ivthreinc - Intuitionistic Logic Explorer

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Theorem ivthreinc 15837

Description: Restating the intermediate value theorem. Given a hypothesis stating the intermediate value theorem (in a strong form which is not provable given our axioms alone), provide a conclusion similar to the theorem as stated in the Metamath Proof Explorer (which is also similar to how we state the theorem for a strictly monotonic function at ivthinc 15835). Being able to have a hypothesis stating the intermediate value theorem will be helpful when it comes time to show that it implies a constructive taboo. This version of the theorem requires that the function 𝐹 is continuous on the entire real line, not just (𝐴[,]𝐵) which may be an unnecessary condition but which is sufficient for the way we want to use it. (Contributed by Jim Kingdon, 7-Jul-2025.)

**Hypotheses**
Ref	Expression
ivthreinc.1	⊢ (𝜑 → 𝐴 ∈ ℝ)
ivthreinc.2	⊢ (𝜑 → 𝐵 ∈ ℝ)
ivthreinc.3	⊢ (𝜑 → 𝑈 ∈ ℝ)
ivthreinc.4	⊢ (𝜑 → 𝐴 < 𝐵)
ivthreinc.7	⊢ (𝜑 → 𝐹 ∈ (ℝ–cn→ℝ))
ivthreinc.9	⊢ (𝜑 → ((𝐹‘𝐴) < 𝑈 ∧ 𝑈 < (𝐹‘𝐵)))
ivthreinc.i	⊢ (𝜑 → ∀𝑓(𝑓 ∈ (ℝ–cn→ℝ) → ∀𝑎 ∈ ℝ ∀𝑏 ∈ ℝ ((𝑎 < 𝑏 ∧ (𝑓‘𝑎) < 0 ∧ 0 < (𝑓‘𝑏)) → ∃𝑥 ∈ ℝ (𝑎 < 𝑥 ∧ 𝑥 < 𝑏 ∧ (𝑓‘𝑥) = 0))))

**Assertion**
Ref	Expression
ivthreinc	⊢ (𝜑 → ∃𝑐 ∈ (𝐴(,)𝐵)(𝐹‘𝑐) = 𝑈)

Distinct variable groups: 𝐴,𝑎,𝑏,𝑥 𝐴,𝑐,𝑥 𝐵,𝑏,𝑥 𝐵,𝑐 𝐹,𝑎,𝑏,𝑓,𝑥 𝐹,𝑐 𝑈,𝑎,𝑏,𝑓,𝑥 𝑈,𝑐 𝜑,𝑥 Allowed substitution hints: 𝜑(𝑓, 𝑎, 𝑏, 𝑐) 𝐴(𝑓) 𝐵(𝑓, 𝑎)

Proof of Theorem ivthreinc Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ivthreinc.4	. . . 4 ⊢ (𝜑 → 𝐴 < 𝐵)
2		eqid 2238	. . . . . 6 ⊢ (𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈)) = (𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))
3		fveq2 5695	. . . . . . 7 ⊢ (𝑟 = 𝐴 → (𝐹‘𝑟) = (𝐹‘𝐴))
4	3	oveq1d 6100	. . . . . 6 ⊢ (𝑟 = 𝐴 → ((𝐹‘𝑟) − 𝑈) = ((𝐹‘𝐴) − 𝑈))
5		ivthreinc.1	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ)
6		ivthreinc.7	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ (ℝ–cn→ℝ))
7		cncff 15769	. . . . . . . . 9 ⊢ (𝐹 ∈ (ℝ–cn→ℝ) → 𝐹:ℝ⟶ℝ)
8	6, 7	syl 14	. . . . . . . 8 ⊢ (𝜑 → 𝐹:ℝ⟶ℝ)
9	8, 5	ffvelcdmd 5844	. . . . . . 7 ⊢ (𝜑 → (𝐹‘𝐴) ∈ ℝ)
10		ivthreinc.3	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ ℝ)
11	9, 10	resubcld 8710	. . . . . 6 ⊢ (𝜑 → ((𝐹‘𝐴) − 𝑈) ∈ ℝ)
12	2, 4, 5, 11	fvmptd3 5799	. . . . 5 ⊢ (𝜑 → ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝐴) = ((𝐹‘𝐴) − 𝑈))
13		ivthreinc.9	. . . . . . 7 ⊢ (𝜑 → ((𝐹‘𝐴) < 𝑈 ∧ 𝑈 < (𝐹‘𝐵)))
14	13	simpld 112	. . . . . 6 ⊢ (𝜑 → (𝐹‘𝐴) < 𝑈)
15	9, 10	sublt0d 8901	. . . . . 6 ⊢ (𝜑 → (((𝐹‘𝐴) − 𝑈) < 0 ↔ (𝐹‘𝐴) < 𝑈))
16	14, 15	mpbird 167	. . . . 5 ⊢ (𝜑 → ((𝐹‘𝐴) − 𝑈) < 0)
17	12, 16	eqbrtrd 4152	. . . 4 ⊢ (𝜑 → ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝐴) < 0)
