ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  limccnp2cntop GIF version

Theorem limccnp2cntop 14543
Description: The image of a convergent sequence under a continuous map is convergent to the image of the original point. Binary operation version. (Contributed by Mario Carneiro, 28-Dec-2016.) (Revised by Jim Kingdon, 14-Nov-2023.)
Hypotheses
Ref Expression
limccnp2.r ((𝜑𝑥𝐴) → 𝑅𝑋)
limccnp2.s ((𝜑𝑥𝐴) → 𝑆𝑌)
limccnp2.x (𝜑𝑋 ⊆ ℂ)
limccnp2.y (𝜑𝑌 ⊆ ℂ)
limccnp2cntop.k 𝐾 = (MetOpen‘(abs ∘ − ))
limccnp2.j 𝐽 = ((𝐾 ×t 𝐾) ↾t (𝑋 × 𝑌))
limccnp2.c (𝜑𝐶 ∈ ((𝑥𝐴𝑅) lim 𝐵))
limccnp2.d (𝜑𝐷 ∈ ((𝑥𝐴𝑆) lim 𝐵))
limccnp2.h (𝜑𝐻 ∈ ((𝐽 CnP 𝐾)‘⟨𝐶, 𝐷⟩))
Assertion
Ref Expression
limccnp2cntop (𝜑 → (𝐶𝐻𝐷) ∈ ((𝑥𝐴 ↦ (𝑅𝐻𝑆)) lim 𝐵))
Distinct variable groups:   𝑥,𝐵   𝑥,𝐶   𝑥,𝐷   𝑥,𝐻   𝑥,𝑋   𝑥,𝐴   𝑥,𝑌   𝜑,𝑥
Allowed substitution hints:   𝑅(𝑥)   𝑆(𝑥)   𝐽(𝑥)   𝐾(𝑥)

Proof of Theorem limccnp2cntop
Dummy variables 𝑑 𝑒 𝑓 𝑔 𝑗 𝑟 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 limccnp2.j . . . . 5 𝐽 = ((𝐾 ×t 𝐾) ↾t (𝑋 × 𝑌))
2 limccnp2cntop.k . . . . . . . 8 𝐾 = (MetOpen‘(abs ∘ − ))
32cntoptopon 14429 . . . . . . 7 𝐾 ∈ (TopOn‘ℂ)
4 txtopon 14159 . . . . . . 7 ((𝐾 ∈ (TopOn‘ℂ) ∧ 𝐾 ∈ (TopOn‘ℂ)) → (𝐾 ×t 𝐾) ∈ (TopOn‘(ℂ × ℂ)))
53, 3, 4mp2an 426 . . . . . 6 (𝐾 ×t 𝐾) ∈ (TopOn‘(ℂ × ℂ))
6 limccnp2.x . . . . . . 7 (𝜑𝑋 ⊆ ℂ)
7 limccnp2.y . . . . . . 7 (𝜑𝑌 ⊆ ℂ)
8 xpss12 4748 . . . . . . 7 ((𝑋 ⊆ ℂ ∧ 𝑌 ⊆ ℂ) → (𝑋 × 𝑌) ⊆ (ℂ × ℂ))
96, 7, 8syl2anc 411 . . . . . 6 (𝜑 → (𝑋 × 𝑌) ⊆ (ℂ × ℂ))
10 resttopon 14068 . . . . . 6 (((𝐾 ×t 𝐾) ∈ (TopOn‘(ℂ × ℂ)) ∧ (𝑋 × 𝑌) ⊆ (ℂ × ℂ)) → ((𝐾 ×t 𝐾) ↾t (𝑋 × 𝑌)) ∈ (TopOn‘(𝑋 × 𝑌)))
115, 9, 10sylancr 414 . . . . 5 (𝜑 → ((𝐾 ×t 𝐾) ↾t (𝑋 × 𝑌)) ∈ (TopOn‘(𝑋 × 𝑌)))
121, 11eqeltrid 2276 . . . 4 (𝜑𝐽 ∈ (TopOn‘(𝑋 × 𝑌)))
133a1i 9 . . . 4 (𝜑𝐾 ∈ (TopOn‘ℂ))
14 limccnp2.h . . . 4 (𝜑𝐻 ∈ ((𝐽 CnP 𝐾)‘⟨𝐶, 𝐷⟩))
15 cnpf2 14104 . . . 4 ((𝐽 ∈ (TopOn‘(𝑋 × 𝑌)) ∧ 𝐾 ∈ (TopOn‘ℂ) ∧ 𝐻 ∈ ((𝐽 CnP 𝐾)‘⟨𝐶, 𝐷⟩)) → 𝐻:(𝑋 × 𝑌)⟶ℂ)
1612, 13, 14, 15syl3anc 1249 . . 3 (𝜑𝐻:(𝑋 × 𝑌)⟶ℂ)
172cntoptop 14430 . . . . . . . . . . 11 𝐾 ∈ Top
1817a1i 9 . . . . . . . . . . 11 (𝜑𝐾 ∈ Top)
19 txtop 14157 . . . . . . . . . . 11 ((𝐾 ∈ Top ∧ 𝐾 ∈ Top) → (𝐾 ×t 𝐾) ∈ Top)
2017, 18, 19sylancr 414 . . . . . . . . . 10 (𝜑 → (𝐾 ×t 𝐾) ∈ Top)
21 cnex 7953 . . . . . . . . . . . . 13 ℂ ∈ V
2221a1i 9 . . . . . . . . . . . 12 (𝜑 → ℂ ∈ V)
2322, 6ssexd 4158 . . . . . . . . . . 11 (𝜑𝑋 ∈ V)
2422, 7ssexd 4158 . . . . . . . . . . 11 (𝜑𝑌 ∈ V)
25 xpexg 4755 . . . . . . . . . . 11 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋 × 𝑌) ∈ V)
