Mathbox for Glauco Siliprandi < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cncfiooicclem1 Structured version   Visualization version   GIF version

Theorem cncfiooicclem1 40605
 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 23834 . . . . . . 7 (𝐹 lim 𝐴) ⊆ ℂ
3 cncfiooicclem1.r . . . . . . 7 (𝜑𝑅 ∈ (𝐹 lim 𝐴))
42, 3sseldi 3738 . . . . . 6 (𝜑𝑅 ∈ ℂ)
54ad2antrr 764 . . . . 5 (((𝜑𝑥 ∈ (𝐴[,]𝐵)) ∧ 𝑥 = 𝐴) → 𝑅 ∈ ℂ)
6 limccl 23834 . . . . . . . 8 (𝐹 lim 𝐵) ⊆ ℂ
7 cncfiooicclem1.l . . . . . . . 8 (𝜑𝐿 ∈ (𝐹 lim 𝐵))
86, 7sseldi 3738 . . . . . . 7 (𝜑𝐿 ∈ ℂ)
98ad3antrrr 768 . . . . . 6 ((((𝜑𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ 𝑥 = 𝐵) → 𝐿 ∈ ℂ)
10 simplll 815 . . . . . . 7 ((((𝜑𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ ¬ 𝑥 = 𝐵) → 𝜑)
11 orel1 396 . . . . . . . . . . 11 𝑥 = 𝐴 → ((𝑥 = 𝐴𝑥 = 𝐵) → 𝑥 = 𝐵))
1211con3dimp 456 . . . . . . . . . 10 ((¬ 𝑥 = 𝐴 ∧ ¬ 𝑥 = 𝐵) → ¬ (𝑥 = 𝐴𝑥 = 𝐵))
13 vex 3339 . . . . . . . . . . 11 𝑥 ∈ V
1413elpr 4339 . . . . . . . . . 10 (𝑥 ∈ {𝐴, 𝐵} ↔ (𝑥 = 𝐴𝑥 = 𝐵))
1512, 14sylnibr 318 . . . . . . . . 9 ((¬ 𝑥 = 𝐴 ∧ ¬ 𝑥 = 𝐵) → ¬ 𝑥 ∈ {𝐴, 𝐵})
1615adantll 752 . . . . . . . 8 ((((𝜑𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ ¬ 𝑥 = 𝐵) → ¬ 𝑥 ∈ {𝐴, 𝐵})
17 simpllr 817 . . . . . . . . . 10 ((((𝜑𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ ¬ 𝑥 = 𝐵) → 𝑥 ∈ (𝐴[,]𝐵))
18 cncfiooicclem1.a . . . . . . . . . . . . 13 (𝜑𝐴 ∈ ℝ)
1918rexrd 10277 . . . . . . . . . . . 12 (𝜑𝐴 ∈ ℝ*)
2010, 19syl 17 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ ¬ 𝑥 = 𝐵) → 𝐴 ∈ ℝ*)
21 cncfiooicclem1.b . . . . . . . . . . . . 13 (𝜑𝐵 ∈ ℝ)
2221rexrd 10277 . . . . . . . . . . . 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 10373 . . . . . . . . . . . 12 (𝜑𝐴𝐵)
2610, 25syl 17 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ ¬ 𝑥 = 𝐵) → 𝐴𝐵)
27 prunioo 12490 . . . . . . . . . . 11 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴𝐵) → ((𝐴(,)𝐵) ∪ {𝐴, 𝐵}) = (𝐴[,]𝐵))
2820, 23, 26, 27syl3anc 1477 . . . . . . . . . 10 ((((𝜑𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ ¬ 𝑥 = 𝐵) → ((𝐴(,)𝐵) ∪ {𝐴, 𝐵}) = (𝐴[,]𝐵))
2917, 28eleqtrrd 2838 . . . . . . . . 9 ((((𝜑𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ ¬ 𝑥 = 𝐵) → 𝑥 ∈ ((𝐴(,)𝐵) ∪ {𝐴, 𝐵}))
30 elun 3892 . . . . . . . . 9 (𝑥 ∈ ((𝐴(,)𝐵) ∪ {𝐴, 𝐵}) ↔ (𝑥 ∈ (𝐴(,)𝐵) ∨ 𝑥 ∈ {𝐴, 𝐵}))
3129, 30sylib 208 . . . . . . . 8 ((((𝜑𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ ¬ 𝑥 = 𝐵) → (𝑥 ∈ (𝐴(,)𝐵) ∨ 𝑥 ∈ {𝐴, 𝐵}))
32 orel2 397 . . . . . . . 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 22893 . . . . . . . . 9 (𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ) → 𝐹:(𝐴(,)𝐵)⟶ℂ)
3634, 35syl 17 . . . . . . . 8 (𝜑𝐹:(𝐴(,)𝐵)⟶ℂ)
3736ffvelrnda 6518 . . . . . . 7 ((𝜑𝑥 ∈ (𝐴(,)𝐵)) → (𝐹𝑥) ∈ ℂ)
3810, 33, 37syl2anc 696 . . . . . 6 ((((𝜑𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ ¬ 𝑥 = 𝐵) → (𝐹𝑥) ∈ ℂ)
399, 38ifclda 4260 . . . . 5 (((𝜑𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) → if(𝑥 = 𝐵, 𝐿, (𝐹𝑥)) ∈ ℂ)
405, 39ifclda 4260 . . . 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 6546 . . 3 (𝜑𝐺:(𝐴[,]𝐵)⟶ℂ)
43 elun 3892 . . . . . . 7 (𝑦 ∈ ((𝐴(,)𝐵) ∪ {𝐴, 𝐵}) ↔ (𝑦 ∈ (𝐴(,)𝐵) ∨ 𝑦 ∈ {𝐴, 𝐵}))
4419, 22, 25, 27syl3anc 1477 . . . . . . . 8 (𝜑 → ((𝐴(,)𝐵) ∪ {𝐴, 𝐵}) = (𝐴[,]𝐵))
4544eleq2d 2821 . . . . . . 7 (𝜑 → (𝑦 ∈ ((𝐴(,)𝐵) ∪ {𝐴, 𝐵}) ↔ 𝑦 ∈ (𝐴[,]𝐵)))
4643, 45syl5bbr 274 . . . . . 6 (𝜑 → ((𝑦 ∈ (𝐴(,)𝐵) ∨ 𝑦 ∈ {𝐴, 𝐵}) ↔ 𝑦 ∈ (𝐴[,]𝐵)))
4746biimpar 503 . . . . 5 ((𝜑𝑦 ∈ (𝐴[,]𝐵)) → (𝑦 ∈ (𝐴(,)𝐵) ∨ 𝑦 ∈ {𝐴, 𝐵}))
48 ioossicc 12448 . . . . . . . . . . . . 13 (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
49 fssres 6227 . . . . . . . . . . . . 13 ((𝐺:(𝐴[,]𝐵)⟶ℂ ∧ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)) → (𝐺 ↾ (𝐴(,)𝐵)):(𝐴(,)𝐵)⟶ℂ)
5042, 48, 49sylancl 697 . . . . . . . . . . . 12 (𝜑 → (𝐺 ↾ (𝐴(,)𝐵)):(𝐴(,)𝐵)⟶ℂ)
5150feqmptd 6407 . . . . . . . . . . 11 (𝜑 → (𝐺 ↾ (𝐴(,)𝐵)) = (𝑦 ∈ (𝐴(,)𝐵) ↦ ((𝐺 ↾ (𝐴(,)𝐵))‘𝑦)))
