ivthALT - Mathbox for Jeff Hankins

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Theorem ivthALT 37215

Description: An alternate proof of the Intermediate Value Theorem ivth 25780 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 1228	. . . . . 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 25219	. . . . . 6 ⊢ (𝐹 ∈ (𝐷–cn→ℂ) → 𝐹:𝐷⟶ℂ)
3	1, 2	syl 18	. . . . 5 ⊢ (((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵)))) → 𝐹:𝐷⟶ℂ)
4		ffun 6712	. . . . 5 ⊢ (𝐹:𝐷⟶ℂ → Fun 𝐹)
5	3, 4	syl 18	. . . 4 ⊢ (((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵)))) → Fun 𝐹)
6	5	3ad2ant3 1153	. . 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 25155	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)) ∈ Conn)
8	7	3adant3 1150	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) → ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)) ∈ Conn)
9	8	3ad2ant1 1151	. . . . . . 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 1213	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵)))) → 𝐹 ∈ (𝐷–cn→ℂ))
11	10, 2	syl 18	. . . . . . . . . . . . 13 ⊢ ((𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵)))) → 𝐹:𝐷⟶ℂ)
12	11	anim2i 629	. . . . . . . . . . . 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 1132	. . . . . . . . . . 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 1153	. . . . . . . . . 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 487	. . . . . . . . . . 11 ⊢ (((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐹:𝐷⟶ℂ) → Fun 𝐹)
16		fdm 6719	. . . . . . . . . . . . 13 ⊢ (𝐹:𝐷⟶ℂ → dom 𝐹 = 𝐷)
17	16	sseq2d 3940	. . . . . . . . . . . 12 ⊢ (𝐹:𝐷⟶ℂ → ((𝐴[,]𝐵) ⊆ dom 𝐹 ↔ (𝐴[,]𝐵) ⊆ 𝐷))
18	17	biimparc 485	. . . . . . . . . . 11 ⊢ (((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐹:𝐷⟶ℂ) → (𝐴[,]𝐵) ⊆ dom 𝐹)
19	15, 18	jca 521	. . . . . . . . . 10 ⊢ (((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐹:𝐷⟶ℂ) → (Fun 𝐹 ∧ (𝐴[,]𝐵) ⊆ dom 𝐹))
20	14, 19	syl 18	. . . . . . . . 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 6806	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ (𝐴[,]𝐵) ⊆ dom 𝐹) → (𝐹 ↾ (𝐴[,]𝐵)):(𝐴[,]𝐵)–onto→(𝐹 “ (𝐴[,]𝐵)))
22	20, 21	syl 18	. . . . . . . 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 25085	. . . . . . . . . 10 ⊢ (topGen‘ran (,)) ∈ Top
24		simp332 1346	. . . . . . . . . 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 25086	. . . . . . . . . . 11 ⊢ ℝ = ∪ (topGen‘ran (,))
26	25	restuni 23485	. . . . . . . . . 10 ⊢ (((topGen‘ran (,)) ∈ Top ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ) → (𝐹 “ (𝐴[,]𝐵)) = ∪ ((topGen‘ran (,)) ↾_t (𝐹 “ (𝐴[,]𝐵))))
27	23, 24, 26	sylancr 599	. . . . . . . . 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 6794	. . . . . . . . 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 18	. . . . . . . 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 235	. . . . . . 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 1345	. . . . . . . . . . . 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 3930	. . . . . . . . . . . . . . 15 ⊢ ℂ ⊆ ℂ
33		eqid 2760	. . . . . . . . . . . . . . . 16 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
34		eqid 2760	. . . . . . . . . . . . . . . 16 ⊢ ((TopOpen‘ℂ_fld) ↾_t 𝐷) = ((TopOpen‘ℂ_fld) ↾_t 𝐷)
35	33	cnfldtop 25107	. . . . . . . . . . . . . . . . . 18 ⊢ (TopOpen‘ℂ_fld) ∈ Top
36	33	cnfldtopon 25106	. . . . . . . . . . . . . . . . . . . 20 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
37	36	toponunii 23239	. . . . . . . . . . . . . . . . . . 19 ⊢ ℂ = ∪ (TopOpen‘ℂ_fld)
38	37	restid 17609	. . . . . . . . . . . . . . . . . 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 2769	. . . . . . . . . . . . . . . 16 ⊢ (TopOpen‘ℂ_fld) = ((TopOpen‘ℂ_fld) ↾_t ℂ)
41	33, 34, 40	cncfcn 25236	. . . . . . . . . . . . . . 15 ⊢ ((𝐷 ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝐷–cn→ℂ) = (((TopOpen‘ℂ_fld) ↾_t 𝐷) Cn (TopOpen‘ℂ_fld)))
42	32, 41	mpan2 704	. . . . . . . . . . . . . 14 ⊢ (𝐷 ⊆ ℂ → (𝐷–cn→ℂ) = (((TopOpen‘ℂ_fld) ↾_t 𝐷) Cn (TopOpen‘ℂ_fld)))
43	42	3ad2ant2 1152	. . . . . . . . . . . . 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 1153	. . . . . . . . . . . 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 2862	. . . . . . . . . . 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 1228	. . . . . . . . . . . 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 1229	. . . . . . . . . . . . . 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 23484	. . . . . . . . . . . . . 14 ⊢ (((TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ) ∧ 𝐷 ⊆ ℂ) → ((TopOpen‘ℂ_fld) ↾_t 𝐷) ∈ (TopOn‘𝐷))
49	36, 47, 48	sylancr 599	. . . . . . . . . . . . 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 23237	. . . . . . . . . . . . 13 ⊢ (((TopOpen‘ℂ_fld) ↾_t 𝐷) ∈ (TopOn‘𝐷) → 𝐷 = ∪ ((TopOpen‘ℂ_fld) ↾_t 𝐷))
