dvne0 - Metamath Proof Explorer

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Theorem dvne0 25923

Description: A function on a closed interval with nonzero derivative is either monotone increasing or monotone decreasing. (Contributed by Mario Carneiro, 19-Feb-2015.)

**Hypotheses**
Ref	Expression
dvne0.a	⊢ (𝜑 → 𝐴 ∈ ℝ)
dvne0.b	⊢ (𝜑 → 𝐵 ∈ ℝ)
dvne0.f	⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ))
dvne0.d	⊢ (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))
dvne0.z	⊢ (𝜑 → ¬ 0 ∈ ran (ℝ D 𝐹))

**Assertion**
Ref	Expression
dvne0	⊢ (𝜑 → (𝐹 Isom < , < ((𝐴[,]𝐵), ran 𝐹) ∨ 𝐹 Isom < , ^◡ < ((𝐴[,]𝐵), ran 𝐹)))

Proof of Theorem dvne0 Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dvne0.z	. . . . . . . . . . . 12 ⊢ (𝜑 → ¬ 0 ∈ ran (ℝ D 𝐹))
2		eleq1 2817	. . . . . . . . . . . . 13 ⊢ (𝑥 = 0 → (𝑥 ∈ ran (ℝ D 𝐹) ↔ 0 ∈ ran (ℝ D 𝐹)))
3	2	notbid 318	. . . . . . . . . . . 12 ⊢ (𝑥 = 0 → (¬ 𝑥 ∈ ran (ℝ D 𝐹) ↔ ¬ 0 ∈ ran (ℝ D 𝐹)))
4	1, 3	syl5ibrcom 247	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 = 0 → ¬ 𝑥 ∈ ran (ℝ D 𝐹)))
5	4	necon2ad 2941	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ ran (ℝ D 𝐹) → 𝑥 ≠ 0))
6	5	imp 406	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (ℝ D 𝐹)) → 𝑥 ≠ 0)
7		dvne0.f	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ))
8		cncff 24793	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ) → 𝐹:(𝐴[,]𝐵)⟶ℝ)
9	7, 8	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹:(𝐴[,]𝐵)⟶ℝ)
10		dvne0.a	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐴 ∈ ℝ)
11		dvne0.b	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐵 ∈ ℝ)
12		iccssre 13397	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
13	10, 11, 12	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
14		dvfre 25862	. . . . . . . . . . . . . 14 ⊢ ((𝐹:(𝐴[,]𝐵)⟶ℝ ∧ (𝐴[,]𝐵) ⊆ ℝ) → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ)
15	9, 13, 14	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (𝜑 → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ)
16	15	frnd 6699	. . . . . . . . . . . 12 ⊢ (𝜑 → ran (ℝ D 𝐹) ⊆ ℝ)
17	16	sselda 3949	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (ℝ D 𝐹)) → 𝑥 ∈ ℝ)
18		0re 11183	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ
19		lttri2 11263	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ ∧ 0 ∈ ℝ) → (𝑥 ≠ 0 ↔ (𝑥 < 0 ∨ 0 < 𝑥)))
20	17, 18, 19	sylancl 586	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (ℝ D 𝐹)) → (𝑥 ≠ 0 ↔ (𝑥 < 0 ∨ 0 < 𝑥)))
21		0xr 11228	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℝ^*
22		elioomnf 13412	. . . . . . . . . . . . . 14 ⊢ (0 ∈ ℝ^* → (𝑥 ∈ (-∞(,)0) ↔ (𝑥 ∈ ℝ ∧ 𝑥 < 0)))