52 nfmpt1 4895 . . . . . . . . . . . . . . . 16 𝑥(𝑥 ∈ (𝐴[,]𝐵) ↦ if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹𝑥))))
5341, 52nfcxfr 2896 . . . . . . . . . . . . . . 15 𝑥𝐺
54 nfcv 2898 . . . . . . . . . . . . . . 15 𝑥(𝐴(,)𝐵)
5553, 54nfres 5549 . . . . . . . . . . . . . 14 𝑥(𝐺 ↾ (𝐴(,)𝐵))
56 nfcv 2898 . . . . . . . . . . . . . 14 𝑥𝑦
5755, 56nffv 6355 . . . . . . . . . . . . 13 𝑥((𝐺 ↾ (𝐴(,)𝐵))‘𝑦)
58 nfcv 2898 . . . . . . . . . . . . . 14 𝑦(𝐺 ↾ (𝐴(,)𝐵))
59 nfcv 2898 . . . . . . . . . . . . . 14 𝑦𝑥
6058, 59nffv 6355 . . . . . . . . . . . . 13 𝑦((𝐺 ↾ (𝐴(,)𝐵))‘𝑥)
61 fveq2 6348 . . . . . . . . . . . . 13 (𝑦 = 𝑥 → ((𝐺 ↾ (𝐴(,)𝐵))‘𝑦) = ((𝐺 ↾ (𝐴(,)𝐵))‘𝑥))
6257, 60, 61cbvmpt 4897 . . . . . . . . . . . 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 6364 . . . . . . . . . . . . . 14 (𝑥 ∈ (𝐴(,)𝐵) → ((𝐺 ↾ (𝐴(,)𝐵))‘𝑥) = (𝐺𝑥))
6564adantl 473 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝐴(,)𝐵)) → ((𝐺 ↾ (𝐴(,)𝐵))‘𝑥) = (𝐺𝑥))
66 simpr 479 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ (𝐴(,)𝐵))
6748, 66sseldi 3738 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ (𝐴[,]𝐵))
684adantr 472 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ (𝐴(,)𝐵)) → 𝑅 ∈ ℂ)
698ad2antrr 764 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝐵) → 𝐿 ∈ ℂ)
7037adantr 472 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (𝐴(,)𝐵)) ∧ ¬ 𝑥 = 𝐵) → (𝐹𝑥) ∈ ℂ)
7169, 70ifclda 4260 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ (𝐴(,)𝐵)) → if(𝑥 = 𝐵, 𝐿, (𝐹𝑥)) ∈ ℂ)
7268, 71ifcld 4271 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (𝐴(,)𝐵)) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹𝑥))) ∈ ℂ)
7341fvmpt2 6449 . . . . . . . . . . . . . 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 696 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝐴(,)𝐵)) → (𝐺𝑥) = if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹𝑥))))
75 elioo4g 12423 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ (𝐴(,)𝐵) ↔ ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝑥 ∈ ℝ) ∧ (𝐴 < 𝑥𝑥 < 𝐵)))
7675biimpi 206 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ (𝐴(,)𝐵) → ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝑥 ∈ ℝ) ∧ (𝐴 < 𝑥𝑥 < 𝐵)))
7776simpld 477 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ (𝐴(,)𝐵) → (𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝑥 ∈ ℝ))
7877simp1d 1137 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ (𝐴(,)𝐵) → 𝐴 ∈ ℝ*)
79 elioore 12394 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ (𝐴(,)𝐵) → 𝑥 ∈ ℝ)
8079rexrd 10277 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ (𝐴(,)𝐵) → 𝑥 ∈ ℝ*)
81 eliooord 12422 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ (𝐴(,)𝐵) → (𝐴 < 𝑥𝑥 < 𝐵))
8281simpld 477 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ (𝐴(,)𝐵) → 𝐴 < 𝑥)
83 xrltne 12183 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ ℝ*𝑥 ∈ ℝ*𝐴 < 𝑥) → 𝑥𝐴)
8478, 80, 82, 83syl3anc 1477 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (𝐴(,)𝐵) → 𝑥𝐴)
8584adantl 473 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ (𝐴(,)𝐵)) → 𝑥𝐴)
8685neneqd 2933 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ (𝐴(,)𝐵)) → ¬ 𝑥 = 𝐴)
8786iffalsed 4237 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (𝐴(,)𝐵)) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹𝑥))) = if(𝑥 = 𝐵, 𝐿, (𝐹𝑥)))
8881simprd 482 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ (𝐴(,)𝐵) → 𝑥 < 𝐵)
8979, 88ltned 10361 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (𝐴(,)𝐵) → 𝑥𝐵)
9089neneqd 2933 . . . . . . . . . . . . . . . 16 (𝑥 ∈ (𝐴(,)𝐵) → ¬ 𝑥 = 𝐵)
9190iffalsed 4237 . . . . . . . . . . . . . . 15 (𝑥 ∈ (𝐴(,)𝐵) → if(𝑥 = 𝐵, 𝐿, (𝐹𝑥)) = (𝐹𝑥))
9291adantl 473 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (𝐴(,)𝐵)) → if(𝑥 = 𝐵, 𝐿, (𝐹𝑥)) = (𝐹𝑥))
9387, 92eqtrd 2790 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝐴(,)𝐵)) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹𝑥))) = (𝐹𝑥))
