ivthALT - Mathbox for Jeff Hankins

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Theorem ivthALT 33687

Description: An alternate proof of the Intermediate Value Theorem ivth 24058 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 1205	. . . . . 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 23504	. . . . . 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 6520	. . . . 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 1131	. . 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 23441	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)) ∈ Conn)
8	7	3adant3 1128	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) → ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)) ∈ Conn)
9	8	3ad2ant1 1129	. . . . . . 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 1190	. . . . . . . . . . . . . 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 618	. . . . . . . . . . . 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 1111	. . . . . . . . . . 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 1131	. . . . . . . . . 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 484	. . . . . . . . . . 11 ⊢ (((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐹:𝐷⟶ℂ) → Fun 𝐹)
16		fdm 6525	. . . . . . . . . . . . 13 ⊢ (𝐹:𝐷⟶ℂ → dom 𝐹 = 𝐷)
17	16	sseq2d 4002	. . . . . . . . . . . 12 ⊢ (𝐹:𝐷⟶ℂ → ((𝐴[,]𝐵) ⊆ dom 𝐹 ↔ (𝐴[,]𝐵) ⊆ 𝐷))
18	17	biimparc 482	. . . . . . . . . . 11 ⊢ (((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐹:𝐷⟶ℂ) → (𝐴[,]𝐵) ⊆ dom 𝐹)
19	15, 18	jca 514	. . . . . . . . . 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 6603	. . . . . . . . 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 23373	. . . . . . . . . 10 ⊢ (topGen‘ran (,)) ∈ Top
24		simp332 1323	. . . . . . . . . 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 23374	. . . . . . . . . . 11 ⊢ ℝ = ∪ (topGen‘ran (,))
26	25	restuni 21773	. . . . . . . . . 10 ⊢ (((topGen‘ran (,)) ∈ Top ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ) → (𝐹 “ (𝐴[,]𝐵)) = ∪ ((topGen‘ran (,)) ↾_t (𝐹 “ (𝐴[,]𝐵))))
27	23, 24, 26	sylancr 589	. . . . . . . . 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 6591	. . . . . . . . 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 234	. . . . . . 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 1322	. . . . . . . . . . . 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 3992	. . . . . . . . . . . . . . 15 ⊢ ℂ ⊆ ℂ
33		eqid 2824	. . . . . . . . . . . . . . . 16 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
34		eqid 2824	. . . . . . . . . . . . . . . 16 ⊢ ((TopOpen‘ℂ_fld) ↾_t 𝐷) = ((TopOpen‘ℂ_fld) ↾_t 𝐷)
35	33	cnfldtop 23395	. . . . . . . . . . . . . . . . . 18 ⊢ (TopOpen‘ℂ_fld) ∈ Top
36	33	cnfldtopon 23394	. . . . . . . . . . . . . . . . . . . 20 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
37	36	toponunii 21527	. . . . . . . . . . . . . . . . . . 19 ⊢ ℂ = ∪ (TopOpen‘ℂ_fld)
38	37	restid 16710	. . . . . . . . . . . . . . . . . 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 2833	. . . . . . . . . . . . . . . 16 ⊢ (TopOpen‘ℂ_fld) = ((TopOpen‘ℂ_fld) ↾_t ℂ)
41	33, 34, 40	cncfcn 23520	. . . . . . . . . . . . . . 15 ⊢ ((𝐷 ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝐷–cn→ℂ) = (((TopOpen‘ℂ_fld) ↾_t 𝐷) Cn (TopOpen‘ℂ_fld)))
42	32, 41	mpan2 689	. . . . . . . . . . . . . 14 ⊢ (𝐷 ⊆ ℂ → (𝐷–cn→ℂ) = (((TopOpen‘ℂ_fld) ↾_t 𝐷) Cn (TopOpen‘ℂ_fld)))
43	42	3ad2ant2 1130	. . . . . . . . . . . . 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 1131	. . . . . . . . . . . 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 2918	. . . . . . . . . . 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 1205	. . . . . . . . . . . 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 1206	. . . . . . . . . . . . . 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 21772	. . . . . . . . . . . . . 14 ⊢ (((TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ) ∧ 𝐷 ⊆ ℂ) → ((TopOpen‘ℂ_fld) ↾_t 𝐷) ∈ (TopOn‘𝐷))
49	36, 47, 48	sylancr 589	. . . . . . . . . . . . 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 21525	. . . . . . . . . . . . 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 4010	. . . . . . . . . . 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 2824	. . . . . . . . . . . 12 ⊢ ∪ ((TopOpen‘ℂ_fld) ↾_t 𝐷) = ∪ ((TopOpen‘ℂ_fld) ↾_t 𝐷)
54	53	cnrest 21896	. . . . . . . . . . 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 586	. . . . . . . . . 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 10621	. . . . . . . . . . . . . 14 ⊢ ℂ ∈ V
58		ssexg 5230	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ⊆ ℂ ∧ ℂ ∈ V) → 𝐷 ∈ V)
59	47, 57, 58	sylancl 588	. . . . . . . . . . . . 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 21776	. . . . . . . . . . . . 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 1367	. . . . . . . . . . . 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 12821	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
