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Theorem dvle 26061

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 2824	. . 3 ⊢ (𝑥 = 𝑌 → (𝐴 ∈ ℝ ↔ 𝑅 ∈ ℝ))
3		dvle.a	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴) ∈ ((𝑀[,]𝑁)–cn→ℝ))
4		cncff 24933	. . . . 5 ⊢ ((𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴) ∈ ((𝑀[,]𝑁)–cn→ℝ) → (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴):(𝑀[,]𝑁)⟶ℝ)
5	3, 4	syl 17	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴):(𝑀[,]𝑁)⟶ℝ)
6		eqid 2735	. . . . 5 ⊢ (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴) = (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴)
7	6	fmpt 7130	. . . 4 ⊢ (∀𝑥 ∈ (𝑀[,]𝑁)𝐴 ∈ ℝ ↔ (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴):(𝑀[,]𝑁)⟶ℝ)
8	5, 7	sylibr 234	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ (𝑀[,]𝑁)𝐴 ∈ ℝ)
9		dvle.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ (𝑀[,]𝑁))
10	2, 8, 9	rspcdva 3623	. 2 ⊢ (𝜑 → 𝑅 ∈ ℝ)
11		dvle.s	. . . . 5 ⊢ (𝑥 = 𝑌 → 𝐶 = 𝑆)
12	11	eleq1d 2824	. . . 4 ⊢ (𝑥 = 𝑌 → (𝐶 ∈ ℝ ↔ 𝑆 ∈ ℝ))
13		dvle.c	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐶) ∈ ((𝑀[,]𝑁)–cn→ℝ))
14		cncff 24933	. . . . . 6 ⊢ ((𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐶) ∈ ((𝑀[,]𝑁)–cn→ℝ) → (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐶):(𝑀[,]𝑁)⟶ℝ)
15	13, 14	syl 17	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐶):(𝑀[,]𝑁)⟶ℝ)
16		eqid 2735	. . . . . 6 ⊢ (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐶) = (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐶)
17	16	fmpt 7130	. . . . 5 ⊢ (∀𝑥 ∈ (𝑀[,]𝑁)𝐶 ∈ ℝ ↔ (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐶):(𝑀[,]𝑁)⟶ℝ)
18	15, 17	sylibr 234	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ (𝑀[,]𝑁)𝐶 ∈ ℝ)
19	12, 18, 9	rspcdva 3623	. . 3 ⊢ (𝜑 → 𝑆 ∈ ℝ)
20		dvle.q	. . . . 5 ⊢ (𝑥 = 𝑋 → 𝐶 = 𝑄)
21	20	eleq1d 2824	. . . 4 ⊢ (𝑥 = 𝑋 → (𝐶 ∈ ℝ ↔ 𝑄 ∈ ℝ))
22		dvle.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝑀[,]𝑁))
23	21, 18, 22	rspcdva 3623	. . 3 ⊢ (𝜑 → 𝑄 ∈ ℝ)
24	19, 23	resubcld 11689	. 2 ⊢ (𝜑 → (𝑆 − 𝑄) ∈ ℝ)
25		dvle.p	. . . 4 ⊢ (𝑥 = 𝑋 → 𝐴 = 𝑃)
26	25	eleq1d 2824	. . 3 ⊢ (𝑥 = 𝑋 → (𝐴 ∈ ℝ ↔ 𝑃 ∈ ℝ))
27	26, 8, 22	rspcdva 3623	. 2 ⊢ (𝜑 → 𝑃 ∈ ℝ)
28	10	recnd 11287	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ ℂ)
29	23	recnd 11287	. . . . . 6 ⊢ (𝜑 → 𝑄 ∈ ℂ)
30	19	recnd 11287	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ ℂ)
31	29, 30	subcld 11618	. . . . 5 ⊢ (𝜑 → (𝑄 − 𝑆) ∈ ℂ)
32	28, 31	addcomd 11461	. . . 4 ⊢ (𝜑 → (𝑅 + (𝑄 − 𝑆)) = ((𝑄 − 𝑆) + 𝑅))
33	28, 30, 29	subsub2d 11647	. . . 4 ⊢ (𝜑 → (𝑅 − (𝑆 − 𝑄)) = (𝑅 + (𝑄 − 𝑆)))
34	29, 30, 28	subsubd 11646	. . . 4 ⊢ (𝜑 → (𝑄 − (𝑆 − 𝑅)) = ((𝑄 − 𝑆) + 𝑅))
35	32, 33, 34	3eqtr4d 2785	. . 3 ⊢ (𝜑 → (𝑅 − (𝑆 − 𝑄)) = (𝑄 − (𝑆 − 𝑅)))
36	19, 10	resubcld 11689	. . . 4 ⊢ (𝜑 → (𝑆 − 𝑅) ∈ ℝ)
37		dvle.m	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℝ)
38		dvle.n	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℝ)
39		eqid 2735	. . . . . . 7 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
40	39	subcn 24902	. . . . . . 7 ⊢ − ∈ (((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld)) Cn (TopOpen‘ℂ_fld))
41		ax-resscn 11210	. . . . . . 7 ⊢ ℝ ⊆ ℂ
42		resubcl 11571	. . . . . . 7 ⊢ ((𝐶 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐶 − 𝐴) ∈ ℝ)
43	39, 40, 13, 3, 41, 42	cncfmpt2ss 24956	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (𝑀[,]𝑁) ↦ (𝐶 − 𝐴)) ∈ ((𝑀[,]𝑁)–cn→ℝ))
44	41	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → ℝ ⊆ ℂ)
45		iccssre 13466	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝑀[,]𝑁) ⊆ ℝ)
