dvne0 - Metamath Proof Explorer

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Theorem dvne0 24296

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 2870	. . . . . . . . . . . . 13 ⊢ (𝑥 = 0 → (𝑥 ∈ ran (ℝ D 𝐹) ↔ 0 ∈ ran (ℝ D 𝐹)))
3	2	notbid 319	. . . . . . . . . . . 12 ⊢ (𝑥 = 0 → (¬ 𝑥 ∈ ran (ℝ D 𝐹) ↔ ¬ 0 ∈ ran (ℝ D 𝐹)))
4	1, 3	syl5ibrcom 248	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 = 0 → ¬ 𝑥 ∈ ran (ℝ D 𝐹)))
5	4	necon2ad 2999	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ ran (ℝ D 𝐹) → 𝑥 ≠ 0))
6	5	imp 407	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (ℝ D 𝐹)) → 𝑥 ≠ 0)
7		dvne0.f	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ))
8		cncff 23189	. . . . . . . . . . . . . . 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 12673	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
13	10, 11, 12	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
14		dvfre 24236	. . . . . . . . . . . . . 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 6394	. . . . . . . . . . . 12 ⊢ (𝜑 → ran (ℝ D 𝐹) ⊆ ℝ)
17	16	sselda 3893	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (ℝ D 𝐹)) → 𝑥 ∈ ℝ)
18		0re 10494	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ
19		lttri2 10575	. . . . . . . . . . 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 10539	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℝ^*
22		elioomnf 12687	. . . . . . . . . . . . . 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 536	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ → (𝑥 ∈ (-∞(,)0) ↔ 𝑥 < 0))
25		elrp 12246	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ⁺ ↔ (𝑥 ∈ ℝ ∧ 0 < 𝑥))
26	25	baib 536	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ → (𝑥 ∈ ℝ⁺ ↔ 0 < 𝑥))
27	24, 26	orbi12d 913	. . . . . . . . . . 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 283	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (ℝ D 𝐹)) → (𝑥 ≠ 0 ↔ (𝑥 ∈ (-∞(,)0) ∨ 𝑥 ∈ ℝ⁺)))
30	6, 29	mpbid 233	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (ℝ D 𝐹)) → (𝑥 ∈ (-∞(,)0) ∨ 𝑥 ∈ ℝ⁺))
31		elun 4050	. . . . . . . 8 ⊢ (𝑥 ∈ ((-∞(,)0) ∪ ℝ⁺) ↔ (𝑥 ∈ (-∞(,)0) ∨ 𝑥 ∈ ℝ⁺))
32	30, 31	sylibr 235	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (ℝ D 𝐹)) → 𝑥 ∈ ((-∞(,)0) ∪ ℝ⁺))
33	32	ex 413	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ran (ℝ D 𝐹) → 𝑥 ∈ ((-∞(,)0) ∪ ℝ⁺)))
34	33	ssrdv 3899	. . . . 5 ⊢ (𝜑 → ran (ℝ D 𝐹) ⊆ ((-∞(,)0) ∪ ℝ⁺))
35		disjssun 4335	. . . . 5 ⊢ ((ran (ℝ D 𝐹) ∩ (-∞(,)0)) = ∅ → (ran (ℝ D 𝐹) ⊆ ((-∞(,)0) ∪ ℝ⁺) ↔ ran (ℝ D 𝐹) ⊆ ℝ⁺))
36	34, 35	syl5ibcom 246	. . . 4 ⊢ (𝜑 → ((ran (ℝ D 𝐹) ∩ (-∞(,)0)) = ∅ → ran (ℝ D 𝐹) ⊆ ℝ⁺))
37	36	imp 407	. . 3 ⊢ ((𝜑 ∧ (ran (ℝ D 𝐹) ∩ (-∞(,)0)) = ∅) → ran (ℝ D 𝐹) ⊆ ℝ⁺)
38	10	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ ran (ℝ D 𝐹) ⊆ ℝ⁺) → 𝐴 ∈ ℝ)
39	11	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ ran (ℝ D 𝐹) ⊆ ℝ⁺) → 𝐵 ∈ ℝ)
40	7	adantr 481	. . . . 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 6373	. . . . . . . . 9 ⊢ (𝜑 → ((ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ ↔ (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℝ))
