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Theorem ivthALT 33768
 Description: An alternate proof of the Intermediate Value Theorem ivth 24067 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) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) → ∃𝑥 ∈ (𝐴(,)𝐵)(𝐹𝑥) = 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.
StepHypRef Expression
1 simp31 1206 . . . . . 6 (((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵)))) → 𝐹 ∈ (𝐷cn→ℂ))
2 cncff 23507 . . . . . 6 (𝐹 ∈ (𝐷cn→ℂ) → 𝐹:𝐷⟶ℂ)
31, 2syl 17 . . . . 5 (((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵)))) → 𝐹:𝐷⟶ℂ)
4 ffun 6508 . . . . 5 (𝐹:𝐷⟶ℂ → Fun 𝐹)
53, 4syl 17 . . . 4 (((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵)))) → Fun 𝐹)
653ad2ant3 1132 . . 3 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) → Fun 𝐹)
7 iccconn 23444 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((topGen‘ran (,)) ↾t (𝐴[,]𝐵)) ∈ Conn)
873adant3 1129 . . . . . . . 8 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) → ((topGen‘ran (,)) ↾t (𝐴[,]𝐵)) ∈ Conn)
983ad2ant1 1130 . . . . . . 7 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) → ((topGen‘ran (,)) ↾t (𝐴[,]𝐵)) ∈ Conn)
10 simpr1 1191 . . . . . . . . . . . . . 14 ((𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵)))) → 𝐹 ∈ (𝐷cn→ℂ))
1110, 2syl 17 . . . . . . . . . . . . 13 ((𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵)))) → 𝐹:𝐷⟶ℂ)
1211anim2i 619 . . . . . . . . . . . 12 (((𝐴[,]𝐵) ⊆ 𝐷 ∧ (𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) → ((𝐴[,]𝐵) ⊆ 𝐷𝐹:𝐷⟶ℂ))
13123impb 1112 . . . . . . . . . . 11 (((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵)))) → ((𝐴[,]𝐵) ⊆ 𝐷𝐹:𝐷⟶ℂ))
14133ad2ant3 1132 . . . . . . . . . 10 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) → ((𝐴[,]𝐵) ⊆ 𝐷𝐹:𝐷⟶ℂ))
154adantl 485 . . . . . . . . . . 11 (((𝐴[,]𝐵) ⊆ 𝐷𝐹:𝐷⟶ℂ) → Fun 𝐹)
16 fdm 6513 . . . . . . . . . . . . 13 (𝐹:𝐷⟶ℂ → dom 𝐹 = 𝐷)
1716sseq2d 3985 . . . . . . . . . . . 12 (𝐹:𝐷⟶ℂ → ((𝐴[,]𝐵) ⊆ dom 𝐹 ↔ (𝐴[,]𝐵) ⊆ 𝐷))
1817biimparc 483 . . . . . . . . . . 11 (((𝐴[,]𝐵) ⊆ 𝐷𝐹:𝐷⟶ℂ) → (𝐴[,]𝐵) ⊆ dom 𝐹)
1915, 18jca 515 . . . . . . . . . 10 (((𝐴[,]𝐵) ⊆ 𝐷𝐹:𝐷⟶ℂ) → (Fun 𝐹 ∧ (𝐴[,]𝐵) ⊆ dom 𝐹))
2014, 19syl 17 . . . . . . . . 9 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) → (Fun 𝐹 ∧ (𝐴[,]𝐵) ⊆ dom 𝐹))
21 fores 6593 . . . . . . . . 9 ((Fun 𝐹 ∧ (𝐴[,]𝐵) ⊆ dom 𝐹) → (𝐹 ↾ (𝐴[,]𝐵)):(𝐴[,]𝐵)–onto→(𝐹 “ (𝐴[,]𝐵)))
2220, 21syl 17 . . . . . . . 8 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) → (𝐹 ↾ (𝐴[,]𝐵)):(𝐴[,]𝐵)–onto→(𝐹 “ (𝐴[,]𝐵)))
23 retop 23376 . . . . . . . . . 10 (topGen‘ran (,)) ∈ Top
24 simp332 1324 . . . . . . . . . 10 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) → (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ)
25 uniretop 23377 . . . . . . . . . . 11 ℝ = (topGen‘ran (,))
2625restuni 21776 . . . . . . . . . 10 (((topGen‘ran (,)) ∈ Top ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ) → (𝐹 “ (𝐴[,]𝐵)) = ((topGen‘ran (,)) ↾t (𝐹 “ (𝐴[,]𝐵))))
