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Theorem ivthALT 36318
Description: An alternate proof of the Intermediate Value Theorem ivth 25503 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 1208 . . . . . 6 (((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵)))) → 𝐹 ∈ (𝐷cn→ℂ))
2 cncff 24933 . . . . . 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 6740 . . . . 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 1134 . . 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 24866 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((topGen‘ran (,)) ↾t (𝐴[,]𝐵)) ∈ Conn)
873adant3 1131 . . . . . . . 8 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) → ((topGen‘ran (,)) ↾t (𝐴[,]𝐵)) ∈ Conn)
983ad2ant1 1132 . . . . . . 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 1193 . . . . . . . . . . . . . 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 617 . . . . . . . . . . . 12 (((𝐴[,]𝐵) ⊆ 𝐷 ∧ (𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) → ((𝐴[,]𝐵) ⊆ 𝐷𝐹:𝐷⟶ℂ))
13123impb 1114 . . . . . . . . . . 11 (((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵)))) → ((𝐴[,]𝐵) ⊆ 𝐷𝐹:𝐷⟶ℂ))
14133ad2ant3 1134 . . . . . . . . . 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 481 . . . . . . . . . . 11 (((𝐴[,]𝐵) ⊆ 𝐷𝐹:𝐷⟶ℂ) → Fun 𝐹)
16 fdm 6746 . . . . . . . . . . . . 13 (𝐹:𝐷⟶ℂ → dom 𝐹 = 𝐷)
1716sseq2d 4028 . . . . . . . . . . . 12 (𝐹:𝐷⟶ℂ → ((𝐴[,]𝐵) ⊆ dom 𝐹 ↔ (𝐴[,]𝐵) ⊆ 𝐷))
1817biimparc 479 . . . . . . . . . . 11 (((𝐴[,]𝐵) ⊆ 𝐷𝐹:𝐷⟶ℂ) → (𝐴[,]𝐵) ⊆ dom 𝐹)
1915, 18jca 511 . . . . . . . . . 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 6831 . . . . . . . . 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 24798 . . . . . . . . . 10 (topGen‘ran (,)) ∈ Top
24 simp332 1326 . . . . . . . . . 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 24799 . . . . . . . . . . 11 ℝ = (topGen‘ran (,))
2625restuni 23186 . . . . . . . . . 10 (((topGen‘ran (,)) ∈ Top ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ) → (𝐹 “ (𝐴[,]𝐵)) = ((topGen‘ran (,)) ↾t (𝐹 “ (𝐴[,]𝐵))))
2723, 24, 26sylancr 587 . . . . . . . . 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 6819 . . . . . . . . 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 232 . . . . . . 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 1325 . . . . . . . . . . . 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 4018 . . . . . . . . . . . . . . 15 ℂ ⊆ ℂ
33 eqid 2735 . . . . . . . . . . . . . . . 16 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
34 eqid 2735 . . . . . . . . . . . . . . . 16 ((TopOpen‘ℂfld) ↾t 𝐷) = ((TopOpen‘ℂfld) ↾t 𝐷)
3533cnfldtop 24820 . . . . . . . . . . . . . . . . . 18 (TopOpen‘ℂfld) ∈ Top
3633cnfldtopon 24819 . . . . . . . . . . . . . . . . . . . 20 (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
3736toponunii 22938 . . . . . . . . . . . . . . . . . . 19 ℂ = (TopOpen‘ℂfld)
3837restid 17480 . . . . . . . . . . . . . . . . . 18 ((TopOpen‘ℂfld) ∈ Top → ((TopOpen‘ℂfld) ↾t ℂ) = (TopOpen‘ℂfld))
3935, 38ax-mp 5 . . . . . . . . . . . . . . . . 17 ((TopOpen‘ℂfld) ↾t ℂ) = (TopOpen‘ℂfld)
4039eqcomi 2744 . . . . . . . . . . . . . . . 16 (TopOpen‘ℂfld) = ((TopOpen‘ℂfld) ↾t ℂ)
4133, 34, 40cncfcn 24950 . . . . . . . . . . . . . . 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 1133 . . . . . . . . . . . . 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 1134 . . . . . . . . . . . 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 1208 . . . . . . . . . . . 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 1209 . . . . . . . . . . . . . 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 23185 . . . . . . . . . . . . . 14 (((TopOpen‘ℂfld) ∈ (TopOn‘ℂ) ∧ 𝐷 ⊆ ℂ) → ((TopOpen‘ℂfld) ↾t 𝐷) ∈ (TopOn‘𝐷))
