ivthALT - Mathbox for Jeff Hankins

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Theorem ivthALT 34451

Description: An alternate proof of the Intermediate Value Theorem ivth 24523 using topology. (Contributed by Jeff Hankins, 17-Aug-2009.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.) (Proof modification is discouraged.)

**Assertion**
Ref	Expression
ivthALT	⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) → ∃𝑥 ∈ (𝐴(,)𝐵)(𝐹‘𝑥) = 𝑈)

Distinct variable groups: 𝑥,𝐴 𝑥,𝐵 𝑥,𝐷 𝑥,𝐹 𝑥,𝑈

Proof of Theorem ivthALT Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp31 1207	. . . . . 6 ⊢ (((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵)))) → 𝐹 ∈ (𝐷–cn→ℂ))
2		cncff 23962	. . . . . 6 ⊢ (𝐹 ∈ (𝐷–cn→ℂ) → 𝐹:𝐷⟶ℂ)
3	1, 2	syl 17	. . . . 5 ⊢ (((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵)))) → 𝐹:𝐷⟶ℂ)
4		ffun 6587	. . . . 5 ⊢ (𝐹:𝐷⟶ℂ → Fun 𝐹)
5	3, 4	syl 17	. . . 4 ⊢ (((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵)))) → Fun 𝐹)
6	5	3ad2ant3 1133	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) → Fun 𝐹)
7		iccconn 23899	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)) ∈ Conn)
8	7	3adant3 1130	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) → ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)) ∈ Conn)
9	8	3ad2ant1 1131	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) → ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)) ∈ Conn)
10		simpr1 1192	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵)))) → 𝐹 ∈ (𝐷–cn→ℂ))
11	10, 2	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵)))) → 𝐹:𝐷⟶ℂ)
12	11	anim2i 616	. . . . . . . . . . . 12 ⊢ (((𝐴[,]𝐵) ⊆ 𝐷 ∧ (𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) → ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐹:𝐷⟶ℂ))
13	12	3impb 1113	. . . . . . . . . . 11 ⊢ (((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵)))) → ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐹:𝐷⟶ℂ))
14	13	3ad2ant3 1133	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) → ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐹:𝐷⟶ℂ))
15	4	adantl 481	. . . . . . . . . . 11 ⊢ (((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐹:𝐷⟶ℂ) → Fun 𝐹)
16		fdm 6593	. . . . . . . . . . . . 13 ⊢ (𝐹:𝐷⟶ℂ → dom 𝐹 = 𝐷)
17	16	sseq2d 3949	. . . . . . . . . . . 12 ⊢ (𝐹:𝐷⟶ℂ → ((𝐴[,]𝐵) ⊆ dom 𝐹 ↔ (𝐴[,]𝐵) ⊆ 𝐷))
18	17	biimparc 479	. . . . . . . . . . 11 ⊢ (((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐹:𝐷⟶ℂ) → (𝐴[,]𝐵) ⊆ dom 𝐹)
19	15, 18	jca 511	. . . . . . . . . 10 ⊢ (((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐹:𝐷⟶ℂ) → (Fun 𝐹 ∧ (𝐴[,]𝐵) ⊆ dom 𝐹))
20	14, 19	syl 17	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) → (Fun 𝐹 ∧ (𝐴[,]𝐵) ⊆ dom 𝐹))
21		fores 6682	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ (𝐴[,]𝐵) ⊆ dom 𝐹) → (𝐹 ↾ (𝐴[,]𝐵)):(𝐴[,]𝐵)–onto→(𝐹 “ (𝐴[,]𝐵)))
22	20, 21	syl 17	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) → (𝐹 ↾ (𝐴[,]𝐵)):(𝐴[,]𝐵)–onto→(𝐹 “ (𝐴[,]𝐵)))
23		retop 23831	. . . . . . . . . 10 ⊢ (topGen‘ran (,)) ∈ Top
24		simp332 1325	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) → (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ)
25		uniretop 23832	. . . . . . . . . . 11 ⊢ ℝ = ∪ (topGen‘ran (,))