2723, 24, 26sylancr 590 . . . . . . . . 9 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) → (𝐹 “ (𝐴[,]𝐵)) = ((topGen‘ran (,)) ↾t (𝐹 “ (𝐴[,]𝐵))))
28 foeq3 6581 . . . . . . . . 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)))))
2927, 28syl 17 . . . . . . . 8 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) → ((𝐹 ↾ (𝐴[,]𝐵)):(𝐴[,]𝐵)–onto→(𝐹 “ (𝐴[,]𝐵)) ↔ (𝐹 ↾ (𝐴[,]𝐵)):(𝐴[,]𝐵)–onto ((topGen‘ran (,)) ↾t (𝐹 “ (𝐴[,]𝐵)))))
3022, 29mpbid 235 . . . . . . 7 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) → (𝐹 ↾ (𝐴[,]𝐵)):(𝐴[,]𝐵)–onto ((topGen‘ran (,)) ↾t (𝐹 “ (𝐴[,]𝐵))))
31 simp331 1323 . . . . . . . . . . . 12 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) → 𝐹 ∈ (𝐷cn→ℂ))
32 ssid 3975 . . . . . . . . . . . . . . 15 ℂ ⊆ ℂ
33 eqid 2824 . . . . . . . . . . . . . . . 16 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
34 eqid 2824 . . . . . . . . . . . . . . . 16 ((TopOpen‘ℂfld) ↾t 𝐷) = ((TopOpen‘ℂfld) ↾t 𝐷)
3533cnfldtop 23398 . . . . . . . . . . . . . . . . . 18 (TopOpen‘ℂfld) ∈ Top
3633cnfldtopon 23397 . . . . . . . . . . . . . . . . . . . 20 (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
3736toponunii 21530 . . . . . . . . . . . . . . . . . . 19 ℂ = (TopOpen‘ℂfld)
3837restid 16709 . . . . . . . . . . . . . . . . . 18 ((TopOpen‘ℂfld) ∈ Top → ((TopOpen‘ℂfld) ↾t ℂ) = (TopOpen‘ℂfld))
3935, 38ax-mp 5 . . . . . . . . . . . . . . . . 17 ((TopOpen‘ℂfld) ↾t ℂ) = (TopOpen‘ℂfld)
4039eqcomi 2833 . . . . . . . . . . . . . . . 16 (TopOpen‘ℂfld) = ((TopOpen‘ℂfld) ↾t ℂ)
4133, 34, 40cncfcn 23524 . . . . . . . . . . . . . . 15 ((𝐷 ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝐷cn→ℂ) = (((TopOpen‘ℂfld) ↾t 𝐷) Cn (TopOpen‘ℂfld)))
4232, 41mpan2 690 . . . . . . . . . . . . . 14 (𝐷 ⊆ ℂ → (𝐷cn→ℂ) = (((TopOpen‘ℂfld) ↾t 𝐷) Cn (TopOpen‘ℂfld)))
43423ad2ant2 1131 . . . . . . . . . . . . 13 (((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵)))) → (𝐷cn→ℂ) = (((TopOpen‘ℂfld) ↾t 𝐷) Cn (TopOpen‘ℂfld)))
44433ad2ant3 1132 . . . . . . . . . . . 12 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) → (𝐷cn→ℂ) = (((TopOpen‘ℂfld) ↾t 𝐷) Cn (TopOpen‘ℂfld)))
4531, 44eleqtrd 2918 . . . . . . . . . . 11 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) → 𝐹 ∈ (((TopOpen‘ℂfld) ↾t 𝐷) Cn (TopOpen‘ℂfld)))
46 simp31 1206 . . . . . . . . . . . 12 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) → (𝐴[,]𝐵) ⊆ 𝐷)
47 simp32 1207 . . . . . . . . . . . . . 14 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) → 𝐷 ⊆ ℂ)
48 resttopon 21775 . . . . . . . . . . . . . 14 (((TopOpen‘ℂfld) ∈ (TopOn‘ℂ) ∧ 𝐷 ⊆ ℂ) → ((TopOpen‘ℂfld) ↾t 𝐷) ∈ (TopOn‘𝐷))
4936, 47, 48sylancr 590 . . . . . . . . . . . . 13 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) → ((TopOpen‘ℂfld) ↾t 𝐷) ∈ (TopOn‘𝐷))
50 toponuni 21528 . . . . . . . . . . . . 13 (((TopOpen‘ℂfld) ↾t 𝐷) ∈ (TopOn‘𝐷) → 𝐷 = ((TopOpen‘ℂfld) ↾t 𝐷))
5149, 50syl 17 . . . . . . . . . . . 12 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) → 𝐷 = ((TopOpen‘ℂfld) ↾t 𝐷))
5246, 51sseqtrd 3993 . . . . . . . . . . 11 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) → (𝐴[,]𝐵) ⊆ ((TopOpen‘ℂfld) ↾t 𝐷))
