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Theorem dvle 26234

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.r	. . . 4 ⊢ (𝑥 = 𝑌 → 𝐴 = 𝑅)
2	1	eleq1d 2845	. . 3 ⊢ (𝑥 = 𝑌 → (𝐴 ∈ ℝ ↔ 𝑅 ∈ ℝ))
3		dvle.a	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴) ∈ ((𝑀[,]𝑁)–cn→ℝ))
4		cncff 25121	. . . . 5 ⊢ ((𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴) ∈ ((𝑀[,]𝑁)–cn→ℝ) → (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴):(𝑀[,]𝑁)⟶ℝ)
5	3, 4	syl 18	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴):(𝑀[,]𝑁)⟶ℝ)
6		eqid 2760	. . . . 5 ⊢ (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴) = (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴)
7	6	fmpt 7103	. . . 4 ⊢ (∀𝑥 ∈ (𝑀[,]𝑁)𝐴 ∈ ℝ ↔ (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴):(𝑀[,]𝑁)⟶ℝ)
8	5, 7	sylibr 237	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ (𝑀[,]𝑁)𝐴 ∈ ℝ)
9		dvle.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ (𝑀[,]𝑁))
10	2, 8, 9	rspcdva 3577	. 2 ⊢ (𝜑 → 𝑅 ∈ ℝ)
11		dvle.s	. . . . 5 ⊢ (𝑥 = 𝑌 → 𝐶 = 𝑆)
12	11	eleq1d 2845	. . . 4 ⊢ (𝑥 = 𝑌 → (𝐶 ∈ ℝ ↔ 𝑆 ∈ ℝ))
13		dvle.c	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐶) ∈ ((𝑀[,]𝑁)–cn→ℝ))
14		cncff 25121	. . . . . 6 ⊢ ((𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐶) ∈ ((𝑀[,]𝑁)–cn→ℝ) → (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐶):(𝑀[,]𝑁)⟶ℝ)
15	13, 14	syl 18	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐶):(𝑀[,]𝑁)⟶ℝ)
16		eqid 2760	. . . . . 6 ⊢ (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐶) = (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐶)
17	16	fmpt 7103	. . . . 5 ⊢ (∀𝑥 ∈ (𝑀[,]𝑁)𝐶 ∈ ℝ ↔ (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐶):(𝑀[,]𝑁)⟶ℝ)
18	15, 17	sylibr 237	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ (𝑀[,]𝑁)𝐶 ∈ ℝ)
19	12, 18, 9	rspcdva 3577	. . 3 ⊢ (𝜑 → 𝑆 ∈ ℝ)
20		dvle.q	. . . . 5 ⊢ (𝑥 = 𝑋 → 𝐶 = 𝑄)
21	20	eleq1d 2845	. . . 4 ⊢ (𝑥 = 𝑋 → (𝐶 ∈ ℝ ↔ 𝑄 ∈ ℝ))
22		dvle.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝑀[,]𝑁))
23	21, 18, 22	rspcdva 3577	. . 3 ⊢ (𝜑 → 𝑄 ∈ ℝ)
24	19, 23	resubcld 11666	. 2 ⊢ (𝜑 → (𝑆 − 𝑄) ∈ ℝ)
25		dvle.p	. . . 4 ⊢ (𝑥 = 𝑋 → 𝐴 = 𝑃)
26	25	eleq1d 2845	. . 3 ⊢ (𝑥 = 𝑋 → (𝐴 ∈ ℝ ↔ 𝑃 ∈ ℝ))
27	26, 8, 22	rspcdva 3577	. 2 ⊢ (𝜑 → 𝑃 ∈ ℝ)
28	10	recnd 11261	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ ℂ)
29	23	recnd 11261	. . . . . 6 ⊢ (𝜑 → 𝑄 ∈ ℂ)
30	19	recnd 11261	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ ℂ)
31	29, 30	subcld 11593	. . . . 5 ⊢ (𝜑 → (𝑄 − 𝑆) ∈ ℂ)
32	28, 31	addcomd 11436	. . . 4 ⊢ (𝜑 → (𝑅 + (𝑄 − 𝑆)) = ((𝑄 − 𝑆) + 𝑅))
33	28, 30, 29	subsub2d 11622	. . . 4 ⊢ (𝜑 → (𝑅 − (𝑆 − 𝑄)) = (𝑅 + (𝑄 − 𝑆)))
34	29, 30, 28	subsubd 11621	. . . 4 ⊢ (𝜑 → (𝑄 − (𝑆 − 𝑅)) = ((𝑄 − 𝑆) + 𝑅))
35	32, 33, 34	3eqtr4d 2805	. . 3 ⊢ (𝜑 → (𝑅 − (𝑆 − 𝑄)) = (𝑄 − (𝑆 − 𝑅)))
36	19, 10	resubcld 11666	. . . 4 ⊢ (𝜑 → (𝑆 − 𝑅) ∈ ℝ)
37		dvle.m	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℝ)
38		dvle.n	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℝ)
39		eqid 2760	. . . . . . 7 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
40	39	subcn 25093	. . . . . . 7 ⊢ − ∈ (((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld)) Cn (TopOpen‘ℂ_fld))
41		ax-resscn 11181	. . . . . . 7 ⊢ ℝ ⊆ ℂ
42		resubcl 11546	. . . . . . 7 ⊢ ((𝐶 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐶 − 𝐴) ∈ ℝ)
43	39, 40, 13, 3, 41, 42	cncfmpt2ss 25144	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (𝑀[,]𝑁) ↦ (𝐶 − 𝐴)) ∈ ((𝑀[,]𝑁)–cn→ℝ))
