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Theorem dvle 23574

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 22504	. . . . 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 2610	. . . . 5 ⊢ (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴) = (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴)
6	5	fmpt 6289	. . . 4 ⊢ (∀𝑥 ∈ (𝑀[,]𝑁)𝐴 ∈ ℝ ↔ (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴):(𝑀[,]𝑁)⟶ℝ)
7	4, 6	sylibr 223	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ (𝑀[,]𝑁)𝐴 ∈ ℝ)
8		dvle.r	. . . . 5 ⊢ (𝑥 = 𝑌 → 𝐴 = 𝑅)
9	8	eleq1d 2672	. . . 4 ⊢ (𝑥 = 𝑌 → (𝐴 ∈ ℝ ↔ 𝑅 ∈ ℝ))
10	9	rspcv 3278	. . 3 ⊢ (𝑌 ∈ (𝑀[,]𝑁) → (∀𝑥 ∈ (𝑀[,]𝑁)𝐴 ∈ ℝ → 𝑅 ∈ ℝ))
11	1, 7, 10	sylc 63	. 2 ⊢ (𝜑 → 𝑅 ∈ ℝ)
12		dvle.c	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐶) ∈ ((𝑀[,]𝑁)–cn→ℝ))
13		cncff 22504	. . . . . 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 2610	. . . . . 6 ⊢ (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐶) = (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐶)
16	15	fmpt 6289	. . . . 5 ⊢ (∀𝑥 ∈ (𝑀[,]𝑁)𝐶 ∈ ℝ ↔ (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐶):(𝑀[,]𝑁)⟶ℝ)
17	14, 16	sylibr 223	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ (𝑀[,]𝑁)𝐶 ∈ ℝ)
18		dvle.s	. . . . . 6 ⊢ (𝑥 = 𝑌 → 𝐶 = 𝑆)
19	18	eleq1d 2672	. . . . 5 ⊢ (𝑥 = 𝑌 → (𝐶 ∈ ℝ ↔ 𝑆 ∈ ℝ))
20	19	rspcv 3278	. . . 4 ⊢ (𝑌 ∈ (𝑀[,]𝑁) → (∀𝑥 ∈ (𝑀[,]𝑁)𝐶 ∈ ℝ → 𝑆 ∈ ℝ))
21	1, 17, 20	sylc 63	. . 3 ⊢ (𝜑 → 𝑆 ∈ ℝ)
22		dvle.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝑀[,]𝑁))
23		dvle.q	. . . . . 6 ⊢ (𝑥 = 𝑋 → 𝐶 = 𝑄)
24	23	eleq1d 2672	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝐶 ∈ ℝ ↔ 𝑄 ∈ ℝ))
25	24	rspcv 3278	. . . 4 ⊢ (𝑋 ∈ (𝑀[,]𝑁) → (∀𝑥 ∈ (𝑀[,]𝑁)𝐶 ∈ ℝ → 𝑄 ∈ ℝ))
26	22, 17, 25	sylc 63	. . 3 ⊢ (𝜑 → 𝑄 ∈ ℝ)
27	21, 26	resubcld 10337	. 2 ⊢ (𝜑 → (𝑆 − 𝑄) ∈ ℝ)
28		dvle.p	. . . . 5 ⊢ (𝑥 = 𝑋 → 𝐴 = 𝑃)
29	28	eleq1d 2672	. . . 4 ⊢ (𝑥 = 𝑋 → (𝐴 ∈ ℝ ↔ 𝑃 ∈ ℝ))
30	29	rspcv 3278	. . 3 ⊢ (𝑋 ∈ (𝑀[,]𝑁) → (∀𝑥 ∈ (𝑀[,]𝑁)𝐴 ∈ ℝ → 𝑃 ∈ ℝ))
31	22, 7, 30	sylc 63	. 2 ⊢ (𝜑 → 𝑃 ∈ ℝ)
32	11	recnd 9947	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ ℂ)
33	26	recnd 9947	. . . . . 6 ⊢ (𝜑 → 𝑄 ∈ ℂ)
34	21	recnd 9947	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ ℂ)
35	33, 34	subcld 10271	. . . . 5 ⊢ (𝜑 → (𝑄 − 𝑆) ∈ ℂ)
36	32, 35	addcomd 10117	. . . 4 ⊢ (𝜑 → (𝑅 + (𝑄 − 𝑆)) = ((𝑄 − 𝑆) + 𝑅))
37	32, 34, 33	subsub2d 10300	. . . 4 ⊢ (𝜑 → (𝑅 − (𝑆 − 𝑄)) = (𝑅 + (𝑄 − 𝑆)))
38	33, 34, 32	subsubd 10299	. . . 4 ⊢ (𝜑 → (𝑄 − (𝑆 − 𝑅)) = ((𝑄 − 𝑆) + 𝑅))
39	36, 37, 38	3eqtr4d 2654	. . 3 ⊢ (𝜑 → (𝑅 − (𝑆 − 𝑄)) = (𝑄 − (𝑆 − 𝑅)))
40	21, 11	resubcld 10337	. . . 4 ⊢ (𝜑 → (𝑆 − 𝑅) ∈ ℝ)
41		dvle.m	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℝ)
42		dvle.n	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℝ)
43		eqid 2610	. . . . . . 7 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
44	43	subcn 22477	. . . . . . 7 ⊢ − ∈ (((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld)) Cn (TopOpen‘ℂ_fld))
45		ax-resscn 9872	. . . . . . 7 ⊢ ℝ ⊆ ℂ