18	13	simprd 114	. . . . . 6 ⊢ (𝜑 → 𝑈 < (𝐹‘𝐵))
19		ivthreinc.2	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℝ)
20	8, 19	ffvelcdmd 5844	. . . . . . 7 ⊢ (𝜑 → (𝐹‘𝐵) ∈ ℝ)
21	10, 20	posdifd 8862	. . . . . 6 ⊢ (𝜑 → (𝑈 < (𝐹‘𝐵) ↔ 0 < ((𝐹‘𝐵) − 𝑈)))
22	18, 21	mpbid 147	. . . . 5 ⊢ (𝜑 → 0 < ((𝐹‘𝐵) − 𝑈))
23		fveq2 5695	. . . . . . 7 ⊢ (𝑟 = 𝐵 → (𝐹‘𝑟) = (𝐹‘𝐵))
24	23	oveq1d 6100	. . . . . 6 ⊢ (𝑟 = 𝐵 → ((𝐹‘𝑟) − 𝑈) = ((𝐹‘𝐵) − 𝑈))
25	20, 10	resubcld 8710	. . . . . 6 ⊢ (𝜑 → ((𝐹‘𝐵) − 𝑈) ∈ ℝ)
26	2, 24, 19, 25	fvmptd3 5799	. . . . 5 ⊢ (𝜑 → ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝐵) = ((𝐹‘𝐵) − 𝑈))
27	22, 26	breqtrrd 4158	. . . 4 ⊢ (𝜑 → 0 < ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝐵))
28	1, 17, 27	3jca 1208	. . 3 ⊢ (𝜑 → (𝐴 < 𝐵 ∧ ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝐴) < 0 ∧ 0 < ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝐵)))
29		breq2 4134	. . . . . 6 ⊢ (𝑏 = 𝐵 → (𝐴 < 𝑏 ↔ 𝐴 < 𝐵))
30		fveq2 5695	. . . . . . 7 ⊢ (𝑏 = 𝐵 → ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝑏) = ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝐵))
31	30	breq2d 4142	. . . . . 6 ⊢ (𝑏 = 𝐵 → (0 < ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝑏) ↔ 0 < ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝐵)))
32	29, 31	3anbi13d 1355	. . . . 5 ⊢ (𝑏 = 𝐵 → ((𝐴 < 𝑏 ∧ ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝐴) < 0 ∧ 0 < ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝑏)) ↔ (𝐴 < 𝐵 ∧ ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝐴) < 0 ∧ 0 < ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝐵))))
33		breq2 4134	. . . . . . 7 ⊢ (𝑏 = 𝐵 → (𝑥 < 𝑏 ↔ 𝑥 < 𝐵))
34	33	3anbi2d 1358	. . . . . 6 ⊢ (𝑏 = 𝐵 → ((𝐴 < 𝑥 ∧ 𝑥 < 𝑏 ∧ ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝑥) = 0) ↔ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵 ∧ ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝑥) = 0)))
35	34	rexbidv 2551	. . . . 5 ⊢ (𝑏 = 𝐵 → (∃𝑥 ∈ ℝ (𝐴 < 𝑥 ∧ 𝑥 < 𝑏 ∧ ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝑥) = 0) ↔ ∃𝑥 ∈ ℝ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵 ∧ ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝑥) = 0)))
36	32, 35	imbi12d 234	. . . 4 ⊢ (𝑏 = 𝐵 → (((𝐴 < 𝑏 ∧ ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝐴) < 0 ∧ 0 < ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝑏)) → ∃𝑥 ∈ ℝ (𝐴 < 𝑥 ∧ 𝑥 < 𝑏 ∧ ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝑥) = 0)) ↔ ((𝐴 < 𝐵 ∧ ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝐴) < 0 ∧ 0 < ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝐵)) → ∃𝑥 ∈ ℝ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵 ∧ ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝑥) = 0))))
37		breq1 4133	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → (𝑎 < 𝑏 ↔ 𝐴 < 𝑏))
38		fveq2 5695	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝑎) = ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝐴))
39	38	breq1d 4140	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → (((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝑎) < 0 ↔ ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝐴) < 0))
40	37, 39	3anbi12d 1354	. . . . . . 7 ⊢ (𝑎 = 𝐴 → ((𝑎 < 𝑏 ∧ ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝑎) < 0 ∧ 0 < ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝑏)) ↔ (𝐴 < 𝑏 ∧ ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝐴) < 0 ∧ 0 < ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝑏))))
41		breq1 4133	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → (𝑎 < 𝑥 ↔ 𝐴 < 𝑥))
42	41	3anbi1d 1357	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → ((𝑎 < 𝑥 ∧ 𝑥 < 𝑏 ∧ ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝑥) = 0) ↔ (𝐴 < 𝑥 ∧ 𝑥 < 𝑏 ∧ ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝑥) = 0)))
43	42	rexbidv 2551	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (∃𝑥 ∈ ℝ (𝑎 < 𝑥 ∧ 𝑥 < 𝑏 ∧ ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝑥) = 0) ↔ ∃𝑥 ∈ ℝ (𝐴 < 𝑥 ∧ 𝑥 < 𝑏 ∧ ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝑥) = 0)))
44	40, 43	imbi12d 234	. . . . . 6 ⊢ (𝑎 = 𝐴 → (((𝑎 < 𝑏 ∧ ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝑎) < 0 ∧ 0 < ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝑏)) → ∃𝑥 ∈ ℝ (𝑎 < 𝑥 ∧ 𝑥 < 𝑏 ∧ ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝑥) = 0)) ↔ ((𝐴 < 𝑏 ∧ ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝐴) < 0 ∧ 0 < ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝑏)) → ∃𝑥 ∈ ℝ (𝐴 < 𝑥 ∧ 𝑥 < 𝑏 ∧ ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝑥) = 0))))
45	44	ralbidv 2550	. . . . 5 ⊢ (𝑎 = 𝐴 → (∀𝑏 ∈ ℝ ((𝑎 < 𝑏 ∧ ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝑎) < 0 ∧ 0 < ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝑏)) → ∃𝑥 ∈ ℝ (𝑎 < 𝑥 ∧ 𝑥 < 𝑏 ∧ ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝑥) = 0)) ↔ ∀𝑏 ∈ ℝ ((𝐴 < 𝑏 ∧ ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝐴) < 0 ∧ 0 < ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝑏)) → ∃𝑥 ∈ ℝ (𝐴 < 𝑥 ∧ 𝑥 < 𝑏 ∧ ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝑥) = 0))))
46	8	ffvelcdmda 5843	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ) → (𝐹‘𝑟) ∈ ℝ)
47	10	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ) → 𝑈 ∈ ℝ)
48	46, 47	resubcld 8710	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ) → ((𝐹‘𝑟) − 𝑈) ∈ ℝ)
49	48	fmpttd 5863	. . . . . . 7 ⊢ (𝜑 → (𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈)):ℝ⟶ℝ)
50		ax-resscn 8272	. . . . . . . . 9 ⊢ ℝ ⊆ ℂ