26	25	restuni 22221	. . . . . . . . . 10 ⊢ (((topGen‘ran (,)) ∈ Top ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ) → (𝐹 “ (𝐴[,]𝐵)) = ∪ ((topGen‘ran (,)) ↾_t (𝐹 “ (𝐴[,]𝐵))))
27	23, 24, 26	sylancr 586	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) → (𝐹 “ (𝐴[,]𝐵)) = ∪ ((topGen‘ran (,)) ↾_t (𝐹 “ (𝐴[,]𝐵))))
28		foeq3 6670	. . . . . . . . 9 ⊢ ((𝐹 “ (𝐴[,]𝐵)) = ∪ ((topGen‘ran (,)) ↾_t (𝐹 “ (𝐴[,]𝐵))) → ((𝐹 ↾ (𝐴[,]𝐵)):(𝐴[,]𝐵)–onto→(𝐹 “ (𝐴[,]𝐵)) ↔ (𝐹 ↾ (𝐴[,]𝐵)):(𝐴[,]𝐵)–onto→∪ ((topGen‘ran (,)) ↾_t (𝐹 “ (𝐴[,]𝐵)))))
29	27, 28	syl 17	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) → ((𝐹 ↾ (𝐴[,]𝐵)):(𝐴[,]𝐵)–onto→(𝐹 “ (𝐴[,]𝐵)) ↔ (𝐹 ↾ (𝐴[,]𝐵)):(𝐴[,]𝐵)–onto→∪ ((topGen‘ran (,)) ↾_t (𝐹 “ (𝐴[,]𝐵)))))
30	22, 29	mpbid 231	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) → (𝐹 ↾ (𝐴[,]𝐵)):(𝐴[,]𝐵)–onto→∪ ((topGen‘ran (,)) ↾_t (𝐹 “ (𝐴[,]𝐵))))
31		simp331 1324	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) → 𝐹 ∈ (𝐷–cn→ℂ))
32		ssid 3939	. . . . . . . . . . . . . . 15 ⊢ ℂ ⊆ ℂ
33		eqid 2738	. . . . . . . . . . . . . . . 16 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
34		eqid 2738	. . . . . . . . . . . . . . . 16 ⊢ ((TopOpen‘ℂ_fld) ↾_t 𝐷) = ((TopOpen‘ℂ_fld) ↾_t 𝐷)
35	33	cnfldtop 23853	. . . . . . . . . . . . . . . . . 18 ⊢ (TopOpen‘ℂ_fld) ∈ Top
36	33	cnfldtopon 23852	. . . . . . . . . . . . . . . . . . . 20 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
37	36	toponunii 21973	. . . . . . . . . . . . . . . . . . 19 ⊢ ℂ = ∪ (TopOpen‘ℂ_fld)
38	37	restid 17061	. . . . . . . . . . . . . . . . . 18 ⊢ ((TopOpen‘ℂ_fld) ∈ Top → ((TopOpen‘ℂ_fld) ↾_t ℂ) = (TopOpen‘ℂ_fld))
39	35, 38	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ ((TopOpen‘ℂ_fld) ↾_t ℂ) = (TopOpen‘ℂ_fld)
40	39	eqcomi 2747	. . . . . . . . . . . . . . . 16 ⊢ (TopOpen‘ℂ_fld) = ((TopOpen‘ℂ_fld) ↾_t ℂ)
41	33, 34, 40	cncfcn 23979	. . . . . . . . . . . . . . 15 ⊢ ((𝐷 ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝐷–cn→ℂ) = (((TopOpen‘ℂ_fld) ↾_t 𝐷) Cn (TopOpen‘ℂ_fld)))
42	32, 41	mpan2 687	. . . . . . . . . . . . . 14 ⊢ (𝐷 ⊆ ℂ → (𝐷–cn→ℂ) = (((TopOpen‘ℂ_fld) ↾_t 𝐷) Cn (TopOpen‘ℂ_fld)))
43	42	3ad2ant2 1132	. . . . . . . . . . . . 13 ⊢ (((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵)))) → (𝐷–cn→ℂ) = (((TopOpen‘ℂ_fld) ↾_t 𝐷) Cn (TopOpen‘ℂ_fld)))
44	43	3ad2ant3 1133	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) → (𝐷–cn→ℂ) = (((TopOpen‘ℂ_fld) ↾_t 𝐷) Cn (TopOpen‘ℂ_fld)))
45	31, 44	eleqtrd 2841	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) → 𝐹 ∈ (((TopOpen‘ℂ_fld) ↾_t 𝐷) Cn (TopOpen‘ℂ_fld)))
46		simp31 1207	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) → (𝐴[,]𝐵) ⊆ 𝐷)
47		simp32 1208	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) → 𝐷 ⊆ ℂ)
48		resttopon 22220	. . . . . . . . . . . . . 14 ⊢ (((TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ) ∧ 𝐷 ⊆ ℂ) → ((TopOpen‘ℂ_fld) ↾_t 𝐷) ∈ (TopOn‘𝐷))
49	36, 47, 48	sylancr 586	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) → ((TopOpen‘ℂ_fld) ↾_t 𝐷) ∈ (TopOn‘𝐷))
50		toponuni 21971	. . . . . . . . . . . . 13 ⊢ (((TopOpen‘ℂ_fld) ↾_t 𝐷) ∈ (TopOn‘𝐷) → 𝐷 = ∪ ((TopOpen‘ℂ_fld) ↾_t 𝐷))
51	49, 50	syl 17	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) → 𝐷 = ∪ ((TopOpen‘ℂ_fld) ↾_t 𝐷))