44	41	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → ℝ ⊆ ℂ)
45		iccssre 13482	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝑀[,]𝑁) ⊆ ℝ)
46	37, 38, 45	syl2anc 596	. . . . . . . . 9 ⊢ (𝜑 → (𝑀[,]𝑁) ⊆ ℝ)
47	15	fvmptelcdm 7106	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → 𝐶 ∈ ℝ)
48	5	fvmptelcdm 7106	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → 𝐴 ∈ ℝ)
49	47, 48	resubcld 11666	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → (𝐶 − 𝐴) ∈ ℝ)
50	49	recnd 11261	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → (𝐶 − 𝐴) ∈ ℂ)
51		tgioo4 25031	. . . . . . . . 9 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
52		iccntr 25048	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(𝑀[,]𝑁)) = (𝑀(,)𝑁))
53	37, 38, 52	syl2anc 596	. . . . . . . . 9 ⊢ (𝜑 → ((int‘(topGen‘ran (,)))‘(𝑀[,]𝑁)) = (𝑀(,)𝑁))
54	44, 46, 50, 51, 39, 53	dvmptntr 26198	. . . . . . . 8 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑀[,]𝑁) ↦ (𝐶 − 𝐴))) = (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ (𝐶 − 𝐴))))
55		reelprrecn 11216	. . . . . . . . . 10 ⊢ ℝ ∈ {ℝ, ℂ}
56	55	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
57		ioossicc 13486	. . . . . . . . . . 11 ⊢ (𝑀(,)𝑁) ⊆ (𝑀[,]𝑁)
58	57	sseli 3927	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝑀(,)𝑁) → 𝑥 ∈ (𝑀[,]𝑁))
59	47	recnd 11261	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → 𝐶 ∈ ℂ)
60	58, 59	sylan2 605	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 𝐶 ∈ ℂ)
61		dvle.f	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 𝐵 ≤ 𝐷)
62		lerel 11297	. . . . . . . . . . 11 ⊢ Rel ≤
63	62	brrelex2i 5712	. . . . . . . . . 10 ⊢ (𝐵 ≤ 𝐷 → 𝐷 ∈ V)
64	61, 63	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 𝐷 ∈ V)
65		dvle.d	. . . . . . . . 9 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐶)) = (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐷))
66	48	recnd 11261	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → 𝐴 ∈ ℂ)
67	58, 66	sylan2 605	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 𝐴 ∈ ℂ)
68	62	brrelex1i 5711	. . . . . . . . . 10 ⊢ (𝐵 ≤ 𝐷 → 𝐵 ∈ V)
69	61, 68	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 𝐵 ∈ V)
70		dvle.b	. . . . . . . . 9 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)) = (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵))
71	56, 60, 64, 65, 67, 69, 70	dvmptsub 26194	. . . . . . . 8 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ (𝐶 − 𝐴))) = (𝑥 ∈ (𝑀(,)𝑁) ↦ (𝐷 − 𝐵)))
72	54, 71	eqtrd 2795	. . . . . . 7 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑀[,]𝑁) ↦ (𝐶 − 𝐴))) = (𝑥 ∈ (𝑀(,)𝑁) ↦ (𝐷 − 𝐵)))
73	58, 47	sylan2 605	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 𝐶 ∈ ℝ)
74	73	fmpttd 7108	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐶):(𝑀(,)𝑁)⟶ℝ)
75		ioossre 13460	. . . . . . . . . . . 12 ⊢ (𝑀(,)𝑁) ⊆ ℝ
76		dvfre 26178	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐶):(𝑀(,)𝑁)⟶ℝ ∧ (𝑀(,)𝑁) ⊆ ℝ) → (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐶)):dom (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐶))⟶ℝ)
77	74, 75, 76	sylancl 598	. . . . . . . . . . 11 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐶)):dom (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐶))⟶ℝ)
78	65	dmeqd 5889	. . . . . . . . . . . . 13 ⊢ (𝜑 → dom (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐶)) = dom (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐷))