4936, 47, 48sylancr 587 . . . . . . . . . . . . 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 22936 . . . . . . . . . . . . 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 4036 . . . . . . . . . . 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 2735 . . . . . . . . . . . 12 ((TopOpen‘ℂfld) ↾t 𝐷) = ((TopOpen‘ℂfld) ↾t 𝐷)
5453cnrest 23309 . . . . . . . . . . 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 584 . . . . . . . . . 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 11234 . . . . . . . . . . . . . 14 ℂ ∈ V
58 ssexg 5329 . . . . . . . . . . . . . 14 ((𝐷 ⊆ ℂ ∧ ℂ ∈ V) → 𝐷 ∈ V)
5947, 57, 58sylancl 586 . . . . . . . . . . . . 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 23189 . . . . . . . . . . . . 13 (((TopOpen‘ℂfld) ∈ Top ∧ (𝐴[,]𝐵) ⊆ 𝐷𝐷 ∈ V) → (((TopOpen‘ℂfld) ↾t 𝐷) ↾t (𝐴[,]𝐵)) = ((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)))
6156, 46, 59, 60syl3anc 1370 . . . . . . . . . . . 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 13466 . . . . . . . . . . . . . . 15 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
63623adant3 1131 . . . . . . . . . . . . . 14 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
64633ad2ant1 1132 . . . . . . . . . . . . 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 2735 . . . . . . . . . . . . . 14 (topGen‘ran (,)) = (topGen‘ran (,))
6633, 65rerest 24840 . . . . . . . . . . . . 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 2775 . . . . . . . . . . 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 7446 . . . . . . . . . 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 5702 . . . . . . . . . . . 12 (𝐹 “ (𝐴[,]𝐵)) = ran (𝐹 ↾ (𝐴[,]𝐵))
7372eqimss2i 4057 . . . . . . . . . . 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 11210 . . . . . . . . . . 11 ℝ ⊆ ℂ
7624, 75sstrdi 4008 . . . . . . . . . 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 23310 . . . . . . . . . 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 1370 . . . . . . . . 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 232 . . . . . . . 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 24840 . . . . . . . . . 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 7447 . . . . . . . 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 2735 . . . . . . . 8 ((topGen‘ran (,)) ↾t (𝐹 “ (𝐴[,]𝐵))) = ((topGen‘ran (,)) ↾t (𝐹 “ (𝐴[,]𝐵)))
8584cnconn 23446 . . . . . . 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 1370 . . . . . 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 24864 . . . . . . . . 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 1133 . . . . . . . 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 1134 . . . . . . 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 1134 . . . . . 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 232 . . . . 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 1202 . . . . . . . . 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 11309 . . . . . . . 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 1203 . . . . . . . . 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 11309 . . . . . . . 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 11347 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵𝐴𝐵))
9796imp 406 . . . . . . . . . 10 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐴 < 𝐵) → 𝐴𝐵)
98973adantl3 1167 . . . . . . . . 9 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵) → 𝐴𝐵)