46		resubcl 10224	. . . . . . 7 ⊢ ((𝐶 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐶 − 𝐴) ∈ ℝ)
47	43, 44, 12, 2, 45, 46	cncfmpt2ss 22526	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (𝑀[,]𝑁) ↦ (𝐶 − 𝐴)) ∈ ((𝑀[,]𝑁)–cn→ℝ))
48		ioossicc 12130	. . . . . . . . . . . . . . . . 17 ⊢ (𝑀(,)𝑁) ⊆ (𝑀[,]𝑁)
49	48	sseli 3564	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (𝑀(,)𝑁) → 𝑥 ∈ (𝑀[,]𝑁))
50	17	r19.21bi 2916	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → 𝐶 ∈ ℝ)
51	49, 50	sylan2 490	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 𝐶 ∈ ℝ)
52		eqid 2610	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐶) = (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐶)
53	51, 52	fmptd 6292	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐶):(𝑀(,)𝑁)⟶ℝ)
54		ioossre 12106	. . . . . . . . . . . . . 14 ⊢ (𝑀(,)𝑁) ⊆ ℝ
55		dvfre 23520	. . . . . . . . . . . . . 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 693	. . . . . . . . . . . . 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 5248	. . . . . . . . . . . . . . 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 9981	. . . . . . . . . . . . . . . . . . 19 ⊢ Rel ≤
61	60	brrelex2i 5083	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐵 ≤ 𝐷 → 𝐷 ∈ V)
62	59, 61	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 𝐷 ∈ V)
63	62	ralrimiva 2949	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ∀𝑥 ∈ (𝑀(,)𝑁)𝐷 ∈ V)
64		dmmptg 5549	. . . . . . . . . . . . . . . 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 2644	. . . . . . . . . . . . . 14 ⊢ (𝜑 → dom (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐶)) = (𝑀(,)𝑁))
67	57, 66	feq12d 5946	. . . . . . . . . . . . 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 221	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐷):(𝑀(,)𝑁)⟶ℝ)
69		eqid 2610	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐷) = (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐷)
70	69	fmpt 6289	. . . . . . . . . . . 12 ⊢ (∀𝑥 ∈ (𝑀(,)𝑁)𝐷 ∈ ℝ ↔ (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐷):(𝑀(,)𝑁)⟶ℝ)
71	68, 70	sylibr 223	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑥 ∈ (𝑀(,)𝑁)𝐷 ∈ ℝ)
72	71	r19.21bi 2916	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 𝐷 ∈ ℝ)
73	7	r19.21bi 2916	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → 𝐴 ∈ ℝ)
74	49, 73	sylan2 490	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 𝐴 ∈ ℝ)
75		eqid 2610	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴) = (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)
76	74, 75	fmptd 6292	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴):(𝑀(,)𝑁)⟶ℝ)
77		dvfre 23520	. . . . . . . . . . . . . 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 693	. . . . . . . . . . . . 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 5248	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → dom (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)) = dom (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵))
