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Theorem tpr2rico 31155
Description: For any point of an open set of the usual topology on (ℝ × ℝ) there is an open square which contains that point and is entirely in the open set. This is square is actually a ball by the (𝑙↑+∞) norm 𝑋. (Contributed by Thierry Arnoux, 21-Sep-2017.)
Hypotheses
Ref Expression
tpr2rico.0 𝐽 = (topGen‘ran (,))
tpr2rico.1 𝐺 = (𝑢 ∈ ℝ, 𝑣 ∈ ℝ ↦ (𝑢 + (i · 𝑣)))
tpr2rico.2 𝐵 = ran (𝑥 ∈ ran (,), 𝑦 ∈ ran (,) ↦ (𝑥 × 𝑦))
Assertion
Ref Expression
tpr2rico ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) → ∃𝑟𝐵 (𝑋𝑟𝑟𝐴))
Distinct variable groups:   𝑣,𝑢,𝑥,𝑦   𝑥,𝑟,𝐴   𝐵,𝑟   𝑥,𝐺   𝑥,𝐽   𝑥,𝑋   𝑦,𝑟,𝑋
Allowed substitution hints:   𝐴(𝑦,𝑣,𝑢)   𝐵(𝑥,𝑦,𝑣,𝑢)   𝐺(𝑦,𝑣,𝑢,𝑟)   𝐽(𝑦,𝑣,𝑢,𝑟)   𝑋(𝑣,𝑢)

Proof of Theorem tpr2rico
Dummy variables 𝑧 𝑚 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ioo 12741 . . . . . . . . . 10 (,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧𝑧 < 𝑦)})
21ixxf 12747 . . . . . . . . 9 (,):(ℝ* × ℝ*)⟶𝒫 ℝ*
3 ffn 6513 . . . . . . . . 9 ((,):(ℝ* × ℝ*)⟶𝒫 ℝ* → (,) Fn (ℝ* × ℝ*))
42, 3mp1i 13 . . . . . . . 8 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → (,) Fn (ℝ* × ℝ*))
5 elssuni 4867 . . . . . . . . . . . . . 14 (𝐴 ∈ (𝐽 ×t 𝐽) → 𝐴 (𝐽 ×t 𝐽))
6 tpr2rico.0 . . . . . . . . . . . . . . . 16 𝐽 = (topGen‘ran (,))
7 retop 23369 . . . . . . . . . . . . . . . 16 (topGen‘ran (,)) ∈ Top
86, 7eqeltri 2909 . . . . . . . . . . . . . . 15 𝐽 ∈ Top
9 uniretop 23370 . . . . . . . . . . . . . . . 16 ℝ = (topGen‘ran (,))
106unieqi 4850 . . . . . . . . . . . . . . . 16 𝐽 = (topGen‘ran (,))
119, 10eqtr4i 2847 . . . . . . . . . . . . . . 15 ℝ = 𝐽
128, 8, 11, 11txunii 22200 . . . . . . . . . . . . . 14 (ℝ × ℝ) = (𝐽 ×t 𝐽)
135, 12sseqtrrdi 4017 . . . . . . . . . . . . 13 (𝐴 ∈ (𝐽 ×t 𝐽) → 𝐴 ⊆ (ℝ × ℝ))
1413ad2antrr 724 . . . . . . . . . . . 12 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → 𝐴 ⊆ (ℝ × ℝ))
15 simplr 767 . . . . . . . . . . . 12 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → 𝑋𝐴)
1614, 15sseldd 3967 . . . . . . . . . . 11 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → 𝑋 ∈ (ℝ × ℝ))
17 xp1st 7720 . . . . . . . . . . 11 (𝑋 ∈ (ℝ × ℝ) → (1st𝑋) ∈ ℝ)
1816, 17syl 17 . . . . . . . . . 10 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → (1st𝑋) ∈ ℝ)
19 simpr 487 . . . . . . . . . . . 12 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → 𝑑 ∈ ℝ+)
2019rpred 12430 . . . . . . . . . . 11 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → 𝑑 ∈ ℝ)
2120rehalfcld 11883 . . . . . . . . . 10 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → (𝑑 / 2) ∈ ℝ)
2218, 21resubcld 11067 . . . . . . . . 9 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → ((1st𝑋) − (𝑑 / 2)) ∈ ℝ)
2322rexrd 10690 . . . . . . . 8 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → ((1st𝑋) − (𝑑 / 2)) ∈ ℝ*)
2418, 21readdcld 10669 . . . . . . . . 9 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → ((1st𝑋) + (𝑑 / 2)) ∈ ℝ)
2524rexrd 10690 . . . . . . . 8 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → ((1st𝑋) + (𝑑 / 2)) ∈ ℝ*)
26 fnovrn 7322 . . . . . . . 8 (((,) Fn (ℝ* × ℝ*) ∧ ((1st𝑋) − (𝑑 / 2)) ∈ ℝ* ∧ ((1st𝑋) + (𝑑 / 2)) ∈ ℝ*) → (((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) ∈ ran (,))
274, 23, 25, 26syl3anc 1367 . . . . . . 7 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → (((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) ∈ ran (,))
28 xp2nd 7721 . . . . . . . . . . 11 (𝑋 ∈ (ℝ × ℝ) → (2nd𝑋) ∈ ℝ)
2916, 28syl 17 . . . . . . . . . 10 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → (2nd𝑋) ∈ ℝ)
3029, 21resubcld 11067 . . . . . . . . 9 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → ((2nd𝑋) − (𝑑 / 2)) ∈ ℝ)
3130rexrd 10690 . . . . . . . 8 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → ((2nd𝑋) − (𝑑 / 2)) ∈ ℝ*)
3229, 21readdcld 10669 . . . . . . . . 9 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → ((2nd𝑋) + (𝑑 / 2)) ∈ ℝ)
3332rexrd 10690 . . . . . . . 8 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → ((2nd𝑋) + (𝑑 / 2)) ∈ ℝ*)
34 fnovrn 7322 . . . . . . . 8 (((,) Fn (ℝ* × ℝ*) ∧ ((2nd𝑋) − (𝑑 / 2)) ∈ ℝ* ∧ ((2nd𝑋) + (𝑑 / 2)) ∈ ℝ*) → (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2))) ∈ ran (,))