9465, 74, 933eqtrd 2794 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝐴(,)𝐵)) → ((𝐺 ↾ (𝐴(,)𝐵))‘𝑥) = (𝐹𝑥))
951, 94mpteq2da 4891 . . . . . . . . . . 11 (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ ((𝐺 ↾ (𝐴(,)𝐵))‘𝑥)) = (𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐹𝑥)))
9651, 63, 953eqtrd 2794 . . . . . . . . . 10 (𝜑 → (𝐺 ↾ (𝐴(,)𝐵)) = (𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐹𝑥)))
9736feqmptd 6407 . . . . . . . . . . 11 (𝜑𝐹 = (𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐹𝑥)))
98 ioosscn 40215 . . . . . . . . . . . . 13 (𝐴(,)𝐵) ⊆ ℂ
9998a1i 11 . . . . . . . . . . . 12 (𝜑 → (𝐴(,)𝐵) ⊆ ℂ)
100 ssid 3761 . . . . . . . . . . . 12 ℂ ⊆ ℂ
101 eqid 2756 . . . . . . . . . . . . 13 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
102 eqid 2756 . . . . . . . . . . . . 13 ((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) = ((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵))
103101cnfldtop 22784 . . . . . . . . . . . . . . 15 (TopOpen‘ℂfld) ∈ Top
104 unicntop 22786 . . . . . . . . . . . . . . . 16 ℂ = (TopOpen‘ℂfld)
105104restid 16292 . . . . . . . . . . . . . . 15 ((TopOpen‘ℂfld) ∈ Top → ((TopOpen‘ℂfld) ↾t ℂ) = (TopOpen‘ℂfld))
106103, 105ax-mp 5 . . . . . . . . . . . . . 14 ((TopOpen‘ℂfld) ↾t ℂ) = (TopOpen‘ℂfld)
107106eqcomi 2765 . . . . . . . . . . . . 13 (TopOpen‘ℂfld) = ((TopOpen‘ℂfld) ↾t ℂ)
108101, 102, 107cncfcn 22909 . . . . . . . . . . . 12 (((𝐴(,)𝐵) ⊆ ℂ ∧ ℂ ⊆ ℂ) → ((𝐴(,)𝐵)–cn→ℂ) = (((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) Cn (TopOpen‘ℂfld)))
10999, 100, 108sylancl 697 . . . . . . . . . . 11 (𝜑 → ((𝐴(,)𝐵)–cn→ℂ) = (((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) Cn (TopOpen‘ℂfld)))
11034, 97, 1093eltr3d 2849 . . . . . . . . . 10 (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐹𝑥)) ∈ (((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) Cn (TopOpen‘ℂfld)))
11196, 110eqeltrd 2835 . . . . . . . . 9 (𝜑 → (𝐺 ↾ (𝐴(,)𝐵)) ∈ (((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) Cn (TopOpen‘ℂfld)))
112104restuni 21164 . . . . . . . . . . 11 (((TopOpen‘ℂfld) ∈ Top ∧ (𝐴(,)𝐵) ⊆ ℂ) → (𝐴(,)𝐵) = ((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)))
113103, 98, 112mp2an 710 . . . . . . . . . 10 (𝐴(,)𝐵) = ((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵))
114113cncnpi 21280 . . . . . . . . 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 489 . . . . . . . 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 6837 . . . . . . . . . . . . 13 (𝐴[,]𝐵) ∈ V
119118a1i 11 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ (𝐴(,)𝐵)) → (𝐴[,]𝐵) ∈ V)
120 restabs 21167 . . . . . . . . . . . 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 1477 . . . . . . . . . . 11 ((𝜑𝑦 ∈ (𝐴(,)𝐵)) → (((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) ↾t (𝐴(,)𝐵)) = ((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)))
122121eqcomd 2762 . . . . . . . . . 10 ((𝜑𝑦 ∈ (𝐴(,)𝐵)) → ((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) = (((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) ↾t (𝐴(,)𝐵)))
123122oveq1d 6824 . . . . . . . . 9 ((𝜑𝑦 ∈ (𝐴(,)𝐵)) → (((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) CnP (TopOpen‘ℂfld)) = ((((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) ↾t (𝐴(,)𝐵)) CnP (TopOpen‘ℂfld)))
124123fveq1d 6350 . . . . . . . 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 2837 . . . . . . 7 ((𝜑𝑦 ∈ (𝐴(,)𝐵)) → (𝐺 ↾ (𝐴(,)𝐵)) ∈ (((((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) ↾t (𝐴(,)𝐵)) CnP (TopOpen‘ℂfld))‘𝑦))
126 resttop 21162 . . . . . . . . . 10 (((TopOpen‘ℂfld) ∈ Top ∧ (𝐴[,]𝐵) ∈ V) → ((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) ∈ Top)
127103, 118, 126mp2an 710 . . . . . . . . 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 40226 . . . . . . . . . . . 12 (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
131 ax-resscn 10181 . . . . . . . . . . . 12 ℝ ⊆ ℂ