53 eqid 2824 . . . . . . . . . . . 12 ((TopOpen‘ℂfld) ↾t 𝐷) = ((TopOpen‘ℂfld) ↾t 𝐷)
5453cnrest 21899 . . . . . . . . . . 11 ((𝐹 ∈ (((TopOpen‘ℂfld) ↾t 𝐷) Cn (TopOpen‘ℂfld)) ∧ (𝐴[,]𝐵) ⊆ ((TopOpen‘ℂfld) ↾t 𝐷)) → (𝐹 ↾ (𝐴[,]𝐵)) ∈ ((((TopOpen‘ℂfld) ↾t 𝐷) ↾t (𝐴[,]𝐵)) Cn (TopOpen‘ℂfld)))
5545, 52, 54syl2anc 587 . . . . . . . . . 10 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) → (𝐹 ↾ (𝐴[,]𝐵)) ∈ ((((TopOpen‘ℂfld) ↾t 𝐷) ↾t (𝐴[,]𝐵)) Cn (TopOpen‘ℂfld)))
5635a1i 11 . . . . . . . . . . . . 13 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) → (TopOpen‘ℂfld) ∈ Top)
57 cnex 10618 . . . . . . . . . . . . . 14 ℂ ∈ V
58 ssexg 5214 . . . . . . . . . . . . . 14 ((𝐷 ⊆ ℂ ∧ ℂ ∈ V) → 𝐷 ∈ V)
5947, 57, 58sylancl 589 . . . . . . . . . . . . 13 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) → 𝐷 ∈ V)
60 restabs 21779 . . . . . . . . . . . . 13 (((TopOpen‘ℂfld) ∈ Top ∧ (𝐴[,]𝐵) ⊆ 𝐷𝐷 ∈ V) → (((TopOpen‘ℂfld) ↾t 𝐷) ↾t (𝐴[,]𝐵)) = ((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)))
6156, 46, 59, 60syl3anc 1368 . . . . . . . . . . . 12 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) → (((TopOpen‘ℂfld) ↾t 𝐷) ↾t (𝐴[,]𝐵)) = ((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)))
62 iccssre 12818 . . . . . . . . . . . . . . 15 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
63623adant3 1129 . . . . . . . . . . . . . 14 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
64633ad2ant1 1130 . . . . . . . . . . . . 13 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) → (𝐴[,]𝐵) ⊆ ℝ)
65 eqid 2824 . . . . . . . . . . . . . 14 (topGen‘ran (,)) = (topGen‘ran (,))
6633, 65rerest 23418 . . . . . . . . . . . . 13 ((𝐴[,]𝐵) ⊆ ℝ → ((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) = ((topGen‘ran (,)) ↾t (𝐴[,]𝐵)))
6764, 66syl 17 . . . . . . . . . . . 12 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) → ((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) = ((topGen‘ran (,)) ↾t (𝐴[,]𝐵)))
6861, 67eqtrd 2859 . . . . . . . . . . 11 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) → (((TopOpen‘ℂfld) ↾t 𝐷) ↾t (𝐴[,]𝐵)) = ((topGen‘ran (,)) ↾t (𝐴[,]𝐵)))
6968oveq1d 7166 . . . . . . . . . 10 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) → ((((TopOpen‘ℂfld) ↾t 𝐷) ↾t (𝐴[,]𝐵)) Cn (TopOpen‘ℂfld)) = (((topGen‘ran (,)) ↾t (𝐴[,]𝐵)) Cn (TopOpen‘ℂfld)))
7055, 69eleqtrd 2918 . . . . . . . . 9 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) → (𝐹 ↾ (𝐴[,]𝐵)) ∈ (((topGen‘ran (,)) ↾t (𝐴[,]𝐵)) Cn (TopOpen‘ℂfld)))
7136a1i 11 . . . . . . . . . 10 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) → (TopOpen‘ℂfld) ∈ (TopOn‘ℂ))
72 df-ima 5556 . . . . . . . . . . . 12 (𝐹 “ (𝐴[,]𝐵)) = ran (𝐹 ↾ (𝐴[,]𝐵))
7372eqimss2i 4012 . . . . . . . . . . 11 ran (𝐹 ↾ (𝐴[,]𝐵)) ⊆ (𝐹 “ (𝐴[,]𝐵))
7473a1i 11 . . . . . . . . . 10 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) → ran (𝐹 ↾ (𝐴[,]𝐵)) ⊆ (𝐹 “ (𝐴[,]𝐵)))
75 ax-resscn 10594 . . . . . . . . . . 11 ℝ ⊆ ℂ
7624, 75sstrdi 3965 . . . . . . . . . 10 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) → (𝐹 “ (𝐴[,]𝐵)) ⊆ ℂ)