63	62	3adant3 1128	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
64	63	3ad2ant1 1129	. . . . . . . . . . . . 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 2824	. . . . . . . . . . . . . 14 ⊢ (topGen‘ran (,)) = (topGen‘ran (,))
66	33, 65	rerest 23415	. . . . . . . . . . . . 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 2859	. . . . . . . . . . 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 7174	. . . . . . . . . 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 2918	. . . . . . . . 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 5571	. . . . . . . . . . . 12 ⊢ (𝐹 “ (𝐴[,]𝐵)) = ran (𝐹 ↾ (𝐴[,]𝐵))
73	72	eqimss2i 4029	. . . . . . . . . . 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 10597	. . . . . . . . . . 11 ⊢ ℝ ⊆ ℂ
76	24, 75	sstrdi 3982	. . . . . . . . . 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 21897	. . . . . . . . . 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 1367	. . . . . . . . 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 234	. . . . . . . 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 23415	. . . . . . . . . 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 7175	. . . . . . . 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 2918	. . . . . . 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 2824	. . . . . . . 8 ⊢ ∪ ((topGen‘ran (,)) ↾_t (𝐹 “ (𝐴[,]𝐵))) = ∪ ((topGen‘ran (,)) ↾_t (𝐹 “ (𝐴[,]𝐵)))
85	84	cnconn 22033	. . . . . . 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 1367	. . . . . 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 23439	. . . . . . . . 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 1130	. . . . . . . 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 1131	. . . . . . 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 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))))) → (((topGen‘ran (,)) ↾_t (𝐹 “ (𝐴[,]𝐵))) ∈ Conn ↔ ∀𝑥 ∈ (𝐹 “ (𝐴[,]𝐵))∀𝑦 ∈ (𝐹 “ (𝐴[,]𝐵))(𝑥[,]𝑦) ⊆ (𝐹 “ (𝐴[,]𝐵))))
91	86, 90	mpbid 234	. . . . 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 1199	. . . . . . . . 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 10694	. . . . . . . 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 1200	. . . . . . . . 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 10694	. . . . . . . 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 10732	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 → 𝐴 ≤ 𝐵))
97	96	imp 409	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐴 < 𝐵) → 𝐴 ≤ 𝐵)
98	97	3adantl3 1164	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵) → 𝐴 ≤ 𝐵)
99	98	3adant3 1128	. . . . . . . 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 12855	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ (𝐴[,]𝐵))
101	93, 95, 99, 100	syl3anc 1367	. . . . . . 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 6996	. . . . . . 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 12856	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ (𝐴[,]𝐵))
105	93, 95, 99, 104	syl3anc 1367	. . . . . . 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 6996	. . . . . . 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 7166	. . . . . . . 8 ⊢ (𝑥 = (𝐹‘𝐴) → (𝑥[,]𝑦) = ((𝐹‘𝐴)[,]𝑦))
109	108	sseq1d 4001	. . . . . . 7 ⊢ (𝑥 = (𝐹‘𝐴) → ((𝑥[,]𝑦) ⊆ (𝐹 “ (𝐴[,]𝐵)) ↔ ((𝐹‘𝐴)[,]𝑦) ⊆ (𝐹 “ (𝐴[,]𝐵))))
110		oveq2 7167	. . . . . . . 8 ⊢ (𝑦 = (𝐹‘𝐵) → ((𝐹‘𝐴)[,]𝑦) = ((𝐹‘𝐴)[,](𝐹‘𝐵)))
111	110	sseq1d 4001	. . . . . . 7 ⊢ (𝑦 = (𝐹‘𝐵) → (((𝐹‘𝐴)[,]𝑦) ⊆ (𝐹 “ (𝐴[,]𝐵)) ↔ ((𝐹‘𝐴)[,](𝐹‘𝐵)) ⊆ (𝐹 “ (𝐴[,]𝐵))))
112	109, 111	rspc2v 3636	. . . . . 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 586	. . . . 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 12825	. . . . . . . 8 ⊢ ((𝐹‘𝐴)(,)(𝐹‘𝐵)) ⊆ ((𝐹‘𝐴)[,](𝐹‘𝐵))
116	115	sseli 3966	. . . . . . 7 ⊢ (𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵)) → 𝑈 ∈ ((𝐹‘𝐴)[,](𝐹‘𝐵)))
117	116	3ad2ant3 1131	. . . . . 6 ⊢ ((𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))) → 𝑈 ∈ ((𝐹‘𝐴)[,](𝐹‘𝐵)))
118	117	3ad2ant3 1131	. . . . 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 1131	. . . 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 3971	. . 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 6734	. . 3 ⊢ ((Fun 𝐹 ∧ 𝑈 ∈ (𝐹 “ (𝐴[,]𝐵))) → ∃𝑥 ∈ (𝐴[,]𝐵)(𝐹‘𝑥) = 𝑈)
