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Theorem dvfsumge 25985

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
dvfsumleOLD.m	⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
dvfsumleOLD.a	⊢ (𝜑 → (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴) ∈ ((𝑀[,]𝑁)–cn→ℝ))
dvfsumleOLD.v	⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 𝐵 ∈ 𝑉)
dvfsumleOLD.b	⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)) = (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵))
dvfsumleOLD.c	⊢ (𝑥 = 𝑀 → 𝐴 = 𝐶)
dvfsumleOLD.d	⊢ (𝑥 = 𝑁 → 𝐴 = 𝐷)
dvfsumleOLD.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		dvfsumleOLD.m	. . . 4 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
2		df-neg 11474	. . . . . 6 ⊢ -𝐴 = (0 − 𝐴)
3	2	mpteq2i 5222	. . . . 5 ⊢ (𝑥 ∈ (𝑀[,]𝑁) ↦ -𝐴) = (𝑥 ∈ (𝑀[,]𝑁) ↦ (0 − 𝐴))
4		eqid 2736	. . . . . 6 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
5	4	subcn 24811	. . . . . 6 ⊢ − ∈ (((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld)) Cn (TopOpen‘ℂ_fld))
6		0red 11243	. . . . . . 7 ⊢ (𝜑 → 0 ∈ ℝ)
7		eluzel2 12862	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑀 ∈ ℤ)
8	1, 7	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℤ)
9	8	zred 12702	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℝ)
10		eluzelz 12867	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑁 ∈ ℤ)
11	1, 10	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℤ)
12	11	zred 12702	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℝ)
13		iccssre 13451	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝑀[,]𝑁) ⊆ ℝ)
14	9, 12, 13	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → (𝑀[,]𝑁) ⊆ ℝ)
15		ax-resscn 11191	. . . . . . . 8 ⊢ ℝ ⊆ ℂ
16	14, 15	sstrdi 3976	. . . . . . 7 ⊢ (𝜑 → (𝑀[,]𝑁) ⊆ ℂ)
17	15	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℝ ⊆ ℂ)
18		cncfmptc 24861	. . . . . . 7 ⊢ ((0 ∈ ℝ ∧ (𝑀[,]𝑁) ⊆ ℂ ∧ ℝ ⊆ ℂ) → (𝑥 ∈ (𝑀[,]𝑁) ↦ 0) ∈ ((𝑀[,]𝑁)–cn→ℝ))
19	6, 16, 17, 18	syl3anc 1373	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (𝑀[,]𝑁) ↦ 0) ∈ ((𝑀[,]𝑁)–cn→ℝ))
20		dvfsumleOLD.a	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴) ∈ ((𝑀[,]𝑁)–cn→ℝ))
21		resubcl 11552	. . . . . 6 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (0 − 𝐴) ∈ ℝ)
22	4, 5, 19, 20, 15, 21	cncfmpt2ss 24865	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (𝑀[,]𝑁) ↦ (0 − 𝐴)) ∈ ((𝑀[,]𝑁)–cn→ℝ))
23	3, 22	eqeltrid 2839	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (𝑀[,]𝑁) ↦ -𝐴) ∈ ((𝑀[,]𝑁)–cn→ℝ))
24		negex 11485	. . . . 5 ⊢ -𝐵 ∈ V
25	24	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀(,)𝑁)) → -𝐵 ∈ V)
26		reelprrecn 11226	. . . . . 6 ⊢ ℝ ∈ {ℝ, ℂ}
27	26	a1i 11	. . . . 5 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
28		ioossicc 13455	. . . . . . . 8 ⊢ (𝑀(,)𝑁) ⊆ (𝑀[,]𝑁)
29	28	sseli 3959	. . . . . . 7 ⊢ (𝑥 ∈ (𝑀(,)𝑁) → 𝑥 ∈ (𝑀[,]𝑁))
30		cncff 24842	. . . . . . . . 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	fvmptelcdm 7108	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → 𝐴 ∈ ℝ)
33	29, 32	sylan2 593	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 𝐴 ∈ ℝ)
34	33	recnd 11268	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 𝐴 ∈ ℂ)
35		dvfsumleOLD.v	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 𝐵 ∈ 𝑉)
