dvne0 - Metamath Proof Explorer

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Theorem dvne0 25892

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 2816	. . . . . . . . . . . . 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 2940	. . . . . . . . . 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 24762	. . . . . . . . . . . . . . 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 13366	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
13	10, 11, 12	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
14		dvfre 25831	. . . . . . . . . . . . . 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 6678	. . . . . . . . . . . 12 ⊢ (𝜑 → ran (ℝ D 𝐹) ⊆ ℝ)
17	16	sselda 3943	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (ℝ D 𝐹)) → 𝑥 ∈ ℝ)
18		0re 11152	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ
19		lttri2 11232	. . . . . . . . . . 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 11197	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℝ^*
22		elioomnf 13381	. . . . . . . . . . . . . 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 12929	. . . . . . . . . . . . 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 4112	. . . . . . . 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 3949	. . . . 5 ⊢ (𝜑 → ran (ℝ D 𝐹) ⊆ ((-∞(,)0) ∪ ℝ⁺))
35		disjssun 4427	. . . . 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 6654	. . . . . . . . 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 6671	. . . . . . 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 6503	. . . . . 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 25885	. . . 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 4312	. . . 4 ⊢ ((ran (ℝ D 𝐹) ∩ (-∞(,)0)) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (ran (ℝ D 𝐹) ∩ (-∞(,)0)))
52		elin 3927	. . . . . 6 ⊢ (𝑥 ∈ (ran (ℝ D 𝐹) ∩ (-∞(,)0)) ↔ (𝑥 ∈ ran (ℝ D 𝐹) ∧ 𝑥 ∈ (-∞(,)0)))
53		fvelrnb 6903	. . . . . . . . 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 7038	. . . . . . . . . . . . . . . 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 13370	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
65		rescncf 24766	. . . . . . . . . . . . . . . . . . . . . . . 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 11101	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ℝ ⊆ ℂ
69	68	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → ℝ ⊆ ℂ)
70		fss 6686	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3955	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℝ)
73		eqid 2729	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
74		tgioo4 24669	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
75	73, 74	dvres 25788	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 24625	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (topGen‘ran (,)) ∈ Top
78		iooretop 24629	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝐴(,)𝐵) ∈ (topGen‘ran (,))
79		isopn3i 22945	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5936	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘(𝐴(,)𝐵))) = ((ℝ D 𝐹) ↾ (𝐴(,)𝐵))
82		fnresdm 6619	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2776	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘(𝐴(,)𝐵))) = (ℝ D 𝐹))
85	76, 84	eqtrd 2764	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → (ℝ D (𝐹 ↾ (𝐴(,)𝐵))) = (ℝ D 𝐹))
86	85	dmeqd 5859	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → dom (ℝ D (𝐹 ↾ (𝐴(,)𝐵))) = dom (ℝ D 𝐹))
87	86, 41	eqtrd 2764	. . . . . . . . . . . . . . . . . . . . . . 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 25891	. . . . . . . . . . . . . . . . . . . . 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 6842	. . . . . . . . . . . . . . . . . . . . . 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 6842	. . . . . . . . . . . . . . . . . . . . . 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 7387	. . . . . . . . . . . . . . . . . . . . 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 5891	. . . . . . . . . . . . . . . . . . . . 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 3998	. . . . . . . . . . . . . . . . . . . 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 13381	. . . . . . . . . . . . . . . . . . . . . . . . 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 11238	. . . . . . . . . . . . . . . . . . . . . . 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 13348	. . . . . . . . . . . . . . . . . . . . . 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 3944	. . . . . . . . . . . . . . . . . . 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 11229	. . . . . . . . . . . . . . . . . 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 13381	. . . . . . . . . . . . . . . . 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 3125	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ ((ℝ D 𝐹)‘𝑦) ∈ (-∞(,)0))) → ∀𝑧 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑧) ∈ (-∞(,)0))
121		ffnfv 7073	. . . . . . . . . . . . . 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 25886	. . . . . . . . . . . 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 2816	. . . . . . . . . . 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 3134	. . . . . . . 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 3012	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 2925 ∀wral 3044 ∃wrex 3053 ∪ cun 3909 ∩ cin 3910 ⊆ wss 3911 ∅c0 4292 class class class wbr 5102 ^◡ccnv 5630 dom cdm 5631 ran crn 5632 ↾ cres 5633 Fn wfn 6494 ⟶wf 6495 ‘cfv 6499 Isom wiso 6500 (class class class)co 7369 ℂcc 11042 ℝcr 11043 0cc0 11044 -∞cmnf 11182 ℝ^*cxr 11183 < clt 11184 ≤ cle 11185 ℝ⁺crp 12927 (,)cioo 13282 [,]cicc 13285 TopOpenctopn 17360 topGenctg 17376 ℂ_fldccnfld 21240 Topctop 22756 intcnt 22880 –cn→ccncf 24745 D cdv 25740

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 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121 ax-pre-sup 11122 ax-addf 11123

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 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-iin 4954 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-isom 6508 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-of 7633 df-om 7823 df-1st 7947 df-2nd 7948 df-supp 8117 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-2o 8412 df-er 8648 df-map 8778 df-pm 8779 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9289 df-fi 9338 df-sup 9369 df-inf 9370 df-oi 9439 df-card 9868 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-div 11812 df-nn 12163 df-2 12225 df-3 12226 df-4 12227 df-5 12228 df-6 12229 df-7 12230 df-8 12231 df-9 12232 df-n0 12419 df-z 12506 df-dec 12626 df-uz 12770 df-q 12884 df-rp 12928 df-xneg 13048 df-xadd 13049 df-xmul 13050 df-ioo 13286 df-ico 13288 df-icc 13289 df-fz 13445 df-fzo 13592 df-seq 13943 df-exp 14003 df-hash 14272 df-cj 15041 df-re 15042 df-im 15043 df-sqrt 15177 df-abs 15178 df-struct 17093 df-sets 17110 df-slot 17128 df-ndx 17140 df-base 17156 df-ress 17177 df-plusg 17209 df-mulr 17210 df-starv 17211 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-hom 17220 df-cco 17221 df-rest 17361 df-topn 17362 df-0g 17380 df-gsum 17381 df-topgen 17382 df-pt 17383 df-prds 17386 df-xrs 17441 df-qtop 17446 df-imas 17447 df-xps 17449 df-mre 17523 df-mrc 17524 df-acs 17526 df-mgm 18543 df-sgrp 18622 df-mnd 18638 df-submnd 18687 df-mulg 18976 df-cntz 19225 df-cmn 19688 df-psmet 21232 df-xmet 21233 df-met 21234 df-bl 21235 df-mopn 21236 df-fbas 21237 df-fg 21238 df-cnfld 21241 df-top 22757 df-topon 22774 df-topsp 22796 df-bases 22809 df-cld 22882 df-ntr 22883 df-cls 22884 df-nei 22961 df-lp 22999 df-perf 23000 df-cn 23090 df-cnp 23091 df-haus 23178 df-cmp 23250 df-tx 23425 df-hmeo 23618 df-fil 23709 df-fm 23801 df-flim 23802 df-flf 23803 df-xms 24184 df-ms 24185 df-tms 24186 df-cncf 24747 df-limc 25743 df-dv 25744

This theorem is referenced by: dvne0f1 25893 dvcnvrelem1 25898

W3C validator