dvne0 - Metamath Proof Explorer

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Theorem dvne0 26338

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 2848	. . . . . . . . . . . . 13 ⊢ (𝑥 = 0 → (𝑥 ∈ ran (ℝ D 𝐹) ↔ 0 ∈ ran (ℝ D 𝐹)))
3	2	notbid 321	. . . . . . . . . . . 12 ⊢ (𝑥 = 0 → (¬ 𝑥 ∈ ran (ℝ D 𝐹) ↔ ¬ 0 ∈ ran (ℝ D 𝐹)))
4	1, 3	syl5ibrcom 250	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 = 0 → ¬ 𝑥 ∈ ran (ℝ D 𝐹)))
5	4	necon2ad 2970	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ ran (ℝ D 𝐹) → 𝑥 ≠ 0))
6	5	imp 412	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (ℝ D 𝐹)) → 𝑥 ≠ 0)
7		dvne0.f	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ))
8		cncff 25221	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ) → 𝐹:(𝐴[,]𝐵)⟶ℝ)
9	7, 8	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹:(𝐴[,]𝐵)⟶ℝ)
10		dvne0.a	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐴 ∈ ℝ)
11		dvne0.b	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐵 ∈ ℝ)
12		iccssre 13565	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
13	10, 11, 12	syl2anc 596	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
14		dvfre 26278	. . . . . . . . . . . . . 14 ⊢ ((𝐹:(𝐴[,]𝐵)⟶ℝ ∧ (𝐴[,]𝐵) ⊆ ℝ) → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ)
15	9, 13, 14	syl2anc 596	. . . . . . . . . . . . 13 ⊢ (𝜑 → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ)
16	15	frnd 6721	. . . . . . . . . . . 12 ⊢ (𝜑 → ran (ℝ D 𝐹) ⊆ ℝ)
17	16	sselda 3909	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (ℝ D 𝐹)) → 𝑥 ∈ ℝ)
18		0re 11315	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ
19		lttri2 11397	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ ∧ 0 ∈ ℝ) → (𝑥 ≠ 0 ↔ (𝑥 < 0 ∨ 0 < 𝑥)))
20	17, 18, 19	sylancl 598	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (ℝ D 𝐹)) → (𝑥 ≠ 0 ↔ (𝑥 < 0 ∨ 0 < 𝑥)))
21		0xr 11361	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℝ^*
22		elioomnf 13580	. . . . . . . . . . . . . 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 545	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ → (𝑥 ∈ (-∞(,)0) ↔ 𝑥 < 0))
25		elrp 13127	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ⁺ ↔ (𝑥 ∈ ℝ ∧ 0 < 𝑥))
26	25	baib 545	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ → (𝑥 ∈ ℝ⁺ ↔ 0 < 𝑥))
27	24, 26	orbi12d 932	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ → ((𝑥 ∈ (-∞(,)0) ∨ 𝑥 ∈ ℝ⁺) ↔ (𝑥 < 0 ∨ 0 < 𝑥)))
28	17, 27	syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (ℝ D 𝐹)) → ((𝑥 ∈ (-∞(,)0) ∨ 𝑥 ∈ ℝ⁺) ↔ (𝑥 < 0 ∨ 0 < 𝑥)))
29	20, 28	bitr4d 285	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (ℝ D 𝐹)) → (𝑥 ≠ 0 ↔ (𝑥 ∈ (-∞(,)0) ∨ 𝑥 ∈ ℝ⁺)))
30	6, 29	mpbid 235	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (ℝ D 𝐹)) → (𝑥 ∈ (-∞(,)0) ∨ 𝑥 ∈ ℝ⁺))
31		elun 4151	. . . . . . . 8 ⊢ (𝑥 ∈ ((-∞(,)0) ∪ ℝ⁺) ↔ (𝑥 ∈ (-∞(,)0) ∨ 𝑥 ∈ ℝ⁺))