52	46, 51	sseqtrd 3957	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) → (𝐴[,]𝐵) ⊆ ∪ ((TopOpen‘ℂ_fld) ↾_t 𝐷))
53		eqid 2738	. . . . . . . . . . . 12 ⊢ ∪ ((TopOpen‘ℂ_fld) ↾_t 𝐷) = ∪ ((TopOpen‘ℂ_fld) ↾_t 𝐷)
54	53	cnrest 22344	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (((TopOpen‘ℂ_fld) ↾_t 𝐷) Cn (TopOpen‘ℂ_fld)) ∧ (𝐴[,]𝐵) ⊆ ∪ ((TopOpen‘ℂ_fld) ↾_t 𝐷)) → (𝐹 ↾ (𝐴[,]𝐵)) ∈ ((((TopOpen‘ℂ_fld) ↾_t 𝐷) ↾_t (𝐴[,]𝐵)) Cn (TopOpen‘ℂ_fld)))
55	45, 52, 54	syl2anc 583	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) → (𝐹 ↾ (𝐴[,]𝐵)) ∈ ((((TopOpen‘ℂ_fld) ↾_t 𝐷) ↾_t (𝐴[,]𝐵)) Cn (TopOpen‘ℂ_fld)))
56	35	a1i 11	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) → (TopOpen‘ℂ_fld) ∈ Top)
57		cnex 10883	. . . . . . . . . . . . . 14 ⊢ ℂ ∈ V
58		ssexg 5242	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ⊆ ℂ ∧ ℂ ∈ V) → 𝐷 ∈ V)
59	47, 57, 58	sylancl 585	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) → 𝐷 ∈ V)
60		restabs 22224	. . . . . . . . . . . . 13 ⊢ (((TopOpen‘ℂ_fld) ∈ Top ∧ (𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ∈ V) → (((TopOpen‘ℂ_fld) ↾_t 𝐷) ↾_t (𝐴[,]𝐵)) = ((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵)))
61	56, 46, 59, 60	syl3anc 1369	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) → (((TopOpen‘ℂ_fld) ↾_t 𝐷) ↾_t (𝐴[,]𝐵)) = ((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵)))
62		iccssre 13090	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
63	62	3adant3 1130	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
64	63	3ad2ant1 1131	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) → (𝐴[,]𝐵) ⊆ ℝ)
65		eqid 2738	. . . . . . . . . . . . . 14 ⊢ (topGen‘ran (,)) = (topGen‘ran (,))
66	33, 65	rerest 23873	. . . . . . . . . . . . 13 ⊢ ((𝐴[,]𝐵) ⊆ ℝ → ((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵)) = ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)))
67	64, 66	syl 17	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) → ((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵)) = ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)))
68	61, 67	eqtrd 2778	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) → (((TopOpen‘ℂ_fld) ↾_t 𝐷) ↾_t (𝐴[,]𝐵)) = ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)))
69	68	oveq1d 7270	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) → ((((TopOpen‘ℂ_fld) ↾_t 𝐷) ↾_t (𝐴[,]𝐵)) Cn (TopOpen‘ℂ_fld)) = (((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)) Cn (TopOpen‘ℂ_fld)))
70	55, 69	eleqtrd 2841	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) → (𝐹 ↾ (𝐴[,]𝐵)) ∈ (((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)) Cn (TopOpen‘ℂ_fld)))
71	36	a1i 11	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) → (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ))
72		df-ima 5593	. . . . . . . . . . . 12 ⊢ (𝐹 “ (𝐴[,]𝐵)) = ran (𝐹 ↾ (𝐴[,]𝐵))
73	72	eqimss2i 3976	. . . . . . . . . . 11 ⊢ ran (𝐹 ↾ (𝐴[,]𝐵)) ⊆ (𝐹 “ (𝐴[,]𝐵))
74	73	a1i 11	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) → ran (𝐹 ↾ (𝐴[,]𝐵)) ⊆ (𝐹 “ (𝐴[,]𝐵)))
75		ax-resscn 10859	. . . . . . . . . . 11 ⊢ ℝ ⊆ ℂ
76	24, 75	sstrdi 3929	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) → (𝐹 “ (𝐴[,]𝐵)) ⊆ ℂ)