23	21, 22	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (-∞(,)0) ↔ (𝑥 ∈ ℝ ∧ 𝑥 < 0))
24	23	baib 535	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ → (𝑥 ∈ (-∞(,)0) ↔ 𝑥 < 0))
25		elrp 12960	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ⁺ ↔ (𝑥 ∈ ℝ ∧ 0 < 𝑥))
26	25	baib 535	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ → (𝑥 ∈ ℝ⁺ ↔ 0 < 𝑥))
27	24, 26	orbi12d 918	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ → ((𝑥 ∈ (-∞(,)0) ∨ 𝑥 ∈ ℝ⁺) ↔ (𝑥 < 0 ∨ 0 < 𝑥)))
28	17, 27	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (ℝ D 𝐹)) → ((𝑥 ∈ (-∞(,)0) ∨ 𝑥 ∈ ℝ⁺) ↔ (𝑥 < 0 ∨ 0 < 𝑥)))
29	20, 28	bitr4d 282	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (ℝ D 𝐹)) → (𝑥 ≠ 0 ↔ (𝑥 ∈ (-∞(,)0) ∨ 𝑥 ∈ ℝ⁺)))
30	6, 29	mpbid 232	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (ℝ D 𝐹)) → (𝑥 ∈ (-∞(,)0) ∨ 𝑥 ∈ ℝ⁺))
31		elun 4119	. . . . . . . 8 ⊢ (𝑥 ∈ ((-∞(,)0) ∪ ℝ⁺) ↔ (𝑥 ∈ (-∞(,)0) ∨ 𝑥 ∈ ℝ⁺))
32	30, 31	sylibr 234	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (ℝ D 𝐹)) → 𝑥 ∈ ((-∞(,)0) ∪ ℝ⁺))
33	32	ex 412	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ran (ℝ D 𝐹) → 𝑥 ∈ ((-∞(,)0) ∪ ℝ⁺)))
34	33	ssrdv 3955	. . . . 5 ⊢ (𝜑 → ran (ℝ D 𝐹) ⊆ ((-∞(,)0) ∪ ℝ⁺))
35		disjssun 4434	. . . . 5 ⊢ ((ran (ℝ D 𝐹) ∩ (-∞(,)0)) = ∅ → (ran (ℝ D 𝐹) ⊆ ((-∞(,)0) ∪ ℝ⁺) ↔ ran (ℝ D 𝐹) ⊆ ℝ⁺))
36	34, 35	syl5ibcom 245	. . . 4 ⊢ (𝜑 → ((ran (ℝ D 𝐹) ∩ (-∞(,)0)) = ∅ → ran (ℝ D 𝐹) ⊆ ℝ⁺))
37	36	imp 406	. . 3 ⊢ ((𝜑 ∧ (ran (ℝ D 𝐹) ∩ (-∞(,)0)) = ∅) → ran (ℝ D 𝐹) ⊆ ℝ⁺)
38	10	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ran (ℝ D 𝐹) ⊆ ℝ⁺) → 𝐴 ∈ ℝ)
39	11	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ran (ℝ D 𝐹) ⊆ ℝ⁺) → 𝐵 ∈ ℝ)
40	7	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ran (ℝ D 𝐹) ⊆ ℝ⁺) → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ))
41		dvne0.d	. . . . . . . . . 10 ⊢ (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))
42	41	feq2d 6675	. . . . . . . . 9 ⊢ (𝜑 → ((ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ ↔ (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℝ))
43	15, 42	mpbid 232	. . . . . . . 8 ⊢ (𝜑 → (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℝ)
44	43	ffnd 6692	. . . . . . 7 ⊢ (𝜑 → (ℝ D 𝐹) Fn (𝐴(,)𝐵))
45	44	anim1i 615	. . . . . 6 ⊢ ((𝜑 ∧ ran (ℝ D 𝐹) ⊆ ℝ⁺) → ((ℝ D 𝐹) Fn (𝐴(,)𝐵) ∧ ran (ℝ D 𝐹) ⊆ ℝ⁺))
46		df-f 6518	. . . . . 6 ⊢ ((ℝ D 𝐹):(𝐴(,)𝐵)⟶ℝ⁺ ↔ ((ℝ D 𝐹) Fn (𝐴(,)𝐵) ∧ ran (ℝ D 𝐹) ⊆ ℝ⁺))
47	45, 46	sylibr 234	. . . . 5 ⊢ ((𝜑 ∧ ran (ℝ D 𝐹) ⊆ ℝ⁺) → (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℝ⁺)