43	15, 42	mpbid 233	. . . . . . . 8 ⊢ (𝜑 → (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℝ)
44	43	ffnd 6388	. . . . . . 7 ⊢ (𝜑 → (ℝ D 𝐹) Fn (𝐴(,)𝐵))
45	44	anim1i 614	. . . . . 6 ⊢ ((𝜑 ∧ ran (ℝ D 𝐹) ⊆ ℝ⁺) → ((ℝ D 𝐹) Fn (𝐴(,)𝐵) ∧ ran (ℝ D 𝐹) ⊆ ℝ⁺))
46		df-f 6234	. . . . . 6 ⊢ ((ℝ D 𝐹):(𝐴(,)𝐵)⟶ℝ⁺ ↔ ((ℝ D 𝐹) Fn (𝐴(,)𝐵) ∧ ran (ℝ D 𝐹) ⊆ ℝ⁺))
47	45, 46	sylibr 235	. . . . 5 ⊢ ((𝜑 ∧ ran (ℝ D 𝐹) ⊆ ℝ⁺) → (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℝ⁺)
48	38, 39, 40, 47	dvgt0 24289	. . . 4 ⊢ ((𝜑 ∧ ran (ℝ D 𝐹) ⊆ ℝ⁺) → 𝐹 Isom < , < ((𝐴[,]𝐵), ran 𝐹))
49	48	orcd 870	. . 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 4234	. . . 4 ⊢ ((ran (ℝ D 𝐹) ∩ (-∞(,)0)) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (ran (ℝ D 𝐹) ∩ (-∞(,)0)))
52		elin 4094	. . . . . 6 ⊢ (𝑥 ∈ (ran (ℝ D 𝐹) ∩ (-∞(,)0)) ↔ (𝑥 ∈ ran (ℝ D 𝐹) ∧ 𝑥 ∈ (-∞(,)0)))
53		fvelrnb 6599	. . . . . . . . 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 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ ((ℝ D 𝐹)‘𝑦) ∈ (-∞(,)0))) → 𝐴 ∈ ℝ)
56	11	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ ((ℝ D 𝐹)‘𝑦) ∈ (-∞(,)0))) → 𝐵 ∈ ℝ)
57	7	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ ((ℝ D 𝐹)‘𝑦) ∈ (-∞(,)0))) → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ))
58	44	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ ((ℝ D 𝐹)‘𝑦) ∈ (-∞(,)0))) → (ℝ D 𝐹) Fn (𝐴(,)𝐵))
59	43	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ ((ℝ D 𝐹)‘𝑦) ∈ (-∞(,)0))) → (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℝ)
60	59	ffvelrnda 6721	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ ((ℝ D 𝐹)‘𝑦) ∈ (-∞(,)0))) ∧ 𝑧 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑧) ∈ ℝ)
61	1	ad2antrr 722	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ ((ℝ D 𝐹)‘𝑦) ∈ (-∞(,)0))) ∧ 𝑧 ∈ (𝐴(,)𝐵)) → ¬ 0 ∈ ran (ℝ D 𝐹))
62		simplrl 773	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ ((ℝ D 𝐹)‘𝑦) ∈ (-∞(,)0))) ∧ (𝑧 ∈ (𝐴(,)𝐵) ∧ 0 ≤ ((ℝ D 𝐹)‘𝑧))) → 𝑦 ∈ (𝐴(,)𝐵))
63		simprl 767	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ ((ℝ D 𝐹)‘𝑦) ∈ (-∞(,)0))) ∧ (𝑧 ∈ (𝐴(,)𝐵) ∧ 0 ≤ ((ℝ D 𝐹)‘𝑧))) → 𝑧 ∈ (𝐴(,)𝐵))
64		ioossicc 12677	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
65		rescncf 23193	. . . . . . . . . . . . . . . . . . . . . . . 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 722	. . . . . . . . . . . . . . . . . . . . . 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 10445	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ℝ ⊆ ℂ
69	68	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → ℝ ⊆ ℂ)
70		fss 6400	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3904	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℝ)
73		eqid 2795	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
74	73	tgioo2 23099	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
75	73, 74	dvres 24197	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 835	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → (ℝ D (𝐹 ↾ (𝐴(,)𝐵))) = ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘(𝐴(,)𝐵))))
