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Theorem cncfiooicclem1 44124
Description: A continuous function 𝐹 on an open interval (𝐴(,)𝐵) can be extended to a continuous function 𝐺 on the corresponding closed interval, if it has a finite right limit 𝑅 in 𝐴 and a finite left limit 𝐿 in 𝐵. 𝐹 can be complex-valued. This lemma assumes 𝐴 < 𝐵, the invoking theorem drops this assumption. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypotheses
Ref Expression
cncfiooicclem1.x 𝑥𝜑
cncfiooicclem1.g 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹𝑥))))
cncfiooicclem1.a (𝜑𝐴 ∈ ℝ)
cncfiooicclem1.b (𝜑𝐵 ∈ ℝ)
cncfiooicclem1.altb (𝜑𝐴 < 𝐵)
cncfiooicclem1.f (𝜑𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ))
cncfiooicclem1.l (𝜑𝐿 ∈ (𝐹 lim 𝐵))
cncfiooicclem1.r (𝜑𝑅 ∈ (𝐹 lim 𝐴))
Assertion
Ref Expression
cncfiooicclem1 (𝜑𝐺 ∈ ((𝐴[,]𝐵)–cn→ℂ))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐹   𝑥,𝐿   𝑥,𝑅
Allowed substitution hints:   𝜑(𝑥)   𝐺(𝑥)

Proof of Theorem cncfiooicclem1
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 cncfiooicclem1.x . . . 4 𝑥𝜑
2 limccl 25239 . . . . . . 7 (𝐹 lim 𝐴) ⊆ ℂ
3 cncfiooicclem1.r . . . . . . 7 (𝜑𝑅 ∈ (𝐹 lim 𝐴))
42, 3sselid 3942 . . . . . 6 (𝜑𝑅 ∈ ℂ)
54ad2antrr 724 . . . . 5 (((𝜑𝑥 ∈ (𝐴[,]𝐵)) ∧ 𝑥 = 𝐴) → 𝑅 ∈ ℂ)
6 limccl 25239 . . . . . . . 8 (𝐹 lim 𝐵) ⊆ ℂ
7 cncfiooicclem1.l . . . . . . . 8 (𝜑𝐿 ∈ (𝐹 lim 𝐵))
86, 7sselid 3942 . . . . . . 7 (𝜑𝐿 ∈ ℂ)
98ad3antrrr 728 . . . . . 6 ((((𝜑𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ 𝑥 = 𝐵) → 𝐿 ∈ ℂ)
10 simplll 773 . . . . . . 7 ((((𝜑𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ ¬ 𝑥 = 𝐵) → 𝜑)
11 orel1 887 . . . . . . . . . . 11 𝑥 = 𝐴 → ((𝑥 = 𝐴𝑥 = 𝐵) → 𝑥 = 𝐵))
1211con3dimp 409 . . . . . . . . . 10 ((¬ 𝑥 = 𝐴 ∧ ¬ 𝑥 = 𝐵) → ¬ (𝑥 = 𝐴𝑥 = 𝐵))
13 vex 3449 . . . . . . . . . . 11 𝑥 ∈ V
1413elpr 4609 . . . . . . . . . 10 (𝑥 ∈ {𝐴, 𝐵} ↔ (𝑥 = 𝐴𝑥 = 𝐵))
1512, 14sylnibr 328 . . . . . . . . 9 ((¬ 𝑥 = 𝐴 ∧ ¬ 𝑥 = 𝐵) → ¬ 𝑥 ∈ {𝐴, 𝐵})
1615adantll 712 . . . . . . . 8 ((((𝜑𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ ¬ 𝑥 = 𝐵) → ¬ 𝑥 ∈ {𝐴, 𝐵})
17 simpllr 774 . . . . . . . . . 10 ((((𝜑𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ ¬ 𝑥 = 𝐵) → 𝑥 ∈ (𝐴[,]𝐵))
18 cncfiooicclem1.a . . . . . . . . . . . . 13 (𝜑𝐴 ∈ ℝ)
1918rexrd 11205 . . . . . . . . . . . 12 (𝜑𝐴 ∈ ℝ*)
2010, 19syl 17 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ ¬ 𝑥 = 𝐵) → 𝐴 ∈ ℝ*)
21 cncfiooicclem1.b . . . . . . . . . . . . 13 (𝜑𝐵 ∈ ℝ)
2221rexrd 11205 . . . . . . . . . . . 12 (𝜑𝐵 ∈ ℝ*)
2310, 22syl 17 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ ¬ 𝑥 = 𝐵) → 𝐵 ∈ ℝ*)
24 cncfiooicclem1.altb . . . . . . . . . . . . 13 (𝜑𝐴 < 𝐵)
2518, 21, 24ltled 11303 . . . . . . . . . . . 12 (𝜑𝐴𝐵)
2610, 25syl 17 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ ¬ 𝑥 = 𝐵) → 𝐴𝐵)
27 prunioo 13398 . . . . . . . . . . 11 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴𝐵) → ((𝐴(,)𝐵) ∪ {𝐴, 𝐵}) = (𝐴[,]𝐵))
2820, 23, 26, 27syl3anc 1371 . . . . . . . . . 10 ((((𝜑𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ ¬ 𝑥 = 𝐵) → ((𝐴(,)𝐵) ∪ {𝐴, 𝐵}) = (𝐴[,]𝐵))
2917, 28eleqtrrd 2841 . . . . . . . . 9 ((((𝜑𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ ¬ 𝑥 = 𝐵) → 𝑥 ∈ ((𝐴(,)𝐵) ∪ {𝐴, 𝐵}))
30 elun 4108 . . . . . . . . 9 (𝑥 ∈ ((𝐴(,)𝐵) ∪ {𝐴, 𝐵}) ↔ (𝑥 ∈ (𝐴(,)𝐵) ∨ 𝑥 ∈ {𝐴, 𝐵}))
3129, 30sylib 217 . . . . . . . 8 ((((𝜑𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ ¬ 𝑥 = 𝐵) → (𝑥 ∈ (𝐴(,)𝐵) ∨ 𝑥 ∈ {𝐴, 𝐵}))
32 orel2 889 . . . . . . . 8 𝑥 ∈ {𝐴, 𝐵} → ((𝑥 ∈ (𝐴(,)𝐵) ∨ 𝑥 ∈ {𝐴, 𝐵}) → 𝑥 ∈ (𝐴(,)𝐵)))
3316, 31, 32sylc 65 . . . . . . 7 ((((𝜑𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ ¬ 𝑥 = 𝐵) → 𝑥 ∈ (𝐴(,)𝐵))
34 cncfiooicclem1.f . . . . . . . . 9 (𝜑𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ))
35 cncff 24256 . . . . . . . . 9 (𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ) → 𝐹:(𝐴(,)𝐵)⟶ℂ)
3634, 35syl 17 . . . . . . . 8 (𝜑𝐹:(𝐴(,)𝐵)⟶ℂ)
3736ffvelcdmda 7035 . . . . . . 7 ((𝜑𝑥 ∈ (𝐴(,)𝐵)) → (𝐹𝑥) ∈ ℂ)