2623, 24, 25syl2anc 411 . . . . . . . . . 10 (𝜑 → (𝑋 × 𝑌) ∈ V)
27 resttop 14067 . . . . . . . . . 10 (((𝐾 ×t 𝐾) ∈ Top ∧ (𝑋 × 𝑌) ∈ V) → ((𝐾 ×t 𝐾) ↾t (𝑋 × 𝑌)) ∈ Top)
2820, 26, 27syl2anc 411 . . . . . . . . 9 (𝜑 → ((𝐾 ×t 𝐾) ↾t (𝑋 × 𝑌)) ∈ Top)
291, 28eqeltrid 2276 . . . . . . . 8 (𝜑𝐽 ∈ Top)
30 toptopon2 13916 . . . . . . . 8 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
3129, 30sylib 122 . . . . . . 7 (𝜑𝐽 ∈ (TopOn‘ 𝐽))
32 cnprcl2k 14103 . . . . . . 7 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ 𝐾 ∈ Top ∧ 𝐻 ∈ ((𝐽 CnP 𝐾)‘⟨𝐶, 𝐷⟩)) → ⟨𝐶, 𝐷⟩ ∈ 𝐽)
3331, 18, 14, 32syl3anc 1249 . . . . . 6 (𝜑 → ⟨𝐶, 𝐷⟩ ∈ 𝐽)
34 toponuni 13912 . . . . . . 7 (𝐽 ∈ (TopOn‘(𝑋 × 𝑌)) → (𝑋 × 𝑌) = 𝐽)
3512, 34syl 14 . . . . . 6 (𝜑 → (𝑋 × 𝑌) = 𝐽)
3633, 35eleqtrrd 2269 . . . . 5 (𝜑 → ⟨𝐶, 𝐷⟩ ∈ (𝑋 × 𝑌))
37 opelxp 4671 . . . . 5 (⟨𝐶, 𝐷⟩ ∈ (𝑋 × 𝑌) ↔ (𝐶𝑋𝐷𝑌))
3836, 37sylib 122 . . . 4 (𝜑 → (𝐶𝑋𝐷𝑌))
3938simpld 112 . . 3 (𝜑𝐶𝑋)
4038simprd 114 . . 3 (𝜑𝐷𝑌)
4116, 39, 40fovcdmd 6036 . 2 (𝜑 → (𝐶𝐻𝐷) ∈ ℂ)
42 txrest 14173 . . . . . . . . . . . . 13 (((𝐾 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑋 ∈ V ∧ 𝑌 ∈ V)) → ((𝐾 ×t 𝐾) ↾t (𝑋 × 𝑌)) = ((𝐾t 𝑋) ×t (𝐾t 𝑌)))
4318, 18, 23, 24, 42syl22anc 1250 . . . . . . . . . . . 12 (𝜑 → ((𝐾 ×t 𝐾) ↾t (𝑋 × 𝑌)) = ((𝐾t 𝑋) ×t (𝐾t 𝑌)))
441, 43eqtrid 2234 . . . . . . . . . . 11 (𝜑𝐽 = ((𝐾t 𝑋) ×t (𝐾t 𝑌)))
45 cnxmet 14428 . . . . . . . . . . . . 13 (abs ∘ − ) ∈ (∞Met‘ℂ)
46 eqid 2189 . . . . . . . . . . . . . 14 ((abs ∘ − ) ↾ (𝑋 × 𝑋)) = ((abs ∘ − ) ↾ (𝑋 × 𝑋))
47 eqid 2189 . . . . . . . . . . . . . 14 (MetOpen‘((abs ∘ − ) ↾ (𝑋 × 𝑋))) = (MetOpen‘((abs ∘ − ) ↾ (𝑋 × 𝑋)))
4846, 2, 47metrest 14403 . . . . . . . . . . . . 13 (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑋 ⊆ ℂ) → (𝐾t 𝑋) = (MetOpen‘((abs ∘ − ) ↾ (𝑋 × 𝑋))))
4945, 6, 48sylancr 414 . . . . . . . . . . . 12 (𝜑 → (𝐾t 𝑋) = (MetOpen‘((abs ∘ − ) ↾ (𝑋 × 𝑋))))
50 eqid 2189 . . . . . . . . . . . . . 14 ((abs ∘ − ) ↾ (𝑌 × 𝑌)) = ((abs ∘ − ) ↾ (𝑌 × 𝑌))
51 eqid 2189 . . . . . . . . . . . . . 14 (MetOpen‘((abs ∘ − ) ↾ (𝑌 × 𝑌))) = (MetOpen‘((abs ∘ − ) ↾ (𝑌 × 𝑌)))
5250, 2, 51metrest 14403 . . . . . . . . . . . . 13 (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑌 ⊆ ℂ) → (𝐾t 𝑌) = (MetOpen‘((abs ∘ − ) ↾ (𝑌 × 𝑌))))
5345, 7, 52sylancr 414 . . . . . . . . . . . 12 (𝜑 → (𝐾t 𝑌) = (MetOpen‘((abs ∘ − ) ↾ (𝑌 × 𝑌))))
5449, 53oveq12d 5909 . . . . . . . . . . 11 (𝜑 → ((𝐾t 𝑋) ×t (𝐾t 𝑌)) = ((MetOpen‘((abs ∘ − ) ↾ (𝑋 × 𝑋))) ×t (MetOpen‘((abs ∘ − ) ↾ (𝑌 × 𝑌)))))
5544, 54eqtrd 2222 . . . . . . . . . 10 (𝜑𝐽 = ((MetOpen‘((abs ∘ − ) ↾ (𝑋 × 𝑋))) ×t (MetOpen‘((abs ∘ − ) ↾ (𝑌 × 𝑌)))))
