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Theorem tpr2rico 32493
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 13268 . . . . . . . . . 10 (,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧𝑧 < 𝑦)})
21ixxf 13274 . . . . . . . . 9 (,):(ℝ* × ℝ*)⟶𝒫 ℝ*
3 ffn 6668 . . . . . . . . 9 ((,):(ℝ* × ℝ*)⟶𝒫 ℝ* → (,) Fn (ℝ* × ℝ*))
42, 3mp1i 13 . . . . . . . 8 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → (,) Fn (ℝ* × ℝ*))
5 elssuni 4898 . . . . . . . . . . . . . 14 (𝐴 ∈ (𝐽 ×t 𝐽) → 𝐴 (𝐽 ×t 𝐽))
6 tpr2rico.0 . . . . . . . . . . . . . . . 16 𝐽 = (topGen‘ran (,))
7 retop 24125 . . . . . . . . . . . . . . . 16 (topGen‘ran (,)) ∈ Top
86, 7eqeltri 2834 . . . . . . . . . . . . . . 15 𝐽 ∈ Top
9 uniretop 24126 . . . . . . . . . . . . . . . 16 ℝ = (topGen‘ran (,))
106unieqi 4878 . . . . . . . . . . . . . . . 16 𝐽 = (topGen‘ran (,))
119, 10eqtr4i 2767 . . . . . . . . . . . . . . 15 ℝ = 𝐽
128, 8, 11, 11txunii 22944 . . . . . . . . . . . . . 14 (ℝ × ℝ) = (𝐽 ×t 𝐽)
135, 12sseqtrrdi 3995 . . . . . . . . . . . . 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 3945 . . . . . . . . . . 11 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → 𝑋 ∈ (ℝ × ℝ))
17 xp1st 7953 . . . . . . . . . . 11 (𝑋 ∈ (ℝ × ℝ) → (1st𝑋) ∈ ℝ)
1816, 17syl 17 . . . . . . . . . 10 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → (1st𝑋) ∈ ℝ)
19 simpr 485 . . . . . . . . . . . 12 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → 𝑑 ∈ ℝ+)
2019rpred 12957 . . . . . . . . . . 11 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → 𝑑 ∈ ℝ)
2120rehalfcld 12400 . . . . . . . . . 10 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → (𝑑 / 2) ∈ ℝ)
2218, 21resubcld 11583 . . . . . . . . 9 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → ((1st𝑋) − (𝑑 / 2)) ∈ ℝ)
2322rexrd 11205 . . . . . . . 8 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → ((1st𝑋) − (𝑑 / 2)) ∈ ℝ*)
2418, 21readdcld 11184 . . . . . . . . 9 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → ((1st𝑋) + (𝑑 / 2)) ∈ ℝ)
2524rexrd 11205 . . . . . . . 8 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → ((1st𝑋) + (𝑑 / 2)) ∈ ℝ*)
26 fnovrn 7529 . . . . . . . 8 (((,) Fn (ℝ* × ℝ*) ∧ ((1st𝑋) − (𝑑 / 2)) ∈ ℝ* ∧ ((1st𝑋) + (𝑑 / 2)) ∈ ℝ*) → (((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) ∈ ran (,))
274, 23, 25, 26syl3anc 1371 . . . . . . 7 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → (((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) ∈ ran (,))
28 xp2nd 7954 . . . . . . . . . . 11 (𝑋 ∈ (ℝ × ℝ) → (2nd𝑋) ∈ ℝ)
2916, 28syl 17 . . . . . . . . . 10 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → (2nd𝑋) ∈ ℝ)
3029, 21resubcld 11583 . . . . . . . . 9 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → ((2nd𝑋) − (𝑑 / 2)) ∈ ℝ)
3130rexrd 11205 . . . . . . . 8 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → ((2nd𝑋) − (𝑑 / 2)) ∈ ℝ*)
3229, 21readdcld 11184 . . . . . . . . 9 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → ((2nd𝑋) + (𝑑 / 2)) ∈ ℝ)
3332rexrd 11205 . . . . . . . 8 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → ((2nd𝑋) + (𝑑 / 2)) ∈ ℝ*)
34 fnovrn 7529 . . . . . . . 8 (((,) Fn (ℝ* × ℝ*) ∧ ((2nd𝑋) − (𝑑 / 2)) ∈ ℝ* ∧ ((2nd𝑋) + (𝑑 / 2)) ∈ ℝ*) → (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2))) ∈ ran (,))
354, 31, 33, 34syl3anc 1371 . . . . . . 7 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2))) ∈ ran (,))
36 eqidd 2737 . . . . . . 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 5647 . . . . . . . . 9 (𝑥 = (((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) → (𝑥 × 𝑦) = ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × 𝑦))
