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Theorem dvle 23770

Description: If 𝐴(𝑥), 𝐶(𝑥) are differentiable functions and 𝐴‘ ≤ 𝐶‘, then for 𝑥 ≤ 𝑦, 𝐴(𝑦) − 𝐴(𝑥) ≤ 𝐶(𝑦) − 𝐶(𝑥). (Contributed by Mario Carneiro, 16-May-2016.)

**Hypotheses**
Ref	Expression
dvle.m	⊢ (𝜑 → 𝑀 ∈ ℝ)
dvle.n	⊢ (𝜑 → 𝑁 ∈ ℝ)
dvle.a	⊢ (𝜑 → (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴) ∈ ((𝑀[,]𝑁)–cn→ℝ))
dvle.b	⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)) = (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵))
dvle.c	⊢ (𝜑 → (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐶) ∈ ((𝑀[,]𝑁)–cn→ℝ))
dvle.d	⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐶)) = (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐷))
dvle.f	⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 𝐵 ≤ 𝐷)
dvle.x	⊢ (𝜑 → 𝑋 ∈ (𝑀[,]𝑁))
dvle.y	⊢ (𝜑 → 𝑌 ∈ (𝑀[,]𝑁))
dvle.l	⊢ (𝜑 → 𝑋 ≤ 𝑌)
dvle.p	⊢ (𝑥 = 𝑋 → 𝐴 = 𝑃)
dvle.q	⊢ (𝑥 = 𝑋 → 𝐶 = 𝑄)
dvle.r	⊢ (𝑥 = 𝑌 → 𝐴 = 𝑅)
dvle.s	⊢ (𝑥 = 𝑌 → 𝐶 = 𝑆)

**Assertion**
Ref	Expression
dvle	⊢ (𝜑 → (𝑅 − 𝑃) ≤ (𝑆 − 𝑄))

Distinct variable groups: 𝑥,𝑀 𝑥,𝑁 𝑥,𝑃 𝑥,𝑄 𝑥,𝑅 𝑥,𝑆 𝑥,𝑋 𝜑,𝑥 𝑥,𝑌 Allowed substitution hints: 𝐴(𝑥) 𝐵(𝑥) 𝐶(𝑥) 𝐷(𝑥)

**Proof of Theorem dvle**
Step	Hyp	Ref	Expression
1		dvle.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ (𝑀[,]𝑁))
2		dvle.a	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴) ∈ ((𝑀[,]𝑁)–cn→ℝ))
3		cncff 22696	. . . . 5 ⊢ ((𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴) ∈ ((𝑀[,]𝑁)–cn→ℝ) → (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴):(𝑀[,]𝑁)⟶ℝ)
4	2, 3	syl 17	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴):(𝑀[,]𝑁)⟶ℝ)
5		eqid 2622	. . . . 5 ⊢ (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴) = (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴)
6	5	fmpt 6381	. . . 4 ⊢ (∀𝑥 ∈ (𝑀[,]𝑁)𝐴 ∈ ℝ ↔ (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴):(𝑀[,]𝑁)⟶ℝ)
7	4, 6	sylibr 224	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ (𝑀[,]𝑁)𝐴 ∈ ℝ)
8		dvle.r	. . . . 5 ⊢ (𝑥 = 𝑌 → 𝐴 = 𝑅)
9	8	eleq1d 2686	. . . 4 ⊢ (𝑥 = 𝑌 → (𝐴 ∈ ℝ ↔ 𝑅 ∈ ℝ))
10	9	rspcv 3305	. . 3 ⊢ (𝑌 ∈ (𝑀[,]𝑁) → (∀𝑥 ∈ (𝑀[,]𝑁)𝐴 ∈ ℝ → 𝑅 ∈ ℝ))
11	1, 7, 10	sylc 65	. 2 ⊢ (𝜑 → 𝑅 ∈ ℝ)
12		dvle.c	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐶) ∈ ((𝑀[,]𝑁)–cn→ℝ))
13		cncff 22696	. . . . . 6 ⊢ ((𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐶) ∈ ((𝑀[,]𝑁)–cn→ℝ) → (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐶):(𝑀[,]𝑁)⟶ℝ)
14	12, 13	syl 17	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐶):(𝑀[,]𝑁)⟶ℝ)
15		eqid 2622	. . . . . 6 ⊢ (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐶) = (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐶)
16	15	fmpt 6381	. . . . 5 ⊢ (∀𝑥 ∈ (𝑀[,]𝑁)𝐶 ∈ ℝ ↔ (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐶):(𝑀[,]𝑁)⟶ℝ)
17	14, 16	sylibr 224	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ (𝑀[,]𝑁)𝐶 ∈ ℝ)
18		dvle.s	. . . . . 6 ⊢ (𝑥 = 𝑌 → 𝐶 = 𝑆)
19	18	eleq1d 2686	. . . . 5 ⊢ (𝑥 = 𝑌 → (𝐶 ∈ ℝ ↔ 𝑆 ∈ ℝ))
20	19	rspcv 3305	. . . 4 ⊢ (𝑌 ∈ (𝑀[,]𝑁) → (∀𝑥 ∈ (𝑀[,]𝑁)𝐶 ∈ ℝ → 𝑆 ∈ ℝ))
21	1, 17, 20	sylc 65	. . 3 ⊢ (𝜑 → 𝑆 ∈ ℝ)