354, 31, 33, 34syl3anc 1367 . . . . . . 7 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2))) ∈ ran (,))
36 eqidd 2822 . . . . . . 7 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) = ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))))
37 xpeq1 5568 . . . . . . . . 9 (𝑥 = (((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) → (𝑥 × 𝑦) = ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × 𝑦))
3837eqeq2d 2832 . . . . . . . 8 (𝑥 = (((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) → (((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) = (𝑥 × 𝑦) ↔ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) = ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × 𝑦)))
39 xpeq2 5575 . . . . . . . . 9 (𝑦 = (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2))) → ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × 𝑦) = ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))))
4039eqeq2d 2832 . . . . . . . 8 (𝑦 = (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2))) → (((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) = ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × 𝑦) ↔ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) = ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2))))))
4138, 40rspc2ev 3634 . . . . . . 7 (((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) ∈ ran (,) ∧ (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2))) ∈ ran (,) ∧ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) = ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2))))) → ∃𝑥 ∈ ran (,)∃𝑦 ∈ ran (,)((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) = (𝑥 × 𝑦))
4227, 35, 36, 41syl3anc 1367 . . . . . 6 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → ∃𝑥 ∈ ran (,)∃𝑦 ∈ ran (,)((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) = (𝑥 × 𝑦))
43 eqid 2821 . . . . . . 7 (𝑥 ∈ ran (,), 𝑦 ∈ ran (,) ↦ (𝑥 × 𝑦)) = (𝑥 ∈ ran (,), 𝑦 ∈ ran (,) ↦ (𝑥 × 𝑦))
44 vex 3497 . . . . . . . 8 𝑥 ∈ V
45 vex 3497 . . . . . . . 8 𝑦 ∈ V
4644, 45xpex 7475 . . . . . . 7 (𝑥 × 𝑦) ∈ V
4743, 46elrnmpo 7286 . . . . . 6 (((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ∈ ran (𝑥 ∈ ran (,), 𝑦 ∈ ran (,) ↦ (𝑥 × 𝑦)) ↔ ∃𝑥 ∈ ran (,)∃𝑦 ∈ ran (,)((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) = (𝑥 × 𝑦))
4842, 47sylibr 236 . . . . 5 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ∈ ran (𝑥 ∈ ran (,), 𝑦 ∈ ran (,) ↦ (𝑥 × 𝑦)))
49 tpr2rico.2 . . . . 5 𝐵 = ran (𝑥 ∈ ran (,), 𝑦 ∈ ran (,) ↦ (𝑥 × 𝑦))
5048, 49eleqtrrdi 2924 . . . 4 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ∈ 𝐵)
5150ralrimiva 3182 . . 3 ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) → ∀𝑑 ∈ ℝ+ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ∈ 𝐵)
52 xpss 5570 . . . . . . 7 (ℝ × ℝ) ⊆ (V × V)
5352, 16sseldi 3964 . . . . . 6 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → 𝑋 ∈ (V × V))
5418rexrd 10690 . . . . . . . 8 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → (1st𝑋) ∈ ℝ*)
5519rphalfcld 12442 . . . . . . . . 9 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → (𝑑 / 2) ∈ ℝ+)
5618, 55ltsubrpd 12462 . . . . . . . 8 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → ((1st𝑋) − (𝑑 / 2)) < (1st𝑋))
5718, 55ltaddrpd 12463 . . . . . . . 8 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → (1st𝑋) < ((1st𝑋) + (𝑑 / 2)))
58 elioo1 12777 . . . . . . . . 9 ((((1st𝑋) − (𝑑 / 2)) ∈ ℝ* ∧ ((1st𝑋) + (𝑑 / 2)) ∈ ℝ*) → ((1st𝑋) ∈ (((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) ↔ ((1st𝑋) ∈ ℝ* ∧ ((1st𝑋) − (𝑑 / 2)) < (1st𝑋) ∧ (1st𝑋) < ((1st𝑋) + (𝑑 / 2)))))
5923, 25, 58syl2anc 586 . . . . . . . 8 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → ((1st𝑋) ∈ (((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) ↔ ((1st𝑋) ∈ ℝ* ∧ ((1st𝑋) − (𝑑 / 2)) < (1st𝑋) ∧ (1st𝑋) < ((1st𝑋) + (𝑑 / 2)))))
6054, 56, 57, 59mpbir3and 1338 . . . . . . 7 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → (1st𝑋) ∈ (((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))))
6129rexrd 10690 . . . . . . . 8 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → (2nd𝑋) ∈ ℝ*)
6229, 55ltsubrpd 12462 . . . . . . . 8 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → ((2nd𝑋) − (𝑑 / 2)) < (2nd𝑋))
6329, 55ltaddrpd 12463 . . . . . . . 8 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → (2nd𝑋) < ((2nd𝑋) + (𝑑 / 2)))