3837eqeq2d 2747 . . . . . . . 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 5654 . . . . . . . . 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 2747 . . . . . . . 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 3592 . . . . . . 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 1371 . . . . . 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 2736 . . . . . . 7 (𝑥 ∈ ran (,), 𝑦 ∈ ran (,) ↦ (𝑥 × 𝑦)) = (𝑥 ∈ ran (,), 𝑦 ∈ ran (,) ↦ (𝑥 × 𝑦))
44 vex 3449 . . . . . . . 8 𝑥 ∈ V
45 vex 3449 . . . . . . . 8 𝑦 ∈ V
4644, 45xpex 7687 . . . . . . 7 (𝑥 × 𝑦) ∈ V
4743, 46elrnmpo 7492 . . . . . 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 233 . . . . 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 2849 . . . 4 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ∈ 𝐵)
5150ralrimiva 3143 . . 3 ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) → ∀𝑑 ∈ ℝ+ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ∈ 𝐵)
52 xpss 5649 . . . . . . 7 (ℝ × ℝ) ⊆ (V × V)
5352, 16sselid 3942 . . . . . 6 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → 𝑋 ∈ (V × V))
5418rexrd 11205 . . . . . . . 8 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → (1st𝑋) ∈ ℝ*)
5519rphalfcld 12969 . . . . . . . . 9 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → (𝑑 / 2) ∈ ℝ+)
5618, 55ltsubrpd 12989 . . . . . . . 8 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → ((1st𝑋) − (𝑑 / 2)) < (1st𝑋))
5718, 55ltaddrpd 12990 . . . . . . . 8 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → (1st𝑋) < ((1st𝑋) + (𝑑 / 2)))
58 elioo1 13304 . . . . . . . . 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 584 . . . . . . . 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 1342 . . . . . . 7 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → (1st𝑋) ∈ (((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))))
6129rexrd 11205 . . . . . . . 8 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → (2nd𝑋) ∈ ℝ*)
6229, 55ltsubrpd 12989 . . . . . . . 8 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → ((2nd𝑋) − (𝑑 / 2)) < (2nd𝑋))
6329, 55ltaddrpd 12990 . . . . . . . 8 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → (2nd𝑋) < ((2nd𝑋) + (𝑑 / 2)))
64 elioo1 13304 . . . . . . . . 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 584 . . . . . . . 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 1342 . . . . . . 7 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → (2nd𝑋) ∈ (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2))))
6760, 66jca 512 . . . . . 6 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → ((1st𝑋) ∈ (((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) ∧ (2nd𝑋) ∈ (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))))
68 elxp7 7956 . . . . . 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 583 . . . . 5 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → 𝑋 ∈ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))))
7069ralrimiva 3143 . . . 4 ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) → ∀𝑑 ∈ ℝ+ 𝑋 ∈ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))))
71 mnfle 13055 . . . . . . . . . . . . . . . . . 18 (((1st𝑋) − (𝑑 / 2)) ∈ ℝ* → -∞ ≤ ((1st𝑋) − (𝑑 / 2)))
7223, 71syl 17 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → -∞ ≤ ((1st𝑋) − (𝑑 / 2)))
73 pnfge 13051 . . . . . . . . . . . . . . . . . 18 (((1st𝑋) + (𝑑 / 2)) ∈ ℝ* → ((1st𝑋) + (𝑑 / 2)) ≤ +∞)