36		dvfsumleOLD.b	. . . . 5 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)) = (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵))
37	27, 34, 35, 36	dvmptneg 25927	. . . 4 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ -𝐴)) = (𝑥 ∈ (𝑀(,)𝑁) ↦ -𝐵))
38		dvfsumleOLD.c	. . . . 5 ⊢ (𝑥 = 𝑀 → 𝐴 = 𝐶)
39	38	negeqd 11481	. . . 4 ⊢ (𝑥 = 𝑀 → -𝐴 = -𝐶)
40		dvfsumleOLD.d	. . . . 5 ⊢ (𝑥 = 𝑁 → 𝐴 = 𝐷)
41	40	negeqd 11481	. . . 4 ⊢ (𝑥 = 𝑁 → -𝐴 = -𝐷)
42		dvfsumleOLD.x	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑋 ∈ ℝ)
43	42	renegcld 11669	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → -𝑋 ∈ ℝ)
44		dvfsumge.l	. . . . 5 ⊢ ((𝜑 ∧ (𝑘 ∈ (𝑀..^𝑁) ∧ 𝑥 ∈ (𝑘(,)(𝑘 + 1)))) → 𝐵 ≤ 𝑋)
45	9	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑀 ∈ ℝ)
46	45	rexrd 11290	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑀 ∈ ℝ^*)
47		elfzole1 13689	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (𝑀..^𝑁) → 𝑀 ≤ 𝑘)
48	47	adantl 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑀 ≤ 𝑘)
49		iooss1 13402	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℝ^* ∧ 𝑀 ≤ 𝑘) → (𝑘(,)(𝑘 + 1)) ⊆ (𝑀(,)(𝑘 + 1)))
50	46, 48, 49	syl2anc 584	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘(,)(𝑘 + 1)) ⊆ (𝑀(,)(𝑘 + 1)))
51	12	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑁 ∈ ℝ)
52	51	rexrd 11290	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑁 ∈ ℝ^*)
53		fzofzp1 13785	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (𝑀..^𝑁) → (𝑘 + 1) ∈ (𝑀...𝑁))
54	53	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘 + 1) ∈ (𝑀...𝑁))
55		elfzle2 13550	. . . . . . . . . . . 12 ⊢ ((𝑘 + 1) ∈ (𝑀...𝑁) → (𝑘 + 1) ≤ 𝑁)
56	54, 55	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘 + 1) ≤ 𝑁)
57		iooss2 13403	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℝ^* ∧ (𝑘 + 1) ≤ 𝑁) → (𝑀(,)(𝑘 + 1)) ⊆ (𝑀(,)𝑁))
58	52, 56, 57	syl2anc 584	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑀(,)(𝑘 + 1)) ⊆ (𝑀(,)𝑁))
59	50, 58	sstrd 3974	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘(,)(𝑘 + 1)) ⊆ (𝑀(,)𝑁))
60	59	sselda 3963	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑘(,)(𝑘 + 1))) → 𝑥 ∈ (𝑀(,)𝑁))
61	32	adantlr 715	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀[,]𝑁)) → 𝐴 ∈ ℝ)
62	29, 61	sylan2 593	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 𝐴 ∈ ℝ)
63	62	fmpttd 7110	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴):(𝑀(,)𝑁)⟶ℝ)
64		ioossre 13429	. . . . . . . . . . 11 ⊢ (𝑀(,)𝑁) ⊆ ℝ
65		dvfre 25912	. . . . . . . . . . 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 586	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)):dom (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴))⟶ℝ)
67	36	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)) = (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵))
68	67	dmeqd 5890	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → dom (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)) = dom (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵))
69	35	adantlr 715	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 𝐵 ∈ 𝑉)
70	69	ralrimiva 3133	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ∀𝑥 ∈ (𝑀(,)𝑁)𝐵 ∈ 𝑉)