77		cnrest2 22345	. . . . . . . . . 10 ⊢ (((TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ) ∧ ran (𝐹 ↾ (𝐴[,]𝐵)) ⊆ (𝐹 “ (𝐴[,]𝐵)) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℂ) → ((𝐹 ↾ (𝐴[,]𝐵)) ∈ (((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)) Cn (TopOpen‘ℂ_fld)) ↔ (𝐹 ↾ (𝐴[,]𝐵)) ∈ (((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)) Cn ((TopOpen‘ℂ_fld) ↾_t (𝐹 “ (𝐴[,]𝐵))))))
78	71, 74, 76, 77	syl3anc 1369	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) → ((𝐹 ↾ (𝐴[,]𝐵)) ∈ (((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)) Cn (TopOpen‘ℂ_fld)) ↔ (𝐹 ↾ (𝐴[,]𝐵)) ∈ (((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)) Cn ((TopOpen‘ℂ_fld) ↾_t (𝐹 “ (𝐴[,]𝐵))))))
79	70, 78	mpbid 231	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) → (𝐹 ↾ (𝐴[,]𝐵)) ∈ (((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)) Cn ((TopOpen‘ℂ_fld) ↾_t (𝐹 “ (𝐴[,]𝐵)))))
80	33, 65	rerest 23873	. . . . . . . . . 10 ⊢ ((𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ → ((TopOpen‘ℂ_fld) ↾_t (𝐹 “ (𝐴[,]𝐵))) = ((topGen‘ran (,)) ↾_t (𝐹 “ (𝐴[,]𝐵))))
81	24, 80	syl 17	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) → ((TopOpen‘ℂ_fld) ↾_t (𝐹 “ (𝐴[,]𝐵))) = ((topGen‘ran (,)) ↾_t (𝐹 “ (𝐴[,]𝐵))))
82	81	oveq2d 7271	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) → (((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)) Cn ((TopOpen‘ℂ_fld) ↾_t (𝐹 “ (𝐴[,]𝐵)))) = (((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)) Cn ((topGen‘ran (,)) ↾_t (𝐹 “ (𝐴[,]𝐵)))))
83	79, 82	eleqtrd 2841	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) → (𝐹 ↾ (𝐴[,]𝐵)) ∈ (((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)) Cn ((topGen‘ran (,)) ↾_t (𝐹 “ (𝐴[,]𝐵)))))
84		eqid 2738	. . . . . . . 8 ⊢ ∪ ((topGen‘ran (,)) ↾_t (𝐹 “ (𝐴[,]𝐵))) = ∪ ((topGen‘ran (,)) ↾_t (𝐹 “ (𝐴[,]𝐵)))
85	84	cnconn 22481	. . . . . . 7 ⊢ ((((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)) ∈ Conn ∧ (𝐹 ↾ (𝐴[,]𝐵)):(𝐴[,]𝐵)–onto→∪ ((topGen‘ran (,)) ↾_t (𝐹 “ (𝐴[,]𝐵))) ∧ (𝐹 ↾ (𝐴[,]𝐵)) ∈ (((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)) Cn ((topGen‘ran (,)) ↾_t (𝐹 “ (𝐴[,]𝐵))))) → ((topGen‘ran (,)) ↾_t (𝐹 “ (𝐴[,]𝐵))) ∈ Conn)
86	9, 30, 83, 85	syl3anc 1369	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) → ((topGen‘ran (,)) ↾_t (𝐹 “ (𝐴[,]𝐵))) ∈ Conn)
87		reconn 23897	. . . . . . . . 9 ⊢ ((𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ → (((topGen‘ran (,)) ↾_t (𝐹 “ (𝐴[,]𝐵))) ∈ Conn ↔ ∀𝑥 ∈ (𝐹 “ (𝐴[,]𝐵))∀𝑦 ∈ (𝐹 “ (𝐴[,]𝐵))(𝑥[,]𝑦) ⊆ (𝐹 “ (𝐴[,]𝐵))))
88	87	3ad2ant2 1132	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))) → (((topGen‘ran (,)) ↾_t (𝐹 “ (𝐴[,]𝐵))) ∈ Conn ↔ ∀𝑥 ∈ (𝐹 “ (𝐴[,]𝐵))∀𝑦 ∈ (𝐹 “ (𝐴[,]𝐵))(𝑥[,]𝑦) ⊆ (𝐹 “ (𝐴[,]𝐵))))
89	88	3ad2ant3 1133	. . . . . . 7 ⊢ (((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵)))) → (((topGen‘ran (,)) ↾_t (𝐹 “ (𝐴[,]𝐵))) ∈ Conn ↔ ∀𝑥 ∈ (𝐹 “ (𝐴[,]𝐵))∀𝑦 ∈ (𝐹 “ (𝐴[,]𝐵))(𝑥[,]𝑦) ⊆ (𝐹 “ (𝐴[,]𝐵))))
90	89	3ad2ant3 1133	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) → (((topGen‘ran (,)) ↾_t (𝐹 “ (𝐴[,]𝐵))) ∈ Conn ↔ ∀𝑥 ∈ (𝐹 “ (𝐴[,]𝐵))∀𝑦 ∈ (𝐹 “ (𝐴[,]𝐵))(𝑥[,]𝑦) ⊆ (𝐹 “ (𝐴[,]𝐵))))
91	86, 90	mpbid 231	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) → ∀𝑥 ∈ (𝐹 “ (𝐴[,]𝐵))∀𝑦 ∈ (𝐹 “ (𝐴[,]𝐵))(𝑥[,]𝑦) ⊆ (𝐹 “ (𝐴[,]𝐵)))
92		simp11 1201	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) → 𝐴 ∈ ℝ)