64 elioo1 12777 . . . . . . . . 9 ((((2nd𝑋) − (𝑑 / 2)) ∈ ℝ* ∧ ((2nd𝑋) + (𝑑 / 2)) ∈ ℝ*) → ((2nd𝑋) ∈ (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2))) ↔ ((2nd𝑋) ∈ ℝ* ∧ ((2nd𝑋) − (𝑑 / 2)) < (2nd𝑋) ∧ (2nd𝑋) < ((2nd𝑋) + (𝑑 / 2)))))
6531, 33, 64syl2anc 586 . . . . . . . 8 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → ((2nd𝑋) ∈ (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2))) ↔ ((2nd𝑋) ∈ ℝ* ∧ ((2nd𝑋) − (𝑑 / 2)) < (2nd𝑋) ∧ (2nd𝑋) < ((2nd𝑋) + (𝑑 / 2)))))
6661, 62, 63, 65mpbir3and 1338 . . . . . . 7 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → (2nd𝑋) ∈ (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2))))
6760, 66jca 514 . . . . . 6 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → ((1st𝑋) ∈ (((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) ∧ (2nd𝑋) ∈ (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))))
68 elxp7 7723 . . . . . 6 (𝑋 ∈ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ↔ (𝑋 ∈ (V × V) ∧ ((1st𝑋) ∈ (((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) ∧ (2nd𝑋) ∈ (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2))))))
6953, 67, 68sylanbrc 585 . . . . 5 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → 𝑋 ∈ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))))
7069ralrimiva 3182 . . . 4 ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) → ∀𝑑 ∈ ℝ+ 𝑋 ∈ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))))
71 mnfle 12528 . . . . . . . . . . . . . . . . . 18 (((1st𝑋) − (𝑑 / 2)) ∈ ℝ* → -∞ ≤ ((1st𝑋) − (𝑑 / 2)))
7223, 71syl 17 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → -∞ ≤ ((1st𝑋) − (𝑑 / 2)))
73 pnfge 12524 . . . . . . . . . . . . . . . . . 18 (((1st𝑋) + (𝑑 / 2)) ∈ ℝ* → ((1st𝑋) + (𝑑 / 2)) ≤ +∞)
7425, 73syl 17 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → ((1st𝑋) + (𝑑 / 2)) ≤ +∞)
75 mnfxr 10697 . . . . . . . . . . . . . . . . . 18 -∞ ∈ ℝ*
76 pnfxr 10694 . . . . . . . . . . . . . . . . . 18 +∞ ∈ ℝ*
77 ioossioo 12828 . . . . . . . . . . . . . . . . . 18 (((-∞ ∈ ℝ* ∧ +∞ ∈ ℝ*) ∧ (-∞ ≤ ((1st𝑋) − (𝑑 / 2)) ∧ ((1st𝑋) + (𝑑 / 2)) ≤ +∞)) → (((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) ⊆ (-∞(,)+∞))
7875, 76, 77mpanl12 700 . . . . . . . . . . . . . . . . 17 ((-∞ ≤ ((1st𝑋) − (𝑑 / 2)) ∧ ((1st𝑋) + (𝑑 / 2)) ≤ +∞) → (((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) ⊆ (-∞(,)+∞))
7972, 74, 78syl2anc 586 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → (((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) ⊆ (-∞(,)+∞))
80 ioomax 12810 . . . . . . . . . . . . . . . 16 (-∞(,)+∞) = ℝ
8179, 80sseqtrdi 4016 . . . . . . . . . . . . . . 15 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → (((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) ⊆ ℝ)
82 mnfle 12528 . . . . . . . . . . . . . . . . . 18 (((2nd𝑋) − (𝑑 / 2)) ∈ ℝ* → -∞ ≤ ((2nd𝑋) − (𝑑 / 2)))
8331, 82syl 17 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → -∞ ≤ ((2nd𝑋) − (𝑑 / 2)))
84 pnfge 12524 . . . . . . . . . . . . . . . . . 18 (((2nd𝑋) + (𝑑 / 2)) ∈ ℝ* → ((2nd𝑋) + (𝑑 / 2)) ≤ +∞)
8533, 84syl 17 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → ((2nd𝑋) + (𝑑 / 2)) ≤ +∞)
86 ioossioo 12828 . . . . . . . . . . . . . . . . . 18 (((-∞ ∈ ℝ* ∧ +∞ ∈ ℝ*) ∧ (-∞ ≤ ((2nd𝑋) − (𝑑 / 2)) ∧ ((2nd𝑋) + (𝑑 / 2)) ≤ +∞)) → (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2))) ⊆ (-∞(,)+∞))
8775, 76, 86mpanl12 700 . . . . . . . . . . . . . . . . 17 ((-∞ ≤ ((2nd𝑋) − (𝑑 / 2)) ∧ ((2nd𝑋) + (𝑑 / 2)) ≤ +∞) → (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2))) ⊆ (-∞(,)+∞))
8883, 85, 87syl2anc 586 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2))) ⊆ (-∞(,)+∞))
8988, 80sseqtrdi 4016 . . . . . . . . . . . . . . 15 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2))) ⊆ ℝ)
90 xpss12 5569 . . . . . . . . . . . . . . 15 (((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) ⊆ ℝ ∧ (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2))) ⊆ ℝ) → ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ⊆ (ℝ × ℝ))
9181, 89, 90syl2anc 586 . . . . . . . . . . . . . 14 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ⊆ (ℝ × ℝ))
9291sselda 3966 . . . . . . . . . . . . 13 ((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2))))) → 𝑥 ∈ (ℝ × ℝ))