132130, 131syl6ss 3752 . . . . . . . . . . 11 (𝜑 → (𝐴[,]𝐵) ⊆ ℂ)
133104restuni 21164 . . . . . . . . . . 11 (((TopOpen‘ℂfld) ∈ Top ∧ (𝐴[,]𝐵) ⊆ ℂ) → (𝐴[,]𝐵) = ((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)))
134103, 132, 133sylancr 698 . . . . . . . . . 10 (𝜑 → (𝐴[,]𝐵) = ((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)))
135129, 134sseqtrd 3778 . . . . . . . . 9 (𝜑 → (𝐴(,)𝐵) ⊆ ((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)))
136135adantr 472 . . . . . . . 8 ((𝜑𝑦 ∈ (𝐴(,)𝐵)) → (𝐴(,)𝐵) ⊆ ((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)))
137 retop 22762 . . . . . . . . . . . . . 14 (topGen‘ran (,)) ∈ Top
138137a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑦 ∈ (𝐴(,)𝐵)) → (topGen‘ran (,)) ∈ Top)
139 ioossre 12424 . . . . . . . . . . . . . . 15 (𝐴(,)𝐵) ⊆ ℝ
140 difss 3876 . . . . . . . . . . . . . . 15 (ℝ ∖ (𝐴[,]𝐵)) ⊆ ℝ
141139, 140unssi 3927 . . . . . . . . . . . . . 14 ((𝐴(,)𝐵) ∪ (ℝ ∖ (𝐴[,]𝐵))) ⊆ ℝ
142141a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑦 ∈ (𝐴(,)𝐵)) → ((𝐴(,)𝐵) ∪ (ℝ ∖ (𝐴[,]𝐵))) ⊆ ℝ)
143 ssun1 3915 . . . . . . . . . . . . . 14 (𝐴(,)𝐵) ⊆ ((𝐴(,)𝐵) ∪ (ℝ ∖ (𝐴[,]𝐵)))
144143a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑦 ∈ (𝐴(,)𝐵)) → (𝐴(,)𝐵) ⊆ ((𝐴(,)𝐵) ∪ (ℝ ∖ (𝐴[,]𝐵))))
145 uniretop 22763 . . . . . . . . . . . . . 14 ℝ = (topGen‘ran (,))
146145ntrss 21057 . . . . . . . . . . . . 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 1477 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ (𝐴(,)𝐵)) → ((int‘(topGen‘ran (,)))‘(𝐴(,)𝐵)) ⊆ ((int‘(topGen‘ran (,)))‘((𝐴(,)𝐵) ∪ (ℝ ∖ (𝐴[,]𝐵)))))
148 simpr 479 . . . . . . . . . . . . 13 ((𝜑𝑦 ∈ (𝐴(,)𝐵)) → 𝑦 ∈ (𝐴(,)𝐵))
149 ioontr 40235 . . . . . . . . . . . . 13 ((int‘(topGen‘ran (,)))‘(𝐴(,)𝐵)) = (𝐴(,)𝐵)
150148, 149syl6eleqr 2846 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ (𝐴(,)𝐵)) → 𝑦 ∈ ((int‘(topGen‘ran (,)))‘(𝐴(,)𝐵)))
151147, 150sseldd 3741 . . . . . . . . . . 11 ((𝜑𝑦 ∈ (𝐴(,)𝐵)) → 𝑦 ∈ ((int‘(topGen‘ran (,)))‘((𝐴(,)𝐵) ∪ (ℝ ∖ (𝐴[,]𝐵)))))
15248, 148sseldi 3738 . . . . . . . . . . 11 ((𝜑𝑦 ∈ (𝐴(,)𝐵)) → 𝑦 ∈ (𝐴[,]𝐵))
153151, 152elind 3937 . . . . . . . . . 10 ((𝜑𝑦 ∈ (𝐴(,)𝐵)) → 𝑦 ∈ (((int‘(topGen‘ran (,)))‘((𝐴(,)𝐵) ∪ (ℝ ∖ (𝐴[,]𝐵)))) ∩ (𝐴[,]𝐵)))
154130adantr 472 . . . . . . . . . . 11 ((𝜑𝑦 ∈ (𝐴(,)𝐵)) → (𝐴[,]𝐵) ⊆ ℝ)
155 eqid 2756 . . . . . . . . . . . 12 ((topGen‘ran (,)) ↾t (𝐴[,]𝐵)) = ((topGen‘ran (,)) ↾t (𝐴[,]𝐵))
156145, 155restntr 21184 . . . . . . . . . . 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 1477 . . . . . . . . . 10 ((𝜑𝑦 ∈ (𝐴(,)𝐵)) → ((int‘((topGen‘ran (,)) ↾t (𝐴[,]𝐵)))‘(𝐴(,)𝐵)) = (((int‘(topGen‘ran (,)))‘((𝐴(,)𝐵) ∪ (ℝ ∖ (𝐴[,]𝐵)))) ∩ (𝐴[,]𝐵)))
158153, 157eleqtrrd 2838 . . . . . . . . 9 ((𝜑𝑦 ∈ (𝐴(,)𝐵)) → 𝑦 ∈ ((int‘((topGen‘ran (,)) ↾t (𝐴[,]𝐵)))‘(𝐴(,)𝐵)))
159101tgioo2 22803 . . . . . . . . . . . . . . 15 (topGen‘ran (,)) = ((TopOpen‘ℂfld) ↾t ℝ)
160159a1i 11 . . . . . . . . . . . . . 14 (𝜑 → (topGen‘ran (,)) = ((TopOpen‘ℂfld) ↾t ℝ))
161160oveq1d 6824 . . . . . . . . . . . . 13 (𝜑 → ((topGen‘ran (,)) ↾t (𝐴[,]𝐵)) = (((TopOpen‘ℂfld) ↾t ℝ) ↾t (𝐴[,]𝐵)))
162103a1i 11 . . . . . . . . . . . . . 14 (𝜑 → (TopOpen‘ℂfld) ∈ Top)
163 reex 10215 . . . . . . . . . . . . . . 15 ℝ ∈ V
164163a1i 11 . . . . . . . . . . . . . 14 (𝜑 → ℝ ∈ V)
165 restabs 21167 . . . . . . . . . . . . . 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 1477 . . . . . . . . . . . . 13 (𝜑 → (((TopOpen‘ℂfld) ↾t ℝ) ↾t (𝐴[,]𝐵)) = ((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)))
167161, 166eqtrd 2790 . . . . . . . . . . . 12 (𝜑 → ((topGen‘ran (,)) ↾t (𝐴[,]𝐵)) = ((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)))
168167fveq2d 6352 . . . . . . . . . . 11 (𝜑 → (int‘((topGen‘ran (,)) ↾t (𝐴[,]𝐵))) = (int‘((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵))))
169168fveq1d 6350 . . . . . . . . . 10 (𝜑 → ((int‘((topGen‘ran (,)) ↾t (𝐴[,]𝐵)))‘(𝐴(,)𝐵)) = ((int‘((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)))‘(𝐴(,)𝐵)))