48	38, 39, 40, 47	dvgt0 25916	. . . 4 ⊢ ((𝜑 ∧ ran (ℝ D 𝐹) ⊆ ℝ⁺) → 𝐹 Isom < , < ((𝐴[,]𝐵), ran 𝐹))
49	48	orcd 873	. . 3 ⊢ ((𝜑 ∧ ran (ℝ D 𝐹) ⊆ ℝ⁺) → (𝐹 Isom < , < ((𝐴[,]𝐵), ran 𝐹) ∨ 𝐹 Isom < , ^◡ < ((𝐴[,]𝐵), ran 𝐹)))
50	37, 49	syldan 591	. 2 ⊢ ((𝜑 ∧ (ran (ℝ D 𝐹) ∩ (-∞(,)0)) = ∅) → (𝐹 Isom < , < ((𝐴[,]𝐵), ran 𝐹) ∨ 𝐹 Isom < , ^◡ < ((𝐴[,]𝐵), ran 𝐹)))
51		n0 4319	. . . 4 ⊢ ((ran (ℝ D 𝐹) ∩ (-∞(,)0)) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (ran (ℝ D 𝐹) ∩ (-∞(,)0)))
52		elin 3933	. . . . . 6 ⊢ (𝑥 ∈ (ran (ℝ D 𝐹) ∩ (-∞(,)0)) ↔ (𝑥 ∈ ran (ℝ D 𝐹) ∧ 𝑥 ∈ (-∞(,)0)))
53		fvelrnb 6924	. . . . . . . . 9 ⊢ ((ℝ D 𝐹) Fn (𝐴(,)𝐵) → (𝑥 ∈ ran (ℝ D 𝐹) ↔ ∃𝑦 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑦) = 𝑥))
54	44, 53	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ ran (ℝ D 𝐹) ↔ ∃𝑦 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑦) = 𝑥))
55	10	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ ((ℝ D 𝐹)‘𝑦) ∈ (-∞(,)0))) → 𝐴 ∈ ℝ)
56	11	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ ((ℝ D 𝐹)‘𝑦) ∈ (-∞(,)0))) → 𝐵 ∈ ℝ)
57	7	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ ((ℝ D 𝐹)‘𝑦) ∈ (-∞(,)0))) → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ))
58	44	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ ((ℝ D 𝐹)‘𝑦) ∈ (-∞(,)0))) → (ℝ D 𝐹) Fn (𝐴(,)𝐵))
59	43	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ ((ℝ D 𝐹)‘𝑦) ∈ (-∞(,)0))) → (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℝ)
60	59	ffvelcdmda 7059	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ ((ℝ D 𝐹)‘𝑦) ∈ (-∞(,)0))) ∧ 𝑧 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑧) ∈ ℝ)
61	1	ad2antrr 726	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ ((ℝ D 𝐹)‘𝑦) ∈ (-∞(,)0))) ∧ 𝑧 ∈ (𝐴(,)𝐵)) → ¬ 0 ∈ ran (ℝ D 𝐹))
62		simplrl 776	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ ((ℝ D 𝐹)‘𝑦) ∈ (-∞(,)0))) ∧ (𝑧 ∈ (𝐴(,)𝐵) ∧ 0 ≤ ((ℝ D 𝐹)‘𝑧))) → 𝑦 ∈ (𝐴(,)𝐵))
63		simprl 770	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ ((ℝ D 𝐹)‘𝑦) ∈ (-∞(,)0))) ∧ (𝑧 ∈ (𝐴(,)𝐵) ∧ 0 ≤ ((ℝ D 𝐹)‘𝑧))) → 𝑧 ∈ (𝐴(,)𝐵))
64		ioossicc 13401	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
65		rescncf 24797	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐴(,)𝐵) ⊆ (𝐴[,]𝐵) → (𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ) → (𝐹 ↾ (𝐴(,)𝐵)) ∈ ((𝐴(,)𝐵)–cn→ℝ)))
66	64, 7, 65	mpsyl 68	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (𝐹 ↾ (𝐴(,)𝐵)) ∈ ((𝐴(,)𝐵)–cn→ℝ))
67	66	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ ((ℝ D 𝐹)‘𝑦) ∈ (-∞(,)0))) ∧ (𝑧 ∈ (𝐴(,)𝐵) ∧ 0 ≤ ((ℝ D 𝐹)‘𝑧))) → (𝐹 ↾ (𝐴(,)𝐵)) ∈ ((𝐴(,)𝐵)–cn→ℝ))
68		ax-resscn 11132	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ℝ ⊆ ℂ