46	37, 38, 45	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → (𝑀[,]𝑁) ⊆ ℝ)
47	15	fvmptelcdm 7133	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → 𝐶 ∈ ℝ)
48	5	fvmptelcdm 7133	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → 𝐴 ∈ ℝ)
49	47, 48	resubcld 11689	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → (𝐶 − 𝐴) ∈ ℝ)
50	49	recnd 11287	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → (𝐶 − 𝐴) ∈ ℂ)
51	39	tgioo2 24839	. . . . . . . . 9 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
52		iccntr 24857	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(𝑀[,]𝑁)) = (𝑀(,)𝑁))
53	37, 38, 52	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → ((int‘(topGen‘ran (,)))‘(𝑀[,]𝑁)) = (𝑀(,)𝑁))
54	44, 46, 50, 51, 39, 53	dvmptntr 26024	. . . . . . . 8 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑀[,]𝑁) ↦ (𝐶 − 𝐴))) = (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ (𝐶 − 𝐴))))
55		reelprrecn 11245	. . . . . . . . . 10 ⊢ ℝ ∈ {ℝ, ℂ}
56	55	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
57		ioossicc 13470	. . . . . . . . . . 11 ⊢ (𝑀(,)𝑁) ⊆ (𝑀[,]𝑁)
58	57	sseli 3991	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝑀(,)𝑁) → 𝑥 ∈ (𝑀[,]𝑁))
59	47	recnd 11287	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → 𝐶 ∈ ℂ)
60	58, 59	sylan2 593	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 𝐶 ∈ ℂ)
61		dvle.f	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 𝐵 ≤ 𝐷)
62		lerel 11323	. . . . . . . . . . 11 ⊢ Rel ≤
63	62	brrelex2i 5746	. . . . . . . . . 10 ⊢ (𝐵 ≤ 𝐷 → 𝐷 ∈ V)
64	61, 63	syl 17	. . . . . . . . 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 11287	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → 𝐴 ∈ ℂ)
67	58, 66	sylan2 593	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 𝐴 ∈ ℂ)
68	62	brrelex1i 5745	. . . . . . . . . 10 ⊢ (𝐵 ≤ 𝐷 → 𝐵 ∈ V)
69	61, 68	syl 17	. . . . . . . . 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 26020	. . . . . . . 8 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ (𝐶 − 𝐴))) = (𝑥 ∈ (𝑀(,)𝑁) ↦ (𝐷 − 𝐵)))
72	54, 71	eqtrd 2775	. . . . . . 7 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑀[,]𝑁) ↦ (𝐶 − 𝐴))) = (𝑥 ∈ (𝑀(,)𝑁) ↦ (𝐷 − 𝐵)))
73	58, 47	sylan2 593	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 𝐶 ∈ ℝ)
74	73	fmpttd 7135	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐶):(𝑀(,)𝑁)⟶ℝ)
75		ioossre 13445	. . . . . . . . . . . 12 ⊢ (𝑀(,)𝑁) ⊆ ℝ
76		dvfre 26004	. . . . . . . . . . . 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 586	. . . . . . . . . . 11 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐶)):dom (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐶))⟶ℝ)
78	65	dmeqd 5919	. . . . . . . . . . . . 13 ⊢ (𝜑 → dom (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐶)) = dom (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐷))
79	64	ralrimiva 3144	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑥 ∈ (𝑀(,)𝑁)𝐷 ∈ V)
80		dmmptg 6264	. . . . . . . . . . . . . 14 ⊢ (∀𝑥 ∈ (𝑀(,)𝑁)𝐷 ∈ V → dom (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐷) = (𝑀(,)𝑁))
81	79, 80	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → dom (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐷) = (𝑀(,)𝑁))
82	78, 81	eqtrd 2775	. . . . . . . . . . . 12 ⊢ (𝜑 → dom (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐶)) = (𝑀(,)𝑁))
83	65, 82	feq12d 6725	. . . . . . . . . . 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 232	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐷):(𝑀(,)𝑁)⟶ℝ)
85	84	fvmptelcdm 7133	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 𝐷 ∈ ℝ)
