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Theorem mvth 25500

Description: The Mean Value Theorem. If 𝐹 is a real continuous function on [𝐴, 𝐵] which is differentiable on (𝐴, 𝐵), then there is some 𝑥 ∈ (𝐴, 𝐵) such that (ℝ D 𝐹)‘𝑥 is equal to the average slope over [𝐴, 𝐵]. This is Metamath 100 proof #75. (Contributed by Mario Carneiro, 1-Sep-2014.) (Proof shortened by Mario Carneiro, 29-Dec-2016.)

**Hypotheses**
Ref	Expression
mvth.a	⊢ (𝜑 → 𝐴 ∈ ℝ)
mvth.b	⊢ (𝜑 → 𝐵 ∈ ℝ)
mvth.lt	⊢ (𝜑 → 𝐴 < 𝐵)
mvth.f	⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ))
mvth.d	⊢ (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))

**Assertion**
Ref	Expression
mvth	⊢ (𝜑 → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = (((𝐹‘𝐵) − (𝐹‘𝐴)) / (𝐵 − 𝐴)))

Distinct variable groups: 𝑥,𝐴 𝑥,𝐵 𝑥,𝐹 𝜑,𝑥

Proof of Theorem mvth Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mvth.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ)
2		mvth.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ)
3		mvth.lt	. . 3 ⊢ (𝜑 → 𝐴 < 𝐵)
4		mvth.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ))
5		mptresid 6048	. . . 4 ⊢ ( I ↾ (𝐴[,]𝐵)) = (𝑧 ∈ (𝐴[,]𝐵) ↦ 𝑧)
6		iccssre 13402	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
7	1, 2, 6	syl2anc 584	. . . . 5 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
8		ax-resscn 11163	. . . . 5 ⊢ ℝ ⊆ ℂ
9		cncfmptid 24420	. . . . 5 ⊢ (((𝐴[,]𝐵) ⊆ ℝ ∧ ℝ ⊆ ℂ) → (𝑧 ∈ (𝐴[,]𝐵) ↦ 𝑧) ∈ ((𝐴[,]𝐵)–cn→ℝ))
10	7, 8, 9	sylancl 586	. . . 4 ⊢ (𝜑 → (𝑧 ∈ (𝐴[,]𝐵) ↦ 𝑧) ∈ ((𝐴[,]𝐵)–cn→ℝ))
11	5, 10	eqeltrid 2837	. . 3 ⊢ (𝜑 → ( I ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℝ))
12		mvth.d	. . 3 ⊢ (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))
13	5	eqcomi 2741	. . . . . . 7 ⊢ (𝑧 ∈ (𝐴[,]𝐵) ↦ 𝑧) = ( I ↾ (𝐴[,]𝐵))
14	13	oveq2i 7416	. . . . . 6 ⊢ (ℝ D (𝑧 ∈ (𝐴[,]𝐵) ↦ 𝑧)) = (ℝ D ( I ↾ (𝐴[,]𝐵)))
15		reelprrecn 11198	. . . . . . . 8 ⊢ ℝ ∈ {ℝ, ℂ}
16	15	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
17		simpr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → 𝑧 ∈ ℝ)
18	17	recnd 11238	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → 𝑧 ∈ ℂ)
19		1red 11211	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → 1 ∈ ℝ)
20	16	dvmptid 25465	. . . . . . 7 ⊢ (𝜑 → (ℝ D (𝑧 ∈ ℝ ↦ 𝑧)) = (𝑧 ∈ ℝ ↦ 1))
21		eqid 2732	. . . . . . . 8 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
22	21	tgioo2 24310	. . . . . . 7 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
23		iccntr 24328	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
24	1, 2, 23	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
25	16, 18, 19, 20, 7, 22, 21, 24	dvmptres2 25470	. . . . . 6 ⊢ (𝜑 → (ℝ D (𝑧 ∈ (𝐴[,]𝐵) ↦ 𝑧)) = (𝑧 ∈ (𝐴(,)𝐵) ↦ 1))
26	14, 25	eqtr3id 2786	. . . . 5 ⊢ (𝜑 → (ℝ D ( I ↾ (𝐴[,]𝐵))) = (𝑧 ∈ (𝐴(,)𝐵) ↦ 1))
27	26	dmeqd 5903	. . . 4 ⊢ (𝜑 → dom (ℝ D ( I ↾ (𝐴[,]𝐵))) = dom (𝑧 ∈ (𝐴(,)𝐵) ↦ 1))
28		1ex 11206	. . . . 5 ⊢ 1 ∈ V
29		eqid 2732	. . . . 5 ⊢ (𝑧 ∈ (𝐴(,)𝐵) ↦ 1) = (𝑧 ∈ (𝐴(,)𝐵) ↦ 1)