5655oveq1d 5906 . . . . . . . . 9 (𝜑 → (𝐽 CnP 𝐾) = (((MetOpen‘((abs ∘ − ) ↾ (𝑋 × 𝑋))) ×t (MetOpen‘((abs ∘ − ) ↾ (𝑌 × 𝑌)))) CnP 𝐾))
5756fveq1d 5532 . . . . . . . 8 (𝜑 → ((𝐽 CnP 𝐾)‘⟨𝐶, 𝐷⟩) = ((((MetOpen‘((abs ∘ − ) ↾ (𝑋 × 𝑋))) ×t (MetOpen‘((abs ∘ − ) ↾ (𝑌 × 𝑌)))) CnP 𝐾)‘⟨𝐶, 𝐷⟩))
5814, 57eleqtrd 2268 . . . . . . 7 (𝜑𝐻 ∈ ((((MetOpen‘((abs ∘ − ) ↾ (𝑋 × 𝑋))) ×t (MetOpen‘((abs ∘ − ) ↾ (𝑌 × 𝑌)))) CnP 𝐾)‘⟨𝐶, 𝐷⟩))
59 xmetres2 14276 . . . . . . . . 9 (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑋 ⊆ ℂ) → ((abs ∘ − ) ↾ (𝑋 × 𝑋)) ∈ (∞Met‘𝑋))
6045, 6, 59sylancr 414 . . . . . . . 8 (𝜑 → ((abs ∘ − ) ↾ (𝑋 × 𝑋)) ∈ (∞Met‘𝑋))
61 xmetres2 14276 . . . . . . . . 9 (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑌 ⊆ ℂ) → ((abs ∘ − ) ↾ (𝑌 × 𝑌)) ∈ (∞Met‘𝑌))
6245, 7, 61sylancr 414 . . . . . . . 8 (𝜑 → ((abs ∘ − ) ↾ (𝑌 × 𝑌)) ∈ (∞Met‘𝑌))
6345a1i 9 . . . . . . . 8 (𝜑 → (abs ∘ − ) ∈ (∞Met‘ℂ))
6447, 51, 2txmetcnp 14415 . . . . . . . 8 (((((abs ∘ − ) ↾ (𝑋 × 𝑋)) ∈ (∞Met‘𝑋) ∧ ((abs ∘ − ) ↾ (𝑌 × 𝑌)) ∈ (∞Met‘𝑌) ∧ (abs ∘ − ) ∈ (∞Met‘ℂ)) ∧ (𝐶𝑋𝐷𝑌)) → (𝐻 ∈ ((((MetOpen‘((abs ∘ − ) ↾ (𝑋 × 𝑋))) ×t (MetOpen‘((abs ∘ − ) ↾ (𝑌 × 𝑌)))) CnP 𝐾)‘⟨𝐶, 𝐷⟩) ↔ (𝐻:(𝑋 × 𝑌)⟶ℂ ∧ ∀𝑒 ∈ ℝ+𝑗 ∈ ℝ+𝑟𝑋𝑠𝑌 (((𝐶((abs ∘ − ) ↾ (𝑋 × 𝑋))𝑟) < 𝑗 ∧ (𝐷((abs ∘ − ) ↾ (𝑌 × 𝑌))𝑠) < 𝑗) → ((𝐶𝐻𝐷)(abs ∘ − )(𝑟𝐻𝑠)) < 𝑒))))
6560, 62, 63, 38, 64syl31anc 1252 . . . . . . 7 (𝜑 → (𝐻 ∈ ((((MetOpen‘((abs ∘ − ) ↾ (𝑋 × 𝑋))) ×t (MetOpen‘((abs ∘ − ) ↾ (𝑌 × 𝑌)))) CnP 𝐾)‘⟨𝐶, 𝐷⟩) ↔ (𝐻:(𝑋 × 𝑌)⟶ℂ ∧ ∀𝑒 ∈ ℝ+𝑗 ∈ ℝ+𝑟𝑋𝑠𝑌 (((𝐶((abs ∘ − ) ↾ (𝑋 × 𝑋))𝑟) < 𝑗 ∧ (𝐷((abs ∘ − ) ↾ (𝑌 × 𝑌))𝑠) < 𝑗) → ((𝐶𝐻𝐷)(abs ∘ − )(𝑟𝐻𝑠)) < 𝑒))))
6658, 65mpbid 147 . . . . . 6 (𝜑 → (𝐻:(𝑋 × 𝑌)⟶ℂ ∧ ∀𝑒 ∈ ℝ+𝑗 ∈ ℝ+𝑟𝑋𝑠𝑌 (((𝐶((abs ∘ − ) ↾ (𝑋 × 𝑋))𝑟) < 𝑗 ∧ (𝐷((abs ∘ − ) ↾ (𝑌 × 𝑌))𝑠) < 𝑗) → ((𝐶𝐻𝐷)(abs ∘ − )(𝑟𝐻𝑠)) < 𝑒)))
6766simprd 114 . . . . 5 (𝜑 → ∀𝑒 ∈ ℝ+𝑗 ∈ ℝ+𝑟𝑋𝑠𝑌 (((𝐶((abs ∘ − ) ↾ (𝑋 × 𝑋))𝑟) < 𝑗 ∧ (𝐷((abs ∘ − ) ↾ (𝑌 × 𝑌))𝑠) < 𝑗) → ((𝐶𝐻𝐷)(abs ∘ − )(𝑟𝐻𝑠)) < 𝑒))
6867r19.21bi 2578 . . . 4 ((𝜑𝑒 ∈ ℝ+) → ∃𝑗 ∈ ℝ+𝑟𝑋𝑠𝑌 (((𝐶((abs ∘ − ) ↾ (𝑋 × 𝑋))𝑟) < 𝑗 ∧ (𝐷((abs ∘ − ) ↾ (𝑌 × 𝑌))𝑠) < 𝑗) → ((𝐶𝐻𝐷)(abs ∘ − )(𝑟𝐻𝑠)) < 𝑒))
69 simpll 527 . . . . . 6 (((𝜑𝑒 ∈ ℝ+) ∧ (𝑗 ∈ ℝ+ ∧ ∀𝑟𝑋𝑠𝑌 (((𝐶((abs ∘ − ) ↾ (𝑋 × 𝑋))𝑟) < 𝑗 ∧ (𝐷((abs ∘ − ) ↾ (𝑌 × 𝑌))𝑠) < 𝑗) → ((𝐶𝐻𝐷)(abs ∘ − )(𝑟𝐻𝑠)) < 𝑒))) → 𝜑)
70 simprl 529 . . . . . 6 (((𝜑𝑒 ∈ ℝ+) ∧ (𝑗 ∈ ℝ+ ∧ ∀𝑟𝑋𝑠𝑌 (((𝐶((abs ∘ − ) ↾ (𝑋 × 𝑋))𝑟) < 𝑗 ∧ (𝐷((abs ∘ − ) ↾ (𝑌 × 𝑌))𝑠) < 𝑗) → ((𝐶𝐻𝐷)(abs ∘ − )(𝑟𝐻𝑠)) < 𝑒))) → 𝑗 ∈ ℝ+)
71 limccnp2.c . . . . . . . . 9 (𝜑𝐶 ∈ ((𝑥𝐴𝑅) lim 𝐵))
72 eqid 2189 . . . . . . . . . . . 12 (𝑥𝐴𝑅) = (𝑥𝐴𝑅)