32	30, 31	sylibr 237	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (ℝ D 𝐹)) → 𝑥 ∈ ((-∞(,)0) ∪ ℝ⁺))
33	32	ex 418	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ran (ℝ D 𝐹) → 𝑥 ∈ ((-∞(,)0) ∪ ℝ⁺)))
34	33	ssrdv 3915	. . . . 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 248	. . . 4 ⊢ (𝜑 → ((ran (ℝ D 𝐹) ∩ (-∞(,)0)) = ∅ → ran (ℝ D 𝐹) ⊆ ℝ⁺))
37	36	imp 412	. . 3 ⊢ ((𝜑 ∧ (ran (ℝ D 𝐹) ∩ (-∞(,)0)) = ∅) → ran (ℝ D 𝐹) ⊆ ℝ⁺)
38	10	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ ran (ℝ D 𝐹) ⊆ ℝ⁺) → 𝐴 ∈ ℝ)
39	11	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ ran (ℝ D 𝐹) ⊆ ℝ⁺) → 𝐵 ∈ ℝ)
40	7	adantr 486	. . . . 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 6696	. . . . . . . . 9 ⊢ (𝜑 → ((ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ ↔ (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℝ))
43	15, 42	mpbid 235	. . . . . . . 8 ⊢ (𝜑 → (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℝ)
44	43	ffnd 6713	. . . . . . 7 ⊢ (𝜑 → (ℝ D 𝐹) Fn (𝐴(,)𝐵))
45	44	anim1i 627	. . . . . 6 ⊢ ((𝜑 ∧ ran (ℝ D 𝐹) ⊆ ℝ⁺) → ((ℝ D 𝐹) Fn (𝐴(,)𝐵) ∧ ran (ℝ D 𝐹) ⊆ ℝ⁺))
46		df-f 6545	. . . . . 6 ⊢ ((ℝ D 𝐹):(𝐴(,)𝐵)⟶ℝ⁺ ↔ ((ℝ D 𝐹) Fn (𝐴(,)𝐵) ∧ ran (ℝ D 𝐹) ⊆ ℝ⁺))
47	45, 46	sylibr 237	. . . . 5 ⊢ ((𝜑 ∧ ran (ℝ D 𝐹) ⊆ ℝ⁺) → (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℝ⁺)
48	38, 39, 40, 47	dvgt0 26331	. . . 4 ⊢ ((𝜑 ∧ ran (ℝ D 𝐹) ⊆ ℝ⁺) → 𝐹 Isom < , < ((𝐴[,]𝐵), ran 𝐹))
49	48	orcd 887	. . 3 ⊢ ((𝜑 ∧ ran (ℝ D 𝐹) ⊆ ℝ⁺) → (𝐹 Isom < , < ((𝐴[,]𝐵), ran 𝐹) ∨ 𝐹 Isom < , ^◡ < ((𝐴[,]𝐵), ran 𝐹)))
50	37, 49	syldan 603	. 2 ⊢ ((𝜑 ∧ (ran (ℝ D 𝐹) ∩ (-∞(,)0)) = ∅) → (𝐹 Isom < , < ((𝐴[,]𝐵), ran 𝐹) ∨ 𝐹 Isom < , ^◡ < ((𝐴[,]𝐵), ran 𝐹)))
51		n0 4299	. . . 4 ⊢ ((ran (ℝ D 𝐹) ∩ (-∞(,)0)) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (ran (ℝ D 𝐹) ∩ (-∞(,)0)))
52		elin 4092	. . . . . 6 ⊢ (𝑥 ∈ (ran (ℝ D 𝐹) ∩ (-∞(,)0)) ↔ (𝑥 ∈ ran (ℝ D 𝐹) ∧ 𝑥 ∈ (-∞(,)0)))
53		fvelrnb 6948	. . . . . . . . 9 ⊢ ((ℝ D 𝐹) Fn (𝐴(,)𝐵) → (𝑥 ∈ ran (ℝ D 𝐹) ↔ ∃𝑦 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑦) = 𝑥))
54	44, 53	syl 18	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ ran (ℝ D 𝐹) ↔ ∃𝑦 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑦) = 𝑥))
55	10	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ ((ℝ D 𝐹)‘𝑦) ∈ (-∞(,)0))) → 𝐴 ∈ ℝ)
56	11	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ ((ℝ D 𝐹)‘𝑦) ∈ (-∞(,)0))) → 𝐵 ∈ ℝ)
57	7	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ ((ℝ D 𝐹)‘𝑦) ∈ (-∞(,)0))) → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ))
58	44	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ ((ℝ D 𝐹)‘𝑦) ∈ (-∞(,)0))) → (ℝ D 𝐹) Fn (𝐴(,)𝐵))
59	43	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ ((ℝ D 𝐹)‘𝑦) ∈ (-∞(,)0))) → (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℝ)
60	59	ffvelcdmda 7087	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ ((ℝ D 𝐹)‘𝑦) ∈ (-∞(,)0))) ∧ 𝑧 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑧) ∈ ℝ)