51	49, 50	syl 18	. . . . . . . . . . . 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 3944	. . . . . . . . . . 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 2760	. . . . . . . . . . . 12 ⊢ ∪ ((TopOpen‘ℂ_fld) ↾_t 𝐷) = ∪ ((TopOpen‘ℂ_fld) ↾_t 𝐷)
54	53	cnrest 23608	. . . . . . . . . . 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 596	. . . . . . . . . 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 11284	. . . . . . . . . . . . . 14 ⊢ ℂ ∈ V
58		ssexg 5280	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ⊆ ℂ ∧ ℂ ∈ V) → 𝐷 ∈ V)
59	47, 57, 58	sylancl 598	. . . . . . . . . . . . 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 23488	. . . . . . . . . . . . 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 1398	. . . . . . . . . . . 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 13563	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
63	62	3adant3 1150	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
64	63	3ad2ant1 1151	. . . . . . . . . . . . 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 2760	. . . . . . . . . . . . . 14 ⊢ (topGen‘ran (,)) = (topGen‘ran (,))
66	33, 65	rerest 25128	. . . . . . . . . . . . 13 ⊢ ((𝐴[,]𝐵) ⊆ ℝ → ((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵)) = ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)))
67	64, 66	syl 18	. . . . . . . . . . . 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 2795	. . . . . . . . . . 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 7435	. . . . . . . . . 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 2862	. . . . . . . . 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 5663	. . . . . . . . . . . 12 ⊢ (𝐹 “ (𝐴[,]𝐵)) = ran (𝐹 ↾ (𝐴[,]𝐵))
73	72	eqimss2i 3969	. . . . . . . . . . 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 11260	. . . . . . . . . . 11 ⊢ ℝ ⊆ ℂ
76	24, 75	sstrdi 3920	. . . . . . . . . 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 23609	. . . . . . . . . 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 1398	. . . . . . . . 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 235	. . . . . . . 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 25128	. . . . . . . . . 10 ⊢ ((𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ → ((TopOpen‘ℂ_fld) ↾_t (𝐹 “ (𝐴[,]𝐵))) = ((topGen‘ran (,)) ↾_t (𝐹 “ (𝐴[,]𝐵))))
81	24, 80	syl 18	. . . . . . . . 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 7436	. . . . . . . 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 2862	. . . . . . 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 2760	. . . . . . . 8 ⊢ ∪ ((topGen‘ran (,)) ↾_t (𝐹 “ (𝐴[,]𝐵))) = ∪ ((topGen‘ran (,)) ↾_t (𝐹 “ (𝐴[,]𝐵)))
85	84	cnconn 23745	. . . . . . 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 1398	. . . . . 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 25153	. . . . . . . . 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 1152	. . . . . . . 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 1153	. . . . . . 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 1153	. . . . . 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 235	. . . . 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 1222	. . . . . . . . 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 11362	. . . . . . . 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 1223	. . . . . . . . 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 11362	. . . . . . . 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 11401	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 → 𝐴 ≤ 𝐵))
97	96	imp 412	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐴 < 𝐵) → 𝐴 ≤ 𝐵)
98	97	3adantl3 1187	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵) → 𝐴 ≤ 𝐵)
99	98	3adant3 1150	. . . . . . . 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 13598	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ (𝐴[,]𝐵))
101	93, 95, 99, 100	syl3anc 1398	. . . . . . 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 7237	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ (𝐴[,]𝐵) ⊆ dom 𝐹) → (𝐴 ∈ (𝐴[,]𝐵) → (𝐹‘𝐴) ∈ (𝐹 “ (𝐴[,]𝐵))))
103	20, 101, 102	sylc 66	. . . . . 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 13599	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ (𝐴[,]𝐵))
105	93, 95, 99, 104	syl3anc 1398	. . . . . . 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 7237	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ (𝐴[,]𝐵) ⊆ dom 𝐹) → (𝐵 ∈ (𝐴[,]𝐵) → (𝐹‘𝐵) ∈ (𝐹 “ (𝐴[,]𝐵))))
107	20, 105, 106	sylc 66	. . . . . 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 7427	. . . . . . . 8 ⊢ (𝑥 = (𝐹‘𝐴) → (𝑥[,]𝑦) = ((𝐹‘𝐴)[,]𝑦))
109	108	sseq1d 3939	. . . . . . 7 ⊢ (𝑥 = (𝐹‘𝐴) → ((𝑥[,]𝑦) ⊆ (𝐹 “ (𝐴[,]𝐵)) ↔ ((𝐹‘𝐴)[,]𝑦) ⊆ (𝐹 “ (𝐴[,]𝐵))))
110		oveq2 7428	. . . . . . . 8 ⊢ (𝑦 = (𝐹‘𝐵) → ((𝐹‘𝐴)[,]𝑦) = ((𝐹‘𝐴)[,](𝐹‘𝐵)))
111	110	sseq1d 3939	. . . . . . 7 ⊢ (𝑦 = (𝐹‘𝐵) → (((𝐹‘𝐴)[,]𝑦) ⊆ (𝐹 “ (𝐴[,]𝐵)) ↔ ((𝐹‘𝐴)[,](𝐹‘𝐵)) ⊆ (𝐹 “ (𝐴[,]𝐵))))