170169adantr 472 . . . . . . . . 9 ((𝜑𝑦 ∈ (𝐴(,)𝐵)) → ((int‘((topGen‘ran (,)) ↾t (𝐴[,]𝐵)))‘(𝐴(,)𝐵)) = ((int‘((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)))‘(𝐴(,)𝐵)))
171158, 170eleqtrd 2837 . . . . . . . 8 ((𝜑𝑦 ∈ (𝐴(,)𝐵)) → 𝑦 ∈ ((int‘((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)))‘(𝐴(,)𝐵)))
172134feq2d 6188 . . . . . . . . . 10 (𝜑 → (𝐺:(𝐴[,]𝐵)⟶ℂ ↔ 𝐺: ((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵))⟶ℂ))
17342, 172mpbid 222 . . . . . . . . 9 (𝜑𝐺: ((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵))⟶ℂ)
174173adantr 472 . . . . . . . 8 ((𝜑𝑦 ∈ (𝐴(,)𝐵)) → 𝐺: ((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵))⟶ℂ)
175 eqid 2756 . . . . . . . . 9 ((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) = ((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵))
176175, 104cnprest 21291 . . . . . . . 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 1478 . . . . . . 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 247 . . . . . 6 ((𝜑𝑦 ∈ (𝐴(,)𝐵)) → 𝐺 ∈ ((((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) CnP (TopOpen‘ℂfld))‘𝑦))
179 elpri 4338 . . . . . . 7 (𝑦 ∈ {𝐴, 𝐵} → (𝑦 = 𝐴𝑦 = 𝐵))
180 lbicc2 12477 . . . . . . . . . . . . . 14 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴𝐵) → 𝐴 ∈ (𝐴[,]𝐵))
18119, 22, 25, 180syl3anc 1477 . . . . . . . . . . . . 13 (𝜑𝐴 ∈ (𝐴[,]𝐵))
182 iftrue 4232 . . . . . . . . . . . . . 14 (𝑥 = 𝐴 → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹𝑥))) = 𝑅)
183182, 41fvmptg 6438 . . . . . . . . . . . . 13 ((𝐴 ∈ (𝐴[,]𝐵) ∧ 𝑅 ∈ (𝐹 lim 𝐴)) → (𝐺𝐴) = 𝑅)
184181, 3, 183syl2anc 696 . . . . . . . . . . . 12 (𝜑 → (𝐺𝐴) = 𝑅)
18597eqcomd 2762 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐹𝑥)) = 𝐹)
18696, 185eqtr2d 2791 . . . . . . . . . . . . . . 15 (𝜑𝐹 = (𝐺 ↾ (𝐴(,)𝐵)))
187186oveq1d 6824 . . . . . . . . . . . . . 14 (𝜑 → (𝐹 lim 𝐴) = ((𝐺 ↾ (𝐴(,)𝐵)) lim 𝐴))
1883, 187eleqtrd 2837 . . . . . . . . . . . . 13 (𝜑𝑅 ∈ ((𝐺 ↾ (𝐴(,)𝐵)) lim 𝐴))
18918, 21, 24, 42limciccioolb 40352 . . . . . . . . . . . . 13 (𝜑 → ((𝐺 ↾ (𝐴(,)𝐵)) lim 𝐴) = (𝐺 lim 𝐴))
190188, 189eleqtrd 2837 . . . . . . . . . . . 12 (𝜑𝑅 ∈ (𝐺 lim 𝐴))
191184, 190eqeltrd 2835 . . . . . . . . . . 11 (𝜑 → (𝐺𝐴) ∈ (𝐺 lim 𝐴))
192 eqid 2756 . . . . . . . . . . . . 13 ((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) = ((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵))
193101, 192cnplimc 23846 . . . . . . . . . . . 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))))
194132, 181, 193syl2anc 696 . . . . . . . . . . 11 (𝜑 → (𝐺 ∈ ((((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) CnP (TopOpen‘ℂfld))‘𝐴) ↔ (𝐺:(𝐴[,]𝐵)⟶ℂ ∧ (𝐺𝐴) ∈ (𝐺 lim 𝐴))))
19542, 191, 194mpbir2and 995 . . . . . . . . . 10 (𝜑𝐺 ∈ ((((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) CnP (TopOpen‘ℂfld))‘𝐴))
196195adantr 472 . . . . . . . . 9 ((𝜑𝑦 = 𝐴) → 𝐺 ∈ ((((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) CnP (TopOpen‘ℂfld))‘𝐴))
197 fveq2 6348 . . . . . . . . . . 11 (𝑦 = 𝐴 → ((((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) CnP (TopOpen‘ℂfld))‘𝑦) = ((((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) CnP (TopOpen‘ℂfld))‘𝐴))
198197eqcomd 2762 . . . . . . . . . 10 (𝑦 = 𝐴 → ((((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) CnP (TopOpen‘ℂfld))‘𝐴) = ((((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) CnP (TopOpen‘ℂfld))‘𝑦))
199198adantl 473 . . . . . . . . 9 ((𝜑𝑦 = 𝐴) → ((((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) CnP (TopOpen‘ℂfld))‘𝐴) = ((((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) CnP (TopOpen‘ℂfld))‘𝑦))
200196, 199eleqtrd 2837 . . . . . . . 8 ((𝜑𝑦 = 𝐴) → 𝐺 ∈ ((((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) CnP (TopOpen‘ℂfld))‘𝑦))
20121leidd 10782 . . . . . . . . . . . . . . 15 (𝜑𝐵𝐵)
20218, 21, 21, 25, 201eliccd 40225 . . . . . . . . . . . . . 14 (𝜑𝐵 ∈ (𝐴[,]𝐵))
203 eqid 2756 . . . . . . . . . . . . . . . . . 18 𝐵 = 𝐵
