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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  ontowfo 6343  cfv 6345  (class class class)co 7151  cc 10535  cr 10536  *cxr 10674   < clt 10675  cle 10676  (,)cioo 12737  [,]cicc 12740  t crest 16696  TopOpenctopn 16697  topGenctg 16713  fldccnfld 20100  Topctop 21507  TopOnctopon 21524   Cn ccn 21838  Conncconn 22025  cnccncf 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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