51	50	a1i 9	. . . . . . . 8 ⊢ (𝜑 → ℝ ⊆ ℂ)
52	8	feqmptd 5756	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 = (𝑟 ∈ ℝ ↦ (𝐹‘𝑟)))
53		ssid 3268	. . . . . . . . . . . 12 ⊢ ℂ ⊆ ℂ
54		cncfss 15775	. . . . . . . . . . . 12 ⊢ ((ℝ ⊆ ℂ ∧ ℂ ⊆ ℂ) → (ℝ–cn→ℝ) ⊆ (ℝ–cn→ℂ))
55	50, 53, 54	mp2an 430	. . . . . . . . . . 11 ⊢ (ℝ–cn→ℝ) ⊆ (ℝ–cn→ℂ)
56	55, 6	sselid 3246	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ (ℝ–cn→ℂ))
57	52, 56	eqeltrrd 2316	. . . . . . . . 9 ⊢ (𝜑 → (𝑟 ∈ ℝ ↦ (𝐹‘𝑟)) ∈ (ℝ–cn→ℂ))
58	10	recnd 8355	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ∈ ℂ)
59	53	a1i 9	. . . . . . . . . 10 ⊢ (𝜑 → ℂ ⊆ ℂ)
60		cncfmptc 15788	. . . . . . . . . 10 ⊢ ((𝑈 ∈ ℂ ∧ ℝ ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑟 ∈ ℝ ↦ 𝑈) ∈ (ℝ–cn→ℂ))
61	58, 51, 59, 60	syl3anc 1278	. . . . . . . . 9 ⊢ (𝜑 → (𝑟 ∈ ℝ ↦ 𝑈) ∈ (ℝ–cn→ℂ))
62	57, 61	subcncf 15805	. . . . . . . 8 ⊢ (𝜑 → (𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈)) ∈ (ℝ–cn→ℂ))
63		cncfcdm 15774	. . . . . . . 8 ⊢ ((ℝ ⊆ ℂ ∧ (𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈)) ∈ (ℝ–cn→ℂ)) → ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈)) ∈ (ℝ–cn→ℝ) ↔ (𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈)):ℝ⟶ℝ))
64	51, 62, 63	syl2anc 415	. . . . . . 7 ⊢ (𝜑 → ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈)) ∈ (ℝ–cn→ℝ) ↔ (𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈)):ℝ⟶ℝ))
65	49, 64	mpbird 167	. . . . . 6 ⊢ (𝜑 → (𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈)) ∈ (ℝ–cn→ℝ))
66		ivthreinc.i	. . . . . . 7 ⊢ (𝜑 → ∀𝑓(𝑓 ∈ (ℝ–cn→ℝ) → ∀𝑎 ∈ ℝ ∀𝑏 ∈ ℝ ((𝑎 < 𝑏 ∧ (𝑓‘𝑎) < 0 ∧ 0 < (𝑓‘𝑏)) → ∃𝑥 ∈ ℝ (𝑎 < 𝑥 ∧ 𝑥 < 𝑏 ∧ (𝑓‘𝑥) = 0))))
67		reex 8314	. . . . . . . . 9 ⊢ ℝ ∈ V
68	67	mptex 5943	. . . . . . . 8 ⊢ (𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈)) ∈ V
69		eleq1 2301	. . . . . . . . 9 ⊢ (𝑓 = (𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈)) → (𝑓 ∈ (ℝ–cn→ℝ) ↔ (𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈)) ∈ (ℝ–cn→ℝ)))
70		fveq1 5694	. . . . . . . . . . . . . 14 ⊢ (𝑓 = (𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈)) → (𝑓‘𝑎) = ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝑎))
71	70	breq1d 4140	. . . . . . . . . . . . 13 ⊢ (𝑓 = (𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈)) → ((𝑓‘𝑎) < 0 ↔ ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝑎) < 0))
72		fveq1 5694	. . . . . . . . . . . . . 14 ⊢ (𝑓 = (𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈)) → (𝑓‘𝑏) = ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝑏))
73	72	breq2d 4142	. . . . . . . . . . . . 13 ⊢ (𝑓 = (𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈)) → (0 < (𝑓‘𝑏) ↔ 0 < ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝑏)))
