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Theorem ivthALT 34287
Description: An alternate proof of the Intermediate Value Theorem ivth 24375 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 1211 . . . . . 6 (((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵)))) → 𝐹 ∈ (𝐷cn→ℂ))
2 cncff 23814 . . . . . 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 6566 . . . . 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 1137 . . 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 23751 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((topGen‘ran (,)) ↾t (𝐴[,]𝐵)) ∈ Conn)
873adant3 1134 . . . . . . . 8 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) → ((topGen‘ran (,)) ↾t (𝐴[,]𝐵)) ∈ Conn)
983ad2ant1 1135 . . . . . . 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 1196 . . . . . . . . . . . . . 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 620 . . . . . . . . . . . 12 (((𝐴[,]𝐵) ⊆ 𝐷 ∧ (𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) → ((𝐴[,]𝐵) ⊆ 𝐷𝐹:𝐷⟶ℂ))
13123impb 1117 . . . . . . . . . . 11 (((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵)))) → ((𝐴[,]𝐵) ⊆ 𝐷𝐹:𝐷⟶ℂ))
14133ad2ant3 1137 . . . . . . . . . 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 6572 . . . . . . . . . . . . 13 (𝐹:𝐷⟶ℂ → dom 𝐹 = 𝐷)
1716sseq2d 3947 . . . . . . . . . . . 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 6661 . . . . . . . . 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 23683 . . . . . . . . . 10 (topGen‘ran (,)) ∈ Top
24 simp332 1329 . . . . . . . . . 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 23684 . . . . . . . . . . 11 ℝ = (topGen‘ran (,))
2625restuni 22083 . . . . . . . . . 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 6649 . . . . . . . . 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 1328 . . . . . . . . . . . 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 3937 . . . . . . . . . . . . . . 15 ℂ ⊆ ℂ
33 eqid 2738 . . . . . . . . . . . . . . . 16 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
34 eqid 2738 . . . . . . . . . . . . . . . 16 ((TopOpen‘ℂfld) ↾t 𝐷) = ((TopOpen‘ℂfld) ↾t 𝐷)
3533cnfldtop 23705 . . . . . . . . . . . . . . . . . 18 (TopOpen‘ℂfld) ∈ Top
3633cnfldtopon 23704 . . . . . . . . . . . . . . . . . . . 20 (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
3736toponunii 21837 . . . . . . . . . . . . . . . . . . 19 ℂ = (TopOpen‘ℂfld)
3837restid 16962 . . . . . . . . . . . . . . . . . 18 ((TopOpen‘ℂfld) ∈ Top → ((TopOpen‘ℂfld) ↾t ℂ) = (TopOpen‘ℂfld))
3935, 38ax-mp 5 . . . . . . . . . . . . . . . . 17 ((TopOpen‘ℂfld) ↾t ℂ) = (TopOpen‘ℂfld)
4039eqcomi 2747 . . . . . . . . . . . . . . . 16 (TopOpen‘ℂfld) = ((TopOpen‘ℂfld) ↾t ℂ)
4133, 34, 40cncfcn 23831 . . . . . . . . . . . . . . 15 ((𝐷 ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝐷cn→ℂ) = (((TopOpen‘ℂfld) ↾t 𝐷) Cn (TopOpen‘ℂfld)))
4232, 41mpan2 691 . . . . . . . . . . . . . 14 (𝐷 ⊆ ℂ → (𝐷cn→ℂ) = (((TopOpen‘ℂfld) ↾t 𝐷) Cn (TopOpen‘ℂfld)))
43423ad2ant2 1136 . . . . . . . . . . . . 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 1137 . . . . . . . . . . . 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 2841 . . . . . . . . . . 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 1211 . . . . . . . . . . . 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 1212 . . . . . . . . . . . . . 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 22082 . . . . . . . . . . . . . 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 21835 . . . . . . . . . . . . 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 3955 . . . . . . . . . . 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 2738 . . . . . . . . . . . 12 ((TopOpen‘ℂfld) ↾t 𝐷) = ((TopOpen‘ℂfld) ↾t 𝐷)
