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Theorem dvfsumge 25386

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 11388	. . . . . 6 ⊢ -𝐴 = (0 − 𝐴)
3	2	mpteq2i 5210	. . . . 5 ⊢ (𝑥 ∈ (𝑀[,]𝑁) ↦ -𝐴) = (𝑥 ∈ (𝑀[,]𝑁) ↦ (0 − 𝐴))
4		eqid 2736	. . . . . 6 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
5	4	subcn 24229	. . . . . 6 ⊢ − ∈ (((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld)) Cn (TopOpen‘ℂ_fld))
6		0red 11158	. . . . . . 7 ⊢ (𝜑 → 0 ∈ ℝ)
7		eluzel2 12768	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑀 ∈ ℤ)
8	1, 7	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℤ)
9	8	zred 12607	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℝ)
10		eluzelz 12773	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑁 ∈ ℤ)
11	1, 10	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℤ)
12	11	zred 12607	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℝ)
13		iccssre 13346	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝑀[,]𝑁) ⊆ ℝ)
14	9, 12, 13	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → (𝑀[,]𝑁) ⊆ ℝ)
15		ax-resscn 11108	. . . . . . . 8 ⊢ ℝ ⊆ ℂ
16	14, 15	sstrdi 3956	. . . . . . 7 ⊢ (𝜑 → (𝑀[,]𝑁) ⊆ ℂ)
17	15	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℝ ⊆ ℂ)
18		cncfmptc 24275	. . . . . . 7 ⊢ ((0 ∈ ℝ ∧ (𝑀[,]𝑁) ⊆ ℂ ∧ ℝ ⊆ ℂ) → (𝑥 ∈ (𝑀[,]𝑁) ↦ 0) ∈ ((𝑀[,]𝑁)–cn→ℝ))
19	6, 16, 17, 18	syl3anc 1371	. . . . . 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 11465	. . . . . 6 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (0 − 𝐴) ∈ ℝ)
22	4, 5, 19, 20, 15, 21	cncfmpt2ss 24279	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (𝑀[,]𝑁) ↦ (0 − 𝐴)) ∈ ((𝑀[,]𝑁)–cn→ℝ))
23	3, 22	eqeltrid 2842	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (𝑀[,]𝑁) ↦ -𝐴) ∈ ((𝑀[,]𝑁)–cn→ℝ))
24		negex 11399	. . . . 5 ⊢ -𝐵 ∈ V
25	24	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀(,)𝑁)) → -𝐵 ∈ V)
26		reelprrecn 11143	. . . . . 6 ⊢ ℝ ∈ {ℝ, ℂ}
27	26	a1i 11	. . . . 5 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
28		ioossicc 13350	. . . . . . . 8 ⊢ (𝑀(,)𝑁) ⊆ (𝑀[,]𝑁)
29	28	sseli 3940	. . . . . . 7 ⊢ (𝑥 ∈ (𝑀(,)𝑁) → 𝑥 ∈ (𝑀[,]𝑁))
30		cncff 24256	. . . . . . . . 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 7061	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → 𝐴 ∈ ℝ)
33	29, 32	sylan2 593	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 𝐴 ∈ ℝ)
34	33	recnd 11183	. . . . 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 25330	. . . 4 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ -𝐴)) = (𝑥 ∈ (𝑀(,)𝑁) ↦ -𝐵))
38		dvfsumle.c	. . . . 5 ⊢ (𝑥 = 𝑀 → 𝐴 = 𝐶)
39	38	negeqd 11395	. . . 4 ⊢ (𝑥 = 𝑀 → -𝐴 = -𝐶)
40		dvfsumle.d	. . . . 5 ⊢ (𝑥 = 𝑁 → 𝐴 = 𝐷)
41	40	negeqd 11395	. . . 4 ⊢ (𝑥 = 𝑁 → -𝐴 = -𝐷)
42		dvfsumle.x	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑋 ∈ ℝ)
43	42	renegcld 11582	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → -𝑋 ∈ ℝ)