22		dvle.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝑀[,]𝑁))
23		dvle.q	. . . . . 6 ⊢ (𝑥 = 𝑋 → 𝐶 = 𝑄)
24	23	eleq1d 2686	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝐶 ∈ ℝ ↔ 𝑄 ∈ ℝ))
25	24	rspcv 3305	. . . 4 ⊢ (𝑋 ∈ (𝑀[,]𝑁) → (∀𝑥 ∈ (𝑀[,]𝑁)𝐶 ∈ ℝ → 𝑄 ∈ ℝ))
26	22, 17, 25	sylc 65	. . 3 ⊢ (𝜑 → 𝑄 ∈ ℝ)
27	21, 26	resubcld 10458	. 2 ⊢ (𝜑 → (𝑆 − 𝑄) ∈ ℝ)
28		dvle.p	. . . . 5 ⊢ (𝑥 = 𝑋 → 𝐴 = 𝑃)
29	28	eleq1d 2686	. . . 4 ⊢ (𝑥 = 𝑋 → (𝐴 ∈ ℝ ↔ 𝑃 ∈ ℝ))
30	29	rspcv 3305	. . 3 ⊢ (𝑋 ∈ (𝑀[,]𝑁) → (∀𝑥 ∈ (𝑀[,]𝑁)𝐴 ∈ ℝ → 𝑃 ∈ ℝ))
31	22, 7, 30	sylc 65	. 2 ⊢ (𝜑 → 𝑃 ∈ ℝ)
32	11	recnd 10068	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ ℂ)
33	26	recnd 10068	. . . . . 6 ⊢ (𝜑 → 𝑄 ∈ ℂ)
34	21	recnd 10068	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ ℂ)
35	33, 34	subcld 10392	. . . . 5 ⊢ (𝜑 → (𝑄 − 𝑆) ∈ ℂ)
36	32, 35	addcomd 10238	. . . 4 ⊢ (𝜑 → (𝑅 + (𝑄 − 𝑆)) = ((𝑄 − 𝑆) + 𝑅))
37	32, 34, 33	subsub2d 10421	. . . 4 ⊢ (𝜑 → (𝑅 − (𝑆 − 𝑄)) = (𝑅 + (𝑄 − 𝑆)))
38	33, 34, 32	subsubd 10420	. . . 4 ⊢ (𝜑 → (𝑄 − (𝑆 − 𝑅)) = ((𝑄 − 𝑆) + 𝑅))
39	36, 37, 38	3eqtr4d 2666	. . 3 ⊢ (𝜑 → (𝑅 − (𝑆 − 𝑄)) = (𝑄 − (𝑆 − 𝑅)))
40	21, 11	resubcld 10458	. . . 4 ⊢ (𝜑 → (𝑆 − 𝑅) ∈ ℝ)
41		dvle.m	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℝ)
42		dvle.n	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℝ)
43		eqid 2622	. . . . . . 7 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
44	43	subcn 22669	. . . . . . 7 ⊢ − ∈ (((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld)) Cn (TopOpen‘ℂ_fld))
45		ax-resscn 9993	. . . . . . 7 ⊢ ℝ ⊆ ℂ
46		resubcl 10345	. . . . . . 7 ⊢ ((𝐶 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐶 − 𝐴) ∈ ℝ)
47	43, 44, 12, 2, 45, 46	cncfmpt2ss 22718	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (𝑀[,]𝑁) ↦ (𝐶 − 𝐴)) ∈ ((𝑀[,]𝑁)–cn→ℝ))
48		ioossicc 12259	. . . . . . . . . . . . . . . . 17 ⊢ (𝑀(,)𝑁) ⊆ (𝑀[,]𝑁)
49	48	sseli 3599	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (𝑀(,)𝑁) → 𝑥 ∈ (𝑀[,]𝑁))
50	17	r19.21bi 2932	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → 𝐶 ∈ ℝ)
51	49, 50	sylan2 491	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 𝐶 ∈ ℝ)
52		eqid 2622	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐶) = (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐶)
53	51, 52	fmptd 6385	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐶):(𝑀(,)𝑁)⟶ℝ)
54		ioossre 12235	. . . . . . . . . . . . . 14 ⊢ (𝑀(,)𝑁) ⊆ ℝ
55		dvfre 23714	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐶):(𝑀(,)𝑁)⟶ℝ ∧ (𝑀(,)𝑁) ⊆ ℝ) → (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐶)):dom (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐶))⟶ℝ)
56	53, 54, 55	sylancl 694	. . . . . . . . . . . . 13 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐶)):dom (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐶))⟶ℝ)
57		dvle.d	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐶)) = (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐷))
58	57	dmeqd 5326	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → dom (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐶)) = dom (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐷))