7425, 73syl 17 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → ((1st𝑋) + (𝑑 / 2)) ≤ +∞)
75 mnfxr 11212 . . . . . . . . . . . . . . . . . 18 -∞ ∈ ℝ*
76 pnfxr 11209 . . . . . . . . . . . . . . . . . 18 +∞ ∈ ℝ*
77 ioossioo 13358 . . . . . . . . . . . . . . . . . 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 584 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → (((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) ⊆ (-∞(,)+∞))
80 ioomax 13339 . . . . . . . . . . . . . . . 16 (-∞(,)+∞) = ℝ
8179, 80sseqtrdi 3994 . . . . . . . . . . . . . . 15 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → (((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) ⊆ ℝ)
82 mnfle 13055 . . . . . . . . . . . . . . . . . 18 (((2nd𝑋) − (𝑑 / 2)) ∈ ℝ* → -∞ ≤ ((2nd𝑋) − (𝑑 / 2)))
8331, 82syl 17 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → -∞ ≤ ((2nd𝑋) − (𝑑 / 2)))
84 pnfge 13051 . . . . . . . . . . . . . . . . . 18 (((2nd𝑋) + (𝑑 / 2)) ∈ ℝ* → ((2nd𝑋) + (𝑑 / 2)) ≤ +∞)
8533, 84syl 17 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → ((2nd𝑋) + (𝑑 / 2)) ≤ +∞)
86 ioossioo 13358 . . . . . . . . . . . . . . . . . 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 584 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2))) ⊆ (-∞(,)+∞))
8988, 80sseqtrdi 3994 . . . . . . . . . . . . . . 15 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2))) ⊆ ℝ)
90 xpss12 5648 . . . . . . . . . . . . . . 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 584 . . . . . . . . . . . . . 14 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ⊆ (ℝ × ℝ))
9291sselda 3944 . . . . . . . . . . . . 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 414 . . . . . . . . . . . 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 551 . . . . . . . . . . 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 570 . . . . . . . . . 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 481 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ (𝑋𝐴𝑑 ∈ ℝ+𝑥 ∈ (ℝ × ℝ))) → 𝐴 ⊆ (ℝ × ℝ))
97 simpr1 1194 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ (𝑋𝐴𝑑 ∈ ℝ+𝑥 ∈ (ℝ × ℝ))) → 𝑋𝐴)
9896, 97sseldd 3945 . . . . . . . . . . . . . . 15 ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ (𝑋𝐴𝑑 ∈ ℝ+𝑥 ∈ (ℝ × ℝ))) → 𝑋 ∈ (ℝ × ℝ))
99983anassrs 1360 . . . . . . . . . . . . . 14 ((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) → 𝑋 ∈ (ℝ × ℝ))
100 simpr 485 . . . . . . . . . . . . . 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 12969 . . . . . . . . . . . . . 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 32492 . . . . . . . . . . . . . 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 1371 . . . . . . . . . . . . 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 2736 . . . . . . . . . . . . . . . . . . . . 21 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
107103, 6, 106cnrehmeo 24316 . . . . . . . . . . . . . . . . . . . 20 𝐺 ∈ ((𝐽 ×t 𝐽)Homeo(TopOpen‘ℂfld))
108106cnfldtopon 24146 . . . . . . . . . . . . . . . . . . . . . 22 (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
109108toponunii 22265 . . . . . . . . . . . . . . . . . . . . 21 ℂ = (TopOpen‘ℂfld)
11012, 109hmeof1o 23115 . . . . . . . . . . . . . . . . . . . 20 (𝐺 ∈ ((𝐽 ×t 𝐽)Homeo(TopOpen‘ℂfld)) → 𝐺:(ℝ × ℝ)–1-1-onto→ℂ)
111 f1of 6784 . . . . . . . . . . . . . . . . . . . 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, 99ffvelcdmd 7036 . . . . . . . . . . . . . . . . 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)
116115ffvelcdmda 7035 . . . . . . . . . . . . . . . . 17 ((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) → (𝐺𝑥) ∈ ℂ)