71		dmmptg 6236	. . . . . . . . . . . . 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 2771	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → dom (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)) = (𝑀(,)𝑁))
74	67, 73	feq12d 6699	. . . . . . . . . 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 232	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵):(𝑀(,)𝑁)⟶ℝ)
76	75	fvmptelcdm 7108	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 𝐵 ∈ ℝ)
77	60, 76	syldan 591	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑘(,)(𝑘 + 1))) → 𝐵 ∈ ℝ)
78	77	anasss 466	. . . . . 6 ⊢ ((𝜑 ∧ (𝑘 ∈ (𝑀..^𝑁) ∧ 𝑥 ∈ (𝑘(,)(𝑘 + 1)))) → 𝐵 ∈ ℝ)
79	42	adantrr 717	. . . . . 6 ⊢ ((𝜑 ∧ (𝑘 ∈ (𝑀..^𝑁) ∧ 𝑥 ∈ (𝑘(,)(𝑘 + 1)))) → 𝑋 ∈ ℝ)
80	78, 79	lenegd 11821	. . . . 5 ⊢ ((𝜑 ∧ (𝑘 ∈ (𝑀..^𝑁) ∧ 𝑥 ∈ (𝑘(,)(𝑘 + 1)))) → (𝐵 ≤ 𝑋 ↔ -𝑋 ≤ -𝐵))
81	44, 80	mpbid 232	. . . 4 ⊢ ((𝜑 ∧ (𝑘 ∈ (𝑀..^𝑁) ∧ 𝑥 ∈ (𝑘(,)(𝑘 + 1)))) → -𝑋 ≤ -𝐵)
82	1, 23, 25, 37, 39, 41, 43, 81	dvfsumle 25983	. . 3 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀..^𝑁)-𝑋 ≤ (-𝐷 − -𝐶))
83		fzofi 13997	. . . . 5 ⊢ (𝑀..^𝑁) ∈ Fin
84	83	a1i 11	. . . 4 ⊢ (𝜑 → (𝑀..^𝑁) ∈ Fin)
85	42	recnd 11268	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑋 ∈ ℂ)
86	84, 85	fsumneg 15808	. . 3 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀..^𝑁)-𝑋 = -Σ𝑘 ∈ (𝑀..^𝑁)𝑋)
87	40	eleq1d 2820	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (𝐴 ∈ ℝ ↔ 𝐷 ∈ ℝ))
88		eqid 2736	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴) = (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴)
89	88	fmpt 7105	. . . . . . . 8 ⊢ (∀𝑥 ∈ (𝑀[,]𝑁)𝐴 ∈ ℝ ↔ (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴):(𝑀[,]𝑁)⟶ℝ)
90	31, 89	sylibr 234	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ (𝑀[,]𝑁)𝐴 ∈ ℝ)
91	9	rexrd 11290	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℝ^*)
92	12	rexrd 11290	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℝ^*)
93		eluzle 12870	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑀 ≤ 𝑁)
94	1, 93	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ≤ 𝑁)
95		ubicc2 13487	. . . . . . . 8 ⊢ ((𝑀 ∈ ℝ^* ∧ 𝑁 ∈ ℝ^* ∧ 𝑀 ≤ 𝑁) → 𝑁 ∈ (𝑀[,]𝑁))
96	91, 92, 94, 95	syl3anc 1373	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ (𝑀[,]𝑁))
97	87, 90, 96	rspcdva 3607	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ ℝ)
98	97	recnd 11268	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ ℂ)
99	38	eleq1d 2820	. . . . . . 7 ⊢ (𝑥 = 𝑀 → (𝐴 ∈ ℝ ↔ 𝐶 ∈ ℝ))
100		lbicc2 13486	. . . . . . . 8 ⊢ ((𝑀 ∈ ℝ^* ∧ 𝑁 ∈ ℝ^* ∧ 𝑀 ≤ 𝑁) → 𝑀 ∈ (𝑀[,]𝑁))
101	91, 92, 94, 100	syl3anc 1373	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ (𝑀[,]𝑁))
102	99, 90, 101	rspcdva 3607	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ ℝ)
103	102	recnd 11268	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℂ)
104	98, 103	neg2subd 11616	. . . 4 ⊢ (𝜑 → (-𝐷 − -𝐶) = (𝐶 − 𝐷))
105	98, 103	negsubdi2d 11615	. . . 4 ⊢ (𝜑 → -(𝐷 − 𝐶) = (𝐶 − 𝐷))
106	104, 105	eqtr4d 2774	. . 3 ⊢ (𝜑 → (-𝐷 − -𝐶) = -(𝐷 − 𝐶))
107	82, 86, 106	3brtr3d 5155	. 2 ⊢ (𝜑 → -Σ𝑘 ∈ (𝑀..^𝑁)𝑋 ≤ -(𝐷 − 𝐶))
108	97, 102	resubcld 11670	. . 3 ⊢ (𝜑 → (𝐷 − 𝐶) ∈ ℝ)
109	84, 42	fsumrecl 15755	. . 3 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀..^𝑁)𝑋 ∈ ℝ)
110	108, 109	lenegd 11821	. 2 ⊢ (𝜑 → ((𝐷 − 𝐶) ≤ Σ𝑘 ∈ (𝑀..^𝑁)𝑋 ↔ -Σ𝑘 ∈ (𝑀..^𝑁)𝑋 ≤ -(𝐷 − 𝐶)))