93	92	rexrd 10956	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) → 𝐴 ∈ ℝ^*)
94		simp12 1202	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) → 𝐵 ∈ ℝ)
95	94	rexrd 10956	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) → 𝐵 ∈ ℝ^*)
96		ltle 10994	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 → 𝐴 ≤ 𝐵))
97	96	imp 406	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐴 < 𝐵) → 𝐴 ≤ 𝐵)
98	97	3adantl3 1166	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵) → 𝐴 ≤ 𝐵)
99	98	3adant3 1130	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) → 𝐴 ≤ 𝐵)
100		lbicc2 13125	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ (𝐴[,]𝐵))
101	93, 95, 99, 100	syl3anc 1369	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) → 𝐴 ∈ (𝐴[,]𝐵))
102		funfvima2 7089	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ (𝐴[,]𝐵) ⊆ dom 𝐹) → (𝐴 ∈ (𝐴[,]𝐵) → (𝐹‘𝐴) ∈ (𝐹 “ (𝐴[,]𝐵))))
103	20, 101, 102	sylc 65	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) → (𝐹‘𝐴) ∈ (𝐹 “ (𝐴[,]𝐵)))
104		ubicc2 13126	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ (𝐴[,]𝐵))
105	93, 95, 99, 104	syl3anc 1369	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) → 𝐵 ∈ (𝐴[,]𝐵))
106		funfvima2 7089	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ (𝐴[,]𝐵) ⊆ dom 𝐹) → (𝐵 ∈ (𝐴[,]𝐵) → (𝐹‘𝐵) ∈ (𝐹 “ (𝐴[,]𝐵))))
107	20, 105, 106	sylc 65	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) → (𝐹‘𝐵) ∈ (𝐹 “ (𝐴[,]𝐵)))
108		oveq1 7262	. . . . . . . 8 ⊢ (𝑥 = (𝐹‘𝐴) → (𝑥[,]𝑦) = ((𝐹‘𝐴)[,]𝑦))
109	108	sseq1d 3948	. . . . . . 7 ⊢ (𝑥 = (𝐹‘𝐴) → ((𝑥[,]𝑦) ⊆ (𝐹 “ (𝐴[,]𝐵)) ↔ ((𝐹‘𝐴)[,]𝑦) ⊆ (𝐹 “ (𝐴[,]𝐵))))
110		oveq2 7263	. . . . . . . 8 ⊢ (𝑦 = (𝐹‘𝐵) → ((𝐹‘𝐴)[,]𝑦) = ((𝐹‘𝐴)[,](𝐹‘𝐵)))
111	110	sseq1d 3948	. . . . . . 7 ⊢ (𝑦 = (𝐹‘𝐵) → (((𝐹‘𝐴)[,]𝑦) ⊆ (𝐹 “ (𝐴[,]𝐵)) ↔ ((𝐹‘𝐴)[,](𝐹‘𝐵)) ⊆ (𝐹 “ (𝐴[,]𝐵))))
112	109, 111	rspc2v 3562	. . . . . 6 ⊢ (((𝐹‘𝐴) ∈ (𝐹 “ (𝐴[,]𝐵)) ∧ (𝐹‘𝐵) ∈ (𝐹 “ (𝐴[,]𝐵))) → (∀𝑥 ∈ (𝐹 “ (𝐴[,]𝐵))∀𝑦 ∈ (𝐹 “ (𝐴[,]𝐵))(𝑥[,]𝑦) ⊆ (𝐹 “ (𝐴[,]𝐵)) → ((𝐹‘𝐴)[,](𝐹‘𝐵)) ⊆ (𝐹 “ (𝐴[,]𝐵))))
113	103, 107, 112	syl2anc 583	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) → (∀𝑥 ∈ (𝐹 “ (𝐴[,]𝐵))∀𝑦 ∈ (𝐹 “ (𝐴[,]𝐵))(𝑥[,]𝑦) ⊆ (𝐹 “ (𝐴[,]𝐵)) → ((𝐹‘𝐴)[,](𝐹‘𝐵)) ⊆ (𝐹 “ (𝐴[,]𝐵))))
114	91, 113	mpd 15	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) → ((𝐹‘𝐴)[,](𝐹‘𝐵)) ⊆ (𝐹 “ (𝐴[,]𝐵)))
115		ioossicc 13094	. . . . . . . 8 ⊢ ((𝐹‘𝐴)(,)(𝐹‘𝐵)) ⊆ ((𝐹‘𝐴)[,](𝐹‘𝐵))
116	115	sseli 3913	. . . . . . 7 ⊢ (𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵)) → 𝑈 ∈ ((𝐹‘𝐴)[,](𝐹‘𝐵)))
117	116	3ad2ant3 1133	. . . . . 6 ⊢ ((𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))) → 𝑈 ∈ ((𝐹‘𝐴)[,](𝐹‘𝐵)))
118	117	3ad2ant3 1133	. . . . 5 ⊢ (((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵)))) → 𝑈 ∈ ((𝐹‘𝐴)[,](𝐹‘𝐵)))
119	118	3ad2ant3 1133	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) → 𝑈 ∈ ((𝐹‘𝐴)[,](𝐹‘𝐵)))
120	114, 119	sseldd 3918	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) → 𝑈 ∈ (𝐹 “ (𝐴[,]𝐵)))
121		fvelima 6817	. . 3 ⊢ ((Fun 𝐹 ∧ 𝑈 ∈ (𝐹 “ (𝐴[,]𝐵))) → ∃𝑥 ∈ (𝐴[,]𝐵)(𝐹‘𝑥) = 𝑈)
122	6, 120, 121	syl2anc 583	. 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) → ∃𝑥 ∈ (𝐴[,]𝐵)(𝐹‘𝑥) = 𝑈)