81	60	brrelexi 5082	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐵 ≤ 𝐷 → 𝐵 ∈ V)
82	59, 81	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 𝐵 ∈ V)
83	82	ralrimiva 2949	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ∀𝑥 ∈ (𝑀(,)𝑁)𝐵 ∈ V)
84		dmmptg 5549	. . . . . . . . . . . . . . . 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 2644	. . . . . . . . . . . . . 14 ⊢ (𝜑 → dom (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)) = (𝑀(,)𝑁))
87	79, 86	feq12d 5946	. . . . . . . . . . . . 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 221	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵):(𝑀(,)𝑁)⟶ℝ)
89		eqid 2610	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵) = (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵)
90	89	fmpt 6289	. . . . . . . . . . . 12 ⊢ (∀𝑥 ∈ (𝑀(,)𝑁)𝐵 ∈ ℝ ↔ (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵):(𝑀(,)𝑁)⟶ℝ)
91	88, 90	sylibr 223	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑥 ∈ (𝑀(,)𝑁)𝐵 ∈ ℝ)
92	91	r19.21bi 2916	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 𝐵 ∈ ℝ)
93	72, 92	resubcld 10337	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀(,)𝑁)) → (𝐷 − 𝐵) ∈ ℝ)
94	72, 92	subge0d 10496	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀(,)𝑁)) → (0 ≤ (𝐷 − 𝐵) ↔ 𝐵 ≤ 𝐷))
95	59, 94	mpbird 246	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 0 ≤ (𝐷 − 𝐵))
96		elrege0 12149	. . . . . . . . 9 ⊢ ((𝐷 − 𝐵) ∈ (0[,)+∞) ↔ ((𝐷 − 𝐵) ∈ ℝ ∧ 0 ≤ (𝐷 − 𝐵)))
97	93, 95, 96	sylanbrc 695	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀(,)𝑁)) → (𝐷 − 𝐵) ∈ (0[,)+∞))
98		eqid 2610	. . . . . . . 8 ⊢ (𝑥 ∈ (𝑀(,)𝑁) ↦ (𝐷 − 𝐵)) = (𝑥 ∈ (𝑀(,)𝑁) ↦ (𝐷 − 𝐵))
99	97, 98	fmptd 6292	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (𝑀(,)𝑁) ↦ (𝐷 − 𝐵)):(𝑀(,)𝑁)⟶(0[,)+∞))
100	45	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → ℝ ⊆ ℂ)
101		iccssre 12126	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝑀[,]𝑁) ⊆ ℝ)
102	41, 42, 101	syl2anc 691	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀[,]𝑁) ⊆ ℝ)
103	50, 73	resubcld 10337	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → (𝐶 − 𝐴) ∈ ℝ)
104	103	recnd 9947	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → (𝐶 − 𝐴) ∈ ℂ)
105	43	tgioo2 22414	. . . . . . . . . 10 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
106		iccntr 22432	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(𝑀[,]𝑁)) = (𝑀(,)𝑁))
107	41, 42, 106	syl2anc 691	. . . . . . . . . 10 ⊢ (𝜑 → ((int‘(topGen‘ran (,)))‘(𝑀[,]𝑁)) = (𝑀(,)𝑁))
108	100, 102, 104, 105, 43, 107	dvmptntr 23540	. . . . . . . . 9 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑀[,]𝑁) ↦ (𝐶 − 𝐴))) = (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ (𝐶 − 𝐴))))
109		reelprrecn 9907	. . . . . . . . . . 11 ⊢ ℝ ∈ {ℝ, ℂ}
110	109	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
111	50	recnd 9947	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → 𝐶 ∈ ℂ)
112	49, 111	sylan2 490	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 𝐶 ∈ ℂ)