5453cnrest 22206 . . . . . . . . . . 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 10834 . . . . . . . . . . . . . 14 ℂ ∈ V
58 ssexg 5230 . . . . . . . . . . . . . 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 22086 . . . . . . . . . . . . 13 (((TopOpen‘ℂfld) ∈ Top ∧ (𝐴[,]𝐵) ⊆ 𝐷𝐷 ∈ V) → (((TopOpen‘ℂfld) ↾t 𝐷) ↾t (𝐴[,]𝐵)) = ((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)))
6156, 46, 59, 60syl3anc 1373 . . . . . . . . . . . 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 13041 . . . . . . . . . . . . . . 15 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
63623adant3 1134 . . . . . . . . . . . . . 14 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
64633ad2ant1 1135 . . . . . . . . . . . . 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 2738 . . . . . . . . . . . . . 14 (topGen‘ran (,)) = (topGen‘ran (,))
6633, 65rerest 23725 . . . . . . . . . . . . 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 2778 . . . . . . . . . . 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 7246 . . . . . . . . . 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 2841 . . . . . . . . 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 5578 . . . . . . . . . . . 12 (𝐹 “ (𝐴[,]𝐵)) = ran (𝐹 ↾ (𝐴[,]𝐵))
7372eqimss2i 3974 . . . . . . . . . . 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 10810 . . . . . . . . . . 11 ℝ ⊆ ℂ
7624, 75sstrdi 3927 . . . . . . . . . 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 22207 . . . . . . . . . 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 1373 . . . . . . . . 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 23725 . . . . . . . . . 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 7247 . . . . . . . 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 2841 . . . . . . 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 2738 . . . . . . . 8 ((topGen‘ran (,)) ↾t (𝐹 “ (𝐴[,]𝐵))) = ((topGen‘ran (,)) ↾t (𝐹 “ (𝐴[,]𝐵)))
8584cnconn 22343 . . . . . . 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 1373 . . . . . 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 23749 . . . . . . . . 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 1136 . . . . . . . 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 1137 . . . . . . 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 1137 . . . . . 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 1205 . . . . . . . . 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 10907 . . . . . . . 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 1206 . . . . . . . . 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 10907 . . . . . . . 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 10945 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵𝐴𝐵))
9796imp 410 . . . . . . . . . 10 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐴 < 𝐵) → 𝐴𝐵)
98973adantl3 1170 . . . . . . . . 9 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵) → 𝐴𝐵)
99983adant3 1134 . . . . . . . 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 13076 . . . . . . . 8 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴𝐵) → 𝐴 ∈ (𝐴[,]𝐵))
10193, 95, 99, 100syl3anc 1373 . . . . . . 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 7065 . . . . . . 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 13077 . . . . . . . 8 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴𝐵) → 𝐵 ∈ (𝐴[,]𝐵))
10593, 95, 99, 104syl3anc 1373 . . . . . . 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 7065 . . . . . . 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 7238 . . . . . . . 8 (𝑥 = (𝐹𝐴) → (𝑥[,]𝑦) = ((𝐹𝐴)[,]𝑦))