3810, 33, 37syl2anc 584 . . . . . 6 ((((𝜑𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ ¬ 𝑥 = 𝐵) → (𝐹𝑥) ∈ ℂ)
399, 38ifclda 4521 . . . . 5 (((𝜑𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) → if(𝑥 = 𝐵, 𝐿, (𝐹𝑥)) ∈ ℂ)
405, 39ifclda 4521 . . . 4 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹𝑥))) ∈ ℂ)
41 cncfiooicclem1.g . . . 4 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹𝑥))))
421, 40, 41fmptdf 7065 . . 3 (𝜑𝐺:(𝐴[,]𝐵)⟶ℂ)
43 elun 4108 . . . . . . 7 (𝑦 ∈ ((𝐴(,)𝐵) ∪ {𝐴, 𝐵}) ↔ (𝑦 ∈ (𝐴(,)𝐵) ∨ 𝑦 ∈ {𝐴, 𝐵}))
4419, 22, 25, 27syl3anc 1371 . . . . . . . 8 (𝜑 → ((𝐴(,)𝐵) ∪ {𝐴, 𝐵}) = (𝐴[,]𝐵))
4544eleq2d 2823 . . . . . . 7 (𝜑 → (𝑦 ∈ ((𝐴(,)𝐵) ∪ {𝐴, 𝐵}) ↔ 𝑦 ∈ (𝐴[,]𝐵)))
4643, 45bitr3id 284 . . . . . 6 (𝜑 → ((𝑦 ∈ (𝐴(,)𝐵) ∨ 𝑦 ∈ {𝐴, 𝐵}) ↔ 𝑦 ∈ (𝐴[,]𝐵)))
4746biimpar 478 . . . . 5 ((𝜑𝑦 ∈ (𝐴[,]𝐵)) → (𝑦 ∈ (𝐴(,)𝐵) ∨ 𝑦 ∈ {𝐴, 𝐵}))
48 ioossicc 13350 . . . . . . . . . . . . 13 (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
49 fssres 6708 . . . . . . . . . . . . 13 ((𝐺:(𝐴[,]𝐵)⟶ℂ ∧ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)) → (𝐺 ↾ (𝐴(,)𝐵)):(𝐴(,)𝐵)⟶ℂ)
5042, 48, 49sylancl 586 . . . . . . . . . . . 12 (𝜑 → (𝐺 ↾ (𝐴(,)𝐵)):(𝐴(,)𝐵)⟶ℂ)
5150feqmptd 6910 . . . . . . . . . . 11 (𝜑 → (𝐺 ↾ (𝐴(,)𝐵)) = (𝑦 ∈ (𝐴(,)𝐵) ↦ ((𝐺 ↾ (𝐴(,)𝐵))‘𝑦)))
52 nfmpt1 5213 . . . . . . . . . . . . . . . 16 𝑥(𝑥 ∈ (𝐴[,]𝐵) ↦ if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹𝑥))))
5341, 52nfcxfr 2905 . . . . . . . . . . . . . . 15 𝑥𝐺
54 nfcv 2907 . . . . . . . . . . . . . . 15 𝑥(𝐴(,)𝐵)
5553, 54nfres 5939 . . . . . . . . . . . . . 14 𝑥(𝐺 ↾ (𝐴(,)𝐵))
56 nfcv 2907 . . . . . . . . . . . . . 14 𝑥𝑦
5755, 56nffv 6852 . . . . . . . . . . . . 13 𝑥((𝐺 ↾ (𝐴(,)𝐵))‘𝑦)
58 nfcv 2907 . . . . . . . . . . . . . 14 𝑦(𝐺 ↾ (𝐴(,)𝐵))
59 nfcv 2907 . . . . . . . . . . . . . 14 𝑦𝑥
6058, 59nffv 6852 . . . . . . . . . . . . 13 𝑦((𝐺 ↾ (𝐴(,)𝐵))‘𝑥)
61 fveq2 6842 . . . . . . . . . . . . 13 (𝑦 = 𝑥 → ((𝐺 ↾ (𝐴(,)𝐵))‘𝑦) = ((𝐺 ↾ (𝐴(,)𝐵))‘𝑥))
6257, 60, 61cbvmpt 5216 . . . . . . . . . . . 12 (𝑦 ∈ (𝐴(,)𝐵) ↦ ((𝐺 ↾ (𝐴(,)𝐵))‘𝑦)) = (𝑥 ∈ (𝐴(,)𝐵) ↦ ((𝐺 ↾ (𝐴(,)𝐵))‘𝑥))
6362a1i 11 . . . . . . . . . . 11 (𝜑 → (𝑦 ∈ (𝐴(,)𝐵) ↦ ((𝐺 ↾ (𝐴(,)𝐵))‘𝑦)) = (𝑥 ∈ (𝐴(,)𝐵) ↦ ((𝐺 ↾ (𝐴(,)𝐵))‘𝑥)))
64 fvres 6861 . . . . . . . . . . . . . 14 (𝑥 ∈ (𝐴(,)𝐵) → ((𝐺 ↾ (𝐴(,)𝐵))‘𝑥) = (𝐺𝑥))
6564adantl 482 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝐴(,)𝐵)) → ((𝐺 ↾ (𝐴(,)𝐵))‘𝑥) = (𝐺𝑥))
66 simpr 485 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ (𝐴(,)𝐵))
6748, 66sselid 3942 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ (𝐴[,]𝐵))
684adantr 481 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ (𝐴(,)𝐵)) → 𝑅 ∈ ℂ)
698ad2antrr 724 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝐵) → 𝐿 ∈ ℂ)
7037adantr 481 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (𝐴(,)𝐵)) ∧ ¬ 𝑥 = 𝐵) → (𝐹𝑥) ∈ ℂ)
7169, 70ifclda 4521 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ (𝐴(,)𝐵)) → if(𝑥 = 𝐵, 𝐿, (𝐹𝑥)) ∈ ℂ)
7268, 71ifcld 4532 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (𝐴(,)𝐵)) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹𝑥))) ∈ ℂ)
7341fvmpt2 6959 . . . . . . . . . . . . . 14 ((𝑥 ∈ (𝐴[,]𝐵) ∧ if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹𝑥))) ∈ ℂ) → (𝐺𝑥) = if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹𝑥))))
7467, 72, 73syl2anc 584 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝐴(,)𝐵)) → (𝐺𝑥) = if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹𝑥))))
75 elioo4g 13324 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ (𝐴(,)𝐵) ↔ ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝑥 ∈ ℝ) ∧ (𝐴 < 𝑥𝑥 < 𝐵)))
7675biimpi 215 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ (𝐴(,)𝐵) → ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝑥 ∈ ℝ) ∧ (𝐴 < 𝑥𝑥 < 𝐵)))
7776simpld 495 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ (𝐴(,)𝐵) → (𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝑥 ∈ ℝ))
7877simp1d 1142 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ (𝐴(,)𝐵) → 𝐴 ∈ ℝ*)
79 elioore 13294 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ (𝐴(,)𝐵) → 𝑥 ∈ ℝ)