123		simpl1 1189	. . . . . . . 8 ⊢ (((𝑥 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵 ∧ (𝑥 = 𝐴 ∨ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵) ∨ 𝑥 = 𝐵)) ∧ (𝐹‘𝑥) = 𝑈) → 𝑥 ∈ ℝ^*)
124	123	a1i 11	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) → (((𝑥 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵 ∧ (𝑥 = 𝐴 ∨ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵) ∨ 𝑥 = 𝐵)) ∧ (𝐹‘𝑥) = 𝑈) → 𝑥 ∈ ℝ^*))
125		simprr 769	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) ∧ ((𝑥 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵 ∧ (𝑥 = 𝐴 ∨ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵) ∨ 𝑥 = 𝐵)) ∧ (𝐹‘𝑥) = 𝑈)) → (𝐹‘𝑥) = 𝑈)
126	24, 103	sseldd 3918	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) → (𝐹‘𝐴) ∈ ℝ)
127		simp333 1326	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) → 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵)))
128	126	rexrd 10956	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) → (𝐹‘𝐴) ∈ ℝ^*)
129	24, 107	sseldd 3918	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) → (𝐹‘𝐵) ∈ ℝ)
130	129	rexrd 10956	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) → (𝐹‘𝐵) ∈ ℝ^*)
131		elioo2 13049	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹‘𝐴) ∈ ℝ^* ∧ (𝐹‘𝐵) ∈ ℝ^*) → (𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵)) ↔ (𝑈 ∈ ℝ ∧ (𝐹‘𝐴) < 𝑈 ∧ 𝑈 < (𝐹‘𝐵))))
132	128, 130, 131	syl2anc 583	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) → (𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵)) ↔ (𝑈 ∈ ℝ ∧ (𝐹‘𝐴) < 𝑈 ∧ 𝑈 < (𝐹‘𝐵))))
133	127, 132	mpbid 231	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) → (𝑈 ∈ ℝ ∧ (𝐹‘𝐴) < 𝑈 ∧ 𝑈 < (𝐹‘𝐵)))
134	133	simp2d 1141	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) → (𝐹‘𝐴) < 𝑈)
135	126, 134	gtned 11040	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) → 𝑈 ≠ (𝐹‘𝐴))
136	135	adantr 480	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) ∧ ((𝑥 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵 ∧ (𝑥 = 𝐴 ∨ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵) ∨ 𝑥 = 𝐵)) ∧ (𝐹‘𝑥) = 𝑈)) → 𝑈 ≠ (𝐹‘𝐴))
137	125, 136	eqnetrd 3010	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) ∧ ((𝑥 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵 ∧ (𝑥 = 𝐴 ∨ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵) ∨ 𝑥 = 𝐵)) ∧ (𝐹‘𝑥) = 𝑈)) → (𝐹‘𝑥) ≠ (𝐹‘𝐴))
138	137	neneqd 2947	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) ∧ ((𝑥 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵 ∧ (𝑥 = 𝐴 ∨ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵) ∨ 𝑥 = 𝐵)) ∧ (𝐹‘𝑥) = 𝑈)) → ¬ (𝐹‘𝑥) = (𝐹‘𝐴))
139		fveq2 6756	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴))
140	138, 139	nsyl 140	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) ∧ ((𝑥 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵 ∧ (𝑥 = 𝐴 ∨ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵) ∨ 𝑥 = 𝐵)) ∧ (𝐹‘𝑥) = 𝑈)) → ¬ 𝑥 = 𝐴)
141		simp13 1203	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) → 𝑈 ∈ ℝ)
142	133	simp3d 1142	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) → 𝑈 < (𝐹‘𝐵))
143	141, 142	ltned 11041	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) → 𝑈 ≠ (𝐹‘𝐵))
144	143	adantr 480	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) ∧ ((𝑥 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵 ∧ (𝑥 = 𝐴 ∨ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵) ∨ 𝑥 = 𝐵)) ∧ (𝐹‘𝑥) = 𝑈)) → 𝑈 ≠ (𝐹‘𝐵))