73 limccnp2.r . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → 𝑅𝑋)
7472, 73dmmptd 5361 . . . . . . . . . . 11 (𝜑 → dom (𝑥𝐴𝑅) = 𝐴)
75 limcrcl 14524 . . . . . . . . . . . . 13 (𝐶 ∈ ((𝑥𝐴𝑅) lim 𝐵) → ((𝑥𝐴𝑅):dom (𝑥𝐴𝑅)⟶ℂ ∧ dom (𝑥𝐴𝑅) ⊆ ℂ ∧ 𝐵 ∈ ℂ))
7671, 75syl 14 . . . . . . . . . . . 12 (𝜑 → ((𝑥𝐴𝑅):dom (𝑥𝐴𝑅)⟶ℂ ∧ dom (𝑥𝐴𝑅) ⊆ ℂ ∧ 𝐵 ∈ ℂ))
7776simp2d 1012 . . . . . . . . . . 11 (𝜑 → dom (𝑥𝐴𝑅) ⊆ ℂ)
7874, 77eqsstrrd 3207 . . . . . . . . . 10 (𝜑𝐴 ⊆ ℂ)
7976simp3d 1013 . . . . . . . . . 10 (𝜑𝐵 ∈ ℂ)
806adantr 276 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → 𝑋 ⊆ ℂ)
8180, 73sseldd 3171 . . . . . . . . . 10 ((𝜑𝑥𝐴) → 𝑅 ∈ ℂ)
8278, 79, 81limcmpted 14529 . . . . . . . . 9 (𝜑 → (𝐶 ∈ ((𝑥𝐴𝑅) lim 𝐵) ↔ (𝐶 ∈ ℂ ∧ ∀𝑗 ∈ ℝ+𝑓 ∈ ℝ+𝑥𝐴 ((𝑥 # 𝐵 ∧ (abs‘(𝑥𝐵)) < 𝑓) → (abs‘(𝑅𝐶)) < 𝑗))))
8371, 82mpbid 147 . . . . . . . 8 (𝜑 → (𝐶 ∈ ℂ ∧ ∀𝑗 ∈ ℝ+𝑓 ∈ ℝ+𝑥𝐴 ((𝑥 # 𝐵 ∧ (abs‘(𝑥𝐵)) < 𝑓) → (abs‘(𝑅𝐶)) < 𝑗)))
8483simprd 114 . . . . . . 7 (𝜑 → ∀𝑗 ∈ ℝ+𝑓 ∈ ℝ+𝑥𝐴 ((𝑥 # 𝐵 ∧ (abs‘(𝑥𝐵)) < 𝑓) → (abs‘(𝑅𝐶)) < 𝑗))
8584r19.21bi 2578 . . . . . 6 ((𝜑𝑗 ∈ ℝ+) → ∃𝑓 ∈ ℝ+𝑥𝐴 ((𝑥 # 𝐵 ∧ (abs‘(𝑥𝐵)) < 𝑓) → (abs‘(𝑅𝐶)) < 𝑗))
8669, 70, 85syl2anc 411 . . . . 5 (((𝜑𝑒 ∈ ℝ+) ∧ (𝑗 ∈ ℝ+ ∧ ∀𝑟𝑋𝑠𝑌 (((𝐶((abs ∘ − ) ↾ (𝑋 × 𝑋))𝑟) < 𝑗 ∧ (𝐷((abs ∘ − ) ↾ (𝑌 × 𝑌))𝑠) < 𝑗) → ((𝐶𝐻𝐷)(abs ∘ − )(𝑟𝐻𝑠)) < 𝑒))) → ∃𝑓 ∈ ℝ+𝑥𝐴 ((𝑥 # 𝐵 ∧ (abs‘(𝑥𝐵)) < 𝑓) → (abs‘(𝑅𝐶)) < 𝑗))
8769adantr 276 . . . . . . 7 ((((𝜑𝑒 ∈ ℝ+) ∧ (𝑗 ∈ ℝ+ ∧ ∀𝑟𝑋𝑠𝑌 (((𝐶((abs ∘ − ) ↾ (𝑋 × 𝑋))𝑟) < 𝑗 ∧ (𝐷((abs ∘ − ) ↾ (𝑌 × 𝑌))𝑠) < 𝑗) → ((𝐶𝐻𝐷)(abs ∘ − )(𝑟𝐻𝑠)) < 𝑒))) ∧ (𝑓 ∈ ℝ+ ∧ ∀𝑥𝐴 ((𝑥 # 𝐵 ∧ (abs‘(𝑥𝐵)) < 𝑓) → (abs‘(𝑅𝐶)) < 𝑗))) → 𝜑)
88 simplrl 535 . . . . . . 7 ((((𝜑𝑒 ∈ ℝ+) ∧ (𝑗 ∈ ℝ+ ∧ ∀𝑟𝑋𝑠𝑌 (((𝐶((abs ∘ − ) ↾ (𝑋 × 𝑋))𝑟) < 𝑗 ∧ (𝐷((abs ∘ − ) ↾ (𝑌 × 𝑌))𝑠) < 𝑗) → ((𝐶𝐻𝐷)(abs ∘ − )(𝑟𝐻𝑠)) < 𝑒))) ∧ (𝑓 ∈ ℝ+ ∧ ∀𝑥𝐴 ((𝑥 # 𝐵 ∧ (abs‘(𝑥𝐵)) < 𝑓) → (abs‘(𝑅𝐶)) < 𝑗))) → 𝑗 ∈ ℝ+)
89 limccnp2.d . . . . . . . . . 10 (𝜑𝐷 ∈ ((𝑥𝐴𝑆) lim 𝐵))
907adantr 276 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → 𝑌 ⊆ ℂ)
91 limccnp2.s . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → 𝑆𝑌)
9290, 91sseldd 3171 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → 𝑆 ∈ ℂ)
9378, 79, 92limcmpted 14529 . . . . . . . . . 10 (𝜑 → (𝐷 ∈ ((𝑥𝐴𝑆) lim 𝐵) ↔ (𝐷 ∈ ℂ ∧ ∀𝑗 ∈ ℝ+𝑔 ∈ ℝ+𝑥𝐴 ((𝑥 # 𝐵 ∧ (abs‘(𝑥𝐵)) < 𝑔) → (abs‘(𝑆𝐷)) < 𝑗))))
9489, 93mpbid 147 . . . . . . . . 9 (𝜑 → (𝐷 ∈ ℂ ∧ ∀𝑗 ∈ ℝ+𝑔 ∈ ℝ+𝑥𝐴 ((𝑥 # 𝐵 ∧ (abs‘(𝑥𝐵)) < 𝑔) → (abs‘(𝑆𝐷)) < 𝑗)))
9594simprd 114 . . . . . . . 8 (𝜑 → ∀𝑗 ∈ ℝ+𝑔 ∈ ℝ+𝑥𝐴 ((𝑥 # 𝐵 ∧ (abs‘(𝑥𝐵)) < 𝑔) → (abs‘(𝑆𝐷)) < 𝑗))
9695r19.21bi 2578 . . . . . . 7 ((𝜑𝑗 ∈ ℝ+) → ∃𝑔 ∈ ℝ+𝑥𝐴 ((𝑥 # 𝐵 ∧ (abs‘(𝑥𝐵)) < 𝑔) → (abs‘(𝑆𝐷)) < 𝑗))