99983adant3 1131 . . . . . . . 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 13501 . . . . . . . 8 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴𝐵) → 𝐴 ∈ (𝐴[,]𝐵))
10193, 95, 99, 100syl3anc 1370 . . . . . . 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 7251 . . . . . . 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 13502 . . . . . . . 8 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴𝐵) → 𝐵 ∈ (𝐴[,]𝐵))
10593, 95, 99, 104syl3anc 1370 . . . . . . 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 7251 . . . . . . 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 7438 . . . . . . . 8 (𝑥 = (𝐹𝐴) → (𝑥[,]𝑦) = ((𝐹𝐴)[,]𝑦))
109108sseq1d 4027 . . . . . . 7 (𝑥 = (𝐹𝐴) → ((𝑥[,]𝑦) ⊆ (𝐹 “ (𝐴[,]𝐵)) ↔ ((𝐹𝐴)[,]𝑦) ⊆ (𝐹 “ (𝐴[,]𝐵))))
110 oveq2 7439 . . . . . . . 8 (𝑦 = (𝐹𝐵) → ((𝐹𝐴)[,]𝑦) = ((𝐹𝐴)[,](𝐹𝐵)))
111110sseq1d 4027 . . . . . . 7 (𝑦 = (𝐹𝐵) → (((𝐹𝐴)[,]𝑦) ⊆ (𝐹 “ (𝐴[,]𝐵)) ↔ ((𝐹𝐴)[,](𝐹𝐵)) ⊆ (𝐹 “ (𝐴[,]𝐵))))
112109, 111rspc2v 3633 . . . . . 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 584 . . . . 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 13470 . . . . . . . 8 ((𝐹𝐴)(,)(𝐹𝐵)) ⊆ ((𝐹𝐴)[,](𝐹𝐵))
116115sseli 3991 . . . . . . 7 (𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵)) → 𝑈 ∈ ((𝐹𝐴)[,](𝐹𝐵)))
1171163ad2ant3 1134 . . . . . 6 ((𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))) → 𝑈 ∈ ((𝐹𝐴)[,](𝐹𝐵)))
1181173ad2ant3 1134 . . . . 5 (((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵)))) → 𝑈 ∈ ((𝐹𝐴)[,](𝐹𝐵)))
1191183ad2ant3 1134 . . . 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 3996 . . 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 6974 . . 3 ((Fun 𝐹𝑈 ∈ (𝐹 “ (𝐴[,]𝐵))) → ∃𝑥 ∈ (𝐴[,]𝐵)(𝐹𝑥) = 𝑈)
1226, 120, 121syl2anc 584 . 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 1190 . . . . . . . 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 3996 . . . . . . . . . . . . . 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 1327 . . . . . . . . . . . . . . . 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 11309 . . . . . . . . . . . . . . . . 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 3996 . . . . . . . . . . . . . . . . . 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 11309 . . . . . . . . . . . . . . . . 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 13425 . . . . . . . . . . . . . . . . 17 (((𝐹𝐴) ∈ ℝ* ∧ (𝐹𝐵) ∈ ℝ*) → (𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵)) ↔ (𝑈 ∈ ℝ ∧ (𝐹𝐴) < 𝑈𝑈 < (𝐹𝐵))))
132128, 130, 131syl2anc 584 . . . . . . . . . . . . . . . 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 232 . . . . . . . . . . . . . . 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 1142 . . . . . . . . . . . . . 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 11394 . . . . . . . . . . . . 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 480 . . . . . . . . . . . 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 3006 . . . . . . . . . . 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 2943 . . . . . . . . . 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 6907 . . . . . . . . . 10 (𝑥 = 𝐴 → (𝐹𝑥) = (𝐹𝐴))
140138, 139nsyl 140 . . . . . . . . 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 1204 . . . . . . . . . . . . . 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 1143 . . . . . . . . . . . . . 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 11395 . . . . . . . . . . . . 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 480 . . . . . . . . . . . 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 3006 . . . . . . . . . . 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 2943 . . . . . . . . . 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 6907 . . . . . . . . . 10 (𝑥 = 𝐵 → (𝐹𝑥) = (𝐹𝐵))
