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Theorem dvfsumge 26077

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 11493	. . . . . 6 ⊢ -𝐴 = (0 − 𝐴)
3	2	mpteq2i 5253	. . . . 5 ⊢ (𝑥 ∈ (𝑀[,]𝑁) ↦ -𝐴) = (𝑥 ∈ (𝑀[,]𝑁) ↦ (0 − 𝐴))
4		eqid 2735	. . . . . 6 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
5	4	subcn 24902	. . . . . 6 ⊢ − ∈ (((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld)) Cn (TopOpen‘ℂ_fld))
6		0red 11262	. . . . . . 7 ⊢ (𝜑 → 0 ∈ ℝ)
7		eluzel2 12881	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑀 ∈ ℤ)
8	1, 7	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℤ)
9	8	zred 12720	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℝ)
10		eluzelz 12886	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑁 ∈ ℤ)
11	1, 10	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℤ)
12	11	zred 12720	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℝ)
13		iccssre 13466	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝑀[,]𝑁) ⊆ ℝ)
14	9, 12, 13	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → (𝑀[,]𝑁) ⊆ ℝ)
15		ax-resscn 11210	. . . . . . . 8 ⊢ ℝ ⊆ ℂ
16	14, 15	sstrdi 4008	. . . . . . 7 ⊢ (𝜑 → (𝑀[,]𝑁) ⊆ ℂ)
17	15	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℝ ⊆ ℂ)
18		cncfmptc 24952	. . . . . . 7 ⊢ ((0 ∈ ℝ ∧ (𝑀[,]𝑁) ⊆ ℂ ∧ ℝ ⊆ ℂ) → (𝑥 ∈ (𝑀[,]𝑁) ↦ 0) ∈ ((𝑀[,]𝑁)–cn→ℝ))
19	6, 16, 17, 18	syl3anc 1370	. . . . . 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 11571	. . . . . 6 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (0 − 𝐴) ∈ ℝ)
22	4, 5, 19, 20, 15, 21	cncfmpt2ss 24956	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (𝑀[,]𝑁) ↦ (0 − 𝐴)) ∈ ((𝑀[,]𝑁)–cn→ℝ))
23	3, 22	eqeltrid 2843	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (𝑀[,]𝑁) ↦ -𝐴) ∈ ((𝑀[,]𝑁)–cn→ℝ))
24		negex 11504	. . . . 5 ⊢ -𝐵 ∈ V
25	24	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀(,)𝑁)) → -𝐵 ∈ V)
26		reelprrecn 11245	. . . . . 6 ⊢ ℝ ∈ {ℝ, ℂ}
27	26	a1i 11	. . . . 5 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
28		ioossicc 13470	. . . . . . . 8 ⊢ (𝑀(,)𝑁) ⊆ (𝑀[,]𝑁)
29	28	sseli 3991	. . . . . . 7 ⊢ (𝑥 ∈ (𝑀(,)𝑁) → 𝑥 ∈ (𝑀[,]𝑁))
30		cncff 24933	. . . . . . . . 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 7133	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → 𝐴 ∈ ℝ)
33	29, 32	sylan2 593	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 𝐴 ∈ ℝ)
34	33	recnd 11287	. . . . 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 26019	. . . 4 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ -𝐴)) = (𝑥 ∈ (𝑀(,)𝑁) ↦ -𝐵))
38		dvfsumleOLD.c	. . . . 5 ⊢ (𝑥 = 𝑀 → 𝐴 = 𝐶)
39	38	negeqd 11500	. . . 4 ⊢ (𝑥 = 𝑀 → -𝐴 = -𝐶)
40		dvfsumleOLD.d	. . . . 5 ⊢ (𝑥 = 𝑁 → 𝐴 = 𝐷)
41	40	negeqd 11500	. . . 4 ⊢ (𝑥 = 𝑁 → -𝐴 = -𝐷)
42		dvfsumleOLD.x	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑋 ∈ ℝ)
43	42	renegcld 11688	. . . 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 11309	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑀 ∈ ℝ^*)
47		elfzole1 13704	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (𝑀..^𝑁) → 𝑀 ≤ 𝑘)
48	47	adantl 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑀 ≤ 𝑘)
49		iooss1 13419	. . . . . . . . . . 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 11309	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑁 ∈ ℝ^*)
53		fzofzp1 13800	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (𝑀..^𝑁) → (𝑘 + 1) ∈ (𝑀...𝑁))
54	53	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘 + 1) ∈ (𝑀...𝑁))
55		elfzle2 13565	. . . . . . . . . . . 12 ⊢ ((𝑘 + 1) ∈ (𝑀...𝑁) → (𝑘 + 1) ≤ 𝑁)
56	54, 55	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘 + 1) ≤ 𝑁)
57		iooss2 13420	. . . . . . . . . . 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 4006	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘(,)(𝑘 + 1)) ⊆ (𝑀(,)𝑁))
60	59	sselda 3995	. . . . . . . 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 7135	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴):(𝑀(,)𝑁)⟶ℝ)
64		ioossre 13445	. . . . . . . . . . 11 ⊢ (𝑀(,)𝑁) ⊆ ℝ
65		dvfre 26004	. . . . . . . . . . 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 5919	. . . . . . . . . . . 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 3144	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ∀𝑥 ∈ (𝑀(,)𝑁)𝐵 ∈ 𝑉)