9392expcom 416 . . . . . . . . . . . 12 (𝑥 ∈ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) → (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → 𝑥 ∈ (ℝ × ℝ)))
9493ancld 553 . . . . . . . . . . 11 (𝑥 ∈ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) → (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ))))
9594imdistanri 572 . . . . . . . . . 10 ((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2))))) → ((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) ∧ 𝑥 ∈ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2))))))
9613adantr 483 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ (𝑋𝐴𝑑 ∈ ℝ+𝑥 ∈ (ℝ × ℝ))) → 𝐴 ⊆ (ℝ × ℝ))
97 simpr1 1190 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ (𝑋𝐴𝑑 ∈ ℝ+𝑥 ∈ (ℝ × ℝ))) → 𝑋𝐴)
9896, 97sseldd 3967 . . . . . . . . . . . . . . 15 ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ (𝑋𝐴𝑑 ∈ ℝ+𝑥 ∈ (ℝ × ℝ))) → 𝑋 ∈ (ℝ × ℝ))
99983anassrs 1356 . . . . . . . . . . . . . 14 ((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) → 𝑋 ∈ (ℝ × ℝ))
100 simpr 487 . . . . . . . . . . . . . 14 ((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) → 𝑥 ∈ (ℝ × ℝ))
101 simplr 767 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) → 𝑑 ∈ ℝ+)
102101rphalfcld 12442 . . . . . . . . . . . . . 14 ((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) → (𝑑 / 2) ∈ ℝ+)
103 tpr2rico.1 . . . . . . . . . . . . . . 15 𝐺 = (𝑢 ∈ ℝ, 𝑣 ∈ ℝ ↦ (𝑢 + (i · 𝑣)))
104103cnre2csqima 31154 . . . . . . . . . . . . . 14 ((𝑋 ∈ (ℝ × ℝ) ∧ 𝑥 ∈ (ℝ × ℝ) ∧ (𝑑 / 2) ∈ ℝ+) → (𝑥 ∈ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) → ((abs‘(ℜ‘((𝐺𝑥) − (𝐺𝑋)))) < (𝑑 / 2) ∧ (abs‘(ℑ‘((𝐺𝑥) − (𝐺𝑋)))) < (𝑑 / 2))))
10599, 100, 102, 104syl3anc 1367 . . . . . . . . . . . . 13 ((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) → (𝑥 ∈ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) → ((abs‘(ℜ‘((𝐺𝑥) − (𝐺𝑋)))) < (𝑑 / 2) ∧ (abs‘(ℑ‘((𝐺𝑥) − (𝐺𝑋)))) < (𝑑 / 2))))
106 eqid 2821 . . . . . . . . . . . . . . . . . . . . 21 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
107103, 6, 106cnrehmeo 23556 . . . . . . . . . . . . . . . . . . . 20 𝐺 ∈ ((𝐽 ×t 𝐽)Homeo(TopOpen‘ℂfld))
108106cnfldtopon 23390 . . . . . . . . . . . . . . . . . . . . . 22 (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
109108toponunii 21523 . . . . . . . . . . . . . . . . . . . . 21 ℂ = (TopOpen‘ℂfld)
11012, 109hmeof1o 22371 . . . . . . . . . . . . . . . . . . . 20 (𝐺 ∈ ((𝐽 ×t 𝐽)Homeo(TopOpen‘ℂfld)) → 𝐺:(ℝ × ℝ)–1-1-onto→ℂ)
111 f1of 6614 . . . . . . . . . . . . . . . . . . . 20 (𝐺:(ℝ × ℝ)–1-1-onto→ℂ → 𝐺:(ℝ × ℝ)⟶ℂ)
112107, 110, 111mp2b 10 . . . . . . . . . . . . . . . . . . 19 𝐺:(ℝ × ℝ)⟶ℂ
113112a1i 11 . . . . . . . . . . . . . . . . . 18 ((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) → 𝐺:(ℝ × ℝ)⟶ℂ)
114113, 99ffvelrnd 6851 . . . . . . . . . . . . . . . . 17 ((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) → (𝐺𝑋) ∈ ℂ)
115112a1i 11 . . . . . . . . . . . . . . . . . 18 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → 𝐺:(ℝ × ℝ)⟶ℂ)
116115ffvelrnda 6850 . . . . . . . . . . . . . . . . 17 ((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) → (𝐺𝑥) ∈ ℂ)
117 sqsscirc2 31152 . . . . . . . . . . . . . . . . 17 ((((𝐺𝑋) ∈ ℂ ∧ (𝐺𝑥) ∈ ℂ) ∧ 𝑑 ∈ ℝ+) → (((abs‘(ℜ‘((𝐺𝑥) − (𝐺𝑋)))) < (𝑑 / 2) ∧ (abs‘(ℑ‘((𝐺𝑥) − (𝐺𝑋)))) < (𝑑 / 2)) → (abs‘((𝐺𝑥) − (𝐺𝑋))) < 𝑑))
118114, 116, 101, 117syl21anc 835 . . . . . . . . . . . . . . . 16 ((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) → (((abs‘(ℜ‘((𝐺𝑥) − (𝐺𝑋)))) < (𝑑 / 2) ∧ (abs‘(ℑ‘((𝐺𝑥) − (𝐺𝑋)))) < (𝑑 / 2)) → (abs‘((𝐺𝑥) − (𝐺𝑋))) < 𝑑))
119118imp 409 . . . . . . . . . . . . . . 15 (((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) ∧ ((abs‘(ℜ‘((𝐺𝑥) − (𝐺𝑋)))) < (𝑑 / 2) ∧ (abs‘(ℑ‘((𝐺𝑥) − (𝐺𝑋)))) < (𝑑 / 2))) → (abs‘((𝐺𝑥) − (𝐺𝑋))) < 𝑑)
120101rpxrd 12431 . . . . . . . . . . . . . . . . . 18 ((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) → 𝑑 ∈ ℝ*)
121120adantr 483 . . . . . . . . . . . . . . . . 17 (((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) ∧ (abs‘((𝐺𝑥) − (𝐺𝑋))) < 𝑑) → 𝑑 ∈ ℝ*)
122 cnxmet 23380 . . . . . . . . . . . . . . . . 17 (abs ∘ − ) ∈ (∞Met‘ℂ)
123121, 122jctil 522 . . . . . . . . . . . . . . . 16 (((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) ∧ (abs‘((𝐺𝑥) − (𝐺𝑋))) < 𝑑) → ((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑑 ∈ ℝ*))