74	71, 73	3anbi23d 1356	. . . . . . . . . . . 12 ⊢ (𝑓 = (𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈)) → ((𝑎 < 𝑏 ∧ (𝑓‘𝑎) < 0 ∧ 0 < (𝑓‘𝑏)) ↔ (𝑎 < 𝑏 ∧ ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝑎) < 0 ∧ 0 < ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝑏))))
75		fveq1 5694	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = (𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈)) → (𝑓‘𝑥) = ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝑥))
76	75	eqeq1d 2247	. . . . . . . . . . . . . 14 ⊢ (𝑓 = (𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈)) → ((𝑓‘𝑥) = 0 ↔ ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝑥) = 0))
77	76	3anbi3d 1359	. . . . . . . . . . . . 13 ⊢ (𝑓 = (𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈)) → ((𝑎 < 𝑥 ∧ 𝑥 < 𝑏 ∧ (𝑓‘𝑥) = 0) ↔ (𝑎 < 𝑥 ∧ 𝑥 < 𝑏 ∧ ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝑥) = 0)))
78	77	rexbidv 2551	. . . . . . . . . . . 12 ⊢ (𝑓 = (𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈)) → (∃𝑥 ∈ ℝ (𝑎 < 𝑥 ∧ 𝑥 < 𝑏 ∧ (𝑓‘𝑥) = 0) ↔ ∃𝑥 ∈ ℝ (𝑎 < 𝑥 ∧ 𝑥 < 𝑏 ∧ ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝑥) = 0)))
79	74, 78	imbi12d 234	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈)) → (((𝑎 < 𝑏 ∧ (𝑓‘𝑎) < 0 ∧ 0 < (𝑓‘𝑏)) → ∃𝑥 ∈ ℝ (𝑎 < 𝑥 ∧ 𝑥 < 𝑏 ∧ (𝑓‘𝑥) = 0)) ↔ ((𝑎 < 𝑏 ∧ ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝑎) < 0 ∧ 0 < ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝑏)) → ∃𝑥 ∈ ℝ (𝑎 < 𝑥 ∧ 𝑥 < 𝑏 ∧ ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝑥) = 0))))
80	79	ralbidv 2550	. . . . . . . . . 10 ⊢ (𝑓 = (𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈)) → (∀𝑏 ∈ ℝ ((𝑎 < 𝑏 ∧ (𝑓‘𝑎) < 0 ∧ 0 < (𝑓‘𝑏)) → ∃𝑥 ∈ ℝ (𝑎 < 𝑥 ∧ 𝑥 < 𝑏 ∧ (𝑓‘𝑥) = 0)) ↔ ∀𝑏 ∈ ℝ ((𝑎 < 𝑏 ∧ ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝑎) < 0 ∧ 0 < ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝑏)) → ∃𝑥 ∈ ℝ (𝑎 < 𝑥 ∧ 𝑥 < 𝑏 ∧ ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝑥) = 0))))
81	80	ralbidv 2550	. . . . . . . . 9 ⊢ (𝑓 = (𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈)) → (∀𝑎 ∈ ℝ ∀𝑏 ∈ ℝ ((𝑎 < 𝑏 ∧ (𝑓‘𝑎) < 0 ∧ 0 < (𝑓‘𝑏)) → ∃𝑥 ∈ ℝ (𝑎 < 𝑥 ∧ 𝑥 < 𝑏 ∧ (𝑓‘𝑥) = 0)) ↔ ∀𝑎 ∈ ℝ ∀𝑏 ∈ ℝ ((𝑎 < 𝑏 ∧ ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝑎) < 0 ∧ 0 < ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝑏)) → ∃𝑥 ∈ ℝ (𝑎 < 𝑥 ∧ 𝑥 < 𝑏 ∧ ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝑥) = 0))))
82	69, 81	imbi12d 234	. . . . . . . 8 ⊢ (𝑓 = (𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈)) → ((𝑓 ∈ (ℝ–cn→ℝ) → ∀𝑎 ∈ ℝ ∀𝑏 ∈ ℝ ((𝑎 < 𝑏 ∧ (𝑓‘𝑎) < 0 ∧ 0 < (𝑓‘𝑏)) → ∃𝑥 ∈ ℝ (𝑎 < 𝑥 ∧ 𝑥 < 𝑏 ∧ (𝑓‘𝑥) = 0))) ↔ ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈)) ∈ (ℝ–cn→ℝ) → ∀𝑎 ∈ ℝ ∀𝑏 ∈ ℝ ((𝑎 < 𝑏 ∧ ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝑎) < 0 ∧ 0 < ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝑏)) → ∃𝑥 ∈ ℝ (𝑎 < 𝑥 ∧ 𝑥 < 𝑏 ∧ ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝑥) = 0)))))
83	68, 82	spcv 2919	. . . . . . 7 ⊢ (∀𝑓(𝑓 ∈ (ℝ–cn→ℝ) → ∀𝑎 ∈ ℝ ∀𝑏 ∈ ℝ ((𝑎 < 𝑏 ∧ (𝑓‘𝑎) < 0 ∧ 0 < (𝑓‘𝑏)) → ∃𝑥 ∈ ℝ (𝑎 < 𝑥 ∧ 𝑥 < 𝑏 ∧ (𝑓‘𝑥) = 0))) → ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈)) ∈ (ℝ–cn→ℝ) → ∀𝑎 ∈ ℝ ∀𝑏 ∈ ℝ ((𝑎 < 𝑏 ∧ ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝑎) < 0 ∧ 0 < ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝑏)) → ∃𝑥 ∈ ℝ (𝑎 < 𝑥 ∧ 𝑥 < 𝑏 ∧ ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝑥) = 0))))
84	66, 83	syl 14	. . . . . 6 ⊢ (𝜑 → ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈)) ∈ (ℝ–cn→ℝ) → ∀𝑎 ∈ ℝ ∀𝑏 ∈ ℝ ((𝑎 < 𝑏 ∧ ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝑎) < 0 ∧ 0 < ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝑏)) → ∃𝑥 ∈ ℝ (𝑎 < 𝑥 ∧ 𝑥 < 𝑏 ∧ ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝑥) = 0))))
85	65, 84	mpd 13	. . . . 5 ⊢ (𝜑 → ∀𝑎 ∈ ℝ ∀𝑏 ∈ ℝ ((𝑎 < 𝑏 ∧ ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝑎) < 0 ∧ 0 < ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝑏)) → ∃𝑥 ∈ ℝ (𝑎 < 𝑥 ∧ 𝑥 < 𝑏 ∧ ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝑥) = 0)))
86	45, 85, 5	rspcdva 2934	. . . 4 ⊢ (𝜑 → ∀𝑏 ∈ ℝ ((𝐴 < 𝑏 ∧ ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝐴) < 0 ∧ 0 < ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝑏)) → ∃𝑥 ∈ ℝ (𝐴 < 𝑥 ∧ 𝑥 < 𝑏 ∧ ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝑥) = 0)))
87	36, 86, 19	rspcdva 2934	. . 3 ⊢ (𝜑 → ((𝐴 < 𝐵 ∧ ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝐴) < 0 ∧ 0 < ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝐵)) → ∃𝑥 ∈ ℝ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵 ∧ ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝑥) = 0)))
88	28, 87	mpd 13	. 2 ⊢ (𝜑 → ∃𝑥 ∈ ℝ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵 ∧ ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝑥) = 0))
89	5	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵 ∧ ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝑥) = 0))) → 𝐴 ∈ ℝ)
90	89	rexrd 8376	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵 ∧ ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝑥) = 0))) → 𝐴 ∈ ℝ^*)
91	19	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵 ∧ ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝑥) = 0))) → 𝐵 ∈ ℝ)
92	91	rexrd 8376	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵 ∧ ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝑥) = 0))) → 𝐵 ∈ ℝ^*)
93		simprl 535	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵 ∧ ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝑥) = 0))) → 𝑥 ∈ ℝ)