9787, 88, 96syl2anc 411 . . . . . 6 ((((𝜑𝑒 ∈ ℝ+) ∧ (𝑗 ∈ ℝ+ ∧ ∀𝑟𝑋𝑠𝑌 (((𝐶((abs ∘ − ) ↾ (𝑋 × 𝑋))𝑟) < 𝑗 ∧ (𝐷((abs ∘ − ) ↾ (𝑌 × 𝑌))𝑠) < 𝑗) → ((𝐶𝐻𝐷)(abs ∘ − )(𝑟𝐻𝑠)) < 𝑒))) ∧ (𝑓 ∈ ℝ+ ∧ ∀𝑥𝐴 ((𝑥 # 𝐵 ∧ (abs‘(𝑥𝐵)) < 𝑓) → (abs‘(𝑅𝐶)) < 𝑗))) → ∃𝑔 ∈ ℝ+𝑥𝐴 ((𝑥 # 𝐵 ∧ (abs‘(𝑥𝐵)) < 𝑔) → (abs‘(𝑆𝐷)) < 𝑗))
98 simp-5l 543 . . . . . . . 8 ((((((𝜑𝑒 ∈ ℝ+) ∧ (𝑗 ∈ ℝ+ ∧ ∀𝑟𝑋𝑠𝑌 (((𝐶((abs ∘ − ) ↾ (𝑋 × 𝑋))𝑟) < 𝑗 ∧ (𝐷((abs ∘ − ) ↾ (𝑌 × 𝑌))𝑠) < 𝑗) → ((𝐶𝐻𝐷)(abs ∘ − )(𝑟𝐻𝑠)) < 𝑒))) ∧ (𝑓 ∈ ℝ+ ∧ ∀𝑥𝐴 ((𝑥 # 𝐵 ∧ (abs‘(𝑥𝐵)) < 𝑓) → (abs‘(𝑅𝐶)) < 𝑗))) ∧ (𝑔 ∈ ℝ+ ∧ ∀𝑥𝐴 ((𝑥 # 𝐵 ∧ (abs‘(𝑥𝐵)) < 𝑔) → (abs‘(𝑆𝐷)) < 𝑗))) ∧ 𝑥𝐴) → 𝜑)
9998, 73sylancom 420 . . . . . . 7 ((((((𝜑𝑒 ∈ ℝ+) ∧ (𝑗 ∈ ℝ+ ∧ ∀𝑟𝑋𝑠𝑌 (((𝐶((abs ∘ − ) ↾ (𝑋 × 𝑋))𝑟) < 𝑗 ∧ (𝐷((abs ∘ − ) ↾ (𝑌 × 𝑌))𝑠) < 𝑗) → ((𝐶𝐻𝐷)(abs ∘ − )(𝑟𝐻𝑠)) < 𝑒))) ∧ (𝑓 ∈ ℝ+ ∧ ∀𝑥𝐴 ((𝑥 # 𝐵 ∧ (abs‘(𝑥𝐵)) < 𝑓) → (abs‘(𝑅𝐶)) < 𝑗))) ∧ (𝑔 ∈ ℝ+ ∧ ∀𝑥𝐴 ((𝑥 # 𝐵 ∧ (abs‘(𝑥𝐵)) < 𝑔) → (abs‘(𝑆𝐷)) < 𝑗))) ∧ 𝑥𝐴) → 𝑅𝑋)
10098, 91sylancom 420 . . . . . . 7 ((((((𝜑𝑒 ∈ ℝ+) ∧ (𝑗 ∈ ℝ+ ∧ ∀𝑟𝑋𝑠𝑌 (((𝐶((abs ∘ − ) ↾ (𝑋 × 𝑋))𝑟) < 𝑗 ∧ (𝐷((abs ∘ − ) ↾ (𝑌 × 𝑌))𝑠) < 𝑗) → ((𝐶𝐻𝐷)(abs ∘ − )(𝑟𝐻𝑠)) < 𝑒))) ∧ (𝑓 ∈ ℝ+ ∧ ∀𝑥𝐴 ((𝑥 # 𝐵 ∧ (abs‘(𝑥𝐵)) < 𝑓) → (abs‘(𝑅𝐶)) < 𝑗))) ∧ (𝑔 ∈ ℝ+ ∧ ∀𝑥𝐴 ((𝑥 # 𝐵 ∧ (abs‘(𝑥𝐵)) < 𝑔) → (abs‘(𝑆𝐷)) < 𝑗))) ∧ 𝑥𝐴) → 𝑆𝑌)
1016ad4antr 494 . . . . . . 7 (((((𝜑𝑒 ∈ ℝ+) ∧ (𝑗 ∈ ℝ+ ∧ ∀𝑟𝑋𝑠𝑌 (((𝐶((abs ∘ − ) ↾ (𝑋 × 𝑋))𝑟) < 𝑗 ∧ (𝐷((abs ∘ − ) ↾ (𝑌 × 𝑌))𝑠) < 𝑗) → ((𝐶𝐻𝐷)(abs ∘ − )(𝑟𝐻𝑠)) < 𝑒))) ∧ (𝑓 ∈ ℝ+ ∧ ∀𝑥𝐴 ((𝑥 # 𝐵 ∧ (abs‘(𝑥𝐵)) < 𝑓) → (abs‘(𝑅𝐶)) < 𝑗))) ∧ (𝑔 ∈ ℝ+ ∧ ∀𝑥𝐴 ((𝑥 # 𝐵 ∧ (abs‘(𝑥𝐵)) < 𝑔) → (abs‘(𝑆𝐷)) < 𝑗))) → 𝑋 ⊆ ℂ)
1027ad4antr 494 . . . . . . 7 (((((𝜑𝑒 ∈ ℝ+) ∧ (𝑗 ∈ ℝ+ ∧ ∀𝑟𝑋𝑠𝑌 (((𝐶((abs ∘ − ) ↾ (𝑋 × 𝑋))𝑟) < 𝑗 ∧ (𝐷((abs ∘ − ) ↾ (𝑌 × 𝑌))𝑠) < 𝑗) → ((𝐶𝐻𝐷)(abs ∘ − )(𝑟𝐻𝑠)) < 𝑒))) ∧ (𝑓 ∈ ℝ+ ∧ ∀𝑥𝐴 ((𝑥 # 𝐵 ∧ (abs‘(𝑥𝐵)) < 𝑓) → (abs‘(𝑅𝐶)) < 𝑗))) ∧ (𝑔 ∈ ℝ+ ∧ ∀𝑥𝐴 ((𝑥 # 𝐵 ∧ (abs‘(𝑥𝐵)) < 𝑔) → (abs‘(𝑆𝐷)) < 𝑗))) → 𝑌 ⊆ ℂ)
10371ad4antr 494 . . . . . . 7 (((((𝜑𝑒 ∈ ℝ+) ∧ (𝑗 ∈ ℝ+ ∧ ∀𝑟𝑋𝑠𝑌 (((𝐶((abs ∘ − ) ↾ (𝑋 × 𝑋))𝑟) < 𝑗 ∧ (𝐷((abs ∘ − ) ↾ (𝑌 × 𝑌))𝑠) < 𝑗) → ((𝐶𝐻𝐷)(abs ∘ − )(𝑟𝐻𝑠)) < 𝑒))) ∧ (𝑓 ∈ ℝ+ ∧ ∀𝑥𝐴 ((𝑥 # 𝐵 ∧ (abs‘(𝑥𝐵)) < 𝑓) → (abs‘(𝑅𝐶)) < 𝑗))) ∧ (𝑔 ∈ ℝ+ ∧ ∀𝑥𝐴 ((𝑥 # 𝐵 ∧ (abs‘(𝑥𝐵)) < 𝑔) → (abs‘(𝑆𝐷)) < 𝑗))) → 𝐶 ∈ ((𝑥𝐴𝑅) lim 𝐵))
10489ad4antr 494 . . . . . . 7 (((((𝜑𝑒 ∈ ℝ+) ∧ (𝑗 ∈ ℝ+ ∧ ∀𝑟𝑋𝑠𝑌 (((𝐶((abs ∘ − ) ↾ (𝑋 × 𝑋))𝑟) < 𝑗 ∧ (𝐷((abs ∘ − ) ↾ (𝑌 × 𝑌))𝑠) < 𝑗) → ((𝐶𝐻𝐷)(abs ∘ − )(𝑟𝐻𝑠)) < 𝑒))) ∧ (𝑓 ∈ ℝ+ ∧ ∀𝑥𝐴 ((𝑥 # 𝐵 ∧ (abs‘(𝑥𝐵)) < 𝑓) → (abs‘(𝑅𝐶)) < 𝑗))) ∧ (𝑔 ∈ ℝ+ ∧ ∀𝑥𝐴 ((𝑥 # 𝐵 ∧ (abs‘(𝑥𝐵)) < 𝑔) → (abs‘(𝑆𝐷)) < 𝑗))) → 𝐷 ∈ ((𝑥𝐴𝑆) lim 𝐵))