122	6, 120, 121	syl2anc 586	. 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 1187	. . . . . . . 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 771	. . . . . . . . . . . 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 3971	. . . . . . . . . . . . . 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 1324	. . . . . . . . . . . . . . . 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 10694	. . . . . . . . . . . . . . . . 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 3971	. . . . . . . . . . . . . . . . . 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 10694	. . . . . . . . . . . . . . . . 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 12782	. . . . . . . . . . . . . . . . 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 586	. . . . . . . . . . . . . . . 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 234	. . . . . . . . . . . . . . 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 1139	. . . . . . . . . . . . . 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 10778	. . . . . . . . . . . . 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 483	. . . . . . . . . . . 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 3086	. . . . . . . . . . 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 3024	. . . . . . . . . 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 6673	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴))
140	138, 139	nsyl 142	. . . . . . . . 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 1201	. . . . . . . . . . . . . 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 1140	. . . . . . . . . . . . . 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 10779	. . . . . . . . . . . . 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 483	. . . . . . . . . . . 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 3086	. . . . . . . . . . 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 3024	. . . . . . . . . 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 6673	. . . . . . . . . 10 ⊢ (𝑥 = 𝐵 → (𝐹‘𝑥) = (𝐹‘𝐵))
148	146, 147	nsyl 142	. . . . . . . . 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 1216	. . . . . . . . 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 33665	. . . . . . . 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 415	. . . . . . 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 515	. . . . . 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 1091	. . . . . 6 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝐴 < 𝑥 ∧ 𝑥 < 𝐵) ↔ (𝑥 ∈ ℝ^* ∧ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)))
154	152, 153	syl6ibr 254	. . . . 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 10690	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ^*)
156		rexr 10690	. . . . . . . . 9 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℝ^*)
157		elicc3 33669	. . . . . . . . 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 597	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑥 ∈ (𝐴[,]𝐵) ↔ (𝑥 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵 ∧ (𝑥 = 𝐴 ∨ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵) ∨ 𝑥 = 𝐵))))
159	158	3adant3 1128	. . . . . . 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 1129	. . . . . 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 631	. . . . 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 12781	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝑥 ∈ (𝐴(,)𝐵) ↔ (𝑥 ∈ ℝ^ ∧ 𝐴 < 𝑥 ∧ 𝑥 < 𝐵)))
163	155, 156, 162	syl2an 597	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑥 ∈ (𝐴(,)𝐵) ↔ (𝑥 ∈ ℝ^* ∧ 𝐴 < 𝑥 ∧ 𝑥 < 𝐵)))
164	163	3adant3 1128	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) → (𝑥 ∈ (𝐴(,)𝐵) ↔ (𝑥 ∈ ℝ^* ∧ 𝐴 < 𝑥 ∧ 𝑥 < 𝐵)))
165	164	3ad2ant1 1129	. . . . 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 296	. . . 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 487	. . . . 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 515	. . 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 3274	. 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 208 ∧ wa 398 ∨ w3o 1082 ∧ w3a 1083 = wceq 1536 ∈ wcel 2113 ≠ wne 3019 ∀wral 3141 ∃wrex 3142 Vcvv 3497 ⊆ wss 3939 ∪ cuni 4841 class class class wbr 5069 dom cdm 5558 ran crn 5559 ↾ cres 5560 “ cima 5561 Fun wfun 6352 ⟶wf 6354 –onto→wfo 6356 ‘cfv 6358 (class class class)co 7159 ℂcc 10538 ℝcr 10539 ℝ^*cxr 10677 < clt 10678 ≤ cle 10679 (,)cioo 12741 [,]cicc 12744 ↾_t crest 16697 TopOpenctopn 16698 topGenctg 16714 ℂ_fldccnfld 20548 Topctop 21504 TopOnctopon 21521 Cn ccn 21835 Conncconn 22022 –cn→ccncf 23487

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-pre-sup 10618

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-oadd 8109 df-er 8292 df-map 8411 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-fi 8878 df-sup 8909 df-inf 8910 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-div 11301 df-nn 11642 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-ioo 12745 df-ico 12747 df-icc 12748 df-fz 12896 df-seq 13373 df-exp 13433 df-cj 14461 df-re 14462 df-im 14463 df-sqrt 14597 df-abs 14598 df-struct 16488 df-ndx 16489 df-slot 16490 df-base 16492 df-plusg 16581 df-mulr 16582 df-starv 16583 df-tset 16587 df-ple 16588 df-ds 16590 df-unif 16591 df-rest 16699 df-topn 16700 df-topgen 16720 df-psmet 20540 df-xmet 20541 df-met 20542 df-bl 20543 df-mopn 20544 df-cnfld 20549 df-top 21505 df-topon 21522 df-topsp 21544 df-bases 21557 df-cld 21630 df-cn 21838 df-cnp 21839 df-conn 22023 df-xms 22933 df-ms 22934 df-cncf 23489

This theorem is referenced by: (None)

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