94	90, 92, 93	3jca 1208	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵 ∧ ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝑥) = 0))) → (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝑥 ∈ ℝ))
95		simprr1 1076	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵 ∧ ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝑥) = 0))) → 𝐴 < 𝑥)
96		simprr2 1077	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵 ∧ ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝑥) = 0))) → 𝑥 < 𝐵)
97	95, 96	jca 306	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵 ∧ ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝑥) = 0))) → (𝐴 < 𝑥 ∧ 𝑥 < 𝐵))
98		elioo4g 10347	. . . 4 ⊢ (𝑥 ∈ (𝐴(,)𝐵) ↔ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝑥 ∈ ℝ) ∧ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)))
99	94, 97, 98	sylanbrc 421	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵 ∧ ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝑥) = 0))) → 𝑥 ∈ (𝐴(,)𝐵))
100	8	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵 ∧ ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝑥) = 0))) → 𝐹:ℝ⟶ℝ)
101	100, 93	ffvelcdmd 5844	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵 ∧ ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝑥) = 0))) → (𝐹‘𝑥) ∈ ℝ)
102	101	recnd 8355	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵 ∧ ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝑥) = 0))) → (𝐹‘𝑥) ∈ ℂ)
103	58	adantr 276	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵 ∧ ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝑥) = 0))) → 𝑈 ∈ ℂ)
104		fveq2 5695	. . . . . . 7 ⊢ (𝑟 = 𝑥 → (𝐹‘𝑟) = (𝐹‘𝑥))
105	104	oveq1d 6100	. . . . . 6 ⊢ (𝑟 = 𝑥 → ((𝐹‘𝑟) − 𝑈) = ((𝐹‘𝑥) − 𝑈))
106	10	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵 ∧ ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝑥) = 0))) → 𝑈 ∈ ℝ)
107	101, 106	resubcld 8710	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵 ∧ ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝑥) = 0))) → ((𝐹‘𝑥) − 𝑈) ∈ ℝ)
108	2, 105, 93, 107	fvmptd3 5799	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵 ∧ ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝑥) = 0))) → ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝑥) = ((𝐹‘𝑥) − 𝑈))
109		simprr3 1078	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵 ∧ ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝑥) = 0))) → ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝑥) = 0)
110	108, 109	eqtr3d 2273	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵 ∧ ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝑥) = 0))) → ((𝐹‘𝑥) − 𝑈) = 0)