77 cnrest2 21900 . . . . . . . . . 10 (((TopOpen‘ℂfld) ∈ (TopOn‘ℂ) ∧ ran (𝐹 ↾ (𝐴[,]𝐵)) ⊆ (𝐹 “ (𝐴[,]𝐵)) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℂ) → ((𝐹 ↾ (𝐴[,]𝐵)) ∈ (((topGen‘ran (,)) ↾t (𝐴[,]𝐵)) Cn (TopOpen‘ℂfld)) ↔ (𝐹 ↾ (𝐴[,]𝐵)) ∈ (((topGen‘ran (,)) ↾t (𝐴[,]𝐵)) Cn ((TopOpen‘ℂfld) ↾t (𝐹 “ (𝐴[,]𝐵))))))
7871, 74, 76, 77syl3anc 1368 . . . . . . . . 9 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) → ((𝐹 ↾ (𝐴[,]𝐵)) ∈ (((topGen‘ran (,)) ↾t (𝐴[,]𝐵)) Cn (TopOpen‘ℂfld)) ↔ (𝐹 ↾ (𝐴[,]𝐵)) ∈ (((topGen‘ran (,)) ↾t (𝐴[,]𝐵)) Cn ((TopOpen‘ℂfld) ↾t (𝐹 “ (𝐴[,]𝐵))))))
7970, 78mpbid 235 . . . . . . . 8 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) → (𝐹 ↾ (𝐴[,]𝐵)) ∈ (((topGen‘ran (,)) ↾t (𝐴[,]𝐵)) Cn ((TopOpen‘ℂfld) ↾t (𝐹 “ (𝐴[,]𝐵)))))
8033, 65rerest 23418 . . . . . . . . . 10 ((𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ → ((TopOpen‘ℂfld) ↾t (𝐹 “ (𝐴[,]𝐵))) = ((topGen‘ran (,)) ↾t (𝐹 “ (𝐴[,]𝐵))))
8124, 80syl 17 . . . . . . . . 9 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) → ((TopOpen‘ℂfld) ↾t (𝐹 “ (𝐴[,]𝐵))) = ((topGen‘ran (,)) ↾t (𝐹 “ (𝐴[,]𝐵))))
8281oveq2d 7167 . . . . . . . 8 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) → (((topGen‘ran (,)) ↾t (𝐴[,]𝐵)) Cn ((TopOpen‘ℂfld) ↾t (𝐹 “ (𝐴[,]𝐵)))) = (((topGen‘ran (,)) ↾t (𝐴[,]𝐵)) Cn ((topGen‘ran (,)) ↾t (𝐹 “ (𝐴[,]𝐵)))))
8379, 82eleqtrd 2918 . . . . . . 7 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) → (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)))
8584cnconn 22036 . . . . . . 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)
869, 30, 83, 85syl3anc 1368 . . . . . 6 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) → ((topGen‘ran (,)) ↾t (𝐹 “ (𝐴[,]𝐵))) ∈ Conn)
87 reconn 23442 . . . . . . . . 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))))
88873ad2ant2 1131 . . . . . . . 8 ((𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))) → (((topGen‘ran (,)) ↾t (𝐹 “ (𝐴[,]𝐵))) ∈ Conn ↔ ∀𝑥 ∈ (𝐹 “ (𝐴[,]𝐵))∀𝑦 ∈ (𝐹 “ (𝐴[,]𝐵))(𝑥[,]𝑦) ⊆ (𝐹 “ (𝐴[,]𝐵))))
89883ad2ant3 1132 . . . . . . 7 (((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵)))) → (((topGen‘ran (,)) ↾t (𝐹 “ (𝐴[,]𝐵))) ∈ Conn ↔ ∀𝑥 ∈ (𝐹 “ (𝐴[,]𝐵))∀𝑦 ∈ (𝐹 “ (𝐴[,]𝐵))(𝑥[,]𝑦) ⊆ (𝐹 “ (𝐴[,]𝐵))))
90893ad2ant3 1132 . . . . . 6 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) → (((topGen‘ran (,)) ↾t (𝐹 “ (𝐴[,]𝐵))) ∈ Conn ↔ ∀𝑥 ∈ (𝐹 “ (𝐴[,]𝐵))∀𝑦 ∈ (𝐹 “ (𝐴[,]𝐵))(𝑥[,]𝑦) ⊆ (𝐹 “ (𝐴[,]𝐵))))
9186, 90mpbid 235 . . . . 5 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) → ∀𝑥 ∈ (𝐹 “ (𝐴[,]𝐵))∀𝑦 ∈ (𝐹 “ (𝐴[,]𝐵))(𝑥[,]𝑦) ⊆ (𝐹 “ (𝐴[,]𝐵)))
92 simp11 1200 . . . . . . . . 9 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) → 𝐴 ∈ ℝ)
9392rexrd 10691 . . . . . . . 8 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) → 𝐴 ∈ ℝ*)
94 simp12 1201 . . . . . . . . 9 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) → 𝐵 ∈ ℝ)
9594rexrd 10691 . . . . . . . 8 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) → 𝐵 ∈ ℝ*)
96 ltle 10729 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵𝐴𝐵))