77		retop 23058	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (topGen‘ran (,)) ∈ Top
78		iooretop 23062	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝐴(,)𝐵) ∈ (topGen‘ran (,))
79		isopn3i 21379	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((topGen‘ran (,)) ∈ Top ∧ (𝐴(,)𝐵) ∈ (topGen‘ran (,))) → ((int‘(topGen‘ran (,)))‘(𝐴(,)𝐵)) = (𝐴(,)𝐵))
80	77, 78, 79	mp2an 688	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((int‘(topGen‘ran (,)))‘(𝐴(,)𝐵)) = (𝐴(,)𝐵)
81	80	reseq2i 5736	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘(𝐴(,)𝐵))) = ((ℝ D 𝐹) ↾ (𝐴(,)𝐵))
82		fnresdm 6341	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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	syl5eq 2843	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘(𝐴(,)𝐵))) = (ℝ D 𝐹))
85	76, 84	eqtrd 2831	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → (ℝ D (𝐹 ↾ (𝐴(,)𝐵))) = (ℝ D 𝐹))
86	85	dmeqd 5665	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → dom (ℝ D (𝐹 ↾ (𝐴(,)𝐵))) = dom (ℝ D 𝐹))
87	86, 41	eqtrd 2831	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → dom (ℝ D (𝐹 ↾ (𝐴(,)𝐵))) = (𝐴(,)𝐵))
88	87	ad2antrr 722	. . . . . . . . . . . . . . . . . . . . . 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 24295	. . . . . . . . . . . . . . . . . . . . 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 722	. . . . . . . . . . . . . . . . . . . . . . 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 6545	. . . . . . . . . . . . . . . . . . . . . 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 6545	. . . . . . . . . . . . . . . . . . . . . 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 7039	. . . . . . . . . . . . . . . . . . . . 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 5695	. . . . . . . . . . . . . . . . . . . . 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 3938	. . . . . . . . . . . . . . . . . . . 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 774	. . . . . . . . . . . . . . . . . . . . . . . 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 12687	. . . . . . . . . . . . . . . . . . . . . . . . 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 219	. . . . . . . . . . . . . . . . . . . . . . 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 496	. . . . . . . . . . . . . . . . . . . . . 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 495	. . . . . . . . . . . . . . . . . . . . . . 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 10581	. . . . . . . . . . . . . . . . . . . . . . 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 769	. . . . . . . . . . . . . . . . . . . . 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 12656	. . . . . . . . . . . . . . . . . . . . . 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 1335	. . . . . . . . . . . . . . . . . . . 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 3894	. . . . . . . . . . . . . . . . . . 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 457	. . . . . . . . . . . . . . . . . 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 199	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ ((ℝ D 𝐹)‘𝑦) ∈ (-∞(,)0))) ∧ 𝑧 ∈ (𝐴(,)𝐵)) → ¬ 0 ≤ ((ℝ D 𝐹)‘𝑧))