8079rexrd 11205 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ (𝐴(,)𝐵) → 𝑥 ∈ ℝ*)
81 eliooord 13323 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ (𝐴(,)𝐵) → (𝐴 < 𝑥𝑥 < 𝐵))
8281simpld 495 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ (𝐴(,)𝐵) → 𝐴 < 𝑥)
83 xrltne 13082 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ ℝ*𝑥 ∈ ℝ*𝐴 < 𝑥) → 𝑥𝐴)
8478, 80, 82, 83syl3anc 1371 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (𝐴(,)𝐵) → 𝑥𝐴)
8584adantl 482 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ (𝐴(,)𝐵)) → 𝑥𝐴)
8685neneqd 2948 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ (𝐴(,)𝐵)) → ¬ 𝑥 = 𝐴)
8786iffalsed 4497 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (𝐴(,)𝐵)) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹𝑥))) = if(𝑥 = 𝐵, 𝐿, (𝐹𝑥)))
8881simprd 496 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ (𝐴(,)𝐵) → 𝑥 < 𝐵)
8979, 88ltned 11291 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (𝐴(,)𝐵) → 𝑥𝐵)
9089neneqd 2948 . . . . . . . . . . . . . . . 16 (𝑥 ∈ (𝐴(,)𝐵) → ¬ 𝑥 = 𝐵)
9190iffalsed 4497 . . . . . . . . . . . . . . 15 (𝑥 ∈ (𝐴(,)𝐵) → if(𝑥 = 𝐵, 𝐿, (𝐹𝑥)) = (𝐹𝑥))
9291adantl 482 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (𝐴(,)𝐵)) → if(𝑥 = 𝐵, 𝐿, (𝐹𝑥)) = (𝐹𝑥))
9387, 92eqtrd 2776 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝐴(,)𝐵)) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹𝑥))) = (𝐹𝑥))
9465, 74, 933eqtrd 2780 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝐴(,)𝐵)) → ((𝐺 ↾ (𝐴(,)𝐵))‘𝑥) = (𝐹𝑥))
951, 94mpteq2da 5203 . . . . . . . . . . 11 (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ ((𝐺 ↾ (𝐴(,)𝐵))‘𝑥)) = (𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐹𝑥)))
9651, 63, 953eqtrd 2780 . . . . . . . . . 10 (𝜑 → (𝐺 ↾ (𝐴(,)𝐵)) = (𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐹𝑥)))
9736feqmptd 6910 . . . . . . . . . . 11 (𝜑𝐹 = (𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐹𝑥)))
98 ioosscn 13326 . . . . . . . . . . . . 13 (𝐴(,)𝐵) ⊆ ℂ
9998a1i 11 . . . . . . . . . . . 12 (𝜑 → (𝐴(,)𝐵) ⊆ ℂ)
100 ssid 3966 . . . . . . . . . . . 12 ℂ ⊆ ℂ
101 eqid 2736 . . . . . . . . . . . . 13 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
102 eqid 2736 . . . . . . . . . . . . 13 ((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) = ((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵))
103101cnfldtop 24147 . . . . . . . . . . . . . . 15 (TopOpen‘ℂfld) ∈ Top
104 unicntop 24149 . . . . . . . . . . . . . . . 16 ℂ = (TopOpen‘ℂfld)
105104restid 17315 . . . . . . . . . . . . . . 15 ((TopOpen‘ℂfld) ∈ Top → ((TopOpen‘ℂfld) ↾t ℂ) = (TopOpen‘ℂfld))
106103, 105ax-mp 5 . . . . . . . . . . . . . 14 ((TopOpen‘ℂfld) ↾t ℂ) = (TopOpen‘ℂfld)
107106eqcomi 2745 . . . . . . . . . . . . 13 (TopOpen‘ℂfld) = ((TopOpen‘ℂfld) ↾t ℂ)
108101, 102, 107cncfcn 24273 . . . . . . . . . . . 12 (((𝐴(,)𝐵) ⊆ ℂ ∧ ℂ ⊆ ℂ) → ((𝐴(,)𝐵)–cn→ℂ) = (((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) Cn (TopOpen‘ℂfld)))
10999, 100, 108sylancl 586 . . . . . . . . . . 11 (𝜑 → ((𝐴(,)𝐵)–cn→ℂ) = (((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) Cn (TopOpen‘ℂfld)))
11034, 97, 1093eltr3d 2852 . . . . . . . . . 10 (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐹𝑥)) ∈ (((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) Cn (TopOpen‘ℂfld)))
11196, 110eqeltrd 2838 . . . . . . . . 9 (𝜑 → (𝐺 ↾ (𝐴(,)𝐵)) ∈ (((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) Cn (TopOpen‘ℂfld)))
112104restuni 22513 . . . . . . . . . . 11 (((TopOpen‘ℂfld) ∈ Top ∧ (𝐴(,)𝐵) ⊆ ℂ) → (𝐴(,)𝐵) = ((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)))
113103, 98, 112mp2an 690 . . . . . . . . . 10 (𝐴(,)𝐵) = ((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵))
114113cncnpi 22629 . . . . . . . . 9 (((𝐺 ↾ (𝐴(,)𝐵)) ∈ (((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) Cn (TopOpen‘ℂfld)) ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (𝐺 ↾ (𝐴(,)𝐵)) ∈ ((((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) CnP (TopOpen‘ℂfld))‘𝑦))