9796imp 410 . . . . . . . . . 10 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐴 < 𝐵) → 𝐴𝐵)
98973adantl3 1165 . . . . . . . . 9 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵) → 𝐴𝐵)
99983adant3 1129 . . . . . . . 8 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) → 𝐴𝐵)
100 lbicc2 12853 . . . . . . . 8 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴𝐵) → 𝐴 ∈ (𝐴[,]𝐵))
10193, 95, 99, 100syl3anc 1368 . . . . . . 7 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) → 𝐴 ∈ (𝐴[,]𝐵))
102 funfvima2 6987 . . . . . . 7 ((Fun 𝐹 ∧ (𝐴[,]𝐵) ⊆ dom 𝐹) → (𝐴 ∈ (𝐴[,]𝐵) → (𝐹𝐴) ∈ (𝐹 “ (𝐴[,]𝐵))))
10320, 101, 102sylc 65 . . . . . 6 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) → (𝐹𝐴) ∈ (𝐹 “ (𝐴[,]𝐵)))
104 ubicc2 12854 . . . . . . . 8 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴𝐵) → 𝐵 ∈ (𝐴[,]𝐵))
10593, 95, 99, 104syl3anc 1368 . . . . . . 7 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) → 𝐵 ∈ (𝐴[,]𝐵))
106 funfvima2 6987 . . . . . . 7 ((Fun 𝐹 ∧ (𝐴[,]𝐵) ⊆ dom 𝐹) → (𝐵 ∈ (𝐴[,]𝐵) → (𝐹𝐵) ∈ (𝐹 “ (𝐴[,]𝐵))))
10720, 105, 106sylc 65 . . . . . 6 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) → (𝐹𝐵) ∈ (𝐹 “ (𝐴[,]𝐵)))
108 oveq1 7158 . . . . . . . 8 (𝑥 = (𝐹𝐴) → (𝑥[,]𝑦) = ((𝐹𝐴)[,]𝑦))
109108sseq1d 3984 . . . . . . 7 (𝑥 = (𝐹𝐴) → ((𝑥[,]𝑦) ⊆ (𝐹 “ (𝐴[,]𝐵)) ↔ ((𝐹𝐴)[,]𝑦) ⊆ (𝐹 “ (𝐴[,]𝐵))))
110 oveq2 7159 . . . . . . . 8 (𝑦 = (𝐹𝐵) → ((𝐹𝐴)[,]𝑦) = ((𝐹𝐴)[,](𝐹𝐵)))
111110sseq1d 3984 . . . . . . 7 (𝑦 = (𝐹𝐵) → (((𝐹𝐴)[,]𝑦) ⊆ (𝐹 “ (𝐴[,]𝐵)) ↔ ((𝐹𝐴)[,](𝐹𝐵)) ⊆ (𝐹 “ (𝐴[,]𝐵))))
112109, 111rspc2v 3619 . . . . . 6 (((𝐹𝐴) ∈ (𝐹 “ (𝐴[,]𝐵)) ∧ (𝐹𝐵) ∈ (𝐹 “ (𝐴[,]𝐵))) → (∀𝑥 ∈ (𝐹 “ (𝐴[,]𝐵))∀𝑦 ∈ (𝐹 “ (𝐴[,]𝐵))(𝑥[,]𝑦) ⊆ (𝐹 “ (𝐴[,]𝐵)) → ((𝐹𝐴)[,](𝐹𝐵)) ⊆ (𝐹 “ (𝐴[,]𝐵))))
113103, 107, 112syl2anc 587 . . . . 5 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) → (∀𝑥 ∈ (𝐹 “ (𝐴[,]𝐵))∀𝑦 ∈ (𝐹 “ (𝐴[,]𝐵))(𝑥[,]𝑦) ⊆ (𝐹 “ (𝐴[,]𝐵)) → ((𝐹𝐴)[,](𝐹𝐵)) ⊆ (𝐹 “ (𝐴[,]𝐵))))
11491, 113mpd 15 . . . 4 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) → ((𝐹𝐴)[,](𝐹𝐵)) ⊆ (𝐹 “ (𝐴[,]𝐵)))
115 ioossicc 12822 . . . . . . . 8 ((𝐹𝐴)(,)(𝐹𝐵)) ⊆ ((𝐹𝐴)[,](𝐹𝐵))
116115sseli 3949 . . . . . . 7 (𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵)) → 𝑈 ∈ ((𝐹𝐴)[,](𝐹𝐵)))
1171163ad2ant3 1132 . . . . . 6 ((𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))) → 𝑈 ∈ ((𝐹𝐴)[,](𝐹𝐵)))
1181173ad2ant3 1132 . . . . 5 (((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵)))) → 𝑈 ∈ ((𝐹𝐴)[,](𝐹𝐵)))
1191183ad2ant3 1132 . . . 4 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) → 𝑈 ∈ ((𝐹𝐴)[,](𝐹𝐵)))
120114, 119sseldd 3954 . . 3 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) → 𝑈 ∈ (𝐹 “ (𝐴[,]𝐵)))
121 fvelima 6724 . . 3 ((Fun 𝐹𝑈 ∈ (𝐹 “ (𝐴[,]𝐵))) → ∃𝑥 ∈ (𝐴[,]𝐵)(𝐹𝑥) = 𝑈)
1226, 120, 121syl2anc 587 . 2 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) → ∃𝑥 ∈ (𝐴[,]𝐵)(𝐹𝑥) = 𝑈)
123 simpl1 1188 . . . . . . . 8 (((𝑥 ∈ ℝ*𝐴𝐵 ∧ (𝑥 = 𝐴 ∨ (𝐴 < 𝑥𝑥 < 𝐵) ∨ 𝑥 = 𝐵)) ∧ (𝐹𝑥) = 𝑈) → 𝑥 ∈ ℝ*)
124123a1i 11 . . . . . . 7 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) → (((𝑥 ∈ ℝ*𝐴𝐵 ∧ (𝑥 = 𝐴 ∨ (𝐴 < 𝑥𝑥 < 𝐵) ∨ 𝑥 = 𝐵)) ∧ (𝐹𝑥) = 𝑈) → 𝑥 ∈ ℝ*))