69	68	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → ℝ ⊆ ℂ)
70		fss 6707	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝐹:(𝐴[,]𝐵)⟶ℝ ∧ ℝ ⊆ ℂ) → 𝐹:(𝐴[,]𝐵)⟶ℂ)
71	9, 68, 70	sylancl 586	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → 𝐹:(𝐴[,]𝐵)⟶ℂ)
72	64, 13	sstrid 3961	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℝ)
73		eqid 2730	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
74		tgioo4 24700	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
75	73, 74	dvres 25819	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((ℝ ⊆ ℂ ∧ 𝐹:(𝐴[,]𝐵)⟶ℂ) ∧ ((𝐴[,]𝐵) ⊆ ℝ ∧ (𝐴(,)𝐵) ⊆ ℝ)) → (ℝ D (𝐹 ↾ (𝐴(,)𝐵))) = ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘(𝐴(,)𝐵))))
76	69, 71, 13, 72, 75	syl22anc 838	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → (ℝ D (𝐹 ↾ (𝐴(,)𝐵))) = ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘(𝐴(,)𝐵))))
77		retop 24656	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (topGen‘ran (,)) ∈ Top
78		iooretop 24660	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝐴(,)𝐵) ∈ (topGen‘ran (,))
79		isopn3i 22976	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((topGen‘ran (,)) ∈ Top ∧ (𝐴(,)𝐵) ∈ (topGen‘ran (,))) → ((int‘(topGen‘ran (,)))‘(𝐴(,)𝐵)) = (𝐴(,)𝐵))
80	77, 78, 79	mp2an 692	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((int‘(topGen‘ran (,)))‘(𝐴(,)𝐵)) = (𝐴(,)𝐵)
81	80	reseq2i 5950	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘(𝐴(,)𝐵))) = ((ℝ D 𝐹) ↾ (𝐴(,)𝐵))
82		fnresdm 6640	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((ℝ D 𝐹) Fn (𝐴(,)𝐵) → ((ℝ D 𝐹) ↾ (𝐴(,)𝐵)) = (ℝ D 𝐹))
83	44, 82	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → ((ℝ D 𝐹) ↾ (𝐴(,)𝐵)) = (ℝ D 𝐹))
84	81, 83	eqtrid 2777	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘(𝐴(,)𝐵))) = (ℝ D 𝐹))
85	76, 84	eqtrd 2765	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → (ℝ D (𝐹 ↾ (𝐴(,)𝐵))) = (ℝ D 𝐹))
86	85	dmeqd 5872	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → dom (ℝ D (𝐹 ↾ (𝐴(,)𝐵))) = dom (ℝ D 𝐹))
87	86, 41	eqtrd 2765	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → dom (ℝ D (𝐹 ↾ (𝐴(,)𝐵))) = (𝐴(,)𝐵))
88	87	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ ((ℝ D 𝐹)‘𝑦) ∈ (-∞(,)0))) ∧ (𝑧 ∈ (𝐴(,)𝐵) ∧ 0 ≤ ((ℝ D 𝐹)‘𝑧))) → dom (ℝ D (𝐹 ↾ (𝐴(,)𝐵))) = (𝐴(,)𝐵))
89	62, 63, 67, 88	dvivth 25922	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ ((ℝ D 𝐹)‘𝑦) ∈ (-∞(,)0))) ∧ (𝑧 ∈ (𝐴(,)𝐵) ∧ 0 ≤ ((ℝ D 𝐹)‘𝑧))) → (((ℝ D (𝐹 ↾ (𝐴(,)𝐵)))‘𝑦)[,]((ℝ D (𝐹 ↾ (𝐴(,)𝐵)))‘𝑧)) ⊆ ran (ℝ D (𝐹 ↾ (𝐴(,)𝐵))))