115111, 114sylan 580 . . . . . . . 8 ((𝜑𝑦 ∈ (𝐴(,)𝐵)) → (𝐺 ↾ (𝐴(,)𝐵)) ∈ ((((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) CnP (TopOpen‘ℂfld))‘𝑦))
116103a1i 11 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ (𝐴(,)𝐵)) → (TopOpen‘ℂfld) ∈ Top)
11748a1i 11 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ (𝐴(,)𝐵)) → (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵))
118 ovex 7390 . . . . . . . . . . . . 13 (𝐴[,]𝐵) ∈ V
119118a1i 11 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ (𝐴(,)𝐵)) → (𝐴[,]𝐵) ∈ V)
120 restabs 22516 . . . . . . . . . . . 12 (((TopOpen‘ℂfld) ∈ Top ∧ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵) ∧ (𝐴[,]𝐵) ∈ V) → (((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) ↾t (𝐴(,)𝐵)) = ((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)))
121116, 117, 119, 120syl3anc 1371 . . . . . . . . . . 11 ((𝜑𝑦 ∈ (𝐴(,)𝐵)) → (((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) ↾t (𝐴(,)𝐵)) = ((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)))
122121eqcomd 2742 . . . . . . . . . 10 ((𝜑𝑦 ∈ (𝐴(,)𝐵)) → ((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) = (((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) ↾t (𝐴(,)𝐵)))
123122oveq1d 7372 . . . . . . . . 9 ((𝜑𝑦 ∈ (𝐴(,)𝐵)) → (((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) CnP (TopOpen‘ℂfld)) = ((((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) ↾t (𝐴(,)𝐵)) CnP (TopOpen‘ℂfld)))
124123fveq1d 6844 . . . . . . . 8 ((𝜑𝑦 ∈ (𝐴(,)𝐵)) → ((((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) CnP (TopOpen‘ℂfld))‘𝑦) = (((((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) ↾t (𝐴(,)𝐵)) CnP (TopOpen‘ℂfld))‘𝑦))
125115, 124eleqtrd 2840 . . . . . . 7 ((𝜑𝑦 ∈ (𝐴(,)𝐵)) → (𝐺 ↾ (𝐴(,)𝐵)) ∈ (((((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) ↾t (𝐴(,)𝐵)) CnP (TopOpen‘ℂfld))‘𝑦))
126 resttop 22511 . . . . . . . . . 10 (((TopOpen‘ℂfld) ∈ Top ∧ (𝐴[,]𝐵) ∈ V) → ((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) ∈ Top)
127103, 118, 126mp2an 690 . . . . . . . . 9 ((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) ∈ Top
128127a1i 11 . . . . . . . 8 ((𝜑𝑦 ∈ (𝐴(,)𝐵)) → ((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) ∈ Top)
12948a1i 11 . . . . . . . . . 10 (𝜑 → (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵))
13018, 21iccssred 13351 . . . . . . . . . . . 12 (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
131 ax-resscn 11108 . . . . . . . . . . . 12 ℝ ⊆ ℂ
132130, 131sstrdi 3956 . . . . . . . . . . 11 (𝜑 → (𝐴[,]𝐵) ⊆ ℂ)
133104restuni 22513 . . . . . . . . . . 11 (((TopOpen‘ℂfld) ∈ Top ∧ (𝐴[,]𝐵) ⊆ ℂ) → (𝐴[,]𝐵) = ((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)))
134103, 132, 133sylancr 587 . . . . . . . . . 10 (𝜑 → (𝐴[,]𝐵) = ((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)))
135129, 134sseqtrd 3984 . . . . . . . . 9 (𝜑 → (𝐴(,)𝐵) ⊆ ((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)))
136135adantr 481 . . . . . . . 8 ((𝜑𝑦 ∈ (𝐴(,)𝐵)) → (𝐴(,)𝐵) ⊆ ((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)))
137 retop 24125 . . . . . . . . . . . . . 14 (topGen‘ran (,)) ∈ Top
138137a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑦 ∈ (𝐴(,)𝐵)) → (topGen‘ran (,)) ∈ Top)
139 ioossre 13325 . . . . . . . . . . . . . . 15 (𝐴(,)𝐵) ⊆ ℝ
140 difss 4091 . . . . . . . . . . . . . . 15 (ℝ ∖ (𝐴[,]𝐵)) ⊆ ℝ
141139, 140unssi 4145 . . . . . . . . . . . . . 14 ((𝐴(,)𝐵) ∪ (ℝ ∖ (𝐴[,]𝐵))) ⊆ ℝ
142141a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑦 ∈ (𝐴(,)𝐵)) → ((𝐴(,)𝐵) ∪ (ℝ ∖ (𝐴[,]𝐵))) ⊆ ℝ)
143 ssun1 4132 . . . . . . . . . . . . . 14 (𝐴(,)𝐵) ⊆ ((𝐴(,)𝐵) ∪ (ℝ ∖ (𝐴[,]𝐵)))
144143a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑦 ∈ (𝐴(,)𝐵)) → (𝐴(,)𝐵) ⊆ ((𝐴(,)𝐵) ∪ (ℝ ∖ (𝐴[,]𝐵))))
145 uniretop 24126 . . . . . . . . . . . . . 14 ℝ = (topGen‘ran (,))
146145ntrss 22406 . . . . . . . . . . . . 13 (((topGen‘ran (,)) ∈ Top ∧ ((𝐴(,)𝐵) ∪ (ℝ ∖ (𝐴[,]𝐵))) ⊆ ℝ ∧ (𝐴(,)𝐵) ⊆ ((𝐴(,)𝐵) ∪ (ℝ ∖ (𝐴[,]𝐵)))) → ((int‘(topGen‘ran (,)))‘(𝐴(,)𝐵)) ⊆ ((int‘(topGen‘ran (,)))‘((𝐴(,)𝐵) ∪ (ℝ ∖ (𝐴[,]𝐵)))))