125 simprr 772 . . . . . . . . . . . 12 ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) ∧ ((𝑥 ∈ ℝ*𝐴𝐵 ∧ (𝑥 = 𝐴 ∨ (𝐴 < 𝑥𝑥 < 𝐵) ∨ 𝑥 = 𝐵)) ∧ (𝐹𝑥) = 𝑈)) → (𝐹𝑥) = 𝑈)
12624, 103sseldd 3954 . . . . . . . . . . . . . 14 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) → (𝐹𝐴) ∈ ℝ)
127 simp333 1325 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) → 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵)))
128126rexrd 10691 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) → (𝐹𝐴) ∈ ℝ*)
12924, 107sseldd 3954 . . . . . . . . . . . . . . . . . 18 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) → (𝐹𝐵) ∈ ℝ)
130129rexrd 10691 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) → (𝐹𝐵) ∈ ℝ*)
131 elioo2 12778 . . . . . . . . . . . . . . . . 17 (((𝐹𝐴) ∈ ℝ* ∧ (𝐹𝐵) ∈ ℝ*) → (𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵)) ↔ (𝑈 ∈ ℝ ∧ (𝐹𝐴) < 𝑈𝑈 < (𝐹𝐵))))
132128, 130, 131syl2anc 587 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) → (𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵)) ↔ (𝑈 ∈ ℝ ∧ (𝐹𝐴) < 𝑈𝑈 < (𝐹𝐵))))
133127, 132mpbid 235 . . . . . . . . . . . . . . 15 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) → (𝑈 ∈ ℝ ∧ (𝐹𝐴) < 𝑈𝑈 < (𝐹𝐵)))
134133simp2d 1140 . . . . . . . . . . . . . 14 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) → (𝐹𝐴) < 𝑈)
135126, 134gtned 10775 . . . . . . . . . . . . 13 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) → 𝑈 ≠ (𝐹𝐴))
136135adantr 484 . . . . . . . . . . . 12 ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) ∧ ((𝑥 ∈ ℝ*𝐴𝐵 ∧ (𝑥 = 𝐴 ∨ (𝐴 < 𝑥𝑥 < 𝐵) ∨ 𝑥 = 𝐵)) ∧ (𝐹𝑥) = 𝑈)) → 𝑈 ≠ (𝐹𝐴))
137125, 136eqnetrd 3081 . . . . . . . . . . 11 ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) ∧ ((𝑥 ∈ ℝ*𝐴𝐵 ∧ (𝑥 = 𝐴 ∨ (𝐴 < 𝑥𝑥 < 𝐵) ∨ 𝑥 = 𝐵)) ∧ (𝐹𝑥) = 𝑈)) → (𝐹𝑥) ≠ (𝐹𝐴))
138137neneqd 3019 . . . . . . . . . 10 ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) ∧ ((𝑥 ∈ ℝ*𝐴𝐵 ∧ (𝑥 = 𝐴 ∨ (𝐴 < 𝑥𝑥 < 𝐵) ∨ 𝑥 = 𝐵)) ∧ (𝐹𝑥) = 𝑈)) → ¬ (𝐹𝑥) = (𝐹𝐴))
139 fveq2 6663 . . . . . . . . . 10 (𝑥 = 𝐴 → (𝐹𝑥) = (𝐹𝐴))
140138, 139nsyl 142 . . . . . . . . 9 ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) ∧ ((𝑥 ∈ ℝ*𝐴𝐵 ∧ (𝑥 = 𝐴 ∨ (𝐴 < 𝑥𝑥 < 𝐵) ∨ 𝑥 = 𝐵)) ∧ (𝐹𝑥) = 𝑈)) → ¬ 𝑥 = 𝐴)
141 simp13 1202 . . . . . . . . . . . . . 14 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) → 𝑈 ∈ ℝ)
142133simp3d 1141 . . . . . . . . . . . . . 14 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) → 𝑈 < (𝐹𝐵))
143141, 142ltned 10776 . . . . . . . . . . . . 13 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) → 𝑈 ≠ (𝐹𝐵))
144143adantr 484 . . . . . . . . . . . 12 ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) ∧ ((𝑥 ∈ ℝ*𝐴𝐵 ∧ (𝑥 = 𝐴 ∨ (𝐴 < 𝑥𝑥 < 𝐵) ∨ 𝑥 = 𝐵)) ∧ (𝐹𝑥) = 𝑈)) → 𝑈 ≠ (𝐹𝐵))
145125, 144eqnetrd 3081 . . . . . . . . . . 11 ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) ∧ ((𝑥 ∈ ℝ*𝐴𝐵 ∧ (𝑥 = 𝐴 ∨ (𝐴 < 𝑥𝑥 < 𝐵) ∨ 𝑥 = 𝐵)) ∧ (𝐹𝑥) = 𝑈)) → (𝐹𝑥) ≠ (𝐹𝐵))
146145neneqd 3019 . . . . . . . . . 10 ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) ∧ ((𝑥 ∈ ℝ*𝐴𝐵 ∧ (𝑥 = 𝐴 ∨ (𝐴 < 𝑥𝑥 < 𝐵) ∨ 𝑥 = 𝐵)) ∧ (𝐹𝑥) = 𝑈)) → ¬ (𝐹𝑥) = (𝐹𝐵))
