dvne0 - Metamath Proof Explorer

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Theorem dvne0 25972

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 2824	. . . . . . . . . . . . 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 2947	. . . . . . . . . 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 24842	. . . . . . . . . . . . . . 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 13345	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
13	10, 11, 12	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
14		dvfre 25911	. . . . . . . . . . . . . 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 6670	. . . . . . . . . . . 12 ⊢ (𝜑 → ran (ℝ D 𝐹) ⊆ ℝ)
17	16	sselda 3933	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (ℝ D 𝐹)) → 𝑥 ∈ ℝ)
18		0re 11134	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ
19		lttri2 11215	. . . . . . . . . . 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 11179	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℝ^*
22		elioomnf 13360	. . . . . . . . . . . . . 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 12907	. . . . . . . . . . . . 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 4105	. . . . . . . 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 3939	. . . . 5 ⊢ (𝜑 → ran (ℝ D 𝐹) ⊆ ((-∞(,)0) ∪ ℝ⁺))
35		disjssun 4420	. . . . 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 6646	. . . . . . . . 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 6663	. . . . . . 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 6496	. . . . . 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 25965	. . . 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 4305	. . . 4 ⊢ ((ran (ℝ D 𝐹) ∩ (-∞(,)0)) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (ran (ℝ D 𝐹) ∩ (-∞(,)0)))
52		elin 3917	. . . . . 6 ⊢ (𝑥 ∈ (ran (ℝ D 𝐹) ∩ (-∞(,)0)) ↔ (𝑥 ∈ ran (ℝ D 𝐹) ∧ 𝑥 ∈ (-∞(,)0)))
53		fvelrnb 6894	. . . . . . . . 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 7029	. . . . . . . . . . . . . . . 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 13349	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
65		rescncf 24846	. . . . . . . . . . . . . . . . . . . . . . . 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 11083	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ℝ ⊆ ℂ
69	68	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → ℝ ⊆ ℂ)
70		fss 6678	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3945	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℝ)
73		eqid 2736	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
74		tgioo4 24749	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
75	73, 74	dvres 25868	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 24705	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (topGen‘ran (,)) ∈ Top
78		iooretop 24709	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝐴(,)𝐵) ∈ (topGen‘ran (,))
79		isopn3i 23026	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5935	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘(𝐴(,)𝐵))) = ((ℝ D 𝐹) ↾ (𝐴(,)𝐵))
82		fnresdm 6611	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2783	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘(𝐴(,)𝐵))) = (ℝ D 𝐹))
85	76, 84	eqtrd 2771	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → (ℝ D (𝐹 ↾ (𝐴(,)𝐵))) = (ℝ D 𝐹))
86	85	dmeqd 5854	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → dom (ℝ D (𝐹 ↾ (𝐴(,)𝐵))) = dom (ℝ D 𝐹))
87	86, 41	eqtrd 2771	. . . . . . . . . . . . . . . . . . . . . . 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 25971	. . . . . . . . . . . . . . . . . . . . 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 6836	. . . . . . . . . . . . . . . . . . . . . 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 6836	. . . . . . . . . . . . . . . . . . . . . 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 7376	. . . . . . . . . . . . . . . . . . . . 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 5887	. . . . . . . . . . . . . . . . . . . . 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 3988	. . . . . . . . . . . . . . . . . . . 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 13360	. . . . . . . . . . . . . . . . . . . . . . . . 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 11221	. . . . . . . . . . . . . . . . . . . . . . 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 13327	. . . . . . . . . . . . . . . . . . . . . 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 3934	. . . . . . . . . . . . . . . . . . 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 11212	. . . . . . . . . . . . . . . . . 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 13360	. . . . . . . . . . . . . . . . 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 3128	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ ((ℝ D 𝐹)‘𝑦) ∈ (-∞(,)0))) → ∀𝑧 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑧) ∈ (-∞(,)0))
121		ffnfv 7064	. . . . . . . . . . . . . 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 25966	. . . . . . . . . . . 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 2824	. . . . . . . . . . 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 3137	. . . . . . . 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 1934	. . . 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 3019	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 1541 ∃wex 1780 ∈ wcel 2113 ≠ wne 2932 ∀wral 3051 ∃wrex 3060 ∪ cun 3899 ∩ cin 3900 ⊆ wss 3901 ∅c0 4285 class class class wbr 5098 ^◡ccnv 5623 dom cdm 5624 ran crn 5625 ↾ cres 5626 Fn wfn 6487 ⟶wf 6488 ‘cfv 6492 Isom wiso 6493 (class class class)co 7358 ℂcc 11024 ℝcr 11025 0cc0 11026 -∞cmnf 11164 ℝ^*cxr 11165 < clt 11166 ≤ cle 11167 ℝ⁺crp 12905 (,)cioo 13261 [,]cicc 13264 TopOpenctopn 17341 topGenctg 17357 ℂ_fldccnfld 21309 Topctop 22837 intcnt 22961 –cn→ccncf 24825 D cdv 25820

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103 ax-pre-sup 11104 ax-addf 11105

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-tp 4585 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-iin 4949 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-om 7809 df-1st 7933 df-2nd 7934 df-supp 8103 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-2o 8398 df-er 8635 df-map 8765 df-pm 8766 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9265 df-fi 9314 df-sup 9345 df-inf 9346 df-oi 9415 df-card 9851 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-div 11795 df-nn 12146 df-2 12208 df-3 12209 df-4 12210 df-5 12211 df-6 12212 df-7 12213 df-8 12214 df-9 12215 df-n0 12402 df-z 12489 df-dec 12608 df-uz 12752 df-q 12862 df-rp 12906 df-xneg 13026 df-xadd 13027 df-xmul 13028 df-ioo 13265 df-ico 13267 df-icc 13268 df-fz 13424 df-fzo 13571 df-seq 13925 df-exp 13985 df-hash 14254 df-cj 15022 df-re 15023 df-im 15024 df-sqrt 15158 df-abs 15159 df-struct 17074 df-sets 17091 df-slot 17109 df-ndx 17121 df-base 17137 df-ress 17158 df-plusg 17190 df-mulr 17191 df-starv 17192 df-sca 17193 df-vsca 17194 df-ip 17195 df-tset 17196 df-ple 17197 df-ds 17199 df-unif 17200 df-hom 17201 df-cco 17202 df-rest 17342 df-topn 17343 df-0g 17361 df-gsum 17362 df-topgen 17363 df-pt 17364 df-prds 17367 df-xrs 17423 df-qtop 17428 df-imas 17429 df-xps 17431 df-mre 17505 df-mrc 17506 df-acs 17508 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-submnd 18709 df-mulg 18998 df-cntz 19246 df-cmn 19711 df-psmet 21301 df-xmet 21302 df-met 21303 df-bl 21304 df-mopn 21305 df-fbas 21306 df-fg 21307 df-cnfld 21310 df-top 22838 df-topon 22855 df-topsp 22877 df-bases 22890 df-cld 22963 df-ntr 22964 df-cls 22965 df-nei 23042 df-lp 23080 df-perf 23081 df-cn 23171 df-cnp 23172 df-haus 23259 df-cmp 23331 df-tx 23506 df-hmeo 23699 df-fil 23790 df-fm 23882 df-flim 23883 df-flf 23884 df-xms 24264 df-ms 24265 df-tms 24266 df-cncf 24827 df-limc 25823 df-dv 25824

This theorem is referenced by: dvne0f1 25973 dvcnvrelem1 25978

W3C validator