59		dvle.f	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 𝐵 ≤ 𝐷)
60		lerel 10102	. . . . . . . . . . . . . . . . . . 19 ⊢ Rel ≤
61	60	brrelex2i 5159	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐵 ≤ 𝐷 → 𝐷 ∈ V)
62	59, 61	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 𝐷 ∈ V)
63	62	ralrimiva 2966	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ∀𝑥 ∈ (𝑀(,)𝑁)𝐷 ∈ V)
64		dmmptg 5632	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑥 ∈ (𝑀(,)𝑁)𝐷 ∈ V → dom (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐷) = (𝑀(,)𝑁))
65	63, 64	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → dom (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐷) = (𝑀(,)𝑁))
66	58, 65	eqtrd 2656	. . . . . . . . . . . . . 14 ⊢ (𝜑 → dom (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐶)) = (𝑀(,)𝑁))
67	57, 66	feq12d 6033	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐶)):dom (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐶))⟶ℝ ↔ (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐷):(𝑀(,)𝑁)⟶ℝ))
68	56, 67	mpbid 222	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐷):(𝑀(,)𝑁)⟶ℝ)
69		eqid 2622	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐷) = (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐷)
70	69	fmpt 6381	. . . . . . . . . . . 12 ⊢ (∀𝑥 ∈ (𝑀(,)𝑁)𝐷 ∈ ℝ ↔ (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐷):(𝑀(,)𝑁)⟶ℝ)
71	68, 70	sylibr 224	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑥 ∈ (𝑀(,)𝑁)𝐷 ∈ ℝ)
72	71	r19.21bi 2932	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 𝐷 ∈ ℝ)
73	7	r19.21bi 2932	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → 𝐴 ∈ ℝ)
74	49, 73	sylan2 491	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 𝐴 ∈ ℝ)
75		eqid 2622	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴) = (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)
76	74, 75	fmptd 6385	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴):(𝑀(,)𝑁)⟶ℝ)
77		dvfre 23714	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴):(𝑀(,)𝑁)⟶ℝ ∧ (𝑀(,)𝑁) ⊆ ℝ) → (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)):dom (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴))⟶ℝ)
78	76, 54, 77	sylancl 694	. . . . . . . . . . . . 13 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)):dom (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴))⟶ℝ)
79		dvle.b	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)) = (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵))
80	79	dmeqd 5326	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → dom (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)) = dom (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵))
81	60	brrelexi 5158	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐵 ≤ 𝐷 → 𝐵 ∈ V)
82	59, 81	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 𝐵 ∈ V)
83	82	ralrimiva 2966	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ∀𝑥 ∈ (𝑀(,)𝑁)𝐵 ∈ V)