90	85	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ ((ℝ D 𝐹)‘𝑦) ∈ (-∞(,)0))) ∧ (𝑧 ∈ (𝐴(,)𝐵) ∧ 0 ≤ ((ℝ D 𝐹)‘𝑧))) → (ℝ D (𝐹 ↾ (𝐴(,)𝐵))) = (ℝ D 𝐹))
91	90	fveq1d 6863	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ ((ℝ D 𝐹)‘𝑦) ∈ (-∞(,)0))) ∧ (𝑧 ∈ (𝐴(,)𝐵) ∧ 0 ≤ ((ℝ D 𝐹)‘𝑧))) → ((ℝ D (𝐹 ↾ (𝐴(,)𝐵)))‘𝑦) = ((ℝ D 𝐹)‘𝑦))
92	90	fveq1d 6863	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ ((ℝ D 𝐹)‘𝑦) ∈ (-∞(,)0))) ∧ (𝑧 ∈ (𝐴(,)𝐵) ∧ 0 ≤ ((ℝ D 𝐹)‘𝑧))) → ((ℝ D (𝐹 ↾ (𝐴(,)𝐵)))‘𝑧) = ((ℝ D 𝐹)‘𝑧))
93	91, 92	oveq12d 7408	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ ((ℝ D 𝐹)‘𝑦) ∈ (-∞(,)0))) ∧ (𝑧 ∈ (𝐴(,)𝐵) ∧ 0 ≤ ((ℝ D 𝐹)‘𝑧))) → (((ℝ D (𝐹 ↾ (𝐴(,)𝐵)))‘𝑦)[,]((ℝ D (𝐹 ↾ (𝐴(,)𝐵)))‘𝑧)) = (((ℝ D 𝐹)‘𝑦)[,]((ℝ D 𝐹)‘𝑧)))
94	90	rneqd 5905	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ ((ℝ D 𝐹)‘𝑦) ∈ (-∞(,)0))) ∧ (𝑧 ∈ (𝐴(,)𝐵) ∧ 0 ≤ ((ℝ D 𝐹)‘𝑧))) → ran (ℝ D (𝐹 ↾ (𝐴(,)𝐵))) = ran (ℝ D 𝐹))
95	89, 93, 94	3sstr3d 4004	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ ((ℝ D 𝐹)‘𝑦) ∈ (-∞(,)0))) ∧ (𝑧 ∈ (𝐴(,)𝐵) ∧ 0 ≤ ((ℝ D 𝐹)‘𝑧))) → (((ℝ D 𝐹)‘𝑦)[,]((ℝ D 𝐹)‘𝑧)) ⊆ ran (ℝ D 𝐹))
96	18	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ ((ℝ D 𝐹)‘𝑦) ∈ (-∞(,)0))) ∧ (𝑧 ∈ (𝐴(,)𝐵) ∧ 0 ≤ ((ℝ D 𝐹)‘𝑧))) → 0 ∈ ℝ)
97		simplrr 777	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ ((ℝ D 𝐹)‘𝑦) ∈ (-∞(,)0))) ∧ (𝑧 ∈ (𝐴(,)𝐵) ∧ 0 ≤ ((ℝ D 𝐹)‘𝑧))) → ((ℝ D 𝐹)‘𝑦) ∈ (-∞(,)0))
98		elioomnf 13412	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (0 ∈ ℝ^* → (((ℝ D 𝐹)‘𝑦) ∈ (-∞(,)0) ↔ (((ℝ D 𝐹)‘𝑦) ∈ ℝ ∧ ((ℝ D 𝐹)‘𝑦) < 0)))
99	21, 98	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((ℝ D 𝐹)‘𝑦) ∈ (-∞(,)0) ↔ (((ℝ D 𝐹)‘𝑦) ∈ ℝ ∧ ((ℝ D 𝐹)‘𝑦) < 0))
100	97, 99	sylib 218	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ ((ℝ D 𝐹)‘𝑦) ∈ (-∞(,)0))) ∧ (𝑧 ∈ (𝐴(,)𝐵) ∧ 0 ≤ ((ℝ D 𝐹)‘𝑧))) → (((ℝ D 𝐹)‘𝑦) ∈ ℝ ∧ ((ℝ D 𝐹)‘𝑦) < 0))
101	100	simprd 495	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ ((ℝ D 𝐹)‘𝑦) ∈ (-∞(,)0))) ∧ (𝑧 ∈ (𝐴(,)𝐵) ∧ 0 ≤ ((ℝ D 𝐹)‘𝑧))) → ((ℝ D 𝐹)‘𝑦) < 0)
102	100	simpld 494	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ ((ℝ D 𝐹)‘𝑦) ∈ (-∞(,)0))) ∧ (𝑧 ∈ (𝐴(,)𝐵) ∧ 0 ≤ ((ℝ D 𝐹)‘𝑧))) → ((ℝ D 𝐹)‘𝑦) ∈ ℝ)
103		ltle 11269	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((ℝ D 𝐹)‘𝑦) ∈ ℝ ∧ 0 ∈ ℝ) → (((ℝ D 𝐹)‘𝑦) < 0 → ((ℝ D 𝐹)‘𝑦) ≤ 0))
104	102, 18, 103	sylancl 586	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ ((ℝ D 𝐹)‘𝑦) ∈ (-∞(,)0))) ∧ (𝑧 ∈ (𝐴(,)𝐵) ∧ 0 ≤ ((ℝ D 𝐹)‘𝑧))) → (((ℝ D 𝐹)‘𝑦) < 0 → ((ℝ D 𝐹)‘𝑦) ≤ 0))
105	101, 104	mpd 15	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ ((ℝ D 𝐹)‘𝑦) ∈ (-∞(,)0))) ∧ (𝑧 ∈ (𝐴(,)𝐵) ∧ 0 ≤ ((ℝ D 𝐹)‘𝑧))) → ((ℝ D 𝐹)‘𝑦) ≤ 0)