124114adantr 483 . . . . . . . . . . . . . . . . 17 (((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) ∧ (abs‘((𝐺𝑥) − (𝐺𝑋))) < 𝑑) → (𝐺𝑋) ∈ ℂ)
125116adantr 483 . . . . . . . . . . . . . . . . 17 (((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) ∧ (abs‘((𝐺𝑥) − (𝐺𝑋))) < 𝑑) → (𝐺𝑥) ∈ ℂ)
126124, 125jca 514 . . . . . . . . . . . . . . . 16 (((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) ∧ (abs‘((𝐺𝑥) − (𝐺𝑋))) < 𝑑) → ((𝐺𝑋) ∈ ℂ ∧ (𝐺𝑥) ∈ ℂ))
127 eqid 2821 . . . . . . . . . . . . . . . . . . 19 (abs ∘ − ) = (abs ∘ − )
128127cnmetdval 23378 . . . . . . . . . . . . . . . . . 18 (((𝐺𝑥) ∈ ℂ ∧ (𝐺𝑋) ∈ ℂ) → ((𝐺𝑥)(abs ∘ − )(𝐺𝑋)) = (abs‘((𝐺𝑥) − (𝐺𝑋))))
129125, 124, 128syl2anc 586 . . . . . . . . . . . . . . . . 17 (((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) ∧ (abs‘((𝐺𝑥) − (𝐺𝑋))) < 𝑑) → ((𝐺𝑥)(abs ∘ − )(𝐺𝑋)) = (abs‘((𝐺𝑥) − (𝐺𝑋))))
130 simpr 487 . . . . . . . . . . . . . . . . 17 (((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) ∧ (abs‘((𝐺𝑥) − (𝐺𝑋))) < 𝑑) → (abs‘((𝐺𝑥) − (𝐺𝑋))) < 𝑑)
131129, 130eqbrtrd 5087 . . . . . . . . . . . . . . . 16 (((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) ∧ (abs‘((𝐺𝑥) − (𝐺𝑋))) < 𝑑) → ((𝐺𝑥)(abs ∘ − )(𝐺𝑋)) < 𝑑)
132 elbl3 23001 . . . . . . . . . . . . . . . . 17 ((((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑑 ∈ ℝ*) ∧ ((𝐺𝑋) ∈ ℂ ∧ (𝐺𝑥) ∈ ℂ)) → ((𝐺𝑥) ∈ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑) ↔ ((𝐺𝑥)(abs ∘ − )(𝐺𝑋)) < 𝑑))
133132biimpar 480 . . . . . . . . . . . . . . . 16 (((((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑑 ∈ ℝ*) ∧ ((𝐺𝑋) ∈ ℂ ∧ (𝐺𝑥) ∈ ℂ)) ∧ ((𝐺𝑥)(abs ∘ − )(𝐺𝑋)) < 𝑑) → (𝐺𝑥) ∈ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑))
134123, 126, 131, 133syl21anc 835 . . . . . . . . . . . . . . 15 (((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) ∧ (abs‘((𝐺𝑥) − (𝐺𝑋))) < 𝑑) → (𝐺𝑥) ∈ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑))
135119, 134syldan 593 . . . . . . . . . . . . . 14 (((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) ∧ ((abs‘(ℜ‘((𝐺𝑥) − (𝐺𝑋)))) < (𝑑 / 2) ∧ (abs‘(ℑ‘((𝐺𝑥) − (𝐺𝑋)))) < (𝑑 / 2))) → (𝐺𝑥) ∈ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑))
136135ex 415 . . . . . . . . . . . . 13 ((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) → (((abs‘(ℜ‘((𝐺𝑥) − (𝐺𝑋)))) < (𝑑 / 2) ∧ (abs‘(ℑ‘((𝐺𝑥) − (𝐺𝑋)))) < (𝑑 / 2)) → (𝐺𝑥) ∈ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑)))
137105, 136syld 47 . . . . . . . . . . . 12 ((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) → (𝑥 ∈ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) → (𝐺𝑥) ∈ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑)))
138 f1ocnv 6626 . . . . . . . . . . . . . . 15 (𝐺:(ℝ × ℝ)–1-1-onto→ℂ → 𝐺:ℂ–1-1-onto→(ℝ × ℝ))
139107, 110, 138mp2b 10 . . . . . . . . . . . . . 14 𝐺:ℂ–1-1-onto→(ℝ × ℝ)
140 f1ofun 6616 . . . . . . . . . . . . . 14 (𝐺:ℂ–1-1-onto→(ℝ × ℝ) → Fun 𝐺)
141139, 140ax-mp 5 . . . . . . . . . . . . 13 Fun 𝐺
142 f1odm 6618 . . . . . . . . . . . . . . 15 (𝐺:ℂ–1-1-onto→(ℝ × ℝ) → dom 𝐺 = ℂ)
143139, 142ax-mp 5 . . . . . . . . . . . . . 14 dom 𝐺 = ℂ
144116, 143eleqtrrdi 2924 . . . . . . . . . . . . 13 ((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) → (𝐺𝑥) ∈ dom 𝐺)
145 funfvima 6991 . . . . . . . . . . . . 13 ((Fun 𝐺 ∧ (𝐺𝑥) ∈ dom 𝐺) → ((𝐺𝑥) ∈ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑) → (𝐺‘(𝐺𝑥)) ∈ (𝐺 “ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑))))
146141, 144, 145sylancr 589 . . . . . . . . . . . 12 ((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) → ((𝐺𝑥) ∈ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑) → (𝐺‘(𝐺𝑥)) ∈ (𝐺 “ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑))))
147107, 110mp1i 13 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) → 𝐺:(ℝ × ℝ)–1-1-onto→ℂ)
148 f1ocnvfv1 7032 . . . . . . . . . . . . . . 15 ((𝐺:(ℝ × ℝ)–1-1-onto→ℂ ∧ 𝑥 ∈ (ℝ × ℝ)) → (𝐺‘(𝐺𝑥)) = 𝑥)
149147, 100, 148syl2anc 586 . . . . . . . . . . . . . 14 ((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) → (𝐺‘(𝐺𝑥)) = 𝑥)
150149eleq1d 2897 . . . . . . . . . . . . 13 ((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) → ((𝐺‘(𝐺𝑥)) ∈ (𝐺 “ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑)) ↔ 𝑥 ∈ (𝐺 “ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑))))