204203iftruei 4233 . . . . . . . . . . . . . . . . 17 if(𝐵 = 𝐵, 𝐿, (𝐹𝐵)) = 𝐿
205204, 8syl5eqel 2839 . . . . . . . . . . . . . . . 16 (𝜑 → if(𝐵 = 𝐵, 𝐿, (𝐹𝐵)) ∈ ℂ)
2064, 205ifcld 4271 . . . . . . . . . . . . . . 15 (𝜑 → if(𝐵 = 𝐴, 𝑅, if(𝐵 = 𝐵, 𝐿, (𝐹𝐵))) ∈ ℂ)
207202, 206jca 555 . . . . . . . . . . . . . 14 (𝜑 → (𝐵 ∈ (𝐴[,]𝐵) ∧ if(𝐵 = 𝐴, 𝑅, if(𝐵 = 𝐵, 𝐿, (𝐹𝐵))) ∈ ℂ))
208 nfv 1988 . . . . . . . . . . . . . . . 16 𝑥(𝐵 ∈ (𝐴[,]𝐵) ∧ if(𝐵 = 𝐴, 𝑅, if(𝐵 = 𝐵, 𝐿, (𝐹𝐵))) ∈ ℂ)
209 nfcv 2898 . . . . . . . . . . . . . . . . . 18 𝑥𝐵
21053, 209nffv 6355 . . . . . . . . . . . . . . . . 17 𝑥(𝐺𝐵)
211210nfeq1 2912 . . . . . . . . . . . . . . . 16 𝑥(𝐺𝐵) = if(𝐵 = 𝐴, 𝑅, if(𝐵 = 𝐵, 𝐿, (𝐹𝐵)))
212208, 211nfim 1970 . . . . . . . . . . . . . . 15 𝑥((𝐵 ∈ (𝐴[,]𝐵) ∧ if(𝐵 = 𝐴, 𝑅, if(𝐵 = 𝐵, 𝐿, (𝐹𝐵))) ∈ ℂ) → (𝐺𝐵) = if(𝐵 = 𝐴, 𝑅, if(𝐵 = 𝐵, 𝐿, (𝐹𝐵))))
213 eleq1 2823 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝐵 → (𝑥 ∈ (𝐴[,]𝐵) ↔ 𝐵 ∈ (𝐴[,]𝐵)))
214182adantl 473 . . . . . . . . . . . . . . . . . . . 20 ((𝑥 = 𝐵𝑥 = 𝐴) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹𝑥))) = 𝑅)
215 eqtr2 2776 . . . . . . . . . . . . . . . . . . . . 21 ((𝑥 = 𝐵𝑥 = 𝐴) → 𝐵 = 𝐴)
216 iftrue 4232 . . . . . . . . . . . . . . . . . . . . . 22 (𝐵 = 𝐴 → if(𝐵 = 𝐴, 𝑅, if(𝐵 = 𝐵, 𝐿, (𝐹𝐵))) = 𝑅)
217216eqcomd 2762 . . . . . . . . . . . . . . . . . . . . 21 (𝐵 = 𝐴𝑅 = if(𝐵 = 𝐴, 𝑅, if(𝐵 = 𝐵, 𝐿, (𝐹𝐵))))
218215, 217syl 17 . . . . . . . . . . . . . . . . . . . 20 ((𝑥 = 𝐵𝑥 = 𝐴) → 𝑅 = if(𝐵 = 𝐴, 𝑅, if(𝐵 = 𝐵, 𝐿, (𝐹𝐵))))
219214, 218eqtrd 2790 . . . . . . . . . . . . . . . . . . 19 ((𝑥 = 𝐵𝑥 = 𝐴) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹𝑥))) = if(𝐵 = 𝐴, 𝑅, if(𝐵 = 𝐵, 𝐿, (𝐹𝐵))))
220 iffalse 4235 . . . . . . . . . . . . . . . . . . . . 21 𝑥 = 𝐴 → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹𝑥))) = if(𝑥 = 𝐵, 𝐿, (𝐹𝑥)))
221220adantl 473 . . . . . . . . . . . . . . . . . . . 20 ((𝑥 = 𝐵 ∧ ¬ 𝑥 = 𝐴) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹𝑥))) = if(𝑥 = 𝐵, 𝐿, (𝐹𝑥)))
222 iftrue 4232 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝐵 → if(𝑥 = 𝐵, 𝐿, (𝐹𝑥)) = 𝐿)
223222adantr 472 . . . . . . . . . . . . . . . . . . . 20 ((𝑥 = 𝐵 ∧ ¬ 𝑥 = 𝐴) → if(𝑥 = 𝐵, 𝐿, (𝐹𝑥)) = 𝐿)
224 df-ne 2929 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥𝐴 ↔ ¬ 𝑥 = 𝐴)
225 pm13.18 3009 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑥 = 𝐵𝑥𝐴) → 𝐵𝐴)
226224, 225sylan2br 494 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑥 = 𝐵 ∧ ¬ 𝑥 = 𝐴) → 𝐵𝐴)
227226neneqd 2933 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑥 = 𝐵 ∧ ¬ 𝑥 = 𝐴) → ¬ 𝐵 = 𝐴)
228227iffalsed 4237 . . . . . . . . . . . . . . . . . . . . 21 ((𝑥 = 𝐵 ∧ ¬ 𝑥 = 𝐴) → if(𝐵 = 𝐴, 𝑅, if(𝐵 = 𝐵, 𝐿, (𝐹𝐵))) = if(𝐵 = 𝐵, 𝐿, (𝐹𝐵)))
229228, 204syl6req 2807 . . . . . . . . . . . . . . . . . . . 20 ((𝑥 = 𝐵 ∧ ¬ 𝑥 = 𝐴) → 𝐿 = if(𝐵 = 𝐴, 𝑅, if(𝐵 = 𝐵, 𝐿, (𝐹𝐵))))
230221, 223, 2293eqtrd 2794 . . . . . . . . . . . . . . . . . . 19 ((𝑥 = 𝐵 ∧ ¬ 𝑥 = 𝐴) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹𝑥))) = if(𝐵 = 𝐴, 𝑅, if(𝐵 = 𝐵, 𝐿, (𝐹𝐵))))
231219, 230pm2.61dan 867 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝐵 → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹𝑥))) = if(𝐵 = 𝐴, 𝑅, if(𝐵 = 𝐵, 𝐿, (𝐹𝐵))))
232231eleq1d 2820 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝐵 → (if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹𝑥))) ∈ ℂ ↔ if(𝐵 = 𝐴, 𝑅, if(𝐵 = 𝐵, 𝐿, (𝐹𝐵))) ∈ ℂ))
233213, 232anbi12d 749 . . . . . . . . . . . . . . . 16 (𝑥 = 𝐵 → ((𝑥 ∈ (𝐴[,]𝐵) ∧ if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹𝑥))) ∈ ℂ) ↔ (𝐵 ∈ (𝐴[,]𝐵) ∧ if(𝐵 = 𝐴, 𝑅, if(𝐵 = 𝐵, 𝐿, (𝐹𝐵))) ∈ ℂ)))
234 fveq2 6348 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝐵 → (𝐺𝑥) = (𝐺𝐵))
235234, 231eqeq12d 2771 . . . . . . . . . . . . . . . 16 (𝑥 = 𝐵 → ((𝐺𝑥) = if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹𝑥))) ↔ (𝐺𝐵) = if(𝐵 = 𝐴, 𝑅, if(𝐵 = 𝐵, 𝐿, (𝐹𝐵)))))