61	1	ad2antrr 739	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ ((ℝ D 𝐹)‘𝑦) ∈ (-∞(,)0))) ∧ 𝑧 ∈ (𝐴(,)𝐵)) → ¬ 0 ∈ ran (ℝ D 𝐹))
62		simplrl 789	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ ((ℝ D 𝐹)‘𝑦) ∈ (-∞(,)0))) ∧ (𝑧 ∈ (𝐴(,)𝐵) ∧ 0 ≤ ((ℝ D 𝐹)‘𝑧))) → 𝑦 ∈ (𝐴(,)𝐵))
63		simprl 783	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ ((ℝ D 𝐹)‘𝑦) ∈ (-∞(,)0))) ∧ (𝑧 ∈ (𝐴(,)𝐵) ∧ 0 ≤ ((ℝ D 𝐹)‘𝑧))) → 𝑧 ∈ (𝐴(,)𝐵))
64		ioossicc 13569	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
65		rescncf 25225	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐴(,)𝐵) ⊆ (𝐴[,]𝐵) → (𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ) → (𝐹 ↾ (𝐴(,)𝐵)) ∈ ((𝐴(,)𝐵)–cn→ℝ)))
66	64, 7, 65	mpsyl 69	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (𝐹 ↾ (𝐴(,)𝐵)) ∈ ((𝐴(,)𝐵)–cn→ℝ))
67	66	ad2antrr 739	. . . . . . . . . . . . . . . . . . . . . 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 11262	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ℝ ⊆ ℂ
69	68	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → ℝ ⊆ ℂ)
70		fss 6729	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝐹:(𝐴[,]𝐵)⟶ℝ ∧ ℝ ⊆ ℂ) → 𝐹:(𝐴[,]𝐵)⟶ℂ)
71	9, 68, 70	sylancl 598	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → 𝐹:(𝐴[,]𝐵)⟶ℂ)
72	64, 13	sstrid 3920	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℝ)
73		eqid 2760	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
74		tgioo4 25131	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
75	73, 74	dvres 26238	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 852	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → (ℝ D (𝐹 ↾ (𝐴(,)𝐵))) = ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘(𝐴(,)𝐵))))
77		retop 25087	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (topGen‘ran (,)) ∈ Top
78		iooretop 25091	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝐴(,)𝐵) ∈ (topGen‘ran (,))
79		isopn3i 23407	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((topGen‘ran (,)) ∈ Top ∧ (𝐴(,)𝐵) ∈ (topGen‘ran (,))) → ((int‘(topGen‘ran (,)))‘(𝐴(,)𝐵)) = (𝐴(,)𝐵))
80	77, 78, 79	mp2an 705	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((int‘(topGen‘ran (,)))‘(𝐴(,)𝐵)) = (𝐴(,)𝐵)
81	80	reseq2i 5967	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘(𝐴(,)𝐵))) = ((ℝ D 𝐹) ↾ (𝐴(,)𝐵))
82		fnresdm 6661	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((ℝ D 𝐹) Fn (𝐴(,)𝐵) → ((ℝ D 𝐹) ↾ (𝐴(,)𝐵)) = (ℝ D 𝐹))
83	44, 82	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → ((ℝ D 𝐹) ↾ (𝐴(,)𝐵)) = (ℝ D 𝐹))
84	81, 83	eqtrid 2807	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘(𝐴(,)𝐵))) = (ℝ D 𝐹))
85	76, 84	eqtrd 2795	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → (ℝ D (𝐹 ↾ (𝐴(,)𝐵))) = (ℝ D 𝐹))
86	85	dmeqd 5887	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → dom (ℝ D (𝐹 ↾ (𝐴(,)𝐵))) = dom (ℝ D 𝐹))