106		simprr 772	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ ((ℝ D 𝐹)‘𝑦) ∈ (-∞(,)0))) ∧ (𝑧 ∈ (𝐴(,)𝐵) ∧ 0 ≤ ((ℝ D 𝐹)‘𝑧))) → 0 ≤ ((ℝ D 𝐹)‘𝑧))
107	63, 60	syldan 591	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ ((ℝ D 𝐹)‘𝑦) ∈ (-∞(,)0))) ∧ (𝑧 ∈ (𝐴(,)𝐵) ∧ 0 ≤ ((ℝ D 𝐹)‘𝑧))) → ((ℝ D 𝐹)‘𝑧) ∈ ℝ)
108		elicc2 13379	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((ℝ D 𝐹)‘𝑦) ∈ ℝ ∧ ((ℝ D 𝐹)‘𝑧) ∈ ℝ) → (0 ∈ (((ℝ D 𝐹)‘𝑦)[,]((ℝ D 𝐹)‘𝑧)) ↔ (0 ∈ ℝ ∧ ((ℝ D 𝐹)‘𝑦) ≤ 0 ∧ 0 ≤ ((ℝ D 𝐹)‘𝑧))))
109	102, 107, 108	syl2anc 584	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ ((ℝ D 𝐹)‘𝑦) ∈ (-∞(,)0))) ∧ (𝑧 ∈ (𝐴(,)𝐵) ∧ 0 ≤ ((ℝ D 𝐹)‘𝑧))) → (0 ∈ (((ℝ D 𝐹)‘𝑦)[,]((ℝ D 𝐹)‘𝑧)) ↔ (0 ∈ ℝ ∧ ((ℝ D 𝐹)‘𝑦) ≤ 0 ∧ 0 ≤ ((ℝ D 𝐹)‘𝑧))))
110	96, 105, 106, 109	mpbir3and 1343	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ ((ℝ D 𝐹)‘𝑦) ∈ (-∞(,)0))) ∧ (𝑧 ∈ (𝐴(,)𝐵) ∧ 0 ≤ ((ℝ D 𝐹)‘𝑧))) → 0 ∈ (((ℝ D 𝐹)‘𝑦)[,]((ℝ D 𝐹)‘𝑧)))
111	95, 110	sseldd 3950	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ ((ℝ D 𝐹)‘𝑦) ∈ (-∞(,)0))) ∧ (𝑧 ∈ (𝐴(,)𝐵) ∧ 0 ≤ ((ℝ D 𝐹)‘𝑧))) → 0 ∈ ran (ℝ D 𝐹))
112	111	expr 456	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ ((ℝ D 𝐹)‘𝑦) ∈ (-∞(,)0))) ∧ 𝑧 ∈ (𝐴(,)𝐵)) → (0 ≤ ((ℝ D 𝐹)‘𝑧) → 0 ∈ ran (ℝ D 𝐹)))
113	61, 112	mtod 198	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ ((ℝ D 𝐹)‘𝑦) ∈ (-∞(,)0))) ∧ 𝑧 ∈ (𝐴(,)𝐵)) → ¬ 0 ≤ ((ℝ D 𝐹)‘𝑧))
114		ltnle 11260	. . . . . . . . . . . . . . . . . 18 ⊢ ((((ℝ D 𝐹)‘𝑧) ∈ ℝ ∧ 0 ∈ ℝ) → (((ℝ D 𝐹)‘𝑧) < 0 ↔ ¬ 0 ≤ ((ℝ D 𝐹)‘𝑧)))
115	60, 18, 114	sylancl 586	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ ((ℝ D 𝐹)‘𝑦) ∈ (-∞(,)0))) ∧ 𝑧 ∈ (𝐴(,)𝐵)) → (((ℝ D 𝐹)‘𝑧) < 0 ↔ ¬ 0 ≤ ((ℝ D 𝐹)‘𝑧)))
116	113, 115	mpbird 257	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ ((ℝ D 𝐹)‘𝑦) ∈ (-∞(,)0))) ∧ 𝑧 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑧) < 0)
117		elioomnf 13412	. . . . . . . . . . . . . . . . 17 ⊢ (0 ∈ ℝ^* → (((ℝ D 𝐹)‘𝑧) ∈ (-∞(,)0) ↔ (((ℝ D 𝐹)‘𝑧) ∈ ℝ ∧ ((ℝ D 𝐹)‘𝑧) < 0)))
118	21, 117	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ (((ℝ D 𝐹)‘𝑧) ∈ (-∞(,)0) ↔ (((ℝ D 𝐹)‘𝑧) ∈ ℝ ∧ ((ℝ D 𝐹)‘𝑧) < 0))
119	60, 116, 118	sylanbrc 583	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ ((ℝ D 𝐹)‘𝑦) ∈ (-∞(,)0))) ∧ 𝑧 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑧) ∈ (-∞(,)0))
120	119	ralrimiva 3126	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ ((ℝ D 𝐹)‘𝑦) ∈ (-∞(,)0))) → ∀𝑧 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑧) ∈ (-∞(,)0))
121		ffnfv 7094	. . . . . . . . . . . . . 14 ⊢ ((ℝ D 𝐹):(𝐴(,)𝐵)⟶(-∞(,)0) ↔ ((ℝ D 𝐹) Fn (𝐴(,)𝐵) ∧ ∀𝑧 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑧) ∈ (-∞(,)0)))