30	28, 29	dmmpti 6691	. . . 4 ⊢ dom (𝑧 ∈ (𝐴(,)𝐵) ↦ 1) = (𝐴(,)𝐵)
31	27, 30	eqtrdi 2788	. . 3 ⊢ (𝜑 → dom (ℝ D ( I ↾ (𝐴[,]𝐵))) = (𝐴(,)𝐵))
32	1, 2, 3, 4, 11, 12, 31	cmvth 25499	. 2 ⊢ (𝜑 → ∃𝑥 ∈ (𝐴(,)𝐵)(((𝐹‘𝐵) − (𝐹‘𝐴)) · ((ℝ D ( I ↾ (𝐴[,]𝐵)))‘𝑥)) = (((( I ↾ (𝐴[,]𝐵))‘𝐵) − (( I ↾ (𝐴[,]𝐵))‘𝐴)) · ((ℝ D 𝐹)‘𝑥)))
33	1	rexrd 11260	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
34	2	rexrd 11260	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
35	1, 2, 3	ltled 11358	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
36		ubicc2 13438	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ (𝐴[,]𝐵))
37	33, 34, 35, 36	syl3anc 1371	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ (𝐴[,]𝐵))
38		fvresi 7167	. . . . . . . . . 10 ⊢ (𝐵 ∈ (𝐴[,]𝐵) → (( I ↾ (𝐴[,]𝐵))‘𝐵) = 𝐵)
39	37, 38	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (( I ↾ (𝐴[,]𝐵))‘𝐵) = 𝐵)
40		lbicc2 13437	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ (𝐴[,]𝐵))
41	33, 34, 35, 40	syl3anc 1371	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ (𝐴[,]𝐵))
42		fvresi 7167	. . . . . . . . . 10 ⊢ (𝐴 ∈ (𝐴[,]𝐵) → (( I ↾ (𝐴[,]𝐵))‘𝐴) = 𝐴)
43	41, 42	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (( I ↾ (𝐴[,]𝐵))‘𝐴) = 𝐴)
44	39, 43	oveq12d 7423	. . . . . . . 8 ⊢ (𝜑 → ((( I ↾ (𝐴[,]𝐵))‘𝐵) − (( I ↾ (𝐴[,]𝐵))‘𝐴)) = (𝐵 − 𝐴))
45	44	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((( I ↾ (𝐴[,]𝐵))‘𝐵) − (( I ↾ (𝐴[,]𝐵))‘𝐴)) = (𝐵 − 𝐴))
46	45	oveq1d 7420	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((( I ↾ (𝐴[,]𝐵))‘𝐵) − (( I ↾ (𝐴[,]𝐵))‘𝐴)) · ((ℝ D 𝐹)‘𝑥)) = ((𝐵 − 𝐴) · ((ℝ D 𝐹)‘𝑥)))
47	26	fveq1d 6890	. . . . . . . . 9 ⊢ (𝜑 → ((ℝ D ( I ↾ (𝐴[,]𝐵)))‘𝑥) = ((𝑧 ∈ (𝐴(,)𝐵) ↦ 1)‘𝑥))
48		eqidd 2733	. . . . . . . . . 10 ⊢ (𝑧 = 𝑥 → 1 = 1)
49	48, 29, 28	fvmpt3i 7000	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐴(,)𝐵) → ((𝑧 ∈ (𝐴(,)𝐵) ↦ 1)‘𝑥) = 1)
50	47, 49	sylan9eq 2792	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((ℝ D ( I ↾ (𝐴[,]𝐵)))‘𝑥) = 1)
51	50	oveq2d 7421	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((𝐹‘𝐵) − (𝐹‘𝐴)) · ((ℝ D ( I ↾ (𝐴[,]𝐵)))‘𝑥)) = (((𝐹‘𝐵) − (𝐹‘𝐴)) · 1))
52		cncff 24400	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ) → 𝐹:(𝐴[,]𝐵)⟶ℝ)
53	4, 52	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹:(𝐴[,]𝐵)⟶ℝ)
54	53, 37	ffvelcdmd 7084	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹‘𝐵) ∈ ℝ)
55	53, 41	ffvelcdmd 7084	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹‘𝐴) ∈ ℝ)
56	54, 55	resubcld 11638	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐹‘𝐵) − (𝐹‘𝐴)) ∈ ℝ)
57	56	recnd 11238	. . . . . . . . 9 ⊢ (𝜑 → ((𝐹‘𝐵) − (𝐹‘𝐴)) ∈ ℂ)
58	57	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((𝐹‘𝐵) − (𝐹‘𝐴)) ∈ ℂ)
59	58	mulridd 11227	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((𝐹‘𝐵) − (𝐹‘𝐴)) · 1) = ((𝐹‘𝐵) − (𝐹‘𝐴)))
60	51, 59	eqtrd 2772	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((𝐹‘𝐵) − (𝐹‘𝐴)) · ((ℝ D ( I ↾ (𝐴[,]𝐵)))‘𝑥)) = ((𝐹‘𝐵) − (𝐹‘𝐴)))
