sstrd 3925	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘(,)(𝑘 + 1)) ⊆ (𝑀(,)𝑁))
60	59	sselda 3915	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑘(,)(𝑘 + 1))) → 𝑥 ∈ (𝑀(,)𝑁))
61	32	adantlr 714	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀[,]𝑁)) → 𝐴 ∈ ℝ)
62	29, 61	sylan2 595	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 𝐴 ∈ ℝ)
63	62	fmpttd 6856	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴):(𝑀(,)𝑁)⟶ℝ)
64		ioossre 12786	. . . . . . . . . . 11 ⊢ (𝑀(,)𝑁) ⊆ ℝ
65		dvfre 24554	. . . . . . . . . . 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 589	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)):dom (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴))⟶ℝ)
67	36	adantr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)) = (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵))
68	67	dmeqd 5738	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → dom (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)) = dom (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵))
69	35	adantlr 714	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 𝐵 ∈ 𝑉)
70	69	ralrimiva 3149	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ∀𝑥 ∈ (𝑀(,)𝑁)𝐵 ∈ 𝑉)
71		dmmptg 6063	. . . . . . . . . . . . 13 ⊢ (∀𝑥 ∈ (𝑀(,)𝑁)𝐵 ∈ 𝑉 → dom (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵) = (𝑀(,)𝑁))
72	70, 71	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → dom (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵) = (𝑀(,)𝑁))
73	68, 72	eqtrd 2833	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → dom (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)) = (𝑀(,)𝑁))
74	67, 73	feq12d 6475	. . . . . . . . . 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	fvmptelrn 6854	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 𝐵 ∈ ℝ)
77	60, 76	syldan 594	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑘(,)(𝑘 + 1))) → 𝐵 ∈ ℝ)
78	77	anasss 470	. . . . . 6 ⊢ ((𝜑 ∧ (𝑘 ∈ (𝑀..^𝑁) ∧ 𝑥 ∈ (𝑘(,)(𝑘 + 1)))) → 𝐵 ∈ ℝ)
79	42	adantrr 716	. . . . . 6 ⊢ ((𝜑 ∧ (𝑘 ∈ (𝑀..^𝑁) ∧ 𝑥 ∈ (𝑘(,)(𝑘 + 1)))) → 𝑋 ∈ ℝ)
80	78, 79	lenegd 11208	. . . . 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 24624	. . 3 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀..^𝑁)-𝑋 ≤ (-𝐷 − -𝐶))
83		fzofi 13337	. . . . 5 ⊢ (𝑀..^𝑁) ∈ Fin
84	83	a1i 11	. . . 4 ⊢ (𝜑 → (𝑀..^𝑁) ∈ Fin)
85	42	recnd 10658	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑋 ∈ ℂ)
86	84, 85	fsumneg 15134	. . 3 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀..^𝑁)-𝑋 = -Σ𝑘 ∈ (𝑀..^𝑁)𝑋)
87	40	eleq1d 2874	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (𝐴 ∈ ℝ ↔ 𝐷 ∈ ℝ))
88		eqid 2798	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴) = (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴)
89	88	fmpt 6851	. . . . . . . 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 10680	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℝ^*)
92	12	rexrd 10680	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℝ^*)
93		eluzle 12244	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑀 ≤ 𝑁)
94	1, 93	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ≤ 𝑁)
95		ubicc2 12843	. . . . . . . 8 ⊢ ((𝑀 ∈ ℝ^* ∧ 𝑁 ∈ ℝ^* ∧ 𝑀 ≤ 𝑁) → 𝑁 ∈ (𝑀[,]𝑁))
96	91, 92, 94, 95	syl3anc 1368	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ (𝑀[,]𝑁))
97	87, 90, 96	rspcdva 3573	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ ℝ)
98	97	recnd 10658	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ ℂ)
99	38	eleq1d 2874	. . . . . . 7 ⊢ (𝑥 = 𝑀 → (𝐴 ∈ ℝ ↔ 𝐶 ∈ ℝ))
100		lbicc2 12842	. . . . . . . 8 ⊢ ((𝑀 ∈ ℝ^* ∧ 𝑁 ∈ ℝ^* ∧ 𝑀 ≤ 𝑁) → 𝑀 ∈ (𝑀[,]𝑁))
101	91, 92, 94, 100	syl3anc 1368	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ (𝑀[,]𝑁))
102	99, 90, 101	rspcdva 3573	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ ℝ)
103	102	recnd 10658	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℂ)
104	98, 103	neg2subd 11003	. . . 4 ⊢ (𝜑 → (-𝐷 − -𝐶) = (𝐶 − 𝐷))
105	98, 103	negsubdi2d 11002	. . . 4 ⊢ (𝜑 → -(𝐷 − 𝐶) = (𝐶 − 𝐷))
106	104, 105	eqtr4d 2836	. . 3 ⊢ (𝜑 → (-𝐷 − -𝐶) = -(𝐷 − 𝐶))
107	82, 86, 106	3brtr3d 5061	. 2 ⊢ (𝜑 → -Σ𝑘 ∈ (𝑀..^𝑁)𝑋 ≤ -(𝐷 − 𝐶))
108	97, 102	resubcld 11057	. . 3 ⊢ (𝜑 → (𝐷 − 𝐶) ∈ ℝ)
109	84, 42	fsumrecl 15083	. . 3 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀..^𝑁)𝑋 ∈ ℝ)
110	108, 109	lenegd 11208	. 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 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 Vcvv 3441 ⊆ wss 3881 {cpr 4527 class class class wbr 5030 ↦ cmpt 5110 dom cdm 5519 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 Fincfn 8492 ℂcc 10524 ℝcr 10525 0cc0 10526 1c1 10527 + caddc 10529 ℝ^*cxr 10663 ≤ cle 10665 − cmin 10859 -cneg 10860 ℤcz 11969 ℤ_≥cuz 12231 (,)cioo 12726 [,]cicc 12729 ...cfz 12885 ..^cfzo 13028 Σcsu 15034 TopOpenctopn 16687 ℂ_fldccnfld 20091 –cn→ccncf 23481 D cdv 24466

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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 ax-mulf 10606

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-pm 8392 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-fi 8859 df-sup 8890 df-inf 8891 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ioo 12730 df-ico 12732 df-icc 12733 df-fz 12886 df-fzo 13029 df-seq 13365 df-exp 13426 df-hash 13687 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-clim 14837 df-sum 15035 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-hom 16581 df-cco 16582 df-rest 16688 df-topn 16689 df-0g 16707 df-gsum 16708 df-topgen 16709 df-pt 16710 df-prds 16713 df-xrs 16767 df-qtop 16772 df-imas 16773 df-xps 16775 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-mulg 18217 df-cntz 18439 df-cmn 18900 df-psmet 20083 df-xmet 20084 df-met 20085 df-bl 20086 df-mopn 20087 df-fbas 20088 df-fg 20089 df-cnfld 20092 df-top 21499 df-topon 21516 df-topsp 21538 df-bases 21551 df-cld 21624 df-ntr 21625 df-cls 21626 df-nei 21703 df-lp 21741 df-perf 21742 df-cn 21832 df-cnp 21833 df-haus 21920 df-cmp 21992 df-tx 22167 df-hmeo 22360 df-fil 22451 df-fm 22543 df-flim 22544 df-flf 22545 df-xms 22927 df-ms 22928 df-tms 22929 df-cncf 23483 df-limc 24469 df-dv 24470

This theorem is referenced by: (None)

W3C validator