147 fveq2 6663 . . . . . . . . . 10 (𝑥 = 𝐵 → (𝐹𝑥) = (𝐹𝐵))
148146, 147nsyl 142 . . . . . . . . 9 ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) ∧ ((𝑥 ∈ ℝ*𝐴𝐵 ∧ (𝑥 = 𝐴 ∨ (𝐴 < 𝑥𝑥 < 𝐵) ∨ 𝑥 = 𝐵)) ∧ (𝐹𝑥) = 𝑈)) → ¬ 𝑥 = 𝐵)
149 simprl3 1217 . . . . . . . . 9 ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) ∧ ((𝑥 ∈ ℝ*𝐴𝐵 ∧ (𝑥 = 𝐴 ∨ (𝐴 < 𝑥𝑥 < 𝐵) ∨ 𝑥 = 𝐵)) ∧ (𝐹𝑥) = 𝑈)) → (𝑥 = 𝐴 ∨ (𝐴 < 𝑥𝑥 < 𝐵) ∨ 𝑥 = 𝐵))
150140, 148, 149ecase13d 33746 . . . . . . . 8 ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) ∧ ((𝑥 ∈ ℝ*𝐴𝐵 ∧ (𝑥 = 𝐴 ∨ (𝐴 < 𝑥𝑥 < 𝐵) ∨ 𝑥 = 𝐵)) ∧ (𝐹𝑥) = 𝑈)) → (𝐴 < 𝑥𝑥 < 𝐵))
151150ex 416 . . . . . . 7 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) → (((𝑥 ∈ ℝ*𝐴𝐵 ∧ (𝑥 = 𝐴 ∨ (𝐴 < 𝑥𝑥 < 𝐵) ∨ 𝑥 = 𝐵)) ∧ (𝐹𝑥) = 𝑈) → (𝐴 < 𝑥𝑥 < 𝐵)))
152124, 151jcad 516 . . . . . 6 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) → (((𝑥 ∈ ℝ*𝐴𝐵 ∧ (𝑥 = 𝐴 ∨ (𝐴 < 𝑥𝑥 < 𝐵) ∨ 𝑥 = 𝐵)) ∧ (𝐹𝑥) = 𝑈) → (𝑥 ∈ ℝ* ∧ (𝐴 < 𝑥𝑥 < 𝐵))))
153 3anass 1092 . . . . . 6 ((𝑥 ∈ ℝ*𝐴 < 𝑥𝑥 < 𝐵) ↔ (𝑥 ∈ ℝ* ∧ (𝐴 < 𝑥𝑥 < 𝐵)))
154152, 153syl6ibr 255 . . . . 5 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) → (((𝑥 ∈ ℝ*𝐴𝐵 ∧ (𝑥 = 𝐴 ∨ (𝐴 < 𝑥𝑥 < 𝐵) ∨ 𝑥 = 𝐵)) ∧ (𝐹𝑥) = 𝑈) → (𝑥 ∈ ℝ*𝐴 < 𝑥𝑥 < 𝐵)))
155 rexr 10687 . . . . . . . . 9 (𝐴 ∈ ℝ → 𝐴 ∈ ℝ*)
156 rexr 10687 . . . . . . . . 9 (𝐵 ∈ ℝ → 𝐵 ∈ ℝ*)
157 elicc3 33750 . . . . . . . . 9 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝑥 ∈ (𝐴[,]𝐵) ↔ (𝑥 ∈ ℝ*𝐴𝐵 ∧ (𝑥 = 𝐴 ∨ (𝐴 < 𝑥𝑥 < 𝐵) ∨ 𝑥 = 𝐵))))
158155, 156, 157syl2an 598 . . . . . . . 8 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑥 ∈ (𝐴[,]𝐵) ↔ (𝑥 ∈ ℝ*𝐴𝐵 ∧ (𝑥 = 𝐴 ∨ (𝐴 < 𝑥𝑥 < 𝐵) ∨ 𝑥 = 𝐵))))
1591583adant3 1129 . . . . . . 7 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) → (𝑥 ∈ (𝐴[,]𝐵) ↔ (𝑥 ∈ ℝ*𝐴𝐵 ∧ (𝑥 = 𝐴 ∨ (𝐴 < 𝑥𝑥 < 𝐵) ∨ 𝑥 = 𝐵))))
1601593ad2ant1 1130 . . . . . 6 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) → (𝑥 ∈ (𝐴[,]𝐵) ↔ (𝑥 ∈ ℝ*𝐴𝐵 ∧ (𝑥 = 𝐴 ∨ (𝐴 < 𝑥𝑥 < 𝐵) ∨ 𝑥 = 𝐵))))
161160anbi1d 632 . . . . 5 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) → ((𝑥 ∈ (𝐴[,]𝐵) ∧ (𝐹𝑥) = 𝑈) ↔ ((𝑥 ∈ ℝ*𝐴𝐵 ∧ (𝑥 = 𝐴 ∨ (𝐴 < 𝑥𝑥 < 𝐵) ∨ 𝑥 = 𝐵)) ∧ (𝐹𝑥) = 𝑈)))
162 elioo1 12777 . . . . . . . 8 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝑥 ∈ (𝐴(,)𝐵) ↔ (𝑥 ∈ ℝ*𝐴 < 𝑥𝑥 < 𝐵)))
163155, 156, 162syl2an 598 . . . . . . 7 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑥 ∈ (𝐴(,)𝐵) ↔ (𝑥 ∈ ℝ*𝐴 < 𝑥𝑥 < 𝐵)))
1641633adant3 1129 . . . . . 6 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) → (𝑥 ∈ (𝐴(,)𝐵) ↔ (𝑥 ∈ ℝ*𝐴 < 𝑥𝑥 < 𝐵)))
1651643ad2ant1 1130 . . . . 5 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) → (𝑥 ∈ (𝐴(,)𝐵) ↔ (𝑥 ∈ ℝ*𝐴 < 𝑥𝑥 < 𝐵)))
166154, 161, 1653imtr4d 297 . . . 4 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) → ((𝑥 ∈ (𝐴[,]𝐵) ∧ (𝐹𝑥) = 𝑈) → 𝑥 ∈ (𝐴(,)𝐵)))
167 simpr 488 . . . . 5 ((𝑥 ∈ (𝐴[,]𝐵) ∧ (𝐹𝑥) = 𝑈) → (𝐹𝑥) = 𝑈)
168167a1i 11 . . . 4 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) → ((𝑥 ∈ (𝐴[,]𝐵) ∧ (𝐹𝑥) = 𝑈) → (𝐹𝑥) = 𝑈))
169166, 168jcad 516 . . 3 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) → ((𝑥 ∈ (𝐴[,]𝐵) ∧ (𝐹𝑥) = 𝑈) → (𝑥 ∈ (𝐴(,)𝐵) ∧ (𝐹𝑥) = 𝑈)))