44		dvfsumge.l	. . . . 5 ⊢ ((𝜑 ∧ (𝑘 ∈ (𝑀..^𝑁) ∧ 𝑥 ∈ (𝑘(,)(𝑘 + 1)))) → 𝐵 ≤ 𝑋)
45	9	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑀 ∈ ℝ)
46	45	rexrd 11205	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑀 ∈ ℝ^*)
47		elfzole1 13580	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (𝑀..^𝑁) → 𝑀 ≤ 𝑘)
48	47	adantl 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑀 ≤ 𝑘)
49		iooss1 13299	. . . . . . . . . . 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 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑁 ∈ ℝ)
52	51	rexrd 11205	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑁 ∈ ℝ^*)
53		fzofzp1 13669	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (𝑀..^𝑁) → (𝑘 + 1) ∈ (𝑀...𝑁))
54	53	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘 + 1) ∈ (𝑀...𝑁))
55		elfzle2 13445	. . . . . . . . . . . 12 ⊢ ((𝑘 + 1) ∈ (𝑀...𝑁) → (𝑘 + 1) ≤ 𝑁)
56	54, 55	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘 + 1) ≤ 𝑁)
57		iooss2 13300	. . . . . . . . . . 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 3954	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘(,)(𝑘 + 1)) ⊆ (𝑀(,)𝑁))
60	59	sselda 3944	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑘(,)(𝑘 + 1))) → 𝑥 ∈ (𝑀(,)𝑁))
61	32	adantlr 713	. . . . . . . . . . . . 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 7063	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴):(𝑀(,)𝑁)⟶ℝ)
64		ioossre 13325	. . . . . . . . . . 11 ⊢ (𝑀(,)𝑁) ⊆ ℝ
65		dvfre 25315	. . . . . . . . . . 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 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)) = (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵))
68	67	dmeqd 5861	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → dom (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)) = dom (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵))
69	35	adantlr 713	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 𝐵 ∈ 𝑉)
70	69	ralrimiva 3143	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ∀𝑥 ∈ (𝑀(,)𝑁)𝐵 ∈ 𝑉)
71		dmmptg 6194	. . . . . . . . . . . . 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 2776	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → dom (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)) = (𝑀(,)𝑁))
74	67, 73	feq12d 6656	. . . . . . . . . 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 231	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵):(𝑀(,)𝑁)⟶ℝ)
76	75	fvmptelcdm 7061	. . . . . . . 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 467	. . . . . 6 ⊢ ((𝜑 ∧ (𝑘 ∈ (𝑀..^𝑁) ∧ 𝑥 ∈ (𝑘(,)(𝑘 + 1)))) → 𝐵 ∈ ℝ)
79	42	adantrr 715	. . . . . 6 ⊢ ((𝜑 ∧ (𝑘 ∈ (𝑀..^𝑁) ∧ 𝑥 ∈ (𝑘(,)(𝑘 + 1)))) → 𝑋 ∈ ℝ)
80	78, 79	lenegd 11734	. . . . 5 ⊢ ((𝜑 ∧ (𝑘 ∈ (𝑀..^𝑁) ∧ 𝑥 ∈ (𝑘(,)(𝑘 + 1)))) → (𝐵 ≤ 𝑋 ↔ -𝑋 ≤ -𝐵))
81	44, 80	mpbid 231	. . . 4 ⊢ ((𝜑 ∧ (𝑘 ∈ (𝑀..^𝑁) ∧ 𝑥 ∈ (𝑘(,)(𝑘 + 1)))) → -𝑋 ≤ -𝐵)