61	46, 60	eqeq12d 2748	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((((( I ↾ (𝐴[,]𝐵))‘𝐵) − (( I ↾ (𝐴[,]𝐵))‘𝐴)) · ((ℝ D 𝐹)‘𝑥)) = (((𝐹‘𝐵) − (𝐹‘𝐴)) · ((ℝ D ( I ↾ (𝐴[,]𝐵)))‘𝑥)) ↔ ((𝐵 − 𝐴) · ((ℝ D 𝐹)‘𝑥)) = ((𝐹‘𝐵) − (𝐹‘𝐴))))
62	2, 1	resubcld 11638	. . . . . . . 8 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ)
63	62	recnd 11238	. . . . . . 7 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℂ)
64	63	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝐵 − 𝐴) ∈ ℂ)
65		dvf 25415	. . . . . . . 8 ⊢ (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℂ
66	12	feq2d 6700	. . . . . . . 8 ⊢ (𝜑 → ((ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℂ ↔ (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℂ))
67	65, 66	mpbii 232	. . . . . . 7 ⊢ (𝜑 → (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℂ)
68	67	ffvelcdmda 7083	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑥) ∈ ℂ)
69	1, 2	posdifd 11797	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ 0 < (𝐵 − 𝐴)))
70	3, 69	mpbid 231	. . . . . . . 8 ⊢ (𝜑 → 0 < (𝐵 − 𝐴))
71	70	gt0ne0d 11774	. . . . . . 7 ⊢ (𝜑 → (𝐵 − 𝐴) ≠ 0)
72	71	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝐵 − 𝐴) ≠ 0)
73	58, 64, 68, 72	divmuld 12008	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((((𝐹‘𝐵) − (𝐹‘𝐴)) / (𝐵 − 𝐴)) = ((ℝ D 𝐹)‘𝑥) ↔ ((𝐵 − 𝐴) · ((ℝ D 𝐹)‘𝑥)) = ((𝐹‘𝐵) − (𝐹‘𝐴))))
74	61, 73	bitr4d 281	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((((( I ↾ (𝐴[,]𝐵))‘𝐵) − (( I ↾ (𝐴[,]𝐵))‘𝐴)) · ((ℝ D 𝐹)‘𝑥)) = (((𝐹‘𝐵) − (𝐹‘𝐴)) · ((ℝ D ( I ↾ (𝐴[,]𝐵)))‘𝑥)) ↔ (((𝐹‘𝐵) − (𝐹‘𝐴)) / (𝐵 − 𝐴)) = ((ℝ D 𝐹)‘𝑥)))
75		eqcom 2739	. . . 4 ⊢ ((((𝐹‘𝐵) − (𝐹‘𝐴)) · ((ℝ D ( I ↾ (𝐴[,]𝐵)))‘𝑥)) = (((( I ↾ (𝐴[,]𝐵))‘𝐵) − (( I ↾ (𝐴[,]𝐵))‘𝐴)) · ((ℝ D 𝐹)‘𝑥)) ↔ (((( I ↾ (𝐴[,]𝐵))‘𝐵) − (( I ↾ (𝐴[,]𝐵))‘𝐴)) · ((ℝ D 𝐹)‘𝑥)) = (((𝐹‘𝐵) − (𝐹‘𝐴)) · ((ℝ D ( I ↾ (𝐴[,]𝐵)))‘𝑥)))
76		eqcom 2739	. . . 4 ⊢ (((ℝ D 𝐹)‘𝑥) = (((𝐹‘𝐵) − (𝐹‘𝐴)) / (𝐵 − 𝐴)) ↔ (((𝐹‘𝐵) − (𝐹‘𝐴)) / (𝐵 − 𝐴)) = ((ℝ D 𝐹)‘𝑥))
77	74, 75, 76	3bitr4g 313	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((((𝐹‘𝐵) − (𝐹‘𝐴)) · ((ℝ D ( I ↾ (𝐴[,]𝐵)))‘𝑥)) = (((( I ↾ (𝐴[,]𝐵))‘𝐵) − (( I ↾ (𝐴[,]𝐵))‘𝐴)) · ((ℝ D 𝐹)‘𝑥)) ↔ ((ℝ D 𝐹)‘𝑥) = (((𝐹‘𝐵) − (𝐹‘𝐴)) / (𝐵 − 𝐴))))
78	77	rexbidva 3176	. 2 ⊢ (𝜑 → (∃𝑥 ∈ (𝐴(,)𝐵)(((𝐹‘𝐵) − (𝐹‘𝐴)) · ((ℝ D ( I ↾ (𝐴[,]𝐵)))‘𝑥)) = (((( I ↾ (𝐴[,]𝐵))‘𝐵) − (( I ↾ (𝐴[,]𝐵))‘𝐴)) · ((ℝ D 𝐹)‘𝑥)) ↔ ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = (((𝐹‘𝐵) − (𝐹‘𝐴)) / (𝐵 − 𝐴))))
79	32, 78	mpbid 231	1 ⊢ (𝜑 → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = (((𝐹‘𝐵) − (𝐹‘𝐴)) / (𝐵 − 𝐴)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ∃wrex 3070 ⊆ wss 3947 {cpr 4629 class class class wbr 5147 ↦ cmpt 5230 I cid 5572 dom cdm 5675 ran crn 5676 ↾ cres 5677 ⟶wf 6536 ‘cfv 6540 (class class class)co 7405 ℂcc 11104 ℝcr 11105 0cc0 11106 1c1 11107 · cmul 11111 ℝ^*cxr 11243 < clt 11244 ≤ cle 11245 − cmin 11440 / cdiv 11867 (,)cioo 13320 [,]cicc 13323 TopOpenctopn 17363 topGenctg 17379 ℂ_fldccnfld 20936 intcnt 22512 –cn→ccncf 24383 D cdv 25371