84		dmmptg 5632	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑥 ∈ (𝑀(,)𝑁)𝐵 ∈ V → dom (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵) = (𝑀(,)𝑁))
85	83, 84	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → dom (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵) = (𝑀(,)𝑁))
86	80, 85	eqtrd 2656	. . . . . . . . . . . . . 14 ⊢ (𝜑 → dom (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)) = (𝑀(,)𝑁))
87	79, 86	feq12d 6033	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)):dom (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴))⟶ℝ ↔ (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵):(𝑀(,)𝑁)⟶ℝ))
88	78, 87	mpbid 222	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵):(𝑀(,)𝑁)⟶ℝ)
89		eqid 2622	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵) = (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵)
90	89	fmpt 6381	. . . . . . . . . . . 12 ⊢ (∀𝑥 ∈ (𝑀(,)𝑁)𝐵 ∈ ℝ ↔ (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵):(𝑀(,)𝑁)⟶ℝ)
91	88, 90	sylibr 224	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑥 ∈ (𝑀(,)𝑁)𝐵 ∈ ℝ)
92	91	r19.21bi 2932	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 𝐵 ∈ ℝ)
93	72, 92	resubcld 10458	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀(,)𝑁)) → (𝐷 − 𝐵) ∈ ℝ)
94	72, 92	subge0d 10617	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀(,)𝑁)) → (0 ≤ (𝐷 − 𝐵) ↔ 𝐵 ≤ 𝐷))
95	59, 94	mpbird 247	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 0 ≤ (𝐷 − 𝐵))
96		elrege0 12278	. . . . . . . . 9 ⊢ ((𝐷 − 𝐵) ∈ (0[,)+∞) ↔ ((𝐷 − 𝐵) ∈ ℝ ∧ 0 ≤ (𝐷 − 𝐵)))
97	93, 95, 96	sylanbrc 698	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀(,)𝑁)) → (𝐷 − 𝐵) ∈ (0[,)+∞))
98		eqid 2622	. . . . . . . 8 ⊢ (𝑥 ∈ (𝑀(,)𝑁) ↦ (𝐷 − 𝐵)) = (𝑥 ∈ (𝑀(,)𝑁) ↦ (𝐷 − 𝐵))
99	97, 98	fmptd 6385	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (𝑀(,)𝑁) ↦ (𝐷 − 𝐵)):(𝑀(,)𝑁)⟶(0[,)+∞))
100	45	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → ℝ ⊆ ℂ)
101		iccssre 12255	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝑀[,]𝑁) ⊆ ℝ)
102	41, 42, 101	syl2anc 693	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀[,]𝑁) ⊆ ℝ)
103	50, 73	resubcld 10458	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → (𝐶 − 𝐴) ∈ ℝ)
104	103	recnd 10068	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → (𝐶 − 𝐴) ∈ ℂ)
105	43	tgioo2 22606	. . . . . . . . . 10 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
106		iccntr 22624	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(𝑀[,]𝑁)) = (𝑀(,)𝑁))
107	41, 42, 106	syl2anc 693	. . . . . . . . . 10 ⊢ (𝜑 → ((int‘(topGen‘ran (,)))‘(𝑀[,]𝑁)) = (𝑀(,)𝑁))
108	100, 102, 104, 105, 43, 107	dvmptntr 23734	. . . . . . . . 9 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑀[,]𝑁) ↦ (𝐶 − 𝐴))) = (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ (𝐶 − 𝐴))))
109		reelprrecn 10028	. . . . . . . . . . 11 ⊢ ℝ ∈ {ℝ, ℂ}
110	109	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
111	50	recnd 10068	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → 𝐶 ∈ ℂ)
112	49, 111	sylan2 491	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 𝐶 ∈ ℂ)
113	73	recnd 10068	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → 𝐴 ∈ ℂ)