111	102, 103, 110	subeq0d 8647	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵 ∧ ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝑥) = 0))) → (𝐹‘𝑥) = 𝑈)
112		fveqeq2 5704	. . . 4 ⊢ (𝑐 = 𝑥 → ((𝐹‘𝑐) = 𝑈 ↔ (𝐹‘𝑥) = 𝑈))
113	112	rspcev 2929	. . 3 ⊢ ((𝑥 ∈ (𝐴(,)𝐵) ∧ (𝐹‘𝑥) = 𝑈) → ∃𝑐 ∈ (𝐴(,)𝐵)(𝐹‘𝑐) = 𝑈)
114	99, 111, 113	syl2anc 415	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵 ∧ ((𝑟 ∈ ℝ ↦ ((𝐹‘𝑟) − 𝑈))‘𝑥) = 0))) → ∃𝑐 ∈ (𝐴(,)𝐵)(𝐹‘𝑐) = 𝑈)
115	88, 114	rexlimddv 2673	1 ⊢ (𝜑 → ∃𝑐 ∈ (𝐴(,)𝐵)(𝐹‘𝑐) = 𝑈)

Colors of variables: wff set class

This proof depends on syntax axioms: → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 1009 ∀wal 1400 = wceq 1402 ∈ wcel 2209 ∀wral 2528 ∃wrex 2529 ⊆ wss 3220 class class class wbr 4130 ↦ cmpt 4192 ⟶wf 5373 ‘cfv 5377 (class class class)co 6085 ℂcc 8178 ℝcr 8179 0cc0 8180 ℝ^*cxr 8360 < clt 8361 − cmin 8499 (,)cioo 10301 –cn→ccncf 15762

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4246 ax-sep 4249 ax-nul 4259 ax-pow 4311 ax-pr 4346 ax-un 4578 ax-setind 4684 ax-iinf 4735 ax-cnex 8271 ax-resscn 8272 ax-1cn 8273 ax-1re 8274 ax-icn 8275 ax-addcl 8276 ax-addrcl 8277 ax-mulcl 8278 ax-mulrcl 8279 ax-addcom 8280 ax-mulcom 8281 ax-addass 8282 ax-mulass 8283 ax-distr 8284 ax-i2m1 8285 ax-0lt1 8286 ax-1rid 8287 ax-0id 8288 ax-rnegex 8289 ax-precex 8290 ax-cnre 8291 ax-pre-ltirr 8292 ax-pre-ltwlin 8293 ax-pre-lttrn 8294 ax-pre-apti 8295 ax-pre-ltadd 8296 ax-pre-mulgt0 8297 ax-pre-mulext 8298 ax-arch 8299 ax-caucvg 8300

This proof depends on definitions: df-bi 117 df-stab 843 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rmo 2536 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3639 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-iun 4014 df-br 4131 df-opab 4193 df-mpt 4194 df-tr 4230 df-id 4438 df-po 4441 df-iso 4442 df-iord 4511 df-on 4513 df-ilim 4514 df-suc 4516 df-iom 4738 df-xp 4780 df-rel 4781 df-cnv 4782 df-co 4783 df-dm 4784 df-rn 4785 df-res 4786 df-ima 4787 df-iota 5337 df-fun 5379 df-fn 5380 df-f 5381 df-f1 5382 df-fo 5383 df-f1o 5384 df-fv 5385 df-isom 5386 df-riota 6038 df-ov 6088 df-oprab 6089 df-mpo 6090 df-1st 6374 df-2nd 6375 df-recs 6576 df-frec 6662 df-map 6924 df-sup 7325 df-inf 7326 df-pnf 8363 df-mnf 8364 df-xr 8365 df-ltxr 8366 df-le 8367 df-sub 8501 df-neg 8502 df-reap 8906 df-ap 8913 df-div 9006 df-inn 9308 df-2 9366 df-3 9367 df-4 9368 df-n0 9569 df-z 9650 df-uz 9932 df-q 10030 df-rp 10066 df-xneg 10185 df-xadd 10186 df-ioo 10305 df-seqfrec 10900 df-exp 10991 df-cj 11623 df-re 11624 df-im 11625 df-rsqrt 11780 df-abs 11781 df-rest 13648 df-topgen 13667 df-psmet 14964 df-xmet 14965 df-met 14966 df-bl 14967 df-mopn 14968 df-top 15190 df-topon 15203 df-bases 15235 df-cn 15380 df-cnp 15381 df-tx 15445 df-cncf 15763

This theorem is used by: ivthdichlem 15843

W3C validator