112	109, 111	rspc2v 3586	. . . . . 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 596	. . . . 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 16	. . . 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 13567	. . . . . . . 8 ⊢ ((𝐹‘𝐴)(,)(𝐹‘𝐵)) ⊆ ((𝐹‘𝐴)[,](𝐹‘𝐵))
116	115	sseli 3904	. . . . . . 7 ⊢ (𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵)) → 𝑈 ∈ ((𝐹‘𝐴)[,](𝐹‘𝐵)))
117	116	3ad2ant3 1153	. . . . . 6 ⊢ ((𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))) → 𝑈 ∈ ((𝐹‘𝐴)[,](𝐹‘𝐵)))
118	117	3ad2ant3 1153	. . . . 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 1153	. . . 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 3909	. . 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 6950	. . 3 ⊢ ((Fun 𝐹 ∧ 𝑈 ∈ (𝐹 “ (𝐴[,]𝐵))) → ∃𝑥 ∈ (𝐴[,]𝐵)(𝐹‘𝑥) = 𝑈)
122	6, 120, 121	syl2anc 596	. 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 1210	. . . . . . . 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 785	. . . . . . . . . . . 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 3909	. . . . . . . . . . . . . 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 1347	. . . . . . . . . . . . . . . 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 11362	. . . . . . . . . . . . . . . . 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 3909	. . . . . . . . . . . . . . . . . 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 11362	. . . . . . . . . . . . . . . . 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 13520	. . . . . . . . . . . . . . . . 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 596	. . . . . . . . . . . . . . . 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 235	. . . . . . . . . . . . . . 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 1161	. . . . . . . . . . . . . 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 11448	. . . . . . . . . . . . 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 486	. . . . . . . . . . . 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 3022	. . . . . . . . . . 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 2960	. . . . . . . . . 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 6885	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴))
140	138, 139	nsyl 141	. . . . . . . . 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 1224	. . . . . . . . . . . . . 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 1162	. . . . . . . . . . . . . 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 11449	. . . . . . . . . . . . 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 486	. . . . . . . . . . . 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 3022	. . . . . . . . . . 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 2960	. . . . . . . . . 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 6885	. . . . . . . . . 10 ⊢ (𝑥 = 𝐵 → (𝐹‘𝑥) = (𝐹‘𝐵))
148	146, 147	nsyl 141	. . . . . . . . 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 1239	. . . . . . . . 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 1502	. . . . . . . 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 418	. . . . . . 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 522	. . . . . 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 1111	. . . . . 6 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝐴 < 𝑥 ∧ 𝑥 < 𝐵) ↔ (𝑥 ∈ ℝ^* ∧ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)))
154	152, 153	imbitrrdi 255	. . . . 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 11358	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ^*)
156		rexr 11358	. . . . . . . . 9 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℝ^*)
157		elicc3 37197	. . . . . . . . 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 608	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑥 ∈ (𝐴[,]𝐵) ↔ (𝑥 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵 ∧ (𝑥 = 𝐴 ∨ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵) ∨ 𝑥 = 𝐵))))
159	158	3adant3 1150	. . . . . . 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 1151	. . . . . 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 643	. . . . 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 13519	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝑥 ∈ (𝐴(,)𝐵) ↔ (𝑥 ∈ ℝ^ ∧ 𝐴 < 𝑥 ∧ 𝑥 < 𝐵)))
163	155, 156, 162	syl2an 608	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑥 ∈ (𝐴(,)𝐵) ↔ (𝑥 ∈ ℝ^* ∧ 𝐴 < 𝑥 ∧ 𝑥 < 𝐵)))
164	163	3adant3 1150	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) → (𝑥 ∈ (𝐴(,)𝐵) ↔ (𝑥 ∈ ℝ^* ∧ 𝐴 < 𝑥 ∧ 𝑥 < 𝐵)))
165	164	3ad2ant1 1151	. . . . 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 297	. . . 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 490	. . . . 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 522	. . 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 3172	. 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 16	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