145	125, 144	eqnetrd 3010	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) ∧ ((𝑥 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵 ∧ (𝑥 = 𝐴 ∨ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵) ∨ 𝑥 = 𝐵)) ∧ (𝐹‘𝑥) = 𝑈)) → (𝐹‘𝑥) ≠ (𝐹‘𝐵))
146	145	neneqd 2947	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) ∧ ((𝑥 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵 ∧ (𝑥 = 𝐴 ∨ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵) ∨ 𝑥 = 𝐵)) ∧ (𝐹‘𝑥) = 𝑈)) → ¬ (𝐹‘𝑥) = (𝐹‘𝐵))
147		fveq2 6756	. . . . . . . . . 10 ⊢ (𝑥 = 𝐵 → (𝐹‘𝑥) = (𝐹‘𝐵))
148	146, 147	nsyl 140	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) ∧ ((𝑥 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵 ∧ (𝑥 = 𝐴 ∨ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵) ∨ 𝑥 = 𝐵)) ∧ (𝐹‘𝑥) = 𝑈)) → ¬ 𝑥 = 𝐵)
149		simprl3 1218	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) ∧ ((𝑥 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵 ∧ (𝑥 = 𝐴 ∨ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵) ∨ 𝑥 = 𝐵)) ∧ (𝐹‘𝑥) = 𝑈)) → (𝑥 = 𝐴 ∨ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵) ∨ 𝑥 = 𝐵))
150	140, 148, 149	ecase13d 34429	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) ∧ ((𝑥 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵 ∧ (𝑥 = 𝐴 ∨ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵) ∨ 𝑥 = 𝐵)) ∧ (𝐹‘𝑥) = 𝑈)) → (𝐴 < 𝑥 ∧ 𝑥 < 𝐵))
151	150	ex 412	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) → (((𝑥 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵 ∧ (𝑥 = 𝐴 ∨ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵) ∨ 𝑥 = 𝐵)) ∧ (𝐹‘𝑥) = 𝑈) → (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)))
152	124, 151	jcad 512	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) → (((𝑥 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵 ∧ (𝑥 = 𝐴 ∨ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵) ∨ 𝑥 = 𝐵)) ∧ (𝐹‘𝑥) = 𝑈) → (𝑥 ∈ ℝ^* ∧ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵))))
153		3anass 1093	. . . . . 6 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝐴 < 𝑥 ∧ 𝑥 < 𝐵) ↔ (𝑥 ∈ ℝ^* ∧ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)))
154	152, 153	syl6ibr 251	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) → (((𝑥 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵 ∧ (𝑥 = 𝐴 ∨ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵) ∨ 𝑥 = 𝐵)) ∧ (𝐹‘𝑥) = 𝑈) → (𝑥 ∈ ℝ^* ∧ 𝐴 < 𝑥 ∧ 𝑥 < 𝐵)))
155		rexr 10952	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ^*)
156		rexr 10952	. . . . . . . . 9 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℝ^*)
157		elicc3 34433	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝑥 ∈ (𝐴[,]𝐵) ↔ (𝑥 ∈ ℝ^ ∧ 𝐴 ≤ 𝐵 ∧ (𝑥 = 𝐴 ∨ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵) ∨ 𝑥 = 𝐵))))
158	155, 156, 157	syl2an 595	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑥 ∈ (𝐴[,]𝐵) ↔ (𝑥 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵 ∧ (𝑥 = 𝐴 ∨ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵) ∨ 𝑥 = 𝐵))))
159	158	3adant3 1130	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) → (𝑥 ∈ (𝐴[,]𝐵) ↔ (𝑥 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵 ∧ (𝑥 = 𝐴 ∨ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵) ∨ 𝑥 = 𝐵))))