170169reximdv2 3263 . 2 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) → (∃𝑥 ∈ (𝐴[,]𝐵)(𝐹𝑥) = 𝑈 → ∃𝑥 ∈ (𝐴(,)𝐵)(𝐹𝑥) = 𝑈))
171122, 170mpd 15 1 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) → ∃𝑥 ∈ (𝐴(,)𝐵)(𝐹𝑥) = 𝑈)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∨ w3o 1083   ∧ w3a 1084   = wceq 1538   ∈ wcel 2115   ≠ wne 3014  ∀wral 3133  ∃wrex 3134  Vcvv 3480   ⊆ wss 3919  ∪ cuni 4824   class class class wbr 5053  dom cdm 5543  ran crn 5544   ↾ cres 5545   “ cima 5546  Fun wfun 6339  ⟶wf 6341  –onto→wfo 6343  ‘cfv 6345  (class class class)co 7151  ℂcc 10535  ℝcr 10536  ℝ*cxr 10674   < clt 10675   ≤ cle 10676  (,)cioo 12737  [,]cicc 12740   ↾t crest 16696  TopOpenctopn 16697  topGenctg 16713  ℂfldccnfld 20100  Topctop 21507  TopOnctopon 21524   Cn ccn 21838  Conncconn 22025  –cn→ccncf 23490 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7457  ax-cnex 10593  ax-resscn 10594  ax-1cn 10595  ax-icn 10596  ax-addcl 10597  ax-addrcl 10598  ax-mulcl 10599  ax-mulrcl 10600  ax-mulcom 10601  ax-addass 10602  ax-mulass 10603  ax-distr 10604  ax-i2m1 10605  ax-1ne0 10606  ax-1rid 10607  ax-rnegex 10608  ax-rrecex 10609  ax-cnre 10610  ax-pre-lttri 10611  ax-pre-lttrn 10612  ax-pre-ltadd 10613  ax-pre-mulgt0 10614  ax-pre-sup 10615 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-br 5054  df-opab 5116  df-mpt 5134  df-tr 5160  df-id 5448  df-eprel 5453  df-po 5462  df-so 5463  df-fr 5502  df-we 5504  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-pred 6137  df-ord 6183  df-on 6184  df-lim 6185  df-suc 6186  df-iota 6304  df-fun 6347  df-fn 6348  df-f 6349  df-f1 6350  df-fo 6351  df-f1o 6352  df-fv 6353  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7577  df-1st 7686  df-2nd 7687  df-wrecs 7945  df-recs 8006  df-rdg 8044  df-1o 8100  df-oadd 8104  df-er 8287  df-map 8406  df-en 8508  df-dom 8509  df-sdom 8510  df-fin 8511  df-fi 8874  df-sup 8905  df-inf 8906  df-pnf 10677  df-mnf 10678  df-xr 10679  df-ltxr 10680  df-le 10681  df-sub 10872  df-neg 10873  df-div 11298  df-nn 11637  df-2 11699  df-3 11700  df-4 11701  df-5 11702  df-6 11703  df-7 11704  df-8 11705  df-9 11706  df-n0 11897  df-z 11981  df-dec 12098  df-uz 12243  df-q 12348  df-rp 12389  df-xneg 12506  df-xadd 12507  df-xmul 12508  df-ioo 12741  df-ico 12743  df-icc 12744  df-fz 12897  df-seq 13376  df-exp 13437  df-cj 14460  df-re 14461  df-im 14462  df-sqrt 14596  df-abs 14597  df-struct 16487  df-ndx 16488  df-slot 16489  df-base 16491  df-plusg 16580  df-mulr 16581  df-starv 16582  df-tset 16586  df-ple 16587  df-ds 16589  df-unif 16590  df-rest 16698  df-topn 16699  df-topgen 16719  df-psmet 20092  df-xmet 20093  df-met 20094  df-bl 20095  df-mopn 20096  df-cnfld 20101  df-top 21508  df-topon 21525  df-topsp 21547  df-bases 21560  df-cld 21633  df-cn 21841  df-cnp 21842  df-conn 22026  df-xms 22936  df-ms 22937  df-cncf 23492 This theorem is referenced by: (None)
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