113	73	recnd 9947	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → 𝐴 ∈ ℂ)
114	49, 113	sylan2 490	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 𝐴 ∈ ℂ)
115	110, 112, 62, 57, 114, 82, 79	dvmptsub 23536	. . . . . . . . 9 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ (𝐶 − 𝐴))) = (𝑥 ∈ (𝑀(,)𝑁) ↦ (𝐷 − 𝐵)))
116	108, 115	eqtrd 2644	. . . . . . . 8 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑀[,]𝑁) ↦ (𝐶 − 𝐴))) = (𝑥 ∈ (𝑀(,)𝑁) ↦ (𝐷 − 𝐵)))
117	116	feq1d 5943	. . . . . . 7 ⊢ (𝜑 → ((ℝ D (𝑥 ∈ (𝑀[,]𝑁) ↦ (𝐶 − 𝐴))):(𝑀(,)𝑁)⟶(0[,)+∞) ↔ (𝑥 ∈ (𝑀(,)𝑁) ↦ (𝐷 − 𝐵)):(𝑀(,)𝑁)⟶(0[,)+∞)))
118	99, 117	mpbird 246	. . . . . 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 23573	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ (𝑀[,]𝑁) ↦ (𝐶 − 𝐴))‘𝑋) ≤ ((𝑥 ∈ (𝑀[,]𝑁) ↦ (𝐶 − 𝐴))‘𝑌))
121	23, 28	oveq12d 6567	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝐶 − 𝐴) = (𝑄 − 𝑃))
122		eqid 2610	. . . . . . 7 ⊢ (𝑥 ∈ (𝑀[,]𝑁) ↦ (𝐶 − 𝐴)) = (𝑥 ∈ (𝑀[,]𝑁) ↦ (𝐶 − 𝐴))
123		ovex 6577	. . . . . . 7 ⊢ (𝐶 − 𝐴) ∈ V
124	121, 122, 123	fvmpt3i 6196	. . . . . 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 6567	. . . . . . 7 ⊢ (𝑥 = 𝑌 → (𝐶 − 𝐴) = (𝑆 − 𝑅))
127	126, 122, 123	fvmpt3i 6196	. . . . . 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 4614	. . . 4 ⊢ (𝜑 → (𝑄 − 𝑃) ≤ (𝑆 − 𝑅))
130	26, 31, 40, 129	subled 10509	. . 3 ⊢ (𝜑 → (𝑄 − (𝑆 − 𝑅)) ≤ 𝑃)
131	39, 130	eqbrtrd 4605	. 2 ⊢ (𝜑 → (𝑅 − (𝑆 − 𝑄)) ≤ 𝑃)
132	11, 27, 31, 131	subled 10509	1 ⊢ (𝜑 → (𝑅 − 𝑃) ≤ (𝑆 − 𝑄))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 Vcvv 3173 ⊆ wss 3540 {cpr 4127 class class class wbr 4583 ↦ cmpt 4643 dom cdm 5038 ran crn 5039 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 ℝcr 9814 0cc0 9815 + caddc 9818 +∞cpnf 9950 ≤ cle 9954 − cmin 10145 (,)cioo 12046 [,)cico 12048 [,]cicc 12049 TopOpenctopn 15905 topGenctg 15921 ℂ_fldccnfld 19567 intcnt 20631 –cn→ccncf 22487 D cdv 23433

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-fi 8200 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-ioo 12050 df-ico 12052 df-icc 12053 df-fz 12198 df-fzo 12335 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-hom 15793 df-cco 15794 df-rest 15906 df-topn 15907 df-0g 15925 df-gsum 15926 df-topgen 15927 df-pt 15928 df-prds 15931 df-xrs 15985 df-qtop 15990 df-imas 15991 df-xps 15993 df-mre 16069 df-mrc 16070 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-mulg 17364 df-cntz 17573 df-cmn 18018 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-fbas 19564 df-fg 19565 df-cnfld 19568 df-top 20521 df-bases 20522 df-topon 20523 df-topsp 20524 df-cld 20633 df-ntr 20634 df-cls 20635 df-nei 20712 df-lp 20750 df-perf 20751 df-cn 20841 df-cnp 20842 df-haus 20929 df-cmp 21000 df-tx 21175 df-hmeo 21368 df-fil 21460 df-fm 21552 df-flim 21553 df-flf 21554 df-xms 21935 df-ms 21936 df-tms 21937 df-cncf 22489 df-limc 23436 df-dv 23437

This theorem is referenced by: dvfsumle 23588 dvfsumlem2 23594 loglesqrt 24299

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