151150biimpd 231 . . . . . . . . . . . 12 ((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) → ((𝐺‘(𝐺𝑥)) ∈ (𝐺 “ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑)) → 𝑥 ∈ (𝐺 “ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑))))
152137, 146, 1513syld 60 . . . . . . . . . . 11 ((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) → (𝑥 ∈ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) → 𝑥 ∈ (𝐺 “ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑))))
153152imp 409 . . . . . . . . . 10 (((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) ∧ 𝑥 ∈ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2))))) → 𝑥 ∈ (𝐺 “ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑)))
15495, 153syl 17 . . . . . . . . 9 ((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2))))) → 𝑥 ∈ (𝐺 “ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑)))
155154ex 415 . . . . . . . 8 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → (𝑥 ∈ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) → 𝑥 ∈ (𝐺 “ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑))))
156155ssrdv 3972 . . . . . . 7 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ⊆ (𝐺 “ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑)))
157156ralrimiva 3182 . . . . . 6 ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) → ∀𝑑 ∈ ℝ+ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ⊆ (𝐺 “ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑)))
158103mpofun 7275 . . . . . . . . . 10 Fun 𝐺
159158a1i 11 . . . . . . . . 9 ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) → Fun 𝐺)
16013sselda 3966 . . . . . . . . . 10 ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) → 𝑋 ∈ (ℝ × ℝ))
161 f1odm 6618 . . . . . . . . . . 11 (𝐺:(ℝ × ℝ)–1-1-onto→ℂ → dom 𝐺 = (ℝ × ℝ))
162107, 110, 161mp2b 10 . . . . . . . . . 10 dom 𝐺 = (ℝ × ℝ)
163160, 162eleqtrrdi 2924 . . . . . . . . 9 ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) → 𝑋 ∈ dom 𝐺)
164 simpr 487 . . . . . . . . 9 ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) → 𝑋𝐴)
165 funfvima 6991 . . . . . . . . . 10 ((Fun 𝐺𝑋 ∈ dom 𝐺) → (𝑋𝐴 → (𝐺𝑋) ∈ (𝐺𝐴)))
166165imp 409 . . . . . . . . 9 (((Fun 𝐺𝑋 ∈ dom 𝐺) ∧ 𝑋𝐴) → (𝐺𝑋) ∈ (𝐺𝐴))
167159, 163, 164, 166syl21anc 835 . . . . . . . 8 ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) → (𝐺𝑋) ∈ (𝐺𝐴))
168 hmeoima 22372 . . . . . . . . . . 11 ((𝐺 ∈ ((𝐽 ×t 𝐽)Homeo(TopOpen‘ℂfld)) ∧ 𝐴 ∈ (𝐽 ×t 𝐽)) → (𝐺𝐴) ∈ (TopOpen‘ℂfld))
169107, 168mpan 688 . . . . . . . . . 10 (𝐴 ∈ (𝐽 ×t 𝐽) → (𝐺𝐴) ∈ (TopOpen‘ℂfld))
170106cnfldtopn 23389 . . . . . . . . . . . . 13 (TopOpen‘ℂfld) = (MetOpen‘(abs ∘ − ))
171170elmopn2 23054 . . . . . . . . . . . 12 ((abs ∘ − ) ∈ (∞Met‘ℂ) → ((𝐺𝐴) ∈ (TopOpen‘ℂfld) ↔ ((𝐺𝐴) ⊆ ℂ ∧ ∀𝑚 ∈ (𝐺𝐴)∃𝑑 ∈ ℝ+ (𝑚(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺𝐴))))
172122, 171ax-mp 5 . . . . . . . . . . 11 ((𝐺𝐴) ∈ (TopOpen‘ℂfld) ↔ ((𝐺𝐴) ⊆ ℂ ∧ ∀𝑚 ∈ (𝐺𝐴)∃𝑑 ∈ ℝ+ (𝑚(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺𝐴)))
173172simprbi 499 . . . . . . . . . 10 ((𝐺𝐴) ∈ (TopOpen‘ℂfld) → ∀𝑚 ∈ (𝐺𝐴)∃𝑑 ∈ ℝ+ (𝑚(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺𝐴))
174169, 173syl 17 . . . . . . . . 9 (𝐴 ∈ (𝐽 ×t 𝐽) → ∀𝑚 ∈ (𝐺𝐴)∃𝑑 ∈ ℝ+ (𝑚(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺𝐴))
175174adantr 483 . . . . . . . 8 ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) → ∀𝑚 ∈ (𝐺𝐴)∃𝑑 ∈ ℝ+ (𝑚(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺𝐴))
176 oveq1 7162 . . . . . . . . . . 11 (𝑚 = (𝐺𝑋) → (𝑚(ball‘(abs ∘ − ))𝑑) = ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑))
177176sseq1d 3997 . . . . . . . . . 10 (𝑚 = (𝐺𝑋) → ((𝑚(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺𝐴) ↔ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺𝐴)))
178177rexbidv 3297 . . . . . . . . 9 (𝑚 = (𝐺𝑋) → (∃𝑑 ∈ ℝ+ (𝑚(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺𝐴) ↔ ∃𝑑 ∈ ℝ+ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺𝐴)))
179178rspcva 3620 . . . . . . . 8 (((𝐺𝑋) ∈ (𝐺𝐴) ∧ ∀𝑚 ∈ (𝐺𝐴)∃𝑑 ∈ ℝ+ (𝑚(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺𝐴)) → ∃𝑑 ∈ ℝ+ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺𝐴))
180167, 175, 179syl2anc 586 . . . . . . 7 ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) → ∃𝑑 ∈ ℝ+ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺𝐴))
181 imass2 5964 . . . . . . . . . 10 (((𝐺𝑋)(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺𝐴) → (𝐺 “ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑)) ⊆ (𝐺 “ (𝐺𝐴)))