147138, 142, 144, 146syl3anc 1371 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ (𝐴(,)𝐵)) → ((int‘(topGen‘ran (,)))‘(𝐴(,)𝐵)) ⊆ ((int‘(topGen‘ran (,)))‘((𝐴(,)𝐵) ∪ (ℝ ∖ (𝐴[,]𝐵)))))
148 simpr 485 . . . . . . . . . . . . 13 ((𝜑𝑦 ∈ (𝐴(,)𝐵)) → 𝑦 ∈ (𝐴(,)𝐵))
149 ioontr 43739 . . . . . . . . . . . . 13 ((int‘(topGen‘ran (,)))‘(𝐴(,)𝐵)) = (𝐴(,)𝐵)
150148, 149eleqtrrdi 2849 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ (𝐴(,)𝐵)) → 𝑦 ∈ ((int‘(topGen‘ran (,)))‘(𝐴(,)𝐵)))
151147, 150sseldd 3945 . . . . . . . . . . 11 ((𝜑𝑦 ∈ (𝐴(,)𝐵)) → 𝑦 ∈ ((int‘(topGen‘ran (,)))‘((𝐴(,)𝐵) ∪ (ℝ ∖ (𝐴[,]𝐵)))))
15248, 148sselid 3942 . . . . . . . . . . 11 ((𝜑𝑦 ∈ (𝐴(,)𝐵)) → 𝑦 ∈ (𝐴[,]𝐵))
153151, 152elind 4154 . . . . . . . . . 10 ((𝜑𝑦 ∈ (𝐴(,)𝐵)) → 𝑦 ∈ (((int‘(topGen‘ran (,)))‘((𝐴(,)𝐵) ∪ (ℝ ∖ (𝐴[,]𝐵)))) ∩ (𝐴[,]𝐵)))
154130adantr 481 . . . . . . . . . . 11 ((𝜑𝑦 ∈ (𝐴(,)𝐵)) → (𝐴[,]𝐵) ⊆ ℝ)
155 eqid 2736 . . . . . . . . . . . 12 ((topGen‘ran (,)) ↾t (𝐴[,]𝐵)) = ((topGen‘ran (,)) ↾t (𝐴[,]𝐵))
156145, 155restntr 22533 . . . . . . . . . . 11 (((topGen‘ran (,)) ∈ Top ∧ (𝐴[,]𝐵) ⊆ ℝ ∧ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)) → ((int‘((topGen‘ran (,)) ↾t (𝐴[,]𝐵)))‘(𝐴(,)𝐵)) = (((int‘(topGen‘ran (,)))‘((𝐴(,)𝐵) ∪ (ℝ ∖ (𝐴[,]𝐵)))) ∩ (𝐴[,]𝐵)))
157138, 154, 117, 156syl3anc 1371 . . . . . . . . . 10 ((𝜑𝑦 ∈ (𝐴(,)𝐵)) → ((int‘((topGen‘ran (,)) ↾t (𝐴[,]𝐵)))‘(𝐴(,)𝐵)) = (((int‘(topGen‘ran (,)))‘((𝐴(,)𝐵) ∪ (ℝ ∖ (𝐴[,]𝐵)))) ∩ (𝐴[,]𝐵)))
158153, 157eleqtrrd 2841 . . . . . . . . 9 ((𝜑𝑦 ∈ (𝐴(,)𝐵)) → 𝑦 ∈ ((int‘((topGen‘ran (,)) ↾t (𝐴[,]𝐵)))‘(𝐴(,)𝐵)))
159101tgioo2 24166 . . . . . . . . . . . . . . 15 (topGen‘ran (,)) = ((TopOpen‘ℂfld) ↾t ℝ)
160159a1i 11 . . . . . . . . . . . . . 14 (𝜑 → (topGen‘ran (,)) = ((TopOpen‘ℂfld) ↾t ℝ))
161160oveq1d 7372 . . . . . . . . . . . . 13 (𝜑 → ((topGen‘ran (,)) ↾t (𝐴[,]𝐵)) = (((TopOpen‘ℂfld) ↾t ℝ) ↾t (𝐴[,]𝐵)))
162103a1i 11 . . . . . . . . . . . . . 14 (𝜑 → (TopOpen‘ℂfld) ∈ Top)
163 reex 11142 . . . . . . . . . . . . . . 15 ℝ ∈ V
164163a1i 11 . . . . . . . . . . . . . 14 (𝜑 → ℝ ∈ V)
165 restabs 22516 . . . . . . . . . . . . . 14 (((TopOpen‘ℂfld) ∈ Top ∧ (𝐴[,]𝐵) ⊆ ℝ ∧ ℝ ∈ V) → (((TopOpen‘ℂfld) ↾t ℝ) ↾t (𝐴[,]𝐵)) = ((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)))
166162, 130, 164, 165syl3anc 1371 . . . . . . . . . . . . 13 (𝜑 → (((TopOpen‘ℂfld) ↾t ℝ) ↾t (𝐴[,]𝐵)) = ((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)))
167161, 166eqtrd 2776 . . . . . . . . . . . 12 (𝜑 → ((topGen‘ran (,)) ↾t (𝐴[,]𝐵)) = ((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)))
168167fveq2d 6846 . . . . . . . . . . 11 (𝜑 → (int‘((topGen‘ran (,)) ↾t (𝐴[,]𝐵))) = (int‘((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵))))
169168fveq1d 6844 . . . . . . . . . 10 (𝜑 → ((int‘((topGen‘ran (,)) ↾t (𝐴[,]𝐵)))‘(𝐴(,)𝐵)) = ((int‘((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)))‘(𝐴(,)𝐵)))
170169adantr 481 . . . . . . . . 9 ((𝜑𝑦 ∈ (𝐴(,)𝐵)) → ((int‘((topGen‘ran (,)) ↾t (𝐴[,]𝐵)))‘(𝐴(,)𝐵)) = ((int‘((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)))‘(𝐴(,)𝐵)))
171158, 170eleqtrd 2840 . . . . . . . 8 ((𝜑𝑦 ∈ (𝐴(,)𝐵)) → 𝑦 ∈ ((int‘((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)))‘(𝐴(,)𝐵)))
172134feq2d 6654 . . . . . . . . . 10 (𝜑 → (𝐺:(𝐴[,]𝐵)⟶ℂ ↔ 𝐺: ((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵))⟶ℂ))
17342, 172mpbid 231 . . . . . . . . 9 (𝜑𝐺: ((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵))⟶ℂ)
174173adantr 481 . . . . . . . 8 ((𝜑𝑦 ∈ (𝐴(,)𝐵)) → 𝐺: ((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵))⟶ℂ)
175 eqid 2736 . . . . . . . . 9 ((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) = ((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵))