79	64	ralrimiva 3154	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑥 ∈ (𝑀(,)𝑁)𝐷 ∈ V)
80		dmmptg 6238	. . . . . . . . . . . . . 14 ⊢ (∀𝑥 ∈ (𝑀(,)𝑁)𝐷 ∈ V → dom (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐷) = (𝑀(,)𝑁))
81	79, 80	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → dom (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐷) = (𝑀(,)𝑁))
82	78, 81	eqtrd 2795	. . . . . . . . . . . 12 ⊢ (𝜑 → dom (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐶)) = (𝑀(,)𝑁))
83	65, 82	feq12d 6690	. . . . . . . . . . 11 ⊢ (𝜑 → ((ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐶)):dom (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐶))⟶ℝ ↔ (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐷):(𝑀(,)𝑁)⟶ℝ))
84	77, 83	mpbid 235	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐷):(𝑀(,)𝑁)⟶ℝ)
85	84	fvmptelcdm 7106	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 𝐷 ∈ ℝ)
86	58, 48	sylan2 605	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 𝐴 ∈ ℝ)
87	86	fmpttd 7108	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴):(𝑀(,)𝑁)⟶ℝ)
88		dvfre 26178	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴):(𝑀(,)𝑁)⟶ℝ ∧ (𝑀(,)𝑁) ⊆ ℝ) → (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)):dom (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴))⟶ℝ)
89	87, 75, 88	sylancl 598	. . . . . . . . . . 11 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)):dom (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴))⟶ℝ)
90	70	dmeqd 5889	. . . . . . . . . . . . 13 ⊢ (𝜑 → dom (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)) = dom (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵))
91	69	ralrimiva 3154	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑥 ∈ (𝑀(,)𝑁)𝐵 ∈ V)
92		dmmptg 6238	. . . . . . . . . . . . . 14 ⊢ (∀𝑥 ∈ (𝑀(,)𝑁)𝐵 ∈ V → dom (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵) = (𝑀(,)𝑁))
93	91, 92	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → dom (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵) = (𝑀(,)𝑁))
94	90, 93	eqtrd 2795	. . . . . . . . . . . 12 ⊢ (𝜑 → dom (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)) = (𝑀(,)𝑁))
95	70, 94	feq12d 6690	. . . . . . . . . . 11 ⊢ (𝜑 → ((ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)):dom (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴))⟶ℝ ↔ (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵):(𝑀(,)𝑁)⟶ℝ))
96	89, 95	mpbid 235	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵):(𝑀(,)𝑁)⟶ℝ)
97	96	fvmptelcdm 7106	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 𝐵 ∈ ℝ)
98	85, 97	resubcld 11666	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀(,)𝑁)) → (𝐷 − 𝐵) ∈ ℝ)
99	85, 97	subge0d 11828	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀(,)𝑁)) → (0 ≤ (𝐷 − 𝐵) ↔ 𝐵 ≤ 𝐷))
100	61, 99	mpbird 260	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 0 ≤ (𝐷 − 𝐵))
101		elrege0 13507	. . . . . . . 8 ⊢ ((𝐷 − 𝐵) ∈ (0[,)+∞) ↔ ((𝐷 − 𝐵) ∈ ℝ ∧ 0 ≤ (𝐷 − 𝐵)))
102	98, 100, 101	sylanbrc 595	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀(,)𝑁)) → (𝐷 − 𝐵) ∈ (0[,)+∞))
103	72, 102	fmpt3d 7109	. . . . . 6 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑀[,]𝑁) ↦ (𝐶 − 𝐴))):(𝑀(,)𝑁)⟶(0[,)+∞))
104		dvle.l	. . . . . 6 ⊢ (𝜑 → 𝑋 ≤ 𝑌)
105	37, 38, 43, 103, 22, 9, 104	dvge0 26233	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ (𝑀[,]𝑁) ↦ (𝐶 − 𝐴))‘𝑋) ≤ ((𝑥 ∈ (𝑀[,]𝑁) ↦ (𝐶 − 𝐴))‘𝑌))