10514ad4antr 494 . . . . . . 7 (((((𝜑𝑒 ∈ ℝ+) ∧ (𝑗 ∈ ℝ+ ∧ ∀𝑟𝑋𝑠𝑌 (((𝐶((abs ∘ − ) ↾ (𝑋 × 𝑋))𝑟) < 𝑗 ∧ (𝐷((abs ∘ − ) ↾ (𝑌 × 𝑌))𝑠) < 𝑗) → ((𝐶𝐻𝐷)(abs ∘ − )(𝑟𝐻𝑠)) < 𝑒))) ∧ (𝑓 ∈ ℝ+ ∧ ∀𝑥𝐴 ((𝑥 # 𝐵 ∧ (abs‘(𝑥𝐵)) < 𝑓) → (abs‘(𝑅𝐶)) < 𝑗))) ∧ (𝑔 ∈ ℝ+ ∧ ∀𝑥𝐴 ((𝑥 # 𝐵 ∧ (abs‘(𝑥𝐵)) < 𝑔) → (abs‘(𝑆𝐷)) < 𝑗))) → 𝐻 ∈ ((𝐽 CnP 𝐾)‘⟨𝐶, 𝐷⟩))
106 nfv 1539 . . . . . . . . 9 𝑥((𝜑𝑒 ∈ ℝ+) ∧ (𝑗 ∈ ℝ+ ∧ ∀𝑟𝑋𝑠𝑌 (((𝐶((abs ∘ − ) ↾ (𝑋 × 𝑋))𝑟) < 𝑗 ∧ (𝐷((abs ∘ − ) ↾ (𝑌 × 𝑌))𝑠) < 𝑗) → ((𝐶𝐻𝐷)(abs ∘ − )(𝑟𝐻𝑠)) < 𝑒)))
107 nfv 1539 . . . . . . . . . 10 𝑥 𝑓 ∈ ℝ+
108 nfra1 2521 . . . . . . . . . 10 𝑥𝑥𝐴 ((𝑥 # 𝐵 ∧ (abs‘(𝑥𝐵)) < 𝑓) → (abs‘(𝑅𝐶)) < 𝑗)
109107, 108nfan 1576 . . . . . . . . 9 𝑥(𝑓 ∈ ℝ+ ∧ ∀𝑥𝐴 ((𝑥 # 𝐵 ∧ (abs‘(𝑥𝐵)) < 𝑓) → (abs‘(𝑅𝐶)) < 𝑗))
110106, 109nfan 1576 . . . . . . . 8 𝑥(((𝜑𝑒 ∈ ℝ+) ∧ (𝑗 ∈ ℝ+ ∧ ∀𝑟𝑋𝑠𝑌 (((𝐶((abs ∘ − ) ↾ (𝑋 × 𝑋))𝑟) < 𝑗 ∧ (𝐷((abs ∘ − ) ↾ (𝑌 × 𝑌))𝑠) < 𝑗) → ((𝐶𝐻𝐷)(abs ∘ − )(𝑟𝐻𝑠)) < 𝑒))) ∧ (𝑓 ∈ ℝ+ ∧ ∀𝑥𝐴 ((𝑥 # 𝐵 ∧ (abs‘(𝑥𝐵)) < 𝑓) → (abs‘(𝑅𝐶)) < 𝑗)))
111 nfv 1539 . . . . . . . . 9 𝑥 𝑔 ∈ ℝ+
112 nfra1 2521 . . . . . . . . 9 𝑥𝑥𝐴 ((𝑥 # 𝐵 ∧ (abs‘(𝑥𝐵)) < 𝑔) → (abs‘(𝑆𝐷)) < 𝑗)
113111, 112nfan 1576 . . . . . . . 8 𝑥(𝑔 ∈ ℝ+ ∧ ∀𝑥𝐴 ((𝑥 # 𝐵 ∧ (abs‘(𝑥𝐵)) < 𝑔) → (abs‘(𝑆𝐷)) < 𝑗))
114110, 113nfan 1576 . . . . . . 7 𝑥((((𝜑𝑒 ∈ ℝ+) ∧ (𝑗 ∈ ℝ+ ∧ ∀𝑟𝑋𝑠𝑌 (((𝐶((abs ∘ − ) ↾ (𝑋 × 𝑋))𝑟) < 𝑗 ∧ (𝐷((abs ∘ − ) ↾ (𝑌 × 𝑌))𝑠) < 𝑗) → ((𝐶𝐻𝐷)(abs ∘ − )(𝑟𝐻𝑠)) < 𝑒))) ∧ (𝑓 ∈ ℝ+ ∧ ∀𝑥𝐴 ((𝑥 # 𝐵 ∧ (abs‘(𝑥𝐵)) < 𝑓) → (abs‘(𝑅𝐶)) < 𝑗))) ∧ (𝑔 ∈ ℝ+ ∧ ∀𝑥𝐴 ((𝑥 # 𝐵 ∧ (abs‘(𝑥𝐵)) < 𝑔) → (abs‘(𝑆𝐷)) < 𝑗)))
115 simp-4r 542 . . . . . . 7 (((((𝜑𝑒 ∈ ℝ+) ∧ (𝑗 ∈ ℝ+ ∧ ∀𝑟𝑋𝑠𝑌 (((𝐶((abs ∘ − ) ↾ (𝑋 × 𝑋))𝑟) < 𝑗 ∧ (𝐷((abs ∘ − ) ↾ (𝑌 × 𝑌))𝑠) < 𝑗) → ((𝐶𝐻𝐷)(abs ∘ − )(𝑟𝐻𝑠)) < 𝑒))) ∧ (𝑓 ∈ ℝ+ ∧ ∀𝑥𝐴 ((𝑥 # 𝐵 ∧ (abs‘(𝑥𝐵)) < 𝑓) → (abs‘(𝑅𝐶)) < 𝑗))) ∧ (𝑔 ∈ ℝ+ ∧ ∀𝑥𝐴 ((𝑥 # 𝐵 ∧ (abs‘(𝑥𝐵)) < 𝑔) → (abs‘(𝑆𝐷)) < 𝑗))) → 𝑒 ∈ ℝ+)
11670ad2antrr 488 . . . . . . 7 (((((𝜑𝑒 ∈ ℝ+) ∧ (𝑗 ∈ ℝ+ ∧ ∀𝑟𝑋𝑠𝑌 (((𝐶((abs ∘ − ) ↾ (𝑋 × 𝑋))𝑟) < 𝑗 ∧ (𝐷((abs ∘ − ) ↾ (𝑌 × 𝑌))𝑠) < 𝑗) → ((𝐶𝐻𝐷)(abs ∘ − )(𝑟𝐻𝑠)) < 𝑒))) ∧ (𝑓 ∈ ℝ+ ∧ ∀𝑥𝐴 ((𝑥 # 𝐵 ∧ (abs‘(𝑥𝐵)) < 𝑓) → (abs‘(𝑅𝐶)) < 𝑗))) ∧ (𝑔 ∈ ℝ+ ∧ ∀𝑥𝐴 ((𝑥 # 𝐵 ∧ (abs‘(𝑥𝐵)) < 𝑔) → (abs‘(𝑆𝐷)) < 𝑗))) → 𝑗 ∈ ℝ+)
117 simprr 531 . . . . . . . 8 (((𝜑𝑒 ∈ ℝ+) ∧ (𝑗 ∈ ℝ+ ∧ ∀𝑟𝑋𝑠𝑌 (((𝐶((abs ∘ − ) ↾ (𝑋 × 𝑋))𝑟) < 𝑗 ∧ (𝐷((abs ∘ − ) ↾ (𝑌 × 𝑌))𝑠) < 𝑗) → ((𝐶𝐻𝐷)(abs ∘ − )(𝑟𝐻𝑠)) < 𝑒))) → ∀𝑟𝑋𝑠𝑌 (((𝐶((abs ∘ − ) ↾ (𝑋 × 𝑋))𝑟) < 𝑗 ∧ (𝐷((abs ∘ − ) ↾ (𝑌 × 𝑌))𝑠) < 𝑗) → ((𝐶𝐻𝐷)(abs ∘ − )(𝑟𝐻𝑠)) < 𝑒))
118117ad2antrr 488 . . . . . . 7 (((((𝜑𝑒 ∈ ℝ+) ∧ (𝑗 ∈ ℝ+ ∧ ∀𝑟𝑋𝑠𝑌 (((𝐶((abs ∘ − ) ↾ (𝑋 × 𝑋))𝑟) < 𝑗 ∧ (𝐷((abs ∘ − ) ↾ (𝑌 × 𝑌))𝑠) < 𝑗) → ((𝐶𝐻𝐷)(abs ∘ − )(𝑟𝐻𝑠)) < 𝑒))) ∧ (𝑓 ∈ ℝ+ ∧ ∀𝑥𝐴 ((𝑥 # 𝐵 ∧ (abs‘(𝑥𝐵)) < 𝑓) → (abs‘(𝑅𝐶)) < 𝑗))) ∧ (𝑔 ∈ ℝ+ ∧ ∀𝑥𝐴 ((𝑥 # 𝐵 ∧ (abs‘(𝑥𝐵)) < 𝑔) → (abs‘(𝑆𝐷)) < 𝑗))) → ∀𝑟𝑋𝑠𝑌 (((𝐶((abs ∘ − ) ↾ (𝑋 × 𝑋))𝑟) < 𝑗 ∧ (𝐷((abs ∘ − ) ↾ (𝑌 × 𝑌))𝑠) < 𝑗) → ((𝐶𝐻𝐷)(abs ∘ − )(𝑟𝐻𝑠)) < 𝑒))