176175, 104cnprest 22640 . . . . . . . 8 (((((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) ∈ Top ∧ (𝐴(,)𝐵) ⊆ ((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵))) ∧ (𝑦 ∈ ((int‘((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)))‘(𝐴(,)𝐵)) ∧ 𝐺: ((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵))⟶ℂ)) → (𝐺 ∈ ((((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) CnP (TopOpen‘ℂfld))‘𝑦) ↔ (𝐺 ↾ (𝐴(,)𝐵)) ∈ (((((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) ↾t (𝐴(,)𝐵)) CnP (TopOpen‘ℂfld))‘𝑦)))
177128, 136, 171, 174, 176syl22anc 837 . . . . . . 7 ((𝜑𝑦 ∈ (𝐴(,)𝐵)) → (𝐺 ∈ ((((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) CnP (TopOpen‘ℂfld))‘𝑦) ↔ (𝐺 ↾ (𝐴(,)𝐵)) ∈ (((((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) ↾t (𝐴(,)𝐵)) CnP (TopOpen‘ℂfld))‘𝑦)))
178125, 177mpbird 256 . . . . . 6 ((𝜑𝑦 ∈ (𝐴(,)𝐵)) → 𝐺 ∈ ((((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) CnP (TopOpen‘ℂfld))‘𝑦))
179 elpri 4608 . . . . . . 7 (𝑦 ∈ {𝐴, 𝐵} → (𝑦 = 𝐴𝑦 = 𝐵))
180 iftrue 4492 . . . . . . . . . . . . 13 (𝑥 = 𝐴 → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹𝑥))) = 𝑅)
181 lbicc2 13381 . . . . . . . . . . . . . 14 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴𝐵) → 𝐴 ∈ (𝐴[,]𝐵))
18219, 22, 25, 181syl3anc 1371 . . . . . . . . . . . . 13 (𝜑𝐴 ∈ (𝐴[,]𝐵))
18341, 180, 182, 3fvmptd3 6971 . . . . . . . . . . . 12 (𝜑 → (𝐺𝐴) = 𝑅)
18497eqcomd 2742 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐹𝑥)) = 𝐹)
18596, 184eqtr2d 2777 . . . . . . . . . . . . . . 15 (𝜑𝐹 = (𝐺 ↾ (𝐴(,)𝐵)))
186185oveq1d 7372 . . . . . . . . . . . . . 14 (𝜑 → (𝐹 lim 𝐴) = ((𝐺 ↾ (𝐴(,)𝐵)) lim 𝐴))
1873, 186eleqtrd 2840 . . . . . . . . . . . . 13 (𝜑𝑅 ∈ ((𝐺 ↾ (𝐴(,)𝐵)) lim 𝐴))
18818, 21, 24, 42limciccioolb 43852 . . . . . . . . . . . . 13 (𝜑 → ((𝐺 ↾ (𝐴(,)𝐵)) lim 𝐴) = (𝐺 lim 𝐴))
189187, 188eleqtrd 2840 . . . . . . . . . . . 12 (𝜑𝑅 ∈ (𝐺 lim 𝐴))
190183, 189eqeltrd 2838 . . . . . . . . . . 11 (𝜑 → (𝐺𝐴) ∈ (𝐺 lim 𝐴))
191 eqid 2736 . . . . . . . . . . . . 13 ((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) = ((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵))
192101, 191cnplimc 25251 . . . . . . . . . . . 12 (((𝐴[,]𝐵) ⊆ ℂ ∧ 𝐴 ∈ (𝐴[,]𝐵)) → (𝐺 ∈ ((((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) CnP (TopOpen‘ℂfld))‘𝐴) ↔ (𝐺:(𝐴[,]𝐵)⟶ℂ ∧ (𝐺𝐴) ∈ (𝐺 lim 𝐴))))
193132, 182, 192syl2anc 584 . . . . . . . . . . 11 (𝜑 → (𝐺 ∈ ((((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) CnP (TopOpen‘ℂfld))‘𝐴) ↔ (𝐺:(𝐴[,]𝐵)⟶ℂ ∧ (𝐺𝐴) ∈ (𝐺 lim 𝐴))))
19442, 190, 193mpbir2and 711 . . . . . . . . . 10 (𝜑𝐺 ∈ ((((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) CnP (TopOpen‘ℂfld))‘𝐴))
195194adantr 481 . . . . . . . . 9 ((𝜑𝑦 = 𝐴) → 𝐺 ∈ ((((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) CnP (TopOpen‘ℂfld))‘𝐴))
196 fveq2 6842 . . . . . . . . . . 11 (𝑦 = 𝐴 → ((((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) CnP (TopOpen‘ℂfld))‘𝑦) = ((((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) CnP (TopOpen‘ℂfld))‘𝐴))
197196eqcomd 2742 . . . . . . . . . 10 (𝑦 = 𝐴 → ((((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) CnP (TopOpen‘ℂfld))‘𝐴) = ((((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) CnP (TopOpen‘ℂfld))‘𝑦))
198197adantl 482 . . . . . . . . 9 ((𝜑𝑦 = 𝐴) → ((((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) CnP (TopOpen‘ℂfld))‘𝐴) = ((((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) CnP (TopOpen‘ℂfld))‘𝑦))
199195, 198eleqtrd 2840 . . . . . . . 8 ((𝜑𝑦 = 𝐴) → 𝐺 ∈ ((((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) CnP (TopOpen‘ℂfld))‘𝑦))
200180adantl 482 . . . . . . . . . . . . . . . 16 ((𝑥 = 𝐵𝑥 = 𝐴) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹𝑥))) = 𝑅)
201 eqtr2 2760 . . . . . . . . . . . . . . . . 17 ((𝑥 = 𝐵𝑥 = 𝐴) → 𝐵 = 𝐴)
202 iftrue 4492 . . . . . . . . . . . . . . . . . 18 (𝐵 = 𝐴 → if(𝐵 = 𝐴, 𝑅, if(𝐵 = 𝐵, 𝐿, (𝐹𝐵))) = 𝑅)
203202eqcomd 2742 . . . . . . . . . . . . . . . . 17 (𝐵 = 𝐴𝑅 = if(𝐵 = 𝐴, 𝑅, if(𝐵 = 𝐵, 𝐿, (𝐹𝐵))))
204201, 203syl 17 . . . . . . . . . . . . . . . 16 ((𝑥 = 𝐵𝑥 = 𝐴) → 𝑅 = if(𝐵 = 𝐴, 𝑅, if(𝐵 = 𝐵, 𝐿, (𝐹𝐵))))
205200, 204eqtrd 2776 . . . . . . . . . . . . . . 15 ((𝑥 = 𝐵𝑥 = 𝐴) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹𝑥))) = if(𝐵 = 𝐴, 𝑅, if(𝐵 = 𝐵, 𝐿, (𝐹𝐵))))