71		dmmptg 6264	. . . . . . . . . . . . 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 2775	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → dom (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)) = (𝑀(,)𝑁))
74	67, 73	feq12d 6725	. . . . . . . . . 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 7133	. . . . . . . 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 11840	. . . . 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 26075	. . 3 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀..^𝑁)-𝑋 ≤ (-𝐷 − -𝐶))
83		fzofi 14012	. . . . 5 ⊢ (𝑀..^𝑁) ∈ Fin
84	83	a1i 11	. . . 4 ⊢ (𝜑 → (𝑀..^𝑁) ∈ Fin)
85	42	recnd 11287	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑋 ∈ ℂ)
86	84, 85	fsumneg 15820	. . 3 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀..^𝑁)-𝑋 = -Σ𝑘 ∈ (𝑀..^𝑁)𝑋)
87	40	eleq1d 2824	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (𝐴 ∈ ℝ ↔ 𝐷 ∈ ℝ))
88		eqid 2735	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴) = (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴)
89	88	fmpt 7130	. . . . . . . 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 11309	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℝ^*)
92	12	rexrd 11309	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℝ^*)
93		eluzle 12889	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑀 ≤ 𝑁)
94	1, 93	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ≤ 𝑁)
95		ubicc2 13502	. . . . . . . 8 ⊢ ((𝑀 ∈ ℝ^* ∧ 𝑁 ∈ ℝ^* ∧ 𝑀 ≤ 𝑁) → 𝑁 ∈ (𝑀[,]𝑁))
96	91, 92, 94, 95	syl3anc 1370	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ (𝑀[,]𝑁))
97	87, 90, 96	rspcdva 3623	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ ℝ)
98	97	recnd 11287	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ ℂ)
99	38	eleq1d 2824	. . . . . . 7 ⊢ (𝑥 = 𝑀 → (𝐴 ∈ ℝ ↔ 𝐶 ∈ ℝ))
100		lbicc2 13501	. . . . . . . 8 ⊢ ((𝑀 ∈ ℝ^* ∧ 𝑁 ∈ ℝ^* ∧ 𝑀 ≤ 𝑁) → 𝑀 ∈ (𝑀[,]𝑁))
101	91, 92, 94, 100	syl3anc 1370	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ (𝑀[,]𝑁))
102	99, 90, 101	rspcdva 3623	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ ℝ)
103	102	recnd 11287	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℂ)
104	98, 103	neg2subd 11635	. . . 4 ⊢ (𝜑 → (-𝐷 − -𝐶) = (𝐶 − 𝐷))
105	98, 103	negsubdi2d 11634	. . . 4 ⊢ (𝜑 → -(𝐷 − 𝐶) = (𝐶 − 𝐷))
106	104, 105	eqtr4d 2778	. . 3 ⊢ (𝜑 → (-𝐷 − -𝐶) = -(𝐷 − 𝐶))
107	82, 86, 106	3brtr3d 5179	. 2 ⊢ (𝜑 → -Σ𝑘 ∈ (𝑀..^𝑁)𝑋 ≤ -(𝐷 − 𝐶))
108	97, 102	resubcld 11689	. . 3 ⊢ (𝜑 → (𝐷 − 𝐶) ∈ ℝ)
109	84, 42	fsumrecl 15767	. . 3 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀..^𝑁)𝑋 ∈ ℝ)
110	108, 109	lenegd 11840	. 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 1537 ∈ wcel 2106 ∀wral 3059 Vcvv 3478 ⊆ wss 3963 {cpr 4633 class class class wbr 5148 ↦ cmpt 5231 dom cdm 5689 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 Fincfn 8984 ℂcc 11151 ℝcr 11152 0cc0 11153 1c1 11154 + caddc 11156 ℝ^*cxr 11292 ≤ cle 11294 − cmin 11490 -cneg 11491 ℤcz 12611 ℤ_≥cuz 12876 (,)cioo 13384 [,]cicc 13387 ...cfz 13544 ..^cfzo 13691 Σcsu 15719 TopOpenctopn 17468 ℂ_fldccnfld 21382 –cn→ccncf 24916 D cdv 25913

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-pm 8868 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-fi 9449 df-sup 9480 df-inf 9481 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-q 12989 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-ioo 13388 df-ico 13390 df-icc 13391 df-fz 13545 df-fzo 13692 df-seq 14040 df-exp 14100 df-hash 14367 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-clim 15521 df-sum 15720 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-rest 17469 df-topn 17470 df-0g 17488 df-gsum 17489 df-topgen 17490 df-pt 17491 df-prds 17494 df-xrs 17549 df-qtop 17554 df-imas 17555 df-xps 17557 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-mulg 19099 df-cntz 19348 df-cmn 19815 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-fbas 21379 df-fg 21380 df-cnfld 21383 df-top 22916 df-topon 22933 df-topsp 22955 df-bases 22969 df-cld 23043 df-ntr 23044 df-cls 23045 df-nei 23122 df-lp 23160 df-perf 23161 df-cn 23251 df-cnp 23252 df-haus 23339 df-cmp 23411 df-tx 23586 df-hmeo 23779 df-fil 23870 df-fm 23962 df-flim 23963 df-flf 23964 df-xms 24346 df-ms 24347 df-tms 24348 df-cncf 24918 df-limc 25916 df-dv 25917

This theorem is referenced by: (None)

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