160	159	3ad2ant1 1131	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) → (𝑥 ∈ (𝐴[,]𝐵) ↔ (𝑥 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵 ∧ (𝑥 = 𝐴 ∨ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵) ∨ 𝑥 = 𝐵))))
161	160	anbi1d 629	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) → ((𝑥 ∈ (𝐴[,]𝐵) ∧ (𝐹‘𝑥) = 𝑈) ↔ ((𝑥 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵 ∧ (𝑥 = 𝐴 ∨ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵) ∨ 𝑥 = 𝐵)) ∧ (𝐹‘𝑥) = 𝑈)))
162		elioo1 13048	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝑥 ∈ (𝐴(,)𝐵) ↔ (𝑥 ∈ ℝ^ ∧ 𝐴 < 𝑥 ∧ 𝑥 < 𝐵)))
163	155, 156, 162	syl2an 595	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑥 ∈ (𝐴(,)𝐵) ↔ (𝑥 ∈ ℝ^* ∧ 𝐴 < 𝑥 ∧ 𝑥 < 𝐵)))
164	163	3adant3 1130	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) → (𝑥 ∈ (𝐴(,)𝐵) ↔ (𝑥 ∈ ℝ^* ∧ 𝐴 < 𝑥 ∧ 𝑥 < 𝐵)))
165	164	3ad2ant1 1131	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) → (𝑥 ∈ (𝐴(,)𝐵) ↔ (𝑥 ∈ ℝ^* ∧ 𝐴 < 𝑥 ∧ 𝑥 < 𝐵)))
166	154, 161, 165	3imtr4d 293	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) → ((𝑥 ∈ (𝐴[,]𝐵) ∧ (𝐹‘𝑥) = 𝑈) → 𝑥 ∈ (𝐴(,)𝐵)))
167		simpr 484	. . . . 5 ⊢ ((𝑥 ∈ (𝐴[,]𝐵) ∧ (𝐹‘𝑥) = 𝑈) → (𝐹‘𝑥) = 𝑈)
168	167	a1i 11	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) → ((𝑥 ∈ (𝐴[,]𝐵) ∧ (𝐹‘𝑥) = 𝑈) → (𝐹‘𝑥) = 𝑈))
169	166, 168	jcad 512	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) → ((𝑥 ∈ (𝐴[,]𝐵) ∧ (𝐹‘𝑥) = 𝑈) → (𝑥 ∈ (𝐴(,)𝐵) ∧ (𝐹‘𝑥) = 𝑈)))
170	169	reximdv2 3198	. 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) → (∃𝑥 ∈ (𝐴[,]𝐵)(𝐹‘𝑥) = 𝑈 → ∃𝑥 ∈ (𝐴(,)𝐵)(𝐹‘𝑥) = 𝑈))
171	122, 170	mpd 15	1 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) → ∃𝑥 ∈ (𝐴(,)𝐵)(𝐹‘𝑥) = 𝑈)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∨ w3o 1084 ∧ w3a 1085 = wceq 1539 ∈ wcel 2108 ≠ wne 2942 ∀wral 3063 ∃wrex 3064 Vcvv 3422 ⊆ wss 3883 ∪ cuni 4836 class class class wbr 5070 dom cdm 5580 ran crn 5581 ↾ cres 5582 “ cima 5583 Fun wfun 6412 ⟶wf 6414 –onto→wfo 6416 ‘cfv 6418 (class class class)co 7255 ℂcc 10800 ℝcr 10801 ℝ^*cxr 10939 < clt 10940 ≤ cle 10941 (,)cioo 13008 [,]cicc 13011 ↾_t crest 17048 TopOpenctopn 17049 topGenctg 17065 ℂ_fldccnfld 20510 Topctop 21950 TopOnctopon 21967 Cn ccn 22283 Conncconn 22470 –cn→ccncf 23945

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-er 8456 df-map 8575 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fi 9100 df-sup 9131 df-inf 9132 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-z 12250 df-dec 12367 df-uz 12512 df-q 12618 df-rp 12660 df-xneg 12777 df-xadd 12778 df-xmul 12779 df-ioo 13012 df-ico 13014 df-icc 13015 df-fz 13169 df-seq 13650 df-exp 13711 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-struct 16776 df-slot 16811 df-ndx 16823 df-base 16841 df-plusg 16901 df-mulr 16902 df-starv 16903 df-tset 16907 df-ple 16908 df-ds 16910 df-unif 16911 df-rest 17050 df-topn 17051 df-topgen 17071 df-psmet 20502 df-xmet 20503 df-met 20504 df-bl 20505 df-mopn 20506 df-cnfld 20511 df-top 21951 df-topon 21968 df-topsp 21990 df-bases 22004 df-cld 22078 df-cn 22286 df-cnp 22287 df-conn 22471 df-xms 23381 df-ms 23382 df-cncf 23947

This theorem is referenced by: (None)

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