117 sqsscirc2 32490 . . . . . . . . . . . . . . . . 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 836 . . . . . . . . . . . . . . . 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 407 . . . . . . . . . . . . . . 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 12958 . . . . . . . . . . . . . . . . . 18 ((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) → 𝑑 ∈ ℝ*)
121120adantr 481 . . . . . . . . . . . . . . . . 17 (((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) ∧ (abs‘((𝐺𝑥) − (𝐺𝑋))) < 𝑑) → 𝑑 ∈ ℝ*)
122 cnxmet 24136 . . . . . . . . . . . . . . . . 17 (abs ∘ − ) ∈ (∞Met‘ℂ)
123121, 122jctil 520 . . . . . . . . . . . . . . . 16 (((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) ∧ (abs‘((𝐺𝑥) − (𝐺𝑋))) < 𝑑) → ((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑑 ∈ ℝ*))
124114adantr 481 . . . . . . . . . . . . . . . . 17 (((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) ∧ (abs‘((𝐺𝑥) − (𝐺𝑋))) < 𝑑) → (𝐺𝑋) ∈ ℂ)
125116adantr 481 . . . . . . . . . . . . . . . . 17 (((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) ∧ (abs‘((𝐺𝑥) − (𝐺𝑋))) < 𝑑) → (𝐺𝑥) ∈ ℂ)
126124, 125jca 512 . . . . . . . . . . . . . . . 16 (((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) ∧ (abs‘((𝐺𝑥) − (𝐺𝑋))) < 𝑑) → ((𝐺𝑋) ∈ ℂ ∧ (𝐺𝑥) ∈ ℂ))
127 eqid 2736 . . . . . . . . . . . . . . . . . . 19 (abs ∘ − ) = (abs ∘ − )
128127cnmetdval 24134 . . . . . . . . . . . . . . . . . 18 (((𝐺𝑥) ∈ ℂ ∧ (𝐺𝑋) ∈ ℂ) → ((𝐺𝑥)(abs ∘ − )(𝐺𝑋)) = (abs‘((𝐺𝑥) − (𝐺𝑋))))
129125, 124, 128syl2anc 584 . . . . . . . . . . . . . . . . 17 (((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) ∧ (abs‘((𝐺𝑥) − (𝐺𝑋))) < 𝑑) → ((𝐺𝑥)(abs ∘ − )(𝐺𝑋)) = (abs‘((𝐺𝑥) − (𝐺𝑋))))
130 simpr 485 . . . . . . . . . . . . . . . . 17 (((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) ∧ (abs‘((𝐺𝑥) − (𝐺𝑋))) < 𝑑) → (abs‘((𝐺𝑥) − (𝐺𝑋))) < 𝑑)
131129, 130eqbrtrd 5127 . . . . . . . . . . . . . . . 16 (((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) ∧ (abs‘((𝐺𝑥) − (𝐺𝑋))) < 𝑑) → ((𝐺𝑥)(abs ∘ − )(𝐺𝑋)) < 𝑑)
132 elbl3 23745 . . . . . . . . . . . . . . . . 17 ((((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑑 ∈ ℝ*) ∧ ((𝐺𝑋) ∈ ℂ ∧ (𝐺𝑥) ∈ ℂ)) → ((𝐺𝑥) ∈ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑) ↔ ((𝐺𝑥)(abs ∘ − )(𝐺𝑋)) < 𝑑))
133132biimpar 478 . . . . . . . . . . . . . . . 16 (((((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑑 ∈ ℝ*) ∧ ((𝐺𝑋) ∈ ℂ ∧ (𝐺𝑥) ∈ ℂ)) ∧ ((𝐺𝑥)(abs ∘ − )(𝐺𝑋)) < 𝑑) → (𝐺𝑥) ∈ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑))
134123, 126, 131, 133syl21anc 836 . . . . . . . . . . . . . . 15 (((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) ∧ (abs‘((𝐺𝑥) − (𝐺𝑋))) < 𝑑) → (𝐺𝑥) ∈ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑))
135119, 134syldan 591 . . . . . . . . . . . . . 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 413 . . . . . . . . . . . . 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 6796 . . . . . . . . . . . . . . 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 6786 . . . . . . . . . . . . . 14 (𝐺:ℂ–1-1-onto→(ℝ × ℝ) → Fun 𝐺)
141139, 140ax-mp 5 . . . . . . . . . . . . 13 Fun 𝐺
142 f1odm 6788 . . . . . . . . . . . . . . 15 (𝐺:ℂ–1-1-onto→(ℝ × ℝ) → dom 𝐺 = ℂ)
143139, 142ax-mp 5 . . . . . . . . . . . . . 14 dom 𝐺 = ℂ
144116, 143eleqtrrdi 2849 . . . . . . . . . . . . 13 ((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) → (𝐺𝑥) ∈ dom 𝐺)
145 funfvima 7180 . . . . . . . . . . . . 13 ((Fun 𝐺 ∧ (𝐺𝑥) ∈ dom 𝐺) → ((𝐺𝑥) ∈ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑) → (𝐺‘(𝐺𝑥)) ∈ (𝐺 “ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑))))