114		ltnle 10572	. . . . . . . . . . . . . . . . . 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 258	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ ((ℝ D 𝐹)‘𝑦) ∈ (-∞(,)0))) ∧ 𝑧 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑧) < 0)
117		elioomnf 12687	. . . . . . . . . . . . . . . . 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 3149	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ ((ℝ D 𝐹)‘𝑦) ∈ (-∞(,)0))) → ∀𝑧 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑧) ∈ (-∞(,)0))
121		ffnfv 6750	. . . . . . . . . . . . . 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 24290	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ ((ℝ D 𝐹)‘𝑦) ∈ (-∞(,)0))) → 𝐹 Isom < , ^◡ < ((𝐴[,]𝐵), ran 𝐹))
124	123	olcd 871	. . . . . . . . . . 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 457	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (((ℝ D 𝐹)‘𝑦) ∈ (-∞(,)0) → (𝐹 Isom < , < ((𝐴[,]𝐵), ran 𝐹) ∨ 𝐹 Isom < , ^◡ < ((𝐴[,]𝐵), ran 𝐹))))
126		eleq1 2870	. . . . . . . . . . 11 ⊢ (((ℝ D 𝐹)‘𝑦) = 𝑥 → (((ℝ D 𝐹)‘𝑦) ∈ (-∞(,)0) ↔ 𝑥 ∈ (-∞(,)0)))
127	126	imbi1d 343	. . . . . . . . . 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 246	. . . . . . . . 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 3247	. . . . . . . 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 241	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ ran (ℝ D 𝐹) → (𝑥 ∈ (-∞(,)0) → (𝐹 Isom < , < ((𝐴[,]𝐵), ran 𝐹) ∨ 𝐹 Isom < , ^◡ < ((𝐴[,]𝐵), ran 𝐹)))))
131	130	impd 411	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ ran (ℝ D 𝐹) ∧ 𝑥 ∈ (-∞(,)0)) → (𝐹 Isom < , < ((𝐴[,]𝐵), ran 𝐹) ∨ 𝐹 Isom < , ^◡ < ((𝐴[,]𝐵), ran 𝐹))))
132	52, 131	syl5bi 243	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (ran (ℝ D 𝐹) ∩ (-∞(,)0)) → (𝐹 Isom < , < ((𝐴[,]𝐵), ran 𝐹) ∨ 𝐹 Isom < , ^◡ < ((𝐴[,]𝐵), ran 𝐹))))
133	132	exlimdv 1911	. . . 4 ⊢ (𝜑 → (∃𝑥 𝑥 ∈ (ran (ℝ D 𝐹) ∩ (-∞(,)0)) → (𝐹 Isom < , < ((𝐴[,]𝐵), ran 𝐹) ∨ 𝐹 Isom < , ^◡ < ((𝐴[,]𝐵), ran 𝐹))))
134	51, 133	syl5bi 243	. . 3 ⊢ (𝜑 → ((ran (ℝ D 𝐹) ∩ (-∞(,)0)) ≠ ∅ → (𝐹 Isom < , < ((𝐴[,]𝐵), ran 𝐹) ∨ 𝐹 Isom < , ^◡ < ((𝐴[,]𝐵), ran 𝐹))))
135	134	imp 407	. 2 ⊢ ((𝜑 ∧ (ran (ℝ D 𝐹) ∩ (-∞(,)0)) ≠ ∅) → (𝐹 Isom < , < ((𝐴[,]𝐵), ran 𝐹) ∨ 𝐹 Isom < , ^◡ < ((𝐴[,]𝐵), ran 𝐹)))
136	50, 135	pm2.61dane 3072	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 207 ∧ wa 396 ∨ wo 842 ∧ w3a 1080 = wceq 1522 ∃wex 1761 ∈ wcel 2081 ≠ wne 2984 ∀wral 3105 ∃wrex 3106 ∪ cun 3861 ∩ cin 3862 ⊆ wss 3863 ∅c0 4215 class class class wbr 4966 ^◡ccnv 5447 dom cdm 5448 ran crn 5449 ↾ cres 5450 Fn wfn 6225 ⟶wf 6226 ‘cfv 6230 Isom wiso 6231 (class class class)co 7021 ℂcc 10386 ℝcr 10387 0cc0 10388 -∞cmnf 10524 ℝ^*cxr 10525 < clt 10526 ≤ cle 10527 ℝ⁺crp 12244 (,)cioo 12593 [,]cicc 12596 TopOpenctopn 16529 topGenctg 16545 ℂ_fldccnfld 20232 Topctop 21190 intcnt 21314 –cn→ccncf 23172 D cdv 24149