206 iffalse 4495 . . . . . . . . . . . . . . . . 17 𝑥 = 𝐴 → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹𝑥))) = if(𝑥 = 𝐵, 𝐿, (𝐹𝑥)))
207206adantl 482 . . . . . . . . . . . . . . . 16 ((𝑥 = 𝐵 ∧ ¬ 𝑥 = 𝐴) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹𝑥))) = if(𝑥 = 𝐵, 𝐿, (𝐹𝑥)))
208 iftrue 4492 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝐵 → if(𝑥 = 𝐵, 𝐿, (𝐹𝑥)) = 𝐿)
209208adantr 481 . . . . . . . . . . . . . . . 16 ((𝑥 = 𝐵 ∧ ¬ 𝑥 = 𝐴) → if(𝑥 = 𝐵, 𝐿, (𝐹𝑥)) = 𝐿)
210 df-ne 2944 . . . . . . . . . . . . . . . . . . . 20 (𝑥𝐴 ↔ ¬ 𝑥 = 𝐴)
211 pm13.18 3025 . . . . . . . . . . . . . . . . . . . 20 ((𝑥 = 𝐵𝑥𝐴) → 𝐵𝐴)
212210, 211sylan2br 595 . . . . . . . . . . . . . . . . . . 19 ((𝑥 = 𝐵 ∧ ¬ 𝑥 = 𝐴) → 𝐵𝐴)
213212neneqd 2948 . . . . . . . . . . . . . . . . . 18 ((𝑥 = 𝐵 ∧ ¬ 𝑥 = 𝐴) → ¬ 𝐵 = 𝐴)
214213iffalsed 4497 . . . . . . . . . . . . . . . . 17 ((𝑥 = 𝐵 ∧ ¬ 𝑥 = 𝐴) → if(𝐵 = 𝐴, 𝑅, if(𝐵 = 𝐵, 𝐿, (𝐹𝐵))) = if(𝐵 = 𝐵, 𝐿, (𝐹𝐵)))
215 eqid 2736 . . . . . . . . . . . . . . . . . 18 𝐵 = 𝐵
216215iftruei 4493 . . . . . . . . . . . . . . . . 17 if(𝐵 = 𝐵, 𝐿, (𝐹𝐵)) = 𝐿
217214, 216eqtr2di 2793 . . . . . . . . . . . . . . . 16 ((𝑥 = 𝐵 ∧ ¬ 𝑥 = 𝐴) → 𝐿 = if(𝐵 = 𝐴, 𝑅, if(𝐵 = 𝐵, 𝐿, (𝐹𝐵))))
218207, 209, 2173eqtrd 2780 . . . . . . . . . . . . . . 15 ((𝑥 = 𝐵 ∧ ¬ 𝑥 = 𝐴) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹𝑥))) = if(𝐵 = 𝐴, 𝑅, if(𝐵 = 𝐵, 𝐿, (𝐹𝐵))))
219205, 218pm2.61dan 811 . . . . . . . . . . . . . 14 (𝑥 = 𝐵 → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹𝑥))) = if(𝐵 = 𝐴, 𝑅, if(𝐵 = 𝐵, 𝐿, (𝐹𝐵))))
22021leidd 11721 . . . . . . . . . . . . . . 15 (𝜑𝐵𝐵)
22118, 21, 21, 25, 220eliccd 43732 . . . . . . . . . . . . . 14 (𝜑𝐵 ∈ (𝐴[,]𝐵))
222216, 8eqeltrid 2842 . . . . . . . . . . . . . . 15 (𝜑 → if(𝐵 = 𝐵, 𝐿, (𝐹𝐵)) ∈ ℂ)
2234, 222ifcld 4532 . . . . . . . . . . . . . 14 (𝜑 → if(𝐵 = 𝐴, 𝑅, if(𝐵 = 𝐵, 𝐿, (𝐹𝐵))) ∈ ℂ)
22441, 219, 221, 223fvmptd3 6971 . . . . . . . . . . . . 13 (𝜑 → (𝐺𝐵) = if(𝐵 = 𝐴, 𝑅, if(𝐵 = 𝐵, 𝐿, (𝐹𝐵))))
22518, 24gtned 11290 . . . . . . . . . . . . . . 15 (𝜑𝐵𝐴)
226225neneqd 2948 . . . . . . . . . . . . . 14 (𝜑 → ¬ 𝐵 = 𝐴)
227226iffalsed 4497 . . . . . . . . . . . . 13 (𝜑 → if(𝐵 = 𝐴, 𝑅, if(𝐵 = 𝐵, 𝐿, (𝐹𝐵))) = if(𝐵 = 𝐵, 𝐿, (𝐹𝐵)))
228216a1i 11 . . . . . . . . . . . . 13 (𝜑 → if(𝐵 = 𝐵, 𝐿, (𝐹𝐵)) = 𝐿)
229224, 227, 2283eqtrd 2780 . . . . . . . . . . . 12 (𝜑 → (𝐺𝐵) = 𝐿)
230185oveq1d 7372 . . . . . . . . . . . . . 14 (𝜑 → (𝐹 lim 𝐵) = ((𝐺 ↾ (𝐴(,)𝐵)) lim 𝐵))
2317, 230eleqtrd 2840 . . . . . . . . . . . . 13 (𝜑𝐿 ∈ ((𝐺 ↾ (𝐴(,)𝐵)) lim 𝐵))
23218, 21, 24, 42limcicciooub 43868 . . . . . . . . . . . . 13 (𝜑 → ((𝐺 ↾ (𝐴(,)𝐵)) lim 𝐵) = (𝐺 lim 𝐵))
233231, 232eleqtrd 2840 . . . . . . . . . . . 12 (𝜑𝐿 ∈ (𝐺 lim 𝐵))
234229, 233eqeltrd 2838 . . . . . . . . . . 11 (𝜑 → (𝐺𝐵) ∈ (𝐺 lim 𝐵))
235101, 191cnplimc 25251 . . . . . . . . . . . 12 (((𝐴[,]𝐵) ⊆ ℂ ∧ 𝐵 ∈ (𝐴[,]𝐵)) → (𝐺 ∈ ((((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) CnP (TopOpen‘ℂfld))‘𝐵) ↔ (𝐺:(𝐴[,]𝐵)⟶ℂ ∧ (𝐺𝐵) ∈ (𝐺 lim 𝐵))))
236132, 221, 235syl2anc 584 . . . . . . . . . . 11 (𝜑 → (𝐺 ∈ ((((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) CnP (TopOpen‘ℂfld))‘𝐵) ↔ (𝐺:(𝐴[,]𝐵)⟶ℂ ∧ (𝐺𝐵) ∈ (𝐺 lim 𝐵))))
23742, 234, 236mpbir2and 711 . . . . . . . . . 10 (𝜑𝐺 ∈ ((((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) CnP (TopOpen‘ℂfld))‘𝐵))
238237adantr 481 . . . . . . . . 9 ((𝜑𝑦 = 𝐵) → 𝐺 ∈ ((((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) CnP (TopOpen‘ℂfld))‘𝐵))
239 fveq2 6842 . . . . . . . . . . 11 (𝑦 = 𝐵 → ((((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) CnP (TopOpen‘ℂfld))‘𝑦) = ((((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) CnP (TopOpen‘ℂfld))‘𝐵))
240239eqcomd 2742 . . . . . . . . . 10 (𝑦 = 𝐵 → ((((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) CnP (TopOpen‘ℂfld))‘𝐵) = ((((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) CnP (TopOpen‘ℂfld))‘𝑦))
241240adantl 482 . . . . . . . . 9 ((𝜑𝑦 = 𝐵) → ((((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) CnP (TopOpen‘ℂfld))‘𝐵) = ((((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) CnP (TopOpen‘ℂfld))‘𝑦))