114	49, 113	sylan2 491	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 𝐴 ∈ ℂ)
115	110, 112, 62, 57, 114, 82, 79	dvmptsub 23730	. . . . . . . . 9 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ (𝐶 − 𝐴))) = (𝑥 ∈ (𝑀(,)𝑁) ↦ (𝐷 − 𝐵)))
116	108, 115	eqtrd 2656	. . . . . . . 8 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑀[,]𝑁) ↦ (𝐶 − 𝐴))) = (𝑥 ∈ (𝑀(,)𝑁) ↦ (𝐷 − 𝐵)))
117	116	feq1d 6030	. . . . . . 7 ⊢ (𝜑 → ((ℝ D (𝑥 ∈ (𝑀[,]𝑁) ↦ (𝐶 − 𝐴))):(𝑀(,)𝑁)⟶(0[,)+∞) ↔ (𝑥 ∈ (𝑀(,)𝑁) ↦ (𝐷 − 𝐵)):(𝑀(,)𝑁)⟶(0[,)+∞)))
118	99, 117	mpbird 247	. . . . . 6 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑀[,]𝑁) ↦ (𝐶 − 𝐴))):(𝑀(,)𝑁)⟶(0[,)+∞))
119		dvle.l	. . . . . 6 ⊢ (𝜑 → 𝑋 ≤ 𝑌)
120	41, 42, 47, 118, 22, 1, 119	dvge0 23769	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ (𝑀[,]𝑁) ↦ (𝐶 − 𝐴))‘𝑋) ≤ ((𝑥 ∈ (𝑀[,]𝑁) ↦ (𝐶 − 𝐴))‘𝑌))
121	23, 28	oveq12d 6668	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝐶 − 𝐴) = (𝑄 − 𝑃))
122		eqid 2622	. . . . . . 7 ⊢ (𝑥 ∈ (𝑀[,]𝑁) ↦ (𝐶 − 𝐴)) = (𝑥 ∈ (𝑀[,]𝑁) ↦ (𝐶 − 𝐴))
123		ovex 6678	. . . . . . 7 ⊢ (𝐶 − 𝐴) ∈ V
124	121, 122, 123	fvmpt3i 6287	. . . . . 6 ⊢ (𝑋 ∈ (𝑀[,]𝑁) → ((𝑥 ∈ (𝑀[,]𝑁) ↦ (𝐶 − 𝐴))‘𝑋) = (𝑄 − 𝑃))
125	22, 124	syl 17	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ (𝑀[,]𝑁) ↦ (𝐶 − 𝐴))‘𝑋) = (𝑄 − 𝑃))
126	18, 8	oveq12d 6668	. . . . . . 7 ⊢ (𝑥 = 𝑌 → (𝐶 − 𝐴) = (𝑆 − 𝑅))
127	126, 122, 123	fvmpt3i 6287	. . . . . 6 ⊢ (𝑌 ∈ (𝑀[,]𝑁) → ((𝑥 ∈ (𝑀[,]𝑁) ↦ (𝐶 − 𝐴))‘𝑌) = (𝑆 − 𝑅))
128	1, 127	syl 17	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ (𝑀[,]𝑁) ↦ (𝐶 − 𝐴))‘𝑌) = (𝑆 − 𝑅))
129	120, 125, 128	3brtr3d 4684	. . . 4 ⊢ (𝜑 → (𝑄 − 𝑃) ≤ (𝑆 − 𝑅))
130	26, 31, 40, 129	subled 10630	. . 3 ⊢ (𝜑 → (𝑄 − (𝑆 − 𝑅)) ≤ 𝑃)
131	39, 130	eqbrtrd 4675	. 2 ⊢ (𝜑 → (𝑅 − (𝑆 − 𝑄)) ≤ 𝑃)
132	11, 27, 31, 131	subled 10630	1 ⊢ (𝜑 → (𝑅 − 𝑃) ≤ (𝑆 − 𝑄))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 Vcvv 3200 ⊆ wss 3574 {cpr 4179 class class class wbr 4653 ↦ cmpt 4729 dom cdm 5114 ran crn 5115 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 ℝcr 9935 0cc0 9936 + caddc 9939 +∞cpnf 10071 ≤ cle 10075 − cmin 10266 (,)cioo 12175 [,)cico 12177 [,]cicc 12178 TopOpenctopn 16082 topGenctg 16098 ℂ_fldccnfld 19746 intcnt 20821 –cn→ccncf 22679 D cdv 23627

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ioo 12179 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-pt 16105 df-prds 16108 df-xrs 16162 df-qtop 16167 df-imas 16168 df-xps 16170 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-mulg 17541 df-cntz 17750 df-cmn 18195 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-fbas 19743 df-fg 19744 df-cnfld 19747 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cld 20823 df-ntr 20824 df-cls 20825 df-nei 20902 df-lp 20940 df-perf 20941 df-cn 21031 df-cnp 21032 df-haus 21119 df-cmp 21190 df-tx 21365 df-hmeo 21558 df-fil 21650 df-fm 21742 df-flim 21743 df-flf 21744 df-xms 22125 df-ms 22126 df-tms 22127 df-cncf 22681 df-limc 23630 df-dv 23631

This theorem is referenced by: dvfsumle 23784 dvfsumlem2 23790 loglesqrt 24499

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