87	86, 41	eqtrd 2795	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → dom (ℝ D (𝐹 ↾ (𝐴(,)𝐵))) = (𝐴(,)𝐵))
88	87	ad2antrr 739	. . . . . . . . . . . . . . . . . . . . . 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 26337	. . . . . . . . . . . . . . . . . . . . 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 739	. . . . . . . . . . . . . . . . . . . . . . 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 6890	. . . . . . . . . . . . . . . . . . . . . 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 6890	. . . . . . . . . . . . . . . . . . . . . 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 7440	. . . . . . . . . . . . . . . . . . . . 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 5920	. . . . . . . . . . . . . . . . . . . . 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 3963	. . . . . . . . . . . . . . . . . . . 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 790	. . . . . . . . . . . . . . . . . . . . . . . 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 13580	. . . . . . . . . . . . . . . . . . . . . . . . 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 221	. . . . . . . . . . . . . . . . . . . . . . 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 501	. . . . . . . . . . . . . . . . . . . . . 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 500	. . . . . . . . . . . . . . . . . . . . . . 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 11403	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((ℝ D 𝐹)‘𝑦) ∈ ℝ ∧ 0 ∈ ℝ) → (((ℝ D 𝐹)‘𝑦) < 0 → ((ℝ D 𝐹)‘𝑦) ≤ 0))
104	102, 18, 103	sylancl 598	. . . . . . . . . . . . . . . . . . . . . 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 16	. . . . . . . . . . . . . . . . . . . . 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 785	. . . . . . . . . . . . . . . . . . . . 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 603	. . . . . . . . . . . . . . . . . . . . . 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 13547	. . . . . . . . . . . . . . . . . . . . . 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 596	. . . . . . . . . . . . . . . . . . . . 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 1361	. . . . . . . . . . . . . . . . . . . 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 3910	. . . . . . . . . . . . . . . . . . 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 462	. . . . . . . . . . . . . . . . . 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 201	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ ((ℝ D 𝐹)‘𝑦) ∈ (-∞(,)0))) ∧ 𝑧 ∈ (𝐴(,)𝐵)) → ¬ 0 ≤ ((ℝ D 𝐹)‘𝑧))
114		ltnle 11394	. . . . . . . . . . . . . . . . . 18 ⊢ ((((ℝ D 𝐹)‘𝑧) ∈ ℝ ∧ 0 ∈ ℝ) → (((ℝ D 𝐹)‘𝑧) < 0 ↔ ¬ 0 ≤ ((ℝ D 𝐹)‘𝑧)))