119 simplrl 535 . . . . . . 7 (((((𝜑𝑒 ∈ ℝ+) ∧ (𝑗 ∈ ℝ+ ∧ ∀𝑟𝑋𝑠𝑌 (((𝐶((abs ∘ − ) ↾ (𝑋 × 𝑋))𝑟) < 𝑗 ∧ (𝐷((abs ∘ − ) ↾ (𝑌 × 𝑌))𝑠) < 𝑗) → ((𝐶𝐻𝐷)(abs ∘ − )(𝑟𝐻𝑠)) < 𝑒))) ∧ (𝑓 ∈ ℝ+ ∧ ∀𝑥𝐴 ((𝑥 # 𝐵 ∧ (abs‘(𝑥𝐵)) < 𝑓) → (abs‘(𝑅𝐶)) < 𝑗))) ∧ (𝑔 ∈ ℝ+ ∧ ∀𝑥𝐴 ((𝑥 # 𝐵 ∧ (abs‘(𝑥𝐵)) < 𝑔) → (abs‘(𝑆𝐷)) < 𝑗))) → 𝑓 ∈ ℝ+)
120 simplrr 536 . . . . . . 7 (((((𝜑𝑒 ∈ ℝ+) ∧ (𝑗 ∈ ℝ+ ∧ ∀𝑟𝑋𝑠𝑌 (((𝐶((abs ∘ − ) ↾ (𝑋 × 𝑋))𝑟) < 𝑗 ∧ (𝐷((abs ∘ − ) ↾ (𝑌 × 𝑌))𝑠) < 𝑗) → ((𝐶𝐻𝐷)(abs ∘ − )(𝑟𝐻𝑠)) < 𝑒))) ∧ (𝑓 ∈ ℝ+ ∧ ∀𝑥𝐴 ((𝑥 # 𝐵 ∧ (abs‘(𝑥𝐵)) < 𝑓) → (abs‘(𝑅𝐶)) < 𝑗))) ∧ (𝑔 ∈ ℝ+ ∧ ∀𝑥𝐴 ((𝑥 # 𝐵 ∧ (abs‘(𝑥𝐵)) < 𝑔) → (abs‘(𝑆𝐷)) < 𝑗))) → ∀𝑥𝐴 ((𝑥 # 𝐵 ∧ (abs‘(𝑥𝐵)) < 𝑓) → (abs‘(𝑅𝐶)) < 𝑗))
121 simprl 529 . . . . . . 7 (((((𝜑𝑒 ∈ ℝ+) ∧ (𝑗 ∈ ℝ+ ∧ ∀𝑟𝑋𝑠𝑌 (((𝐶((abs ∘ − ) ↾ (𝑋 × 𝑋))𝑟) < 𝑗 ∧ (𝐷((abs ∘ − ) ↾ (𝑌 × 𝑌))𝑠) < 𝑗) → ((𝐶𝐻𝐷)(abs ∘ − )(𝑟𝐻𝑠)) < 𝑒))) ∧ (𝑓 ∈ ℝ+ ∧ ∀𝑥𝐴 ((𝑥 # 𝐵 ∧ (abs‘(𝑥𝐵)) < 𝑓) → (abs‘(𝑅𝐶)) < 𝑗))) ∧ (𝑔 ∈ ℝ+ ∧ ∀𝑥𝐴 ((𝑥 # 𝐵 ∧ (abs‘(𝑥𝐵)) < 𝑔) → (abs‘(𝑆𝐷)) < 𝑗))) → 𝑔 ∈ ℝ+)
122 simprr 531 . . . . . . 7 (((((𝜑𝑒 ∈ ℝ+) ∧ (𝑗 ∈ ℝ+ ∧ ∀𝑟𝑋𝑠𝑌 (((𝐶((abs ∘ − ) ↾ (𝑋 × 𝑋))𝑟) < 𝑗 ∧ (𝐷((abs ∘ − ) ↾ (𝑌 × 𝑌))𝑠) < 𝑗) → ((𝐶𝐻𝐷)(abs ∘ − )(𝑟𝐻𝑠)) < 𝑒))) ∧ (𝑓 ∈ ℝ+ ∧ ∀𝑥𝐴 ((𝑥 # 𝐵 ∧ (abs‘(𝑥𝐵)) < 𝑓) → (abs‘(𝑅𝐶)) < 𝑗))) ∧ (𝑔 ∈ ℝ+ ∧ ∀𝑥𝐴 ((𝑥 # 𝐵 ∧ (abs‘(𝑥𝐵)) < 𝑔) → (abs‘(𝑆𝐷)) < 𝑗))) → ∀𝑥𝐴 ((𝑥 # 𝐵 ∧ (abs‘(𝑥𝐵)) < 𝑔) → (abs‘(𝑆𝐷)) < 𝑗))
12399, 100, 101, 102, 2, 1, 103, 104, 105, 114, 115, 116, 118, 119, 120, 121, 122limccnp2lem 14542 . . . . . 6 (((((𝜑𝑒 ∈ ℝ+) ∧ (𝑗 ∈ ℝ+ ∧ ∀𝑟𝑋𝑠𝑌 (((𝐶((abs ∘ − ) ↾ (𝑋 × 𝑋))𝑟) < 𝑗 ∧ (𝐷((abs ∘ − ) ↾ (𝑌 × 𝑌))𝑠) < 𝑗) → ((𝐶𝐻𝐷)(abs ∘ − )(𝑟𝐻𝑠)) < 𝑒))) ∧ (𝑓 ∈ ℝ+ ∧ ∀𝑥𝐴 ((𝑥 # 𝐵 ∧ (abs‘(𝑥𝐵)) < 𝑓) → (abs‘(𝑅𝐶)) < 𝑗))) ∧ (𝑔 ∈ ℝ+ ∧ ∀𝑥𝐴 ((𝑥 # 𝐵 ∧ (abs‘(𝑥𝐵)) < 𝑔) → (abs‘(𝑆𝐷)) < 𝑗))) → ∃𝑑 ∈ ℝ+𝑥𝐴 ((𝑥 # 𝐵 ∧ (abs‘(𝑥𝐵)) < 𝑑) → (abs‘((𝑅𝐻𝑆) − (𝐶𝐻𝐷))) < 𝑒))
12497, 123rexlimddv 2612 . . . . 5 ((((𝜑𝑒 ∈ ℝ+) ∧ (𝑗 ∈ ℝ+ ∧ ∀𝑟𝑋𝑠𝑌 (((𝐶((abs ∘ − ) ↾ (𝑋 × 𝑋))𝑟) < 𝑗 ∧ (𝐷((abs ∘ − ) ↾ (𝑌 × 𝑌))𝑠) < 𝑗) → ((𝐶𝐻𝐷)(abs ∘ − )(𝑟𝐻𝑠)) < 𝑒))) ∧ (𝑓 ∈ ℝ+ ∧ ∀𝑥𝐴 ((𝑥 # 𝐵 ∧ (abs‘(𝑥𝐵)) < 𝑓) → (abs‘(𝑅𝐶)) < 𝑗))) → ∃𝑑 ∈ ℝ+𝑥𝐴 ((𝑥 # 𝐵 ∧ (abs‘(𝑥𝐵)) < 𝑑) → (abs‘((𝑅𝐻𝑆) − (𝐶𝐻𝐷))) < 𝑒))
12586, 124rexlimddv 2612 . . . 4 (((𝜑𝑒 ∈ ℝ+) ∧ (𝑗 ∈ ℝ+ ∧ ∀𝑟𝑋𝑠𝑌 (((𝐶((abs ∘ − ) ↾ (𝑋 × 𝑋))𝑟) < 𝑗 ∧ (𝐷((abs ∘ − ) ↾ (𝑌 × 𝑌))𝑠) < 𝑗) → ((𝐶𝐻𝐷)(abs ∘ − )(𝑟𝐻𝑠)) < 𝑒))) → ∃𝑑 ∈ ℝ+𝑥𝐴 ((𝑥 # 𝐵 ∧ (abs‘(𝑥𝐵)) < 𝑑) → (abs‘((𝑅𝐻𝑆) − (𝐶𝐻𝐷))) < 𝑒))
12668, 125rexlimddv 2612 . . 3 ((𝜑𝑒 ∈ ℝ+) → ∃𝑑 ∈ ℝ+𝑥𝐴 ((𝑥 # 𝐵 ∧ (abs‘(𝑥𝐵)) < 𝑑) → (abs‘((𝑅𝐻𝑆) − (𝐶𝐻𝐷))) < 𝑒))
127126ralrimiva 2563 . 2 (𝜑 → ∀𝑒 ∈ ℝ+𝑑 ∈ ℝ+𝑥𝐴 ((𝑥 # 𝐵 ∧ (abs‘(𝑥𝐵)) < 𝑑) → (abs‘((𝑅𝐻𝑆) − (𝐶𝐻𝐷))) < 𝑒))