236233, 235imbi12d 333 . . . . . . . . . . . . . . 15 (𝑥 = 𝐵 → (((𝑥 ∈ (𝐴[,]𝐵) ∧ if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹𝑥))) ∈ ℂ) → (𝐺𝑥) = if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹𝑥)))) ↔ ((𝐵 ∈ (𝐴[,]𝐵) ∧ if(𝐵 = 𝐴, 𝑅, if(𝐵 = 𝐵, 𝐿, (𝐹𝐵))) ∈ ℂ) → (𝐺𝐵) = if(𝐵 = 𝐴, 𝑅, if(𝐵 = 𝐵, 𝐿, (𝐹𝐵))))))
237212, 236, 73vtoclg1f 3401 . . . . . . . . . . . . . 14 (𝐵 ∈ (𝐴[,]𝐵) → ((𝐵 ∈ (𝐴[,]𝐵) ∧ if(𝐵 = 𝐴, 𝑅, if(𝐵 = 𝐵, 𝐿, (𝐹𝐵))) ∈ ℂ) → (𝐺𝐵) = if(𝐵 = 𝐴, 𝑅, if(𝐵 = 𝐵, 𝐿, (𝐹𝐵)))))
238202, 207, 237sylc 65 . . . . . . . . . . . . 13 (𝜑 → (𝐺𝐵) = if(𝐵 = 𝐴, 𝑅, if(𝐵 = 𝐵, 𝐿, (𝐹𝐵))))
23918, 24gtned 10360 . . . . . . . . . . . . . . 15 (𝜑𝐵𝐴)
240239neneqd 2933 . . . . . . . . . . . . . 14 (𝜑 → ¬ 𝐵 = 𝐴)
241240iffalsed 4237 . . . . . . . . . . . . 13 (𝜑 → if(𝐵 = 𝐴, 𝑅, if(𝐵 = 𝐵, 𝐿, (𝐹𝐵))) = if(𝐵 = 𝐵, 𝐿, (𝐹𝐵)))
242204a1i 11 . . . . . . . . . . . . 13 (𝜑 → if(𝐵 = 𝐵, 𝐿, (𝐹𝐵)) = 𝐿)
243238, 241, 2423eqtrd 2794 . . . . . . . . . . . 12 (𝜑 → (𝐺𝐵) = 𝐿)
244186oveq1d 6824 . . . . . . . . . . . . . 14 (𝜑 → (𝐹 lim 𝐵) = ((𝐺 ↾ (𝐴(,)𝐵)) lim 𝐵))
2457, 244eleqtrd 2837 . . . . . . . . . . . . 13 (𝜑𝐿 ∈ ((𝐺 ↾ (𝐴(,)𝐵)) lim 𝐵))
24618, 21, 24, 42limcicciooub 40368 . . . . . . . . . . . . 13 (𝜑 → ((𝐺 ↾ (𝐴(,)𝐵)) lim 𝐵) = (𝐺 lim 𝐵))
247245, 246eleqtrd 2837 . . . . . . . . . . . 12 (𝜑𝐿 ∈ (𝐺 lim 𝐵))
248243, 247eqeltrd 2835 . . . . . . . . . . 11 (𝜑 → (𝐺𝐵) ∈ (𝐺 lim 𝐵))
249101, 192cnplimc 23846 . . . . . . . . . . . 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))))
250132, 202, 249syl2anc 696 . . . . . . . . . . 11 (𝜑 → (𝐺 ∈ ((((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) CnP (TopOpen‘ℂfld))‘𝐵) ↔ (𝐺:(𝐴[,]𝐵)⟶ℂ ∧ (𝐺𝐵) ∈ (𝐺 lim 𝐵))))
25142, 248, 250mpbir2and 995 . . . . . . . . . 10 (𝜑𝐺 ∈ ((((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) CnP (TopOpen‘ℂfld))‘𝐵))
252251adantr 472 . . . . . . . . 9 ((𝜑𝑦 = 𝐵) → 𝐺 ∈ ((((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) CnP (TopOpen‘ℂfld))‘𝐵))
253 fveq2 6348 . . . . . . . . . . 11 (𝑦 = 𝐵 → ((((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) CnP (TopOpen‘ℂfld))‘𝑦) = ((((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) CnP (TopOpen‘ℂfld))‘𝐵))
254253eqcomd 2762 . . . . . . . . . 10 (𝑦 = 𝐵 → ((((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) CnP (TopOpen‘ℂfld))‘𝐵) = ((((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) CnP (TopOpen‘ℂfld))‘𝑦))
255254adantl 473 . . . . . . . . 9 ((𝜑𝑦 = 𝐵) → ((((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) CnP (TopOpen‘ℂfld))‘𝐵) = ((((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) CnP (TopOpen‘ℂfld))‘𝑦))
256252, 255eleqtrd 2837 . . . . . . . 8 ((𝜑𝑦 = 𝐵) → 𝐺 ∈ ((((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) CnP (TopOpen‘ℂfld))‘𝑦))
257200, 256jaodan 861 . . . . . . 7 ((𝜑 ∧ (𝑦 = 𝐴𝑦 = 𝐵)) → 𝐺 ∈ ((((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) CnP (TopOpen‘ℂfld))‘𝑦))
258179, 257sylan2 492 . . . . . 6 ((𝜑𝑦 ∈ {𝐴, 𝐵}) → 𝐺 ∈ ((((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) CnP (TopOpen‘ℂfld))‘𝑦))
259178, 258jaodan 861 . . . . 5 ((𝜑 ∧ (𝑦 ∈ (𝐴(,)𝐵) ∨ 𝑦 ∈ {𝐴, 𝐵})) → 𝐺 ∈ ((((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) CnP (TopOpen‘ℂfld))‘𝑦))
26047, 259syldan 488 . . . 4 ((𝜑𝑦 ∈ (𝐴[,]𝐵)) → 𝐺 ∈ ((((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) CnP (TopOpen‘ℂfld))‘𝑦))
261260ralrimiva 3100 . . 3 (𝜑 → ∀𝑦 ∈ (𝐴[,]𝐵)𝐺 ∈ ((((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) CnP (TopOpen‘ℂfld))‘𝑦))
262101cnfldtopon 22783 . . . . 5 (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
263 resttopon 21163 . . . . 5 (((TopOpen‘ℂfld) ∈ (TopOn‘ℂ) ∧ (𝐴[,]𝐵) ⊆ ℂ) → ((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) ∈ (TopOn‘(𝐴[,]𝐵)))
264262, 132, 263sylancr 698 . . . 4 (𝜑 → ((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) ∈ (TopOn‘(𝐴[,]𝐵)))
265 cncnp 21282 . . . 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))))
266264, 262, 265sylancl 697 . . 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))))