115	60, 18, 114	sylancl 598	. . . . . . . . . . . . . . . . 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 260	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ ((ℝ D 𝐹)‘𝑦) ∈ (-∞(,)0))) ∧ 𝑧 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑧) < 0)
117		elioomnf 13580	. . . . . . . . . . . . . . . . 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 595	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ ((ℝ D 𝐹)‘𝑦) ∈ (-∞(,)0))) ∧ 𝑧 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑧) ∈ (-∞(,)0))
120	119	ralrimiva 3154	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ ((ℝ D 𝐹)‘𝑦) ∈ (-∞(,)0))) → ∀𝑧 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑧) ∈ (-∞(,)0))
121		ffnfv 7122	. . . . . . . . . . . . . 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 595	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ ((ℝ D 𝐹)‘𝑦) ∈ (-∞(,)0))) → (ℝ D 𝐹):(𝐴(,)𝐵)⟶(-∞(,)0))
123	55, 56, 57, 122	dvlt0 26332	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ ((ℝ D 𝐹)‘𝑦) ∈ (-∞(,)0))) → 𝐹 Isom < , ^◡ < ((𝐴[,]𝐵), ran 𝐹))
124	123	olcd 888	. . . . . . . . . . 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 462	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (((ℝ D 𝐹)‘𝑦) ∈ (-∞(,)0) → (𝐹 Isom < , < ((𝐴[,]𝐵), ran 𝐹) ∨ 𝐹 Isom < , ^◡ < ((𝐴[,]𝐵), ran 𝐹))))
126		eleq1 2848	. . . . . . . . . . 11 ⊢ (((ℝ D 𝐹)‘𝑦) = 𝑥 → (((ℝ D 𝐹)‘𝑦) ∈ (-∞(,)0) ↔ 𝑥 ∈ (-∞(,)0)))
127	126	imbi1d 344	. . . . . . . . . 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 248	. . . . . . . . 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 3163	. . . . . . . 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 243	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ ran (ℝ D 𝐹) → (𝑥 ∈ (-∞(,)0) → (𝐹 Isom < , < ((𝐴[,]𝐵), ran 𝐹) ∨ 𝐹 Isom < , ^◡ < ((𝐴[,]𝐵), ran 𝐹)))))
131	130	impd 416	. . . . . 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 245	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (ran (ℝ D 𝐹) ∩ (-∞(,)0)) → (𝐹 Isom < , < ((𝐴[,]𝐵), ran 𝐹) ∨ 𝐹 Isom < , ^◡ < ((𝐴[,]𝐵), ran 𝐹))))
133	132	exlimdv 1965	. . . 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 245	. . 3 ⊢ (𝜑 → ((ran (ℝ D 𝐹) ∩ (-∞(,)0)) ≠ ∅ → (𝐹 Isom < , < ((𝐴[,]𝐵), ran 𝐹) ∨ 𝐹 Isom < , ^◡ < ((𝐴[,]𝐵), ran 𝐹))))
135	134	imp 412	. 2 ⊢ ((𝜑 ∧ (ran (ℝ D 𝐹) ∩ (-∞(,)0)) ≠ ∅) → (𝐹 Isom < , < ((𝐴[,]𝐵), ran 𝐹) ∨ 𝐹 Isom < , ^◡ < ((𝐴[,]𝐵), ran 𝐹)))
136	50, 135	pm2.61dane 3042	1 ⊢ (𝜑 → (𝐹 Isom < , < ((𝐴[,]𝐵), ran 𝐹) ∨ 𝐹 Isom < , ^◡ < ((𝐴[,]𝐵), ran 𝐹)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 ∧ w3a 1103 = wceq 1569 ∃wex 1811 ∈ wcel 2144 ≠ wne 2955 ∀wral 3076 ∃wrex 3086 ⊆ wss 3896 ∩ cin 4088 ∪ cun 4148 ∅c0 4277 class class class wbr 5102 ^◡ccnv 5649 dom cdm 5650 ran crn 5651 ↾ cres 5652 Fn wfn 6536 ⟶wf 6537 ‘cfv 6541 Isom wiso 6542 (class class class)co 7422 ℂcc 11203 ℝcr 11204 0cc0 11205 -∞cmnf 11346 ℝ^*cxr 11347 < clt 11348 ≤ cle 11349 ℝ⁺crp 13125 (,)cioo 13481 [,]cicc 13484 TopOpenctopn 17599 topGenctg 17615 ℂ_fldccnfld 21685 Topctop 23218 intcnt 23342 –cn→ccncf 25204 D cdv 26190