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 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185 ax-mulf 11186

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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8143 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-er 8699 df-map 8818 df-pm 8819 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-fi 9402 df-sup 9433 df-inf 9434 df-oi 9501 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-ioo 13324 df-ico 13326 df-icc 13327 df-fz 13481 df-fzo 13624 df-seq 13963 df-exp 14024 df-hash 14287 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-starv 17208 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-unif 17216 df-hom 17217 df-cco 17218 df-rest 17364 df-topn 17365 df-0g 17383 df-gsum 17384 df-topgen 17385 df-pt 17386 df-prds 17389 df-xrs 17444 df-qtop 17449 df-imas 17450 df-xps 17452 df-mre 17526 df-mrc 17527 df-acs 17529 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-submnd 18668 df-mulg 18945 df-cntz 19175 df-cmn 19644 df-psmet 20928 df-xmet 20929 df-met 20930 df-bl 20931 df-mopn 20932 df-fbas 20933 df-fg 20934 df-cnfld 20937 df-top 22387 df-topon 22404 df-topsp 22426 df-bases 22440 df-cld 22514 df-ntr 22515 df-cls 22516 df-nei 22593 df-lp 22631 df-perf 22632 df-cn 22722 df-cnp 22723 df-haus 22810 df-cmp 22882 df-tx 23057 df-hmeo 23250 df-fil 23341 df-fm 23433 df-flim 23434 df-flf 23435 df-xms 23817 df-ms 23818 df-tms 23819 df-cncf 24385 df-limc 25374 df-dv 25375

This theorem is referenced by: dvlip 25501 c1liplem1 25504 dvgt0lem1 25510 dvcvx 25528 dvbdfbdioolem1 44630

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