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Theorem dvle 25371

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 2822	. . 3 ⊢ (𝑥 = 𝑌 → (𝐴 ∈ ℝ ↔ 𝑅 ∈ ℝ))
3		dvle.a	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴) ∈ ((𝑀[,]𝑁)–cn→ℝ))
4		cncff 24256	. . . . 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 2736	. . . . 5 ⊢ (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴) = (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴)
7	6	fmpt 7058	. . . 4 ⊢ (∀𝑥 ∈ (𝑀[,]𝑁)𝐴 ∈ ℝ ↔ (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴):(𝑀[,]𝑁)⟶ℝ)
8	5, 7	sylibr 233	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ (𝑀[,]𝑁)𝐴 ∈ ℝ)
9		dvle.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ (𝑀[,]𝑁))
10	2, 8, 9	rspcdva 3582	. 2 ⊢ (𝜑 → 𝑅 ∈ ℝ)
11		dvle.s	. . . . 5 ⊢ (𝑥 = 𝑌 → 𝐶 = 𝑆)
12	11	eleq1d 2822	. . . 4 ⊢ (𝑥 = 𝑌 → (𝐶 ∈ ℝ ↔ 𝑆 ∈ ℝ))
13		dvle.c	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐶) ∈ ((𝑀[,]𝑁)–cn→ℝ))
14		cncff 24256	. . . . . 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 2736	. . . . . 6 ⊢ (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐶) = (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐶)
17	16	fmpt 7058	. . . . 5 ⊢ (∀𝑥 ∈ (𝑀[,]𝑁)𝐶 ∈ ℝ ↔ (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐶):(𝑀[,]𝑁)⟶ℝ)
18	15, 17	sylibr 233	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ (𝑀[,]𝑁)𝐶 ∈ ℝ)
19	12, 18, 9	rspcdva 3582	. . 3 ⊢ (𝜑 → 𝑆 ∈ ℝ)
20		dvle.q	. . . . 5 ⊢ (𝑥 = 𝑋 → 𝐶 = 𝑄)
21	20	eleq1d 2822	. . . 4 ⊢ (𝑥 = 𝑋 → (𝐶 ∈ ℝ ↔ 𝑄 ∈ ℝ))
22		dvle.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝑀[,]𝑁))
23	21, 18, 22	rspcdva 3582	. . 3 ⊢ (𝜑 → 𝑄 ∈ ℝ)
24	19, 23	resubcld 11583	. 2 ⊢ (𝜑 → (𝑆 − 𝑄) ∈ ℝ)
25		dvle.p	. . . 4 ⊢ (𝑥 = 𝑋 → 𝐴 = 𝑃)
26	25	eleq1d 2822	. . 3 ⊢ (𝑥 = 𝑋 → (𝐴 ∈ ℝ ↔ 𝑃 ∈ ℝ))
27	26, 8, 22	rspcdva 3582	. 2 ⊢ (𝜑 → 𝑃 ∈ ℝ)
28	10	recnd 11183	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ ℂ)
29	23	recnd 11183	. . . . . 6 ⊢ (𝜑 → 𝑄 ∈ ℂ)
30	19	recnd 11183	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ ℂ)
31	29, 30	subcld 11512	. . . . 5 ⊢ (𝜑 → (𝑄 − 𝑆) ∈ ℂ)
32	28, 31	addcomd 11357	. . . 4 ⊢ (𝜑 → (𝑅 + (𝑄 − 𝑆)) = ((𝑄 − 𝑆) + 𝑅))
33	28, 30, 29	subsub2d 11541	. . . 4 ⊢ (𝜑 → (𝑅 − (𝑆 − 𝑄)) = (𝑅 + (𝑄 − 𝑆)))
34	29, 30, 28	subsubd 11540	. . . 4 ⊢ (𝜑 → (𝑄 − (𝑆 − 𝑅)) = ((𝑄 − 𝑆) + 𝑅))
35	32, 33, 34	3eqtr4d 2786	. . 3 ⊢ (𝜑 → (𝑅 − (𝑆 − 𝑄)) = (𝑄 − (𝑆 − 𝑅)))
36	19, 10	resubcld 11583	. . . 4 ⊢ (𝜑 → (𝑆 − 𝑅) ∈ ℝ)
37		dvle.m	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℝ)
38		dvle.n	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℝ)
39		eqid 2736	. . . . . . 7 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
40	39	subcn 24229	. . . . . . 7 ⊢ − ∈ (((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld)) Cn (TopOpen‘ℂ_fld))
41		ax-resscn 11108	. . . . . . 7 ⊢ ℝ ⊆ ℂ
42		resubcl 11465	. . . . . . 7 ⊢ ((𝐶 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐶 − 𝐴) ∈ ℝ)
43	39, 40, 13, 3, 41, 42	cncfmpt2ss 24279	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (𝑀[,]𝑁) ↦ (𝐶 − 𝐴)) ∈ ((𝑀[,]𝑁)–cn→ℝ))
44	41	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → ℝ ⊆ ℂ)
45		iccssre 13346	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝑀[,]𝑁) ⊆ ℝ)
46	37, 38, 45	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → (𝑀[,]𝑁) ⊆ ℝ)
47	15	fvmptelcdm 7061	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → 𝐶 ∈ ℝ)
48	5	fvmptelcdm 7061	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → 𝐴 ∈ ℝ)
49	47, 48	resubcld 11583	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → (𝐶 − 𝐴) ∈ ℝ)
50	49	recnd 11183	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → (𝐶 − 𝐴) ∈ ℂ)
51	39	tgioo2 24166	. . . . . . . . 9 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
52		iccntr 24184	. . . . . . . . . 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 25335	. . . . . . . 8 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑀[,]𝑁) ↦ (𝐶 − 𝐴))) = (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ (𝐶 − 𝐴))))
55		reelprrecn 11143	. . . . . . . . . 10 ⊢ ℝ ∈ {ℝ, ℂ}
56	55	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
57		ioossicc 13350	. . . . . . . . . . 11 ⊢ (𝑀(,)𝑁) ⊆ (𝑀[,]𝑁)
58	57	sseli 3940	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝑀(,)𝑁) → 𝑥 ∈ (𝑀[,]𝑁))
59	47	recnd 11183	. . . . . . . . . 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 11219	. . . . . . . . . . 11 ⊢ Rel ≤