26742, 261, 266mpbir2and 995 . 2 (𝜑𝐺 ∈ (((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) Cn (TopOpen‘ℂfld)))
268101, 192, 107cncfcn 22909 . . 3 (((𝐴[,]𝐵) ⊆ ℂ ∧ ℂ ⊆ ℂ) → ((𝐴[,]𝐵)–cn→ℂ) = (((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) Cn (TopOpen‘ℂfld)))
269132, 100, 268sylancl 697 . 2 (𝜑 → ((𝐴[,]𝐵)–cn→ℂ) = (((TopOpen‘ℂfld) ↾t (𝐴[,]𝐵)) Cn (TopOpen‘ℂfld)))
270267, 269eleqtrrd 2838 1 (𝜑𝐺 ∈ ((𝐴[,]𝐵)–cn→ℂ))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 382   ∧ wa 383   ∧ w3a 1072   = wceq 1628  Ⅎwnf 1853   ∈ wcel 2135   ≠ wne 2928  ∀wral 3046  Vcvv 3336   ∖ cdif 3708   ∪ cun 3709   ∩ cin 3710   ⊆ wss 3711  ifcif 4226  {cpr 4319  ∪ cuni 4584   class class class wbr 4800   ↦ cmpt 4877  ran crn 5263   ↾ cres 5264  ⟶wf 6041  ‘cfv 6045  (class class class)co 6809  ℂcc 10122  ℝcr 10123  ℝ*cxr 10261   < clt 10262   ≤ cle 10263  (,)cioo 12364  [,]cicc 12367   ↾t crest 16279  TopOpenctopn 16280  topGenctg 16296  ℂfldccnfld 19944  Topctop 20896  TopOnctopon 20913  intcnt 21019   Cn ccn 21226   CnP ccnp 21227  –cn→ccncf 22876   limℂ climc 23821 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-8 2137  ax-9 2144  ax-10 2164  ax-11 2179  ax-12 2192  ax-13 2387  ax-ext 2736  ax-rep 4919  ax-sep 4929  ax-nul 4937  ax-pow 4988  ax-pr 5051  ax-un 7110  ax-cnex 10180  ax-resscn 10181  ax-1cn 10182  ax-icn 10183  ax-addcl 10184  ax-addrcl 10185  ax-mulcl 10186  ax-mulrcl 10187  ax-mulcom 10188  ax-addass 10189  ax-mulass 10190  ax-distr 10191  ax-i2m1 10192  ax-1ne0 10193  ax-1rid 10194  ax-rnegex 10195  ax-rrecex 10196  ax-cnre 10197  ax-pre-lttri 10198  ax-pre-lttrn 10199  ax-pre-ltadd 10200  ax-pre-mulgt0 10201  ax-pre-sup 10202 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1631  df-ex 1850  df-nf 1855  df-sb 2043  df-eu 2607  df-mo 2608  df-clab 2743  df-cleq 2749  df-clel 2752  df-nfc 2887  df-ne 2929  df-nel 3032  df-ral 3051  df-rex 3052  df-reu 3053  df-rmo 3054  df-rab 3055  df-v 3338  df-sbc 3573  df-csb 3671  df-dif 3714  df-un 3716  df-in 3718  df-ss 3725  df-pss 3727  df-nul 4055  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4585  df-int 4624  df-iun 4670  df-iin 4671  df-br 4801  df-opab 4861  df-mpt 4878  df-tr 4901  df-id 5170  df-eprel 5175  df-po 5183  df-so 5184  df-fr 5221  df-we 5223  df-xp 5268  df-rel 5269  df-cnv 5270  df-co 5271  df-dm 5272  df-rn 5273  df-res 5274  df-ima 5275  df-pred 5837  df-ord 5883  df-on 5884  df-lim 5885  df-suc 5886  df-iota 6008  df-fun 6047  df-fn 6048  df-f 6049  df-f1 6050  df-fo 6051  df-f1o 6052  df-fv 6053  df-riota 6770  df-ov 6812  df-oprab 6813  df-mpt2 6814  df-om 7227  df-1st 7329  df-2nd 7330  df-wrecs 7572  df-recs 7633  df-rdg 7671  df-1o 7725  df-oadd 7729  df-er 7907  df-map 8021  df-pm 8022  df-en 8118  df-dom 8119  df-sdom 8120  df-fin 8121  df-fi 8478  df-sup 8509  df-inf 8510  df-pnf 10264  df-mnf 10265  df-xr 10266  df-ltxr 10267  df-le 10268  df-sub 10456  df-neg 10457  df-div 10873  df-nn 11209  df-2 11267  df-3 11268  df-4 11269  df-5 11270  df-6 11271  df-7 11272  df-8 11273  df-9 11274  df-n0 11481  df-z 11566  df-dec 11682  df-uz 11876  df-q 11978  df-rp 12022  df-xneg 12135  df-xadd 12136  df-xmul 12137  df-ioo 12368  df-ioc 12369  df-ico 12370  df-icc 12371  df-fz 12516  df-seq 12992  df-exp 13051  df-cj 14034  df-re 14035  df-im 14036  df-sqrt 14170  df-abs 14171  df-struct 16057  df-ndx 16058  df-slot 16059  df-base 16061  df-plusg 16152  df-mulr 16153  df-starv 16154  df-tset 16158  df-ple 16159  df-ds 16162  df-unif 16163  df-rest 16281  df-topn 16282  df-topgen 16302  df-psmet 19936  df-xmet 19937  df-met 19938  df-bl 19939  df-mopn 19940  df-cnfld 19945  df-top 20897  df-topon 20914  df-topsp 20935  df-bases 20948  df-cld 21021  df-ntr 21022  df-cls 21023  df-cn 21229  df-cnp 21230  df-xms 22322  df-ms 22323  df-cncf 22878  df-limc 23825 This theorem is referenced by:  cncfiooicc  40606
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