242238, 241eleqtrd 2840 . . . . . . . 8 ((𝜑𝑦 = 𝐵) → 𝐺 ∈ ((((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) CnP (TopOpen‘ℂfld))‘𝑦))
243199, 242jaodan 956 . . . . . . 7 ((𝜑 ∧ (𝑦 = 𝐴𝑦 = 𝐵)) → 𝐺 ∈ ((((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) CnP (TopOpen‘ℂfld))‘𝑦))
244179, 243sylan2 593 . . . . . 6 ((𝜑𝑦 ∈ {𝐴, 𝐵}) → 𝐺 ∈ ((((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) CnP (TopOpen‘ℂfld))‘𝑦))
245178, 244jaodan 956 . . . . 5 ((𝜑 ∧ (𝑦 ∈ (𝐴(,)𝐵) ∨ 𝑦 ∈ {𝐴, 𝐵})) → 𝐺 ∈ ((((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) CnP (TopOpen‘ℂfld))‘𝑦))
24647, 245syldan 591 . . . 4 ((𝜑𝑦 ∈ (𝐴[,]𝐵)) → 𝐺 ∈ ((((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) CnP (TopOpen‘ℂfld))‘𝑦))
247246ralrimiva 3143 . . 3 (𝜑 → ∀𝑦 ∈ (𝐴[,]𝐵)𝐺 ∈ ((((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) CnP (TopOpen‘ℂfld))‘𝑦))
248101cnfldtopon 24146 . . . . 5 (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
249 resttopon 22512 . . . . 5 (((TopOpen‘ℂfld) ∈ (TopOn‘ℂ) ∧ (𝐴[,]𝐵) ⊆ ℂ) → ((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) ∈ (TopOn‘(𝐴[,]𝐵)))
250248, 132, 249sylancr 587 . . . 4 (𝜑 → ((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) ∈ (TopOn‘(𝐴[,]𝐵)))
251 cncnp 22631 . . . 4 ((((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) ∈ (TopOn‘(𝐴[,]𝐵)) ∧ (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)) → (𝐺 ∈ (((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) Cn (TopOpen‘ℂfld)) ↔ (𝐺:(𝐴[,]𝐵)⟶ℂ ∧ ∀𝑦 ∈ (𝐴[,]𝐵)𝐺 ∈ ((((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) CnP (TopOpen‘ℂfld))‘𝑦))))
252250, 248, 251sylancl 586 . . 3 (𝜑 → (𝐺 ∈ (((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) Cn (TopOpen‘ℂfld)) ↔ (𝐺:(𝐴[,]𝐵)⟶ℂ ∧ ∀𝑦 ∈ (𝐴[,]𝐵)𝐺 ∈ ((((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) CnP (TopOpen‘ℂfld))‘𝑦))))
25342, 247, 252mpbir2and 711 . 2 (𝜑𝐺 ∈ (((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) Cn (TopOpen‘ℂfld)))
254101, 191, 107cncfcn 24273 . . 3 (((𝐴[,]𝐵) ⊆ ℂ ∧ ℂ ⊆ ℂ) → ((𝐴[,]𝐵)–cn→ℂ) = (((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) Cn (TopOpen‘ℂfld)))
255132, 100, 254sylancl 586 . 2 (𝜑 → ((𝐴[,]𝐵)–cn→ℂ) = (((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) Cn (TopOpen‘ℂfld)))
256253, 255eleqtrrd 2841 1 (𝜑𝐺 ∈ ((𝐴[,]𝐵)–cn→ℂ))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  wo 845  w3a 1087   = wceq 1541  wnf 1785  wcel 2106  wne 2943  wral 3064  Vcvv 3445  cdif 3907  cun 3908  cin 3909  wss 3910  ifcif 4486  {cpr 4588   cuni 4865   class class class wbr 5105  cmpt 5188  ran crn 5634  cres 5635  wf 6492  cfv 6496  (class class class)co 7357  cc 11049  cr 11050  *cxr 11188   < clt 11189  cle 11190  (,)cioo 13264  [,]cicc 13267  t crest 17302  TopOpenctopn 17303  topGenctg 17319  fldccnfld 20796  Topctop 22242  TopOnctopon 22259  intcnt 22368   Cn ccn 22575   CnP ccnp 22576  cnccncf 24239   lim climc 25226
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-pm 8768  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fi 9347  df-sup 9378  df-inf 9379  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  df-uz 12764  df-q 12874  df-rp 12916  df-xneg 13033  df-xadd 13034  df-xmul 13035  df-ioo 13268  df-ioc 13269  df-ico 13270  df-icc 13271  df-fz 13425  df-seq 13907  df-exp 13968  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-struct 17019  df-slot 17054  df-ndx 17066  df-base 17084  df-plusg 17146  df-mulr 17147  df-starv 17148  df-tset 17152  df-ple 17153  df-ds 17155  df-unif 17156  df-rest 17304  df-topn 17305  df-topgen 17325  df-psmet 20788  df-xmet 20789  df-met 20790  df-bl 20791  df-mopn 20792  df-cnfld 20797  df-top 22243  df-topon 22260  df-topsp 22282  df-bases 22296  df-cld 22370  df-ntr 22371  df-cls 22372  df-cn 22578  df-cnp 22579  df-xms 23673  df-ms 23674  df-cncf 24241  df-limc 25230
This theorem is referenced by:  cncfiooicc  44125
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