146141, 144, 145sylancr 587 . . . . . . . . . . . 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 7222 . . . . . . . . . . . . . . 15 ((𝐺:(ℝ × ℝ)–1-1-onto→ℂ ∧ 𝑥 ∈ (ℝ × ℝ)) → (𝐺‘(𝐺𝑥)) = 𝑥)
149147, 100, 148syl2anc 584 . . . . . . . . . . . . . 14 ((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) → (𝐺‘(𝐺𝑥)) = 𝑥)
150149eleq1d 2822 . . . . . . . . . . . . 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 228 . . . . . . . . . . . 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 407 . . . . . . . . . 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 413 . . . . . . . 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 3950 . . . . . . 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 3143 . . . . . 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 7480 . . . . . . . . . 10 Fun 𝐺
159158a1i 11 . . . . . . . . 9 ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) → Fun 𝐺)
16013sselda 3944 . . . . . . . . . 10 ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) → 𝑋 ∈ (ℝ × ℝ))
161 f1odm 6788 . . . . . . . . . . 11 (𝐺:(ℝ × ℝ)–1-1-onto→ℂ → dom 𝐺 = (ℝ × ℝ))
162107, 110, 161mp2b 10 . . . . . . . . . 10 dom 𝐺 = (ℝ × ℝ)
163160, 162eleqtrrdi 2849 . . . . . . . . 9 ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) → 𝑋 ∈ dom 𝐺)
164 simpr 485 . . . . . . . . 9 ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) → 𝑋𝐴)
165 funfvima 7180 . . . . . . . . . 10 ((Fun 𝐺𝑋 ∈ dom 𝐺) → (𝑋𝐴 → (𝐺𝑋) ∈ (𝐺𝐴)))
166165imp 407 . . . . . . . . 9 (((Fun 𝐺𝑋 ∈ dom 𝐺) ∧ 𝑋𝐴) → (𝐺𝑋) ∈ (𝐺𝐴))
167159, 163, 164, 166syl21anc 836 . . . . . . . 8 ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) → (𝐺𝑋) ∈ (𝐺𝐴))
168 hmeoima 23116 . . . . . . . . . . 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 24145 . . . . . . . . . . . . 13 (TopOpen‘ℂfld) = (MetOpen‘(abs ∘ − ))
171170elmopn2 23798 . . . . . . . . . . . 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 497 . . . . . . . . . 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 481 . . . . . . . 8 ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) → ∀𝑚 ∈ (𝐺𝐴)∃𝑑 ∈ ℝ+ (𝑚(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺𝐴))
176 oveq1 7364 . . . . . . . . . . 11 (𝑚 = (𝐺𝑋) → (𝑚(ball‘(abs ∘ − ))𝑑) = ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑))
177176sseq1d 3975 . . . . . . . . . 10 (𝑚 = (𝐺𝑋) → ((𝑚(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺𝐴) ↔ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺𝐴)))
178177rexbidv 3175 . . . . . . . . 9 (𝑚 = (𝐺𝑋) → (∃𝑑 ∈ ℝ+ (𝑚(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺𝐴) ↔ ∃𝑑 ∈ ℝ+ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺𝐴)))
179178rspcva 3579 . . . . . . . 8 (((𝐺𝑋) ∈ (𝐺𝐴) ∧ ∀𝑚 ∈ (𝐺𝐴)∃𝑑 ∈ ℝ+ (𝑚(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺𝐴)) → ∃𝑑 ∈ ℝ+ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺𝐴))
180167, 175, 179syl2anc 584 . . . . . . 7 ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) → ∃𝑑 ∈ ℝ+ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺𝐴))
181 imass2 6054 . . . . . . . . . 10 (((𝐺𝑋)(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺𝐴) → (𝐺 “ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑)) ⊆ (𝐺 “ (𝐺𝐴)))
182 f1of1 6783 . . . . . . . . . . . . 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 6800 . . . . . . . . . . . 12 ((𝐺:(ℝ × ℝ)–1-1→ℂ ∧ 𝐴 ⊆ (ℝ × ℝ)) → (𝐺 “ (𝐺𝐴)) = 𝐴)
185183, 13, 184sylancr 587 . . . . . . . . . . 11 (𝐴 ∈ (𝐽 ×t 𝐽) → (𝐺 “ (𝐺𝐴)) = 𝐴)