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∨ w3o 1102 ∧ w3a 1103 = wceq 1569 ∈ wcel 2144 ≠ wne 2955 ∀wral 3076 ∃wrex 3086 Vcvv 3450 ⊆ wss 3895 ∪ cuni 4866 class class class wbr 5102 dom cdm 5650 ran crn 5651 ↾ cres 5652 “ cima 5653 Fun wfun 6532 ⟶wf 6534 –onto→wfo 6536 ‘cfv 6538 (class class class)co 7420 ℂcc 11201 ℝcr 11202 ℝ^*cxr 11345 < clt 11346 ≤ cle 11347 (,)cioo 13479 [,]cicc 13482 ↾_t crest 17596 TopOpenctopn 17597 topGenctg 17613 ℂ_fldccnfld 21683 Topctop 23216 TopOnctopon 23233 Cn ccn 23547 Conncconn 23734 –cn→ccncf 25202

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 7751 ax-cnex 11259 ax-resscn 11260 ax-1cn 11261 ax-icn 11262 ax-addcl 11263 ax-addrcl 11264 ax-mulcl 11265 ax-mulrcl 11266 ax-mulcom 11267 ax-addass 11268 ax-mulass 11269 ax-distr 11270 ax-i2m1 11271 ax-1ne0 11272 ax-1rid 11273 ax-rnegex 11274 ax-rrecex 11275 ax-cnre 11276 ax-pre-lttri 11277 ax-pre-lttrn 11278 ax-pre-ltadd 11279 ax-pre-mulgt0 11280 ax-pre-sup 11281

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 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 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-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-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 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7878 df-1st 8001 df-2nd 8002 df-frecs 8302 df-wrecs 8333 df-recs 8382 df-rdg 8421 df-1o 8479 df-er 8720 df-map 8852 df-en 8977 df-dom 8978 df-sdom 8979 df-fin 8980 df-fi 9406 df-sup 9437 df-inf 9438 df-pnf 11348 df-mnf 11349 df-xr 11350 df-ltxr 11351 df-le 11352 df-sub 11546 df-neg 11547 df-div 11977 df-nn 12339 df-2 12408 df-3 12409 df-4 12410 df-5 12411 df-6 12412 df-7 12413 df-8 12414 df-9 12415 df-n0 12610 df-z 12697 df-dec 12818 df-uz 12969 df-q 13079 df-rp 13124 df-xneg 13244 df-xadd 13245 df-xmul 13246 df-ioo 13483 df-ico 13485 df-icc 13486 df-fz 13643 df-seq 14148 df-exp 14208 df-cj 15271 df-re 15272 df-im 15273 df-sqrt 15407 df-abs 15408 df-struct 17330 df-slot 17365 df-ndx 17377 df-base 17393 df-plusg 17446 df-mulr 17447 df-starv 17448 df-tset 17452 df-ple 17453 df-ds 17455 df-unif 17456 df-rest 17598 df-topn 17599 df-topgen 17619 df-psmet 21675 df-xmet 21676 df-met 21677 df-bl 21678 df-mopn 21679 df-cnfld 21684 df-top 23217 df-topon 23234 df-topsp 23256 df-bases 23269 df-cld 23342 df-cn 23550 df-cnp 23551 df-conn 23735 df-xms 24644 df-ms 24645 df-cncf 25204

This theorem is used by: (None)

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