148146, 147nsyl 140 . . . . . . . . 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 1219 . . . . . . . . 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 36296 . . . . . . . 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 412 . . . . . . 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 512 . . . . . 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 1094 . . . . . 6 ((𝑥 ∈ ℝ*𝐴 < 𝑥𝑥 < 𝐵) ↔ (𝑥 ∈ ℝ* ∧ (𝐴 < 𝑥𝑥 < 𝐵)))
154152, 153imbitrrdi 252 . . . . 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 11305 . . . . . . . . 9 (𝐴 ∈ ℝ → 𝐴 ∈ ℝ*)
156 rexr 11305 . . . . . . . . 9 (𝐵 ∈ ℝ → 𝐵 ∈ ℝ*)
157 elicc3 36300 . . . . . . . . 9 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝑥 ∈ (𝐴[,]𝐵) ↔ (𝑥 ∈ ℝ*𝐴𝐵 ∧ (𝑥 = 𝐴 ∨ (𝐴 < 𝑥𝑥 < 𝐵) ∨ 𝑥 = 𝐵))))
158155, 156, 157syl2an 596 . . . . . . . 8 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑥 ∈ (𝐴[,]𝐵) ↔ (𝑥 ∈ ℝ*𝐴𝐵 ∧ (𝑥 = 𝐴 ∨ (𝐴 < 𝑥𝑥 < 𝐵) ∨ 𝑥 = 𝐵))))
1591583adant3 1131 . . . . . . 7 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) → (𝑥 ∈ (𝐴[,]𝐵) ↔ (𝑥 ∈ ℝ*𝐴𝐵 ∧ (𝑥 = 𝐴 ∨ (𝐴 < 𝑥𝑥 < 𝐵) ∨ 𝑥 = 𝐵))))
1601593ad2ant1 1132 . . . . . 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 631 . . . . 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 13424 . . . . . . . 8 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝑥 ∈ (𝐴(,)𝐵) ↔ (𝑥 ∈ ℝ*𝐴 < 𝑥𝑥 < 𝐵)))
163155, 156, 162syl2an 596 . . . . . . 7 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑥 ∈ (𝐴(,)𝐵) ↔ (𝑥 ∈ ℝ*𝐴 < 𝑥𝑥 < 𝐵)))
1641633adant3 1131 . . . . . 6 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) → (𝑥 ∈ (𝐴(,)𝐵) ↔ (𝑥 ∈ ℝ*𝐴 < 𝑥𝑥 < 𝐵)))
1651643ad2ant1 1132 . . . . 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 294 . . . 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 484 . . . . 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 512 . . 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 3162 . 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 206  wa 395  w3o 1085  w3a 1086   = wceq 1537  wcel 2106  wne 2938  wral 3059  wrex 3068  Vcvv 3478  wss 3963   cuni 4912   class class class wbr 5148  dom cdm 5689  ran crn 5690  cres 5691  cima 5692  Fun wfun 6557  wf 6559  ontowfo 6561  cfv 6563  (class class class)co 7431  cc 11151  cr 11152  *cxr 11292   < clt 11293  cle 11294  (,)cioo 13384  [,]cicc 13387  t crest 17467  TopOpenctopn 17468  topGenctg 17484  fldccnfld 21382  Topctop 22915  TopOnctopon 22932   Cn ccn 23248  Conncconn 23435  cnccncf 24916
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-fi 9449  df-sup 9480  df-inf 9481  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12612  df-dec 12732  df-uz 12877  df-q 12989  df-rp 13033  df-xneg 13152  df-xadd 13153  df-xmul 13154  df-ioo 13388  df-ico 13390  df-icc 13391  df-fz 13545  df-seq 14040  df-exp 14100  df-cj 15135  df-re 15136  df-im 15137  df-sqrt 15271  df-abs 15272  df-struct 17181  df-slot 17216  df-ndx 17228  df-base 17246  df-plusg 17311  df-mulr 17312  df-starv 17313  df-tset 17317  df-ple 17318  df-ds 17320  df-unif 17321  df-rest 17469  df-topn 17470  df-topgen 17490  df-psmet 21374  df-xmet 21375  df-met 21376  df-bl 21377  df-mopn 21378  df-cnfld 21383  df-top 22916  df-topon 22933  df-topsp 22955  df-bases 22969  df-cld 23043  df-cn 23251  df-cnp 23252  df-conn 23436  df-xms 24346  df-ms 24347  df-cncf 24918
This theorem is referenced by: (None)
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