106	20, 25	oveq12d 7431	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝐶 − 𝐴) = (𝑄 − 𝑃))
107		eqid 2760	. . . . . . 7 ⊢ (𝑥 ∈ (𝑀[,]𝑁) ↦ (𝐶 − 𝐴)) = (𝑥 ∈ (𝑀[,]𝑁) ↦ (𝐶 − 𝐴))
108		ovex 7446	. . . . . . 7 ⊢ (𝐶 − 𝐴) ∈ V
109	106, 107, 108	fvmpt3i 6992	. . . . . 6 ⊢ (𝑋 ∈ (𝑀[,]𝑁) → ((𝑥 ∈ (𝑀[,]𝑁) ↦ (𝐶 − 𝐴))‘𝑋) = (𝑄 − 𝑃))
110	22, 109	syl 18	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ (𝑀[,]𝑁) ↦ (𝐶 − 𝐴))‘𝑋) = (𝑄 − 𝑃))
111	11, 1	oveq12d 7431	. . . . . . 7 ⊢ (𝑥 = 𝑌 → (𝐶 − 𝐴) = (𝑆 − 𝑅))
112	111, 107, 108	fvmpt3i 6992	. . . . . 6 ⊢ (𝑌 ∈ (𝑀[,]𝑁) → ((𝑥 ∈ (𝑀[,]𝑁) ↦ (𝐶 − 𝐴))‘𝑌) = (𝑆 − 𝑅))
113	9, 112	syl 18	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ (𝑀[,]𝑁) ↦ (𝐶 − 𝐴))‘𝑌) = (𝑆 − 𝑅))
114	105, 110, 113	3brtr3d 5136	. . . 4 ⊢ (𝜑 → (𝑄 − 𝑃) ≤ (𝑆 − 𝑅))
115	23, 27, 36, 114	subled 11841	. . 3 ⊢ (𝜑 → (𝑄 − (𝑆 − 𝑅)) ≤ 𝑃)
116	35, 115	eqbrtrd 5127	. 2 ⊢ (𝜑 → (𝑅 − (𝑆 − 𝑄)) ≤ 𝑃)
117	10, 24, 27, 116	subled 11841	1 ⊢ (𝜑 → (𝑅 − 𝑃) ≤ (𝑆 − 𝑄))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ∀wral 3076 Vcvv 3450 ⊆ wss 3899 {cpr 4586 class class class wbr 5103 ↦ cmpt 5186 dom cdm 5655 ran crn 5656 ⟶wf 6529 ‘cfv 6533 (class class class)co 7413 ℂcc 11122 ℝcr 11123 0cc0 11124 + caddc 11127 +∞cpnf 11264 ≤ cle 11268 − cmin 11465 (,)cioo 13398 [,)cico 13400 [,]cicc 13401 TopOpenctopn 17506 topGenctg 17522 ℂ_fldccnfld 21585 intcnt 23242 –cn→ccncf 25104 D cdv 26090

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7736 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 ax-pre-sup 11202 ax-addf 11203

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-of 7678 df-om 7863 df-1st 7986 df-2nd 7987 df-supp 8159 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-1o 8455 df-2o 8456 df-er 8696 df-map 8828 df-pm 8829 df-ixp 8905 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-fsupp 9332 df-fi 9381 df-sup 9412 df-inf 9413 df-oi 9482 df-card 9944 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11467 df-neg 11468 df-div 11896 df-nn 12258 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 12529 df-z 12616 df-dec 12737 df-uz 12888 df-q 12998 df-rp 13043 df-xneg 13163 df-xadd 13164 df-xmul 13165 df-ioo 13402 df-ico 13404 df-icc 13405 df-fz 13562 df-fzo 13710 df-seq 14066 df-exp 14126 df-hash 14395 df-cj 15186 df-re 15187 df-im 15188 df-sqrt 15322 df-abs 15323 df-struct 17239 df-sets 17256 df-slot 17274 df-ndx 17286 df-base 17302 df-ress 17323 df-plusg 17355 df-mulr 17356 df-starv 17357 df-sca 17358 df-vsca 17359 df-ip 17360 df-tset 17361 df-ple 17362 df-ds 17364 df-unif 17365 df-hom 17366 df-cco 17367 df-rest 17507 df-topn 17508 df-0g 17526 df-gsum 17527 df-topgen 17528 df-pt 17529 df-prds 17532 df-xrs 17588 df-qtop 17593 df-imas 17594 df-xps 17596 df-mre 17670 df-mrc 17671 df-acs 17673 df-mgm 18730 df-sgrp 18821 df-mnd 18837 df-submnd 18892 df-mulg 19191 df-cntz 19444 df-cmn 19909 df-psmet 21577 df-xmet 21578 df-met 21579 df-bl 21580 df-mopn 21581 df-fbas 21582 df-fg 21583 df-cnfld 21586 df-top 23119 df-topon 23136 df-topsp 23158 df-bases 23171 df-cld 23244 df-ntr 23245 df-cls 23246 df-nei 23323 df-lp 23361 df-perf 23362 df-cn 23452 df-cnp 23453 df-haus 23540 df-cmp 23612 df-tx 23788 df-hmeo 23981 df-fil 24072 df-fm 24164 df-flim 24165 df-flf 24166 df-xms 24546 df-ms 24547 df-tms 24548 df-cncf 25106 df-limc 26093 df-dv 26094

This theorem is used by: dvfsumle 26248 dvfsumlem2 26254 loglesqrt 26998 dvle2 42938

W3C validator