82	1, 23, 25, 37, 39, 41, 43, 81	dvfsumle 25385	. . 3 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀..^𝑁)-𝑋 ≤ (-𝐷 − -𝐶))
83		fzofi 13879	. . . . 5 ⊢ (𝑀..^𝑁) ∈ Fin
84	83	a1i 11	. . . 4 ⊢ (𝜑 → (𝑀..^𝑁) ∈ Fin)
85	42	recnd 11183	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑋 ∈ ℂ)
86	84, 85	fsumneg 15672	. . 3 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀..^𝑁)-𝑋 = -Σ𝑘 ∈ (𝑀..^𝑁)𝑋)
87	40	eleq1d 2822	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (𝐴 ∈ ℝ ↔ 𝐷 ∈ ℝ))
88		eqid 2736	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴) = (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴)
89	88	fmpt 7058	. . . . . . . 8 ⊢ (∀𝑥 ∈ (𝑀[,]𝑁)𝐴 ∈ ℝ ↔ (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴):(𝑀[,]𝑁)⟶ℝ)
90	31, 89	sylibr 233	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ (𝑀[,]𝑁)𝐴 ∈ ℝ)
91	9	rexrd 11205	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℝ^*)
92	12	rexrd 11205	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℝ^*)
93		eluzle 12776	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑀 ≤ 𝑁)
94	1, 93	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ≤ 𝑁)
95		ubicc2 13382	. . . . . . . 8 ⊢ ((𝑀 ∈ ℝ^* ∧ 𝑁 ∈ ℝ^* ∧ 𝑀 ≤ 𝑁) → 𝑁 ∈ (𝑀[,]𝑁))
96	91, 92, 94, 95	syl3anc 1371	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ (𝑀[,]𝑁))
97	87, 90, 96	rspcdva 3582	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ ℝ)
98	97	recnd 11183	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ ℂ)
99	38	eleq1d 2822	. . . . . . 7 ⊢ (𝑥 = 𝑀 → (𝐴 ∈ ℝ ↔ 𝐶 ∈ ℝ))
100		lbicc2 13381	. . . . . . . 8 ⊢ ((𝑀 ∈ ℝ^* ∧ 𝑁 ∈ ℝ^* ∧ 𝑀 ≤ 𝑁) → 𝑀 ∈ (𝑀[,]𝑁))
101	91, 92, 94, 100	syl3anc 1371	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ (𝑀[,]𝑁))
102	99, 90, 101	rspcdva 3582	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ ℝ)
103	102	recnd 11183	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℂ)
104	98, 103	neg2subd 11529	. . . 4 ⊢ (𝜑 → (-𝐷 − -𝐶) = (𝐶 − 𝐷))
105	98, 103	negsubdi2d 11528	. . . 4 ⊢ (𝜑 → -(𝐷 − 𝐶) = (𝐶 − 𝐷))
106	104, 105	eqtr4d 2779	. . 3 ⊢ (𝜑 → (-𝐷 − -𝐶) = -(𝐷 − 𝐶))
107	82, 86, 106	3brtr3d 5136	. 2 ⊢ (𝜑 → -Σ𝑘 ∈ (𝑀..^𝑁)𝑋 ≤ -(𝐷 − 𝐶))
108	97, 102	resubcld 11583	. . 3 ⊢ (𝜑 → (𝐷 − 𝐶) ∈ ℝ)
109	84, 42	fsumrecl 15619	. . 3 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀..^𝑁)𝑋 ∈ ℝ)
110	108, 109	lenegd 11734	. 2 ⊢ (𝜑 → ((𝐷 − 𝐶) ≤ Σ𝑘 ∈ (𝑀..^𝑁)𝑋 ↔ -Σ𝑘 ∈ (𝑀..^𝑁)𝑋 ≤ -(𝐷 − 𝐶)))
111	107, 110	mpbird 256	1 ⊢ (𝜑 → (𝐷 − 𝐶) ≤ Σ𝑘 ∈ (𝑀..^𝑁)𝑋)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3064 Vcvv 3445 ⊆ wss 3910 {cpr 4588 class class class wbr 5105 ↦ cmpt 5188 dom cdm 5633 ⟶wf 6492 ‘cfv 6496 (class class class)co 7357 Fincfn 8883 ℂcc 11049 ℝcr 11050 0cc0 11051 1c1 11052 + caddc 11054 ℝ^*cxr 11188 ≤ cle 11190 − cmin 11385 -cneg 11386 ℤcz 12499 ℤ_≥cuz 12763 (,)cioo 13264 [,]cicc 13267 ...cfz 13424 ..^cfzo 13567 Σcsu 15570 TopOpenctopn 17303 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: (None)

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