122	58, 120, 121	sylanbrc 583	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ ((ℝ D 𝐹)‘𝑦) ∈ (-∞(,)0))) → (ℝ D 𝐹):(𝐴(,)𝐵)⟶(-∞(,)0))
123	55, 56, 57, 122	dvlt0 25917	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ ((ℝ D 𝐹)‘𝑦) ∈ (-∞(,)0))) → 𝐹 Isom < , ^◡ < ((𝐴[,]𝐵), ran 𝐹))
124	123	olcd 874	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ ((ℝ D 𝐹)‘𝑦) ∈ (-∞(,)0))) → (𝐹 Isom < , < ((𝐴[,]𝐵), ran 𝐹) ∨ 𝐹 Isom < , ^◡ < ((𝐴[,]𝐵), ran 𝐹)))
125	124	expr 456	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (((ℝ D 𝐹)‘𝑦) ∈ (-∞(,)0) → (𝐹 Isom < , < ((𝐴[,]𝐵), ran 𝐹) ∨ 𝐹 Isom < , ^◡ < ((𝐴[,]𝐵), ran 𝐹))))
126		eleq1 2817	. . . . . . . . . . 11 ⊢ (((ℝ D 𝐹)‘𝑦) = 𝑥 → (((ℝ D 𝐹)‘𝑦) ∈ (-∞(,)0) ↔ 𝑥 ∈ (-∞(,)0)))
127	126	imbi1d 341	. . . . . . . . . 10 ⊢ (((ℝ D 𝐹)‘𝑦) = 𝑥 → ((((ℝ D 𝐹)‘𝑦) ∈ (-∞(,)0) → (𝐹 Isom < , < ((𝐴[,]𝐵), ran 𝐹) ∨ 𝐹 Isom < , ^◡ < ((𝐴[,]𝐵), ran 𝐹))) ↔ (𝑥 ∈ (-∞(,)0) → (𝐹 Isom < , < ((𝐴[,]𝐵), ran 𝐹) ∨ 𝐹 Isom < , ^◡ < ((𝐴[,]𝐵), ran 𝐹)))))
128	125, 127	syl5ibcom 245	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (((ℝ D 𝐹)‘𝑦) = 𝑥 → (𝑥 ∈ (-∞(,)0) → (𝐹 Isom < , < ((𝐴[,]𝐵), ran 𝐹) ∨ 𝐹 Isom < , ^◡ < ((𝐴[,]𝐵), ran 𝐹)))))
129	128	rexlimdva 3135	. . . . . . . 8 ⊢ (𝜑 → (∃𝑦 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑦) = 𝑥 → (𝑥 ∈ (-∞(,)0) → (𝐹 Isom < , < ((𝐴[,]𝐵), ran 𝐹) ∨ 𝐹 Isom < , ^◡ < ((𝐴[,]𝐵), ran 𝐹)))))
130	54, 129	sylbid 240	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ ran (ℝ D 𝐹) → (𝑥 ∈ (-∞(,)0) → (𝐹 Isom < , < ((𝐴[,]𝐵), ran 𝐹) ∨ 𝐹 Isom < , ^◡ < ((𝐴[,]𝐵), ran 𝐹)))))
131	130	impd 410	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ ran (ℝ D 𝐹) ∧ 𝑥 ∈ (-∞(,)0)) → (𝐹 Isom < , < ((𝐴[,]𝐵), ran 𝐹) ∨ 𝐹 Isom < , ^◡ < ((𝐴[,]𝐵), ran 𝐹))))
132	52, 131	biimtrid 242	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (ran (ℝ D 𝐹) ∩ (-∞(,)0)) → (𝐹 Isom < , < ((𝐴[,]𝐵), ran 𝐹) ∨ 𝐹 Isom < , ^◡ < ((𝐴[,]𝐵), ran 𝐹))))
133	132	exlimdv 1933	. . . 4 ⊢ (𝜑 → (∃𝑥 𝑥 ∈ (ran (ℝ D 𝐹) ∩ (-∞(,)0)) → (𝐹 Isom < , < ((𝐴[,]𝐵), ran 𝐹) ∨ 𝐹 Isom < , ^◡ < ((𝐴[,]𝐵), ran 𝐹))))
134	51, 133	biimtrid 242	. . 3 ⊢ (𝜑 → ((ran (ℝ D 𝐹) ∩ (-∞(,)0)) ≠ ∅ → (𝐹 Isom < , < ((𝐴[,]𝐵), ran 𝐹) ∨ 𝐹 Isom < , ^◡ < ((𝐴[,]𝐵), ran 𝐹))))
135	134	imp 406	. 2 ⊢ ((𝜑 ∧ (ran (ℝ D 𝐹) ∩ (-∞(,)0)) ≠ ∅) → (𝐹 Isom < , < ((𝐴[,]𝐵), ran 𝐹) ∨ 𝐹 Isom < , ^◡ < ((𝐴[,]𝐵), ran 𝐹)))