12816adantr 276 . . . 4 ((𝜑𝑥𝐴) → 𝐻:(𝑋 × 𝑌)⟶ℂ)
129128, 73, 91fovcdmd 6036 . . 3 ((𝜑𝑥𝐴) → (𝑅𝐻𝑆) ∈ ℂ)
13078, 79, 129limcmpted 14529 . 2 (𝜑 → ((𝐶𝐻𝐷) ∈ ((𝑥𝐴 ↦ (𝑅𝐻𝑆)) lim 𝐵) ↔ ((𝐶𝐻𝐷) ∈ ℂ ∧ ∀𝑒 ∈ ℝ+𝑑 ∈ ℝ+𝑥𝐴 ((𝑥 # 𝐵 ∧ (abs‘(𝑥𝐵)) < 𝑑) → (abs‘((𝑅𝐻𝑆) − (𝐶𝐻𝐷))) < 𝑒))))
13141, 127, 130mpbir2and 946 1 (𝜑 → (𝐶𝐻𝐷) ∈ ((𝑥𝐴 ↦ (𝑅𝐻𝑆)) lim 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  w3a 980   = wceq 1364  wcel 2160  wral 2468  wrex 2469  Vcvv 2752  wss 3144  cop 3610   cuni 3824   class class class wbr 4018  cmpt 4079   × cxp 4639  dom cdm 4641  cres 4643  ccom 4645  wf 5227  cfv 5231  (class class class)co 5891  cc 7827   < clt 8010  cmin 8146   # cap 8556  +crp 9671  abscabs 11024  t crest 12710  ∞Metcxmet 13810  MetOpencmopn 13815  Topctop 13894  TopOnctopon 13907   CnP ccnp 14083   ×t ctx 14149   lim climc 14520
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-iinf 4602  ax-cnex 7920  ax-resscn 7921  ax-1cn 7922  ax-1re 7923  ax-icn 7924  ax-addcl 7925  ax-addrcl 7926  ax-mulcl 7927  ax-mulrcl 7928  ax-addcom 7929  ax-mulcom 7930  ax-addass 7931  ax-mulass 7932  ax-distr 7933  ax-i2m1 7934  ax-0lt1 7935  ax-1rid 7936  ax-0id 7937  ax-rnegex 7938  ax-precex 7939  ax-cnre 7940  ax-pre-ltirr 7941  ax-pre-ltwlin 7942  ax-pre-lttrn 7943  ax-pre-apti 7944  ax-pre-ltadd 7945  ax-pre-mulgt0 7946  ax-pre-mulext 7947  ax-arch 7948  ax-caucvg 7949
This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4308  df-po 4311  df-iso 4312  df-iord 4381  df-on 4383  df-ilim 4384  df-suc 4386  df-iom 4605  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-f1 5236  df-fo 5237  df-f1o 5238  df-fv 5239  df-isom 5240  df-riota 5847  df-ov 5894  df-oprab 5895  df-mpo 5896  df-1st 6159  df-2nd 6160  df-recs 6324  df-frec 6410  df-map 6668  df-pm 6669  df-sup 7001  df-inf 7002  df-pnf 8012  df-mnf 8013  df-xr 8014  df-ltxr 8015  df-le 8016  df-sub 8148  df-neg 8149  df-reap 8550  df-ap 8557  df-div 8648  df-inn 8938  df-2 8996  df-3 8997  df-4 8998  df-n0 9195  df-z 9272  df-uz 9547  df-q 9638  df-rp 9672  df-xneg 9790  df-xadd 9791  df-seqfrec 10464  df-exp 10538  df-cj 10869  df-re 10870  df-im 10871  df-rsqrt 11025  df-abs 11026  df-rest 12712  df-topgen 12731  df-psmet 13817  df-xmet 13818  df-met 13819  df-bl 13820  df-mopn 13821  df-top 13895  df-topon 13908  df-bases 13940  df-cnp 14086  df-tx 14150  df-limced 14522
This theorem is referenced by:  dvcnp2cntop  14560  dvaddxxbr  14562  dvmulxxbr  14563  dvcoapbr  14568
  Copyright terms: Public domain W3C validator