182 f1of1 6613 . . . . . . . . . . . . 13 (𝐺:(ℝ × ℝ)–1-1-onto→ℂ → 𝐺:(ℝ × ℝ)–1-1→ℂ)
183107, 110, 182mp2b 10 . . . . . . . . . . . 12 𝐺:(ℝ × ℝ)–1-1→ℂ
184 f1imacnv 6630 . . . . . . . . . . . 12 ((𝐺:(ℝ × ℝ)–1-1→ℂ ∧ 𝐴 ⊆ (ℝ × ℝ)) → (𝐺 “ (𝐺𝐴)) = 𝐴)
185183, 13, 184sylancr 589 . . . . . . . . . . 11 (𝐴 ∈ (𝐽 ×t 𝐽) → (𝐺 “ (𝐺𝐴)) = 𝐴)
186185sseq2d 3998 . . . . . . . . . 10 (𝐴 ∈ (𝐽 ×t 𝐽) → ((𝐺 “ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑)) ⊆ (𝐺 “ (𝐺𝐴)) ↔ (𝐺 “ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑)) ⊆ 𝐴))
187181, 186syl5ib 246 . . . . . . . . 9 (𝐴 ∈ (𝐽 ×t 𝐽) → (((𝐺𝑋)(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺𝐴) → (𝐺 “ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑)) ⊆ 𝐴))
188187reximdv 3273 . . . . . . . 8 (𝐴 ∈ (𝐽 ×t 𝐽) → (∃𝑑 ∈ ℝ+ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺𝐴) → ∃𝑑 ∈ ℝ+ (𝐺 “ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑)) ⊆ 𝐴))
189188adantr 483 . . . . . . 7 ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) → (∃𝑑 ∈ ℝ+ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺𝐴) → ∃𝑑 ∈ ℝ+ (𝐺 “ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑)) ⊆ 𝐴))
190180, 189mpd 15 . . . . . 6 ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) → ∃𝑑 ∈ ℝ+ (𝐺 “ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑)) ⊆ 𝐴)
191 r19.29 3254 . . . . . 6 ((∀𝑑 ∈ ℝ+ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ⊆ (𝐺 “ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑)) ∧ ∃𝑑 ∈ ℝ+ (𝐺 “ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑)) ⊆ 𝐴) → ∃𝑑 ∈ ℝ+ (((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ⊆ (𝐺 “ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑)) ∧ (𝐺 “ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑)) ⊆ 𝐴))
192157, 190, 191syl2anc 586 . . . . 5 ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) → ∃𝑑 ∈ ℝ+ (((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ⊆ (𝐺 “ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑)) ∧ (𝐺 “ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑)) ⊆ 𝐴))
193 sstr 3974 . . . . . 6 ((((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ⊆ (𝐺 “ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑)) ∧ (𝐺 “ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑)) ⊆ 𝐴) → ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ⊆ 𝐴)
194193reximi 3243 . . . . 5 (∃𝑑 ∈ ℝ+ (((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ⊆ (𝐺 “ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑)) ∧ (𝐺 “ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑)) ⊆ 𝐴) → ∃𝑑 ∈ ℝ+ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ⊆ 𝐴)
195192, 194syl 17 . . . 4 ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) → ∃𝑑 ∈ ℝ+ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ⊆ 𝐴)
196 r19.29 3254 . . . 4 ((∀𝑑 ∈ ℝ+ 𝑋 ∈ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ∧ ∃𝑑 ∈ ℝ+ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ⊆ 𝐴) → ∃𝑑 ∈ ℝ+ (𝑋 ∈ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ∧ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ⊆ 𝐴))
19770, 195, 196syl2anc 586 . . 3 ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) → ∃𝑑 ∈ ℝ+ (𝑋 ∈ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ∧ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ⊆ 𝐴))
198 r19.29 3254 . . 3 ((∀𝑑 ∈ ℝ+ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ∈ 𝐵 ∧ ∃𝑑 ∈ ℝ+ (𝑋 ∈ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ∧ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ⊆ 𝐴)) → ∃𝑑 ∈ ℝ+ (((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ∈ 𝐵 ∧ (𝑋 ∈ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ∧ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ⊆ 𝐴)))
19951, 197, 198syl2anc 586 . 2 ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) → ∃𝑑 ∈ ℝ+ (((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ∈ 𝐵 ∧ (𝑋 ∈ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ∧ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ⊆ 𝐴)))
200 eleq2 2901 . . . . 5 (𝑟 = ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) → (𝑋𝑟𝑋 ∈ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2))))))
201 sseq1 3991 . . . . 5 (𝑟 = ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) → (𝑟𝐴 ↔ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ⊆ 𝐴))