63	62	brrelex2i 5689	. . . . . . . . . 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 11183	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → 𝐴 ∈ ℂ)
67	58, 66	sylan2 593	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 𝐴 ∈ ℂ)
68	62	brrelex1i 5688	. . . . . . . . . 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 25331	. . . . . . . 8 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ (𝐶 − 𝐴))) = (𝑥 ∈ (𝑀(,)𝑁) ↦ (𝐷 − 𝐵)))
72	54, 71	eqtrd 2776	. . . . . . 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 7063	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐶):(𝑀(,)𝑁)⟶ℝ)
75		ioossre 13325	. . . . . . . . . . . 12 ⊢ (𝑀(,)𝑁) ⊆ ℝ
76		dvfre 25315	. . . . . . . . . . . 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 5861	. . . . . . . . . . . . 13 ⊢ (𝜑 → dom (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐶)) = dom (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐷))
79	64	ralrimiva 3143	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑥 ∈ (𝑀(,)𝑁)𝐷 ∈ V)
80		dmmptg 6194	. . . . . . . . . . . . . 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 2776	. . . . . . . . . . . 12 ⊢ (𝜑 → dom (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐶)) = (𝑀(,)𝑁))
83	65, 82	feq12d 6656	. . . . . . . . . . 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 231	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐷):(𝑀(,)𝑁)⟶ℝ)
85	84	fvmptelcdm 7061	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 𝐷 ∈ ℝ)
86	58, 48	sylan2 593	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 𝐴 ∈ ℝ)
87	86	fmpttd 7063	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴):(𝑀(,)𝑁)⟶ℝ)
88		dvfre 25315	. . . . . . . . . . . 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 5861	. . . . . . . . . . . . 13 ⊢ (𝜑 → dom (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)) = dom (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵))
91	69	ralrimiva 3143	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑥 ∈ (𝑀(,)𝑁)𝐵 ∈ V)
92		dmmptg 6194	. . . . . . . . . . . . . 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 2776	. . . . . . . . . . . 12 ⊢ (𝜑 → dom (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)) = (𝑀(,)𝑁))
95	70, 94	feq12d 6656	. . . . . . . . . . 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 231	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵):(𝑀(,)𝑁)⟶ℝ)
97	96	fvmptelcdm 7061	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 𝐵 ∈ ℝ)
98	85, 97	resubcld 11583	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀(,)𝑁)) → (𝐷 − 𝐵) ∈ ℝ)
99	85, 97	subge0d 11745	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀(,)𝑁)) → (0 ≤ (𝐷 − 𝐵) ↔ 𝐵 ≤ 𝐷))
100	61, 99	mpbird 256	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 0 ≤ (𝐷 − 𝐵))
101		elrege0 13371	. . . . . . . 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 7064	. . . . . 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 25370	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ (𝑀[,]𝑁) ↦ (𝐶 − 𝐴))‘𝑋) ≤ ((𝑥 ∈ (𝑀[,]𝑁) ↦ (𝐶 − 𝐴))‘𝑌))
106	20, 25	oveq12d 7375	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝐶 − 𝐴) = (𝑄 − 𝑃))
107		eqid 2736	. . . . . . 7 ⊢ (𝑥 ∈ (𝑀[,]𝑁) ↦ (𝐶 − 𝐴)) = (𝑥 ∈ (𝑀[,]𝑁) ↦ (𝐶 − 𝐴))
108		ovex 7390	. . . . . . 7 ⊢ (𝐶 − 𝐴) ∈ V
109	106, 107, 108	fvmpt3i 6953	. . . . . 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 7375	. . . . . . 7 ⊢ (𝑥 = 𝑌 → (𝐶 − 𝐴) = (𝑆 − 𝑅))
112	111, 107, 108	fvmpt3i 6953	. . . . . 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 5136	. . . 4 ⊢ (𝜑 → (𝑄 − 𝑃) ≤ (𝑆 − 𝑅))
115	23, 27, 36, 114	subled 11758	. . 3 ⊢ (𝜑 → (𝑄 − (𝑆 − 𝑅)) ≤ 𝑃)
116	35, 115	eqbrtrd 5127	. 2 ⊢ (𝜑 → (𝑅 − (𝑆 − 𝑄)) ≤ 𝑃)
117	10, 24, 27, 116	subled 11758	1 ⊢ (𝜑 → (𝑅 − 𝑃) ≤ (𝑆 − 𝑄))

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 ran crn 5634 ⟶wf 6492 ‘cfv 6496 (class class class)co 7357 ℂcc 11049 ℝcr 11050 0cc0 11051 + caddc 11054 +∞cpnf 11186 ≤ cle 11190 − cmin 11385 (,)cioo 13264 [,)cico 13266 [,]cicc 13267 TopOpenctopn 17303 topGenctg 17319 ℂ_fldccnfld 20796 intcnt 22368 –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-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-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: dvfsumle 25385 dvfsumlem2 25391 loglesqrt 26111 dvle2 40529

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