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5086 ax-sep 5099 ax-nul 5106 ax-pow 5162 ax-pr 5226 ax-un 7324 ax-cnex 10444 ax-resscn 10445 ax-1cn 10446 ax-icn 10447 ax-addcl 10448 ax-addrcl 10449 ax-mulcl 10450 ax-mulrcl 10451 ax-mulcom 10452 ax-addass 10453 ax-mulass 10454 ax-distr 10455 ax-i2m1 10456 ax-1ne0 10457 ax-1rid 10458 ax-rnegex 10459 ax-rrecex 10460 ax-cnre 10461 ax-pre-lttri 10462 ax-pre-lttrn 10463 ax-pre-ltadd 10464 ax-pre-mulgt0 10465 ax-pre-sup 10466 ax-addf 10467 ax-mulf 10468

This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3710 df-csb 3816 df-dif 3866 df-un 3868 df-in 3870 df-ss 3878 df-pss 3880 df-nul 4216 df-if 4386 df-pw 4459 df-sn 4477 df-pr 4479 df-tp 4481 df-op 4483 df-uni 4750 df-int 4787 df-iun 4831 df-iin 4832 df-br 4967 df-opab 5029 df-mpt 5046 df-tr 5069 df-id 5353 df-eprel 5358 df-po 5367 df-so 5368 df-fr 5407 df-se 5408 df-we 5409 df-xp 5454 df-rel 5455 df-cnv 5456 df-co 5457 df-dm 5458 df-rn 5459 df-res 5460 df-ima 5461 df-pred 6028 df-ord 6074 df-on 6075 df-lim 6076 df-suc 6077 df-iota 6194 df-fun 6232 df-fn 6233 df-f 6234 df-f1 6235 df-fo 6236 df-f1o 6237 df-fv 6238 df-isom 6239 df-riota 6982 df-ov 7024 df-oprab 7025 df-mpo 7026 df-of 7272 df-om 7442 df-1st 7550 df-2nd 7551 df-supp 7687 df-wrecs 7803 df-recs 7865 df-rdg 7903 df-1o 7958 df-2o 7959 df-oadd 7962 df-er 8144 df-map 8263 df-pm 8264 df-ixp 8316 df-en 8363 df-dom 8364 df-sdom 8365 df-fin 8366 df-fsupp 8685 df-fi 8726 df-sup 8757 df-inf 8758 df-oi 8825 df-card 9219 df-pnf 10528 df-mnf 10529 df-xr 10530 df-ltxr 10531 df-le 10532 df-sub 10724 df-neg 10725 df-div 11151 df-nn 11492 df-2 11553 df-3 11554 df-4 11555 df-5 11556 df-6 11557 df-7 11558 df-8 11559 df-9 11560 df-n0 11751 df-z 11835 df-dec 11953 df-uz 12099 df-q 12203 df-rp 12245 df-xneg 12362 df-xadd 12363 df-xmul 12364 df-ioo 12597 df-ico 12599 df-icc 12600 df-fz 12748 df-fzo 12889 df-seq 13225 df-exp 13285 df-hash 13546 df-cj 14297 df-re 14298 df-im 14299 df-sqrt 14433 df-abs 14434 df-struct 16319 df-ndx 16320 df-slot 16321 df-base 16323 df-sets 16324 df-ress 16325 df-plusg 16412 df-mulr 16413 df-starv 16414 df-sca 16415 df-vsca 16416 df-ip 16417 df-tset 16418 df-ple 16419 df-ds 16421 df-unif 16422 df-hom 16423 df-cco 16424 df-rest 16530 df-topn 16531 df-0g 16549 df-gsum 16550 df-topgen 16551 df-pt 16552 df-prds 16555 df-xrs 16609 df-qtop 16614 df-imas 16615 df-xps 16617 df-mre 16691 df-mrc 16692 df-acs 16694 df-mgm 17686 df-sgrp 17728 df-mnd 17739 df-submnd 17780 df-mulg 17987 df-cntz 18193 df-cmn 18640 df-psmet 20224 df-xmet 20225 df-met 20226 df-bl 20227 df-mopn 20228 df-fbas 20229 df-fg 20230 df-cnfld 20233 df-top 21191 df-topon 21208 df-topsp 21230 df-bases 21243 df-cld 21316 df-ntr 21317 df-cls 21318 df-nei 21395 df-lp 21433 df-perf 21434 df-cn 21524 df-cnp 21525 df-haus 21612 df-cmp 21684 df-tx 21859 df-hmeo 22052 df-fil 22143 df-fm 22235 df-flim 22236 df-flf 22237 df-xms 22618 df-ms 22619 df-tms 22620 df-cncf 23174 df-limc 24152 df-dv 24153

This theorem is referenced by: dvne0f1 24297 dvcnvrelem1 24302

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