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Theorem tpr2rico 30426
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 12386 . . . . . . . . . 10 (,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧𝑧 < 𝑦)})
21ixxf 12392 . . . . . . . . 9 (,):(ℝ* × ℝ*)⟶𝒫 ℝ*
3 ffn 6225 . . . . . . . . 9 ((,):(ℝ* × ℝ*)⟶𝒫 ℝ* → (,) Fn (ℝ* × ℝ*))
42, 3mp1i 13 . . . . . . . 8 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → (,) Fn (ℝ* × ℝ*))
5 elssuni 4627 . . . . . . . . . . . . . 14 (𝐴 ∈ (𝐽 ×t 𝐽) → 𝐴 (𝐽 ×t 𝐽))
6 tpr2rico.0 . . . . . . . . . . . . . . . 16 𝐽 = (topGen‘ran (,))
7 retop 22858 . . . . . . . . . . . . . . . 16 (topGen‘ran (,)) ∈ Top
86, 7eqeltri 2840 . . . . . . . . . . . . . . 15 𝐽 ∈ Top
9 uniretop 22859 . . . . . . . . . . . . . . . 16 ℝ = (topGen‘ran (,))
106unieqi 4605 . . . . . . . . . . . . . . . 16 𝐽 = (topGen‘ran (,))
119, 10eqtr4i 2790 . . . . . . . . . . . . . . 15 ℝ = 𝐽
128, 8, 11, 11txunii 21690 . . . . . . . . . . . . . 14 (ℝ × ℝ) = (𝐽 ×t 𝐽)
135, 12syl6sseqr 3814 . . . . . . . . . . . . 13 (𝐴 ∈ (𝐽 ×t 𝐽) → 𝐴 ⊆ (ℝ × ℝ))
1413ad2antrr 717 . . . . . . . . . . . 12 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → 𝐴 ⊆ (ℝ × ℝ))
15 simplr 785 . . . . . . . . . . . 12 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → 𝑋𝐴)
1614, 15sseldd 3764 . . . . . . . . . . 11 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → 𝑋 ∈ (ℝ × ℝ))
17 xp1st 7402 . . . . . . . . . . 11 (𝑋 ∈ (ℝ × ℝ) → (1st𝑋) ∈ ℝ)
1816, 17syl 17 . . . . . . . . . 10 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → (1st𝑋) ∈ ℝ)
19 simpr 477 . . . . . . . . . . . 12 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → 𝑑 ∈ ℝ+)
2019rpred 12075 . . . . . . . . . . 11 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → 𝑑 ∈ ℝ)
2120rehalfcld 11529 . . . . . . . . . 10 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → (𝑑 / 2) ∈ ℝ)
2218, 21resubcld 10716 . . . . . . . . 9 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → ((1st𝑋) − (𝑑 / 2)) ∈ ℝ)
2322rexrd 10347 . . . . . . . 8 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → ((1st𝑋) − (𝑑 / 2)) ∈ ℝ*)
2418, 21readdcld 10327 . . . . . . . . 9 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → ((1st𝑋) + (𝑑 / 2)) ∈