86	58, 48	sylan2 593	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 𝐴 ∈ ℝ)
87	86	fmpttd 7135	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴):(𝑀(,)𝑁)⟶ℝ)
88		dvfre 26004	. . . . . . . . . . . 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 586	. . . . . . . . . . 11 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)):dom (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴))⟶ℝ)
90	70	dmeqd 5919	. . . . . . . . . . . . 13 ⊢ (𝜑 → dom (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)) = dom (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵))
91	69	ralrimiva 3144	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑥 ∈ (𝑀(,)𝑁)𝐵 ∈ V)
92		dmmptg 6264	. . . . . . . . . . . . . 14 ⊢ (∀𝑥 ∈ (𝑀(,)𝑁)𝐵 ∈ V → dom (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵) = (𝑀(,)𝑁))
93	91, 92	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → dom (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵) = (𝑀(,)𝑁))
94	90, 93	eqtrd 2775	. . . . . . . . . . . 12 ⊢ (𝜑 → dom (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)) = (𝑀(,)𝑁))
95	70, 94	feq12d 6725	. . . . . . . . . . 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 232	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵):(𝑀(,)𝑁)⟶ℝ)
97	96	fvmptelcdm 7133	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 𝐵 ∈ ℝ)
98	85, 97	resubcld 11689	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀(,)𝑁)) → (𝐷 − 𝐵) ∈ ℝ)
99	85, 97	subge0d 11851	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀(,)𝑁)) → (0 ≤ (𝐷 − 𝐵) ↔ 𝐵 ≤ 𝐷))
100	61, 99	mpbird 257	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 0 ≤ (𝐷 − 𝐵))
101		elrege0 13491	. . . . . . . 8 ⊢ ((𝐷 − 𝐵) ∈ (0[,)+∞) ↔ ((𝐷 − 𝐵) ∈ ℝ ∧ 0 ≤ (𝐷 − 𝐵)))
102	98, 100, 101	sylanbrc 583	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀(,)𝑁)) → (𝐷 − 𝐵) ∈ (0[,)+∞))
103	72, 102	fmpt3d 7136	. . . . . 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 26060	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ (𝑀[,]𝑁) ↦ (𝐶 − 𝐴))‘𝑋) ≤ ((𝑥 ∈ (𝑀[,]𝑁) ↦ (𝐶 − 𝐴))‘𝑌))
106	20, 25	oveq12d 7449	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝐶 − 𝐴) = (𝑄 − 𝑃))
107		eqid 2735	. . . . . . 7 ⊢ (𝑥 ∈ (𝑀[,]𝑁) ↦ (𝐶 − 𝐴)) = (𝑥 ∈ (𝑀[,]𝑁) ↦ (𝐶 − 𝐴))
108		ovex 7464	. . . . . . 7 ⊢ (𝐶 − 𝐴) ∈ V
109	106, 107, 108	fvmpt3i 7021	. . . . . 6 ⊢ (𝑋 ∈ (𝑀[,]𝑁) → ((𝑥 ∈ (𝑀[,]𝑁) ↦ (𝐶 − 𝐴))‘𝑋) = (𝑄 − 𝑃))
110	22, 109	syl 17	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ (𝑀[,]𝑁) ↦ (𝐶 − 𝐴))‘𝑋) = (𝑄 − 𝑃))
111	11, 1	oveq12d 7449	. . . . . . 7 ⊢ (𝑥 = 𝑌 → (𝐶 − 𝐴) = (𝑆 − 𝑅))
112	111, 107, 108	fvmpt3i 7021	. . . . . 6 ⊢ (𝑌 ∈ (𝑀[,]𝑁) → ((𝑥 ∈ (𝑀[,]𝑁) ↦ (𝐶 − 𝐴))‘𝑌) = (𝑆 − 𝑅))
113	9, 112	syl 17	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ (𝑀[,]𝑁) ↦ (𝐶 − 𝐴))‘𝑌) = (𝑆 − 𝑅))
114	105, 110, 113	3brtr3d 5179	. . . 4 ⊢ (𝜑 → (𝑄 − 𝑃) ≤ (𝑆 − 𝑅))
115	23, 27, 36, 114	subled 11864	. . 3 ⊢ (𝜑 → (𝑄 − (𝑆 − 𝑅)) ≤ 𝑃)
116	35, 115	eqbrtrd 5170	. 2 ⊢ (𝜑 → (𝑅 − (𝑆 − 𝑄)) ≤ 𝑃)
117	10, 24, 27, 116	subled 11864	1 ⊢ (𝜑 → (𝑅 − 𝑃) ≤ (𝑆 − 𝑄))

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 ran crn 5690 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 ℝcr 11152 0cc0 11153 + caddc 11156 +∞cpnf 11290 ≤ cle 11294 − cmin 11490 (,)cioo 13384 [,)cico 13386 [,]cicc 13387 TopOpenctopn 17468 topGenctg 17484 ℂ_fldccnfld 21382 intcnt 23041 –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-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-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: dvfsumle 26075 dvfsumleOLD 26076 dvfsumlem2 26082 dvfsumlem2OLD 26083 loglesqrt 26819 dvle2 42054

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