136	50, 135	pm2.61dane 3013	1 ⊢ (𝜑 → (𝐹 Isom < , < ((𝐴[,]𝐵), ran 𝐹) ∨ 𝐹 Isom < , ^◡ < ((𝐴[,]𝐵), ran 𝐹)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 = wceq 1540 ∃wex 1779 ∈ wcel 2109 ≠ wne 2926 ∀wral 3045 ∃wrex 3054 ∪ cun 3915 ∩ cin 3916 ⊆ wss 3917 ∅c0 4299 class class class wbr 5110 ^◡ccnv 5640 dom cdm 5641 ran crn 5642 ↾ cres 5643 Fn wfn 6509 ⟶wf 6510 ‘cfv 6514 Isom wiso 6515 (class class class)co 7390 ℂcc 11073 ℝcr 11074 0cc0 11075 -∞cmnf 11213 ℝ^*cxr 11214 < clt 11215 ≤ cle 11216 ℝ⁺crp 12958 (,)cioo 13313 [,]cicc 13316 TopOpenctopn 17391 topGenctg 17407 ℂ_fldccnfld 21271 Topctop 22787 intcnt 22911 –cn→ccncf 24776 D cdv 25771

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153 ax-addf 11154

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-tp 4597 df-op 4599 df-uni 4875 df-int 4914 df-iun 4960 df-iin 4961 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-se 5595 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-isom 6523 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-of 7656 df-om 7846 df-1st 7971 df-2nd 7972 df-supp 8143 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-2o 8438 df-er 8674 df-map 8804 df-pm 8805 df-ixp 8874 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-fsupp 9320 df-fi 9369 df-sup 9400 df-inf 9401 df-oi 9470 df-card 9899 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-div 11843 df-nn 12194 df-2 12256 df-3 12257 df-4 12258 df-5 12259 df-6 12260 df-7 12261 df-8 12262 df-9 12263 df-n0 12450 df-z 12537 df-dec 12657 df-uz 12801 df-q 12915 df-rp 12959 df-xneg 13079 df-xadd 13080 df-xmul 13081 df-ioo 13317 df-ico 13319 df-icc 13320 df-fz 13476 df-fzo 13623 df-seq 13974 df-exp 14034 df-hash 14303 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-struct 17124 df-sets 17141 df-slot 17159 df-ndx 17171 df-base 17187 df-ress 17208 df-plusg 17240 df-mulr 17241 df-starv 17242 df-sca 17243 df-vsca 17244 df-ip 17245 df-tset 17246 df-ple 17247 df-ds 17249 df-unif 17250 df-hom 17251 df-cco 17252 df-rest 17392 df-topn 17393 df-0g 17411 df-gsum 17412 df-topgen 17413 df-pt 17414 df-prds 17417 df-xrs 17472 df-qtop 17477 df-imas 17478 df-xps 17480 df-mre 17554 df-mrc 17555 df-acs 17557 df-mgm 18574 df-sgrp 18653 df-mnd 18669 df-submnd 18718 df-mulg 19007 df-cntz 19256 df-cmn 19719 df-psmet 21263 df-xmet 21264 df-met 21265 df-bl 21266 df-mopn 21267 df-fbas 21268 df-fg 21269 df-cnfld 21272 df-top 22788 df-topon 22805 df-topsp 22827 df-bases 22840 df-cld 22913 df-ntr 22914 df-cls 22915 df-nei 22992 df-lp 23030 df-perf 23031 df-cn 23121 df-cnp 23122 df-haus 23209 df-cmp 23281 df-tx 23456 df-hmeo 23649 df-fil 23740 df-fm 23832 df-flim 23833 df-flf 23834 df-xms 24215 df-ms 24216 df-tms 24217 df-cncf 24778 df-limc 25774 df-dv 25775

This theorem is referenced by: dvne0f1 25924 dvcnvrelem1 25929

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