186185sseq2d 3976 . . . . . . . . . 10 (𝐴 ∈ (𝐽 ×t 𝐽) → ((𝐺 “ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑)) ⊆ (𝐺 “ (𝐺𝐴)) ↔ (𝐺 “ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑)) ⊆ 𝐴))
187181, 186imbitrid 243 . . . . . . . . 9 (𝐴 ∈ (𝐽 ×t 𝐽) → (((𝐺𝑋)(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺𝐴) → (𝐺 “ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑)) ⊆ 𝐴))
188187reximdv 3167 . . . . . . . 8 (𝐴 ∈ (𝐽 ×t 𝐽) → (∃𝑑 ∈ ℝ+ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺𝐴) → ∃𝑑 ∈ ℝ+ (𝐺 “ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑)) ⊆ 𝐴))
189188adantr 481 . . . . . . 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 3117 . . . . . 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 584 . . . . 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 3952 . . . . . 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 3087 . . . . 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 3117 . . . 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 584 . . 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 3117 . . 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 584 . 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 2826 . . . . 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 3969 . . . . 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 631 . . . 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 3581 . . 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 3148 . 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 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3064  wrex 3073  Vcvv 3445  wss 3910  𝒫 cpw 4560   cuni 4865   class class class wbr 5105   × cxp 5631  ccnv 5632  dom cdm 5633  ran crn 5634  cima 5636  ccom 5637  Fun wfun 6490   Fn wfn 6491  wf 6492  1-1wf1 6493  1-1-ontowf1o 6495  cfv 6496  (class class class)co 7357  cmpo 7359  1st c1st 7919  2nd c2nd 7920  cc 11049  cr 11050  ici 11053   + caddc 11054   · cmul 11056  +∞cpnf 11186  -∞cmnf 11187  *cxr 11188   < clt 11189  cle 11190  cmin 11385   / cdiv 11812  2c2 12208  +crp 12915  (,)cioo 13264  cre 14982  cim 14983  abscabs 15119  TopOpenctopn 17303  topGenctg 17319  ∞Metcxmet 20781  ballcbl 20783  fldccnfld 20796  Topctop 22242   ×t ctx 22911  Homeochmeo 23104
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  ax-addf 11130  ax-mulf 11131
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-se 5589  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-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-er 8648  df-map 8767  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9306  df-fi 9347  df-sup 9378  df-inf 9379  df-oi 9446  df-card 9875  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-icc 13271  df-fz 13425  df-fzo 13568  df-seq 13907  df-exp 13968  df-hash 14231  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-starv 17148  df-sca 17149  df-vsca 17150  df-ip 17151  df-tset 17152  df-ple 17153  df-ds 17155  df-unif 17156  df-hom 17157  df-cco 17158  df-rest 17304  df-topn 17305  df-0g 17323  df-gsum 17324  df-topgen 17325  df-pt 17326  df-prds 17329  df-xrs 17384  df-qtop 17389  df-imas 17390  df-xps 17392  df-mre 17466  df-mrc 17467  df-acs 17469  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-submnd 18602  df-mulg 18873  df-cntz 19097  df-cmn 19564  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-cn 22578  df-cnp 22579  df-tx 22913  df-hmeo 23106  df-xms 23673  df-ms 23674  df-tms 23675  df-cncf 24241
This theorem is referenced by:  dya2iocnei  32882
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