109108sseq1d 3946 . . . . . . 7 (𝑥 = (𝐹𝐴) → ((𝑥[,]𝑦) ⊆ (𝐹 “ (𝐴[,]𝐵)) ↔ ((𝐹𝐴)[,]𝑦) ⊆ (𝐹 “ (𝐴[,]𝐵))))
110 oveq2 7239 . . . . . . . 8 (𝑦 = (𝐹𝐵) → ((𝐹𝐴)[,]𝑦) = ((𝐹𝐴)[,](𝐹𝐵)))
111110sseq1d 3946 . . . . . . 7 (𝑦 = (𝐹𝐵) → (((𝐹𝐴)[,]𝑦) ⊆ (𝐹 “ (𝐴[,]𝐵)) ↔ ((𝐹𝐴)[,](𝐹𝐵)) ⊆ (𝐹 “ (𝐴[,]𝐵))))
112109, 111rspc2v 3559 . . . . . 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 13045 . . . . . . . 8 ((𝐹𝐴)(,)(𝐹𝐵)) ⊆ ((𝐹𝐴)[,](𝐹𝐵))
116115sseli 3910 . . . . . . 7 (𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵)) → 𝑈 ∈ ((𝐹𝐴)[,](𝐹𝐵)))
1171163ad2ant3 1137 . . . . . 6 ((𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))) → 𝑈 ∈ ((𝐹𝐴)[,](𝐹𝐵)))
1181173ad2ant3 1137 . . . . 5 (((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵)))) → 𝑈 ∈ ((𝐹𝐴)[,](𝐹𝐵)))
1191183ad2ant3 1137 . . . 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 3916 . . 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 6796 . . 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 1193 . . . . . . . 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 773 . . . . . . . . . . . 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 3916 . . . . . . . . . . . . . 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 1330 . . . . . . . . . . . . . . . 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 10907 . . . . . . . . . . . . . . . . 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 3916 . . . . . . . . . . . . . . . . . 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 10907 . . . . . . . . . . . . . . . . 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 13000 . . . . . . . . . . . . . . . . 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 1145 . . . . . . . . . . . . . 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 10991 . . . . . . . . . . . . 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 3009 . . . . . . . . . . 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 2946 . . . . . . . . . 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 6735 . . . . . . . . . 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 1207 . . . . . . . . . . . . . 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 1146 . . . . . . . . . . . . . 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 10992 . . . . . . . . . . . . 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 3009 . . . . . . . . . . 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 2946 . . . . . . . . . 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 6735 . . . . . . . . . 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 1222 . . . . . . . . 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 34265 . . . . . . . 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 1097 . . . . . 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 10903 . . . . . . . . 9 (𝐴 ∈ ℝ → 𝐴 ∈ ℝ*)
156 rexr 10903 . . . . . . . . 9 (𝐵 ∈ ℝ → 𝐵 ∈ ℝ*)
157 elicc3 34269 . . . . . . . . 9 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝑥 ∈ (𝐴[,]𝐵) ↔ (𝑥 ∈ ℝ*𝐴𝐵 ∧ (𝑥 = 𝐴 ∨ (𝐴 < 𝑥𝑥 < 𝐵) ∨ 𝑥 = 𝐵))))
158155, 156, 157syl2an 599 . . . . . . . 8 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑥 ∈ (𝐴[,]𝐵) ↔ (𝑥 ∈ ℝ*𝐴𝐵 ∧ (𝑥 = 𝐴 ∨ (𝐴 < 𝑥𝑥 < 𝐵) ∨ 𝑥 = 𝐵))))
1591583adant3 1134 . . . . . . 7 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) → (𝑥 ∈ (𝐴[,]𝐵) ↔ (𝑥 ∈ ℝ*𝐴𝐵 ∧ (𝑥 = 𝐴 ∨ (𝐴 < 𝑥𝑥 < 𝐵) ∨ 𝑥 = 𝐵))))
1601593ad2ant1 1135 . . . . . 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 633 . . . . 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 12999 . . . . . . . 8 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝑥 ∈ (𝐴(,)𝐵) ↔ (𝑥 ∈ ℝ*𝐴 < 𝑥𝑥 < 𝐵)))