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7753 ax-cnex 11261 ax-resscn 11262 ax-1cn 11263 ax-icn 11264 ax-addcl 11265 ax-addrcl 11266 ax-mulcl 11267 ax-mulrcl 11268 ax-mulcom 11269 ax-addass 11270 ax-mulass 11271 ax-distr 11272 ax-i2m1 11273 ax-1ne0 11274 ax-1rid 11275 ax-rnegex 11276 ax-rrecex 11277 ax-cnre 11278 ax-pre-lttri 11279 ax-pre-lttrn 11280 ax-pre-ltadd 11281 ax-pre-mulgt0 11282 ax-pre-sup 11283 ax-addf 11284

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3848 df-ss 3897 df-pss 4013 df-dif 4046 df-in 4090 df-un 4150 df-nul 4278 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4835 df-uni 4867 df-int 4907 df-iun 4952 df-iin 4953 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-se 5604 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-isom 6550 df-riota 7379 df-ov 7425 df-oprab 7426 df-mpo 7427 df-of 7695 df-om 7880 df-1st 8003 df-2nd 8004 df-supp 8182 df-frecs 8304 df-wrecs 8335 df-recs 8384 df-rdg 8423 df-1o 8481 df-2o 8482 df-er 8722 df-map 8854 df-pm 8855 df-ixp 8931 df-en 8979 df-dom 8980 df-sdom 8981 df-fin 8982 df-fsupp 9359 df-fi 9408 df-sup 9439 df-inf 9440 df-oi 9509 df-card 10025 df-pnf 11350 df-mnf 11351 df-xr 11352 df-ltxr 11353 df-le 11354 df-sub 11548 df-neg 11549 df-div 11979 df-nn 12341 df-2 12410 df-3 12411 df-4 12412 df-5 12413 df-6 12414 df-7 12415 df-8 12416 df-9 12417 df-n0 12612 df-z 12699 df-dec 12820 df-uz 12971 df-q 13081 df-rp 13126 df-xneg 13246 df-xadd 13247 df-xmul 13248 df-ioo 13485 df-ico 13487 df-icc 13488 df-fz 13645 df-fzo 13794 df-seq 14150 df-exp 14210 df-hash 14480 df-cj 15273 df-re 15274 df-im 15275 df-sqrt 15409 df-abs 15410 df-struct 17332 df-sets 17349 df-slot 17367 df-ndx 17379 df-base 17395 df-ress 17416 df-plusg 17448 df-mulr 17449 df-starv 17450 df-sca 17451 df-vsca 17452 df-ip 17453 df-tset 17454 df-ple 17455 df-ds 17457 df-unif 17458 df-hom 17459 df-cco 17460 df-rest 17600 df-topn 17601 df-0g 17619 df-gsum 17620 df-topgen 17621 df-pt 17622 df-prds 17625 df-xrs 17681 df-qtop 17686 df-imas 17687 df-xps 17689 df-mre 17763 df-mrc 17764 df-acs 17766 df-mgm 18823 df-sgrp 18915 df-mnd 18931 df-submnd 18986 df-mulg 19285 df-cntz 19538 df-cmn 20003 df-psmet 21677 df-xmet 21678 df-met 21679 df-bl 21680 df-mopn 21681 df-fbas 21682 df-fg 21683 df-cnfld 21686 df-top 23219 df-topon 23236 df-topsp 23258 df-bases 23271 df-cld 23344 df-ntr 23345 df-cls 23346 df-nei 23423 df-lp 23461 df-perf 23462 df-cn 23552 df-cnp 23553 df-haus 23640 df-cmp 23712 df-tx 23888 df-hmeo 24081 df-fil 24172 df-fm 24264 df-flim 24265 df-flf 24266 df-xms 24646 df-ms 24647 df-tms 24648 df-cncf 25206 df-limc 26193 df-dv 26194

This theorem is used by: dvne0f1 26339 dvcnvrelem1 26344

W3C validator