202200, 201anbi12d 632 . . . 4 (𝑟 = ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) → ((𝑋𝑟𝑟𝐴) ↔ (𝑋 ∈ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ∧ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ⊆ 𝐴)))
203202rspcev 3622 . . 3 ((((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ∈ 𝐵 ∧ (𝑋 ∈ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ∧ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ⊆ 𝐴)) → ∃𝑟𝐵 (𝑋𝑟𝑟𝐴))
204203rexlimivw 3282 . 2 (∃𝑑 ∈ ℝ+ (((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ∈ 𝐵 ∧ (𝑋 ∈ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ∧ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ⊆ 𝐴)) → ∃𝑟𝐵 (𝑋𝑟𝑟𝐴))
205199, 204syl 17 1 ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) → ∃𝑟𝐵 (𝑋𝑟𝑟𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  w3a 1083   = wceq 1533  wcel 2110  wral 3138  wrex 3139  Vcvv 3494  wss 3935  𝒫 cpw 4538   cuni 4837   class class class wbr 5065   × cxp 5552  ccnv 5553  dom cdm 5554  ran crn 5555  cima 5557  ccom 5558  Fun wfun 6348   Fn wfn 6349  wf 6350  1-1wf1 6351  1-1-ontowf1o 6353  cfv 6354  (class class class)co 7155  cmpo 7157  1st c1st 7686  2nd c2nd 7687  cc 10534  cr 10535  ici 10538   + caddc 10539   · cmul 10541  +∞cpnf 10671  -∞cmnf 10672  *cxr 10673   < clt 10674  cle 10675  cmin 10869   / cdiv 11296  2c2 11691  +crp 12388  (,)cioo 12737  cre 14455  cim 14456  abscabs 14592  TopOpenctopn 16694  topGenctg 16710  ∞Metcxmet 20529  ballcbl 20531  fldccnfld 20544  Topctop 21500   ×t ctx 22167  Homeochmeo 22360
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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5189  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460  ax-cnex 10592  ax-resscn 10593  ax-1cn 10594  ax-icn 10595  ax-addcl 10596  ax-addrcl 10597  ax-mulcl 10598  ax-mulrcl 10599  ax-mulcom 10600  ax-addass 10601  ax-mulass 10602  ax-distr 10603  ax-i2m1 10604  ax-1ne0 10605  ax-1rid 10606  ax-rnegex 10607  ax-rrecex 10608  ax-cnre 10609  ax-pre-lttri 10610  ax-pre-lttrn 10611  ax-pre-ltadd 10612  ax-pre-mulgt0 10613  ax-pre-sup 10614  ax-addf 10615  ax-mulf 10616
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4838  df-int 4876  df-iun 4920  df-iin 4921  df-br 5066  df-opab 5128  df-mpt 5146  df-tr 5172  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-se 5514  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-isom 6363  df-riota 7113  df-ov 7158  df-oprab 7159  df-mpo 7160  df-of 7408  df-om 7580  df-1st 7688  df-2nd 7689  df-supp 7830  df-wrecs 7946  df-recs 8007  df-rdg 8045  df-1o 8101  df-2o 8102  df-oadd 8105  df-er 8288  df-map 8407  df-ixp 8461  df-en 8509  df-dom 8510  df-sdom 8511  df-fin 8512  df-fsupp 8833  df-fi 8874  df-sup 8905  df-inf 8906  df-oi 8973  df-card 9367  df-pnf 10676  df-mnf 10677  df-xr 10678  df-ltxr 10679  df-le 10680  df-sub 10871  df-neg 10872  df-div 11297  df-nn 11638  df-2 11699  df-3 11700  df-4 11701  df-5 11702  df-6 11703  df-7 11704  df-8 11705  df-9 11706  df-n0 11897  df-z 11981  df-dec 12098  df-uz 12243  df-q 12348  df-rp 12389  df-xneg 12506  df-xadd 12507  df-xmul 12508  df-ioo 12741  df-icc 12744  df-fz 12892  df-fzo 13033  df-seq 13369  df-exp 13429  df-hash 13690  df-cj 14457  df-re 14458  df-im 14459  df-sqrt 14593  df-abs 14594  df-struct 16484  df-ndx 16485  df-slot 16486  df-base 16488  df-sets 16489  df-ress 16490  df-plusg 16577  df-mulr 16578  df-starv 16579  df-sca 16580  df-vsca 16581  df-ip 16582  df-tset 16583  df-ple 16584  df-ds 16586  df-unif 16587  df-hom 16588  df-cco 16589  df-rest 16695  df-topn 16696  df-0g 16714  df-gsum 16715  df-topgen 16716  df-pt 16717  df-prds 16720  df-xrs 16774  df-qtop 16779  df-imas 16780  df-xps 16782  df-mre 16856  df-mrc 16857  df-acs 16859  df-mgm 17851  df-sgrp 17900  df-mnd 17911  df-submnd 17956  df-mulg 18224  df-cntz 18446  df-cmn 18907  df-psmet 20536  df-xmet 20537  df-met 20538  df-bl 20539  df-mopn 20540  df-cnfld 20545  df-top 21501  df-topon 21518  df-topsp 21540  df-bases 21553  df-cn 21834  df-cnp 21835  df-tx 22169  df-hmeo 22362  df-xms 22929  df-ms 22930  df-tms 22931  df-cncf 23485
This theorem is referenced by:  dya2iocnei  31540
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