111	107, 110	mpbird 257	1 ⊢ (𝜑 → (𝐷 − 𝐶) ≤ Σ𝑘 ∈ (𝑀..^𝑁)𝑋)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ∀wral 3052 Vcvv 3464 ⊆ wss 3931 {cpr 4608 class class class wbr 5124 ↦ cmpt 5206 dom cdm 5659 ⟶wf 6532 ‘cfv 6536 (class class class)co 7410 Fincfn 8964 ℂcc 11132 ℝcr 11133 0cc0 11134 1c1 11135 + caddc 11137 ℝ^*cxr 11273 ≤ cle 11275 − cmin 11471 -cneg 11472 ℤcz 12593 ℤ_≥cuz 12857 (,)cioo 13367 [,]cicc 13370 ...cfz 13529 ..^cfzo 13676 Σcsu 15707 TopOpenctopn 17440 ℂ_fldccnfld 21320 –cn→ccncf 24825 D cdv 25821

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2708 ax-rep 5254 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-inf2 9660 ax-cnex 11190 ax-resscn 11191 ax-1cn 11192 ax-icn 11193 ax-addcl 11194 ax-addrcl 11195 ax-mulcl 11196 ax-mulrcl 11197 ax-mulcom 11198 ax-addass 11199 ax-mulass 11200 ax-distr 11201 ax-i2m1 11202 ax-1ne0 11203 ax-1rid 11204 ax-rnegex 11205 ax-rrecex 11206 ax-cnre 11207 ax-pre-lttri 11208 ax-pre-lttrn 11209 ax-pre-ltadd 11210 ax-pre-mulgt0 11211 ax-pre-sup 11212 ax-addf 11213

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3364 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-tp 4611 df-op 4613 df-uni 4889 df-int 4928 df-iun 4974 df-iin 4975 df-br 5125 df-opab 5187 df-mpt 5207 df-tr 5235 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-se 5612 df-we 5613 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6295 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-of 7676 df-om 7867 df-1st 7993 df-2nd 7994 df-supp 8165 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-2o 8486 df-er 8724 df-map 8847 df-pm 8848 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9379 df-fi 9428 df-sup 9459 df-inf 9460 df-oi 9529 df-card 9958 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280 df-sub 11473 df-neg 11474 df-div 11900 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12507 df-z 12594 df-dec 12714 df-uz 12858 df-q 12970 df-rp 13014 df-xneg 13133 df-xadd 13134 df-xmul 13135 df-ioo 13371 df-ico 13373 df-icc 13374 df-fz 13530 df-fzo 13677 df-seq 14025 df-exp 14085 df-hash 14354 df-cj 15123 df-re 15124 df-im 15125 df-sqrt 15259 df-abs 15260 df-clim 15509 df-sum 15708 df-struct 17171 df-sets 17188 df-slot 17206 df-ndx 17218 df-base 17234 df-ress 17257 df-plusg 17289 df-mulr 17290 df-starv 17291 df-sca 17292 df-vsca 17293 df-ip 17294 df-tset 17295 df-ple 17296 df-ds 17298 df-unif 17299 df-hom 17300 df-cco 17301 df-rest 17441 df-topn 17442 df-0g 17460 df-gsum 17461 df-topgen 17462 df-pt 17463 df-prds 17466 df-xrs 17521 df-qtop 17526 df-imas 17527 df-xps 17529 df-mre 17603 df-mrc 17604 df-acs 17606 df-mgm 18623 df-sgrp 18702 df-mnd 18718 df-submnd 18767 df-mulg 19056 df-cntz 19305 df-cmn 19768 df-psmet 21312 df-xmet 21313 df-met 21314 df-bl 21315 df-mopn 21316 df-fbas 21317 df-fg 21318 df-cnfld 21321 df-top 22837 df-topon 22854 df-topsp 22876 df-bases 22889 df-cld 22962 df-ntr 22963 df-cls 22964 df-nei 23041 df-lp 23079 df-perf 23080 df-cn 23170 df-cnp 23171 df-haus 23258 df-cmp 23330 df-tx 23505 df-hmeo 23698 df-fil 23789 df-fm 23881 df-flim 23882 df-flf 23883 df-xms 24264 df-ms 24265 df-tms 24266 df-cncf 24827 df-limc 25824 df-dv 25825

This theorem is referenced by: (None)

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