163155, 156, 162syl2an 599 . . . . . . 7 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑥 ∈ (𝐴(,)𝐵) ↔ (𝑥 ∈ ℝ*𝐴 < 𝑥𝑥 < 𝐵)))
1641633adant3 1134 . . . . . 6 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) → (𝑥 ∈ (𝐴(,)𝐵) ↔ (𝑥 ∈ ℝ*𝐴 < 𝑥𝑥 < 𝐵)))
1651643ad2ant1 1135 . . . . 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 3197 . 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 1088  w3a 1089   = wceq 1543  wcel 2111  wne 2941  wral 3062  wrex 3063  Vcvv 3420  wss 3880   cuni 4833   class class class wbr 5067  dom cdm 5565  ran crn 5566  cres 5567  cima 5568  Fun wfun 6391  wf 6393  ontowfo 6395  cfv 6397  (class class class)co 7231  cc 10751  cr 10752  *cxr 10890   < clt 10891  cle 10892  (,)cioo 12959  [,]cicc 12962  t crest 16949  TopOpenctopn 16950  topGenctg 16966  fldccnfld 20387  Topctop 21814  TopOnctopon 21831   Cn ccn 22145  Conncconn 22332  cnccncf 23797
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2159  ax-12 2176  ax-ext 2709  ax-rep 5193  ax-sep 5206  ax-nul 5213  ax-pow 5272  ax-pr 5336  ax-un 7541  ax-cnex 10809  ax-resscn 10810  ax-1cn 10811  ax-icn 10812  ax-addcl 10813  ax-addrcl 10814  ax-mulcl 10815  ax-mulrcl 10816  ax-mulcom 10817  ax-addass 10818  ax-mulass 10819  ax-distr 10820  ax-i2m1 10821  ax-1ne0 10822  ax-1rid 10823  ax-rnegex 10824  ax-rrecex 10825  ax-cnre 10826  ax-pre-lttri 10827  ax-pre-lttrn 10828  ax-pre-ltadd 10829  ax-pre-mulgt0 10830  ax-pre-sup 10831
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2072  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3067  df-rex 3068  df-reu 3069  df-rmo 3070  df-rab 3071  df-v 3422  df-sbc 3709  df-csb 3826  df-dif 3883  df-un 3885  df-in 3887  df-ss 3897  df-pss 3899  df-nul 4252  df-if 4454  df-pw 4529  df-sn 4556  df-pr 4558  df-tp 4560  df-op 4562  df-uni 4834  df-int 4874  df-iun 4920  df-br 5068  df-opab 5130  df-mpt 5150  df-tr 5176  df-id 5469  df-eprel 5474  df-po 5482  df-so 5483  df-fr 5523  df-we 5525  df-xp 5571  df-rel 5572  df-cnv 5573  df-co 5574  df-dm 5575  df-rn 5576  df-res 5577  df-ima 5578  df-pred 6175  df-ord 6233  df-on 6234  df-lim 6235  df-suc 6236  df-iota 6355  df-fun 6399  df-fn 6400  df-f 6401  df-f1 6402  df-fo 6403  df-f1o 6404  df-fv 6405  df-riota 7188  df-ov 7234  df-oprab 7235  df-mpo 7236  df-om 7663  df-1st 7779  df-2nd 7780  df-wrecs 8067  df-recs 8128  df-rdg 8166  df-1o 8222  df-er 8411  df-map 8530  df-en 8647  df-dom 8648  df-sdom 8649  df-fin 8650  df-fi 9051  df-sup 9082  df-inf 9083  df-pnf 10893  df-mnf 10894  df-xr 10895  df-ltxr 10896  df-le 10897  df-sub 11088  df-neg 11089  df-div 11514  df-nn 11855  df-2 11917  df-3 11918  df-4 11919  df-5 11920  df-6 11921  df-7 11922  df-8 11923  df-9 11924  df-n0 12115  df-z 12201  df-dec 12318  df-uz 12463  df-q 12569  df-rp 12611  df-xneg 12728  df-xadd 12729  df-xmul 12730  df-ioo 12963  df-ico 12965  df-icc 12966  df-fz 13120  df-seq 13599  df-exp 13660  df-cj 14686  df-re 14687  df-im 14688  df-sqrt 14822  df-abs 14823  df-struct 16724  df-slot 16759  df-ndx 16769  df-base 16785  df-plusg 16839  df-mulr 16840  df-starv 16841  df-tset 16845  df-ple 16846  df-ds 16848  df-unif 16849  df-rest 16951  df-topn 16952  df-topgen 16972  df-psmet 20379  df-xmet 20380  df-met 20381  df-bl 20382  df-mopn 20383  df-cnfld 20388  df-top 21815  df-topon 21832  df-topsp 21854  df-bases 21867  df-cld 21940  df-cn 22148  df-cnp 22149  df-conn 22333  df-xms 23242  df-ms 23243  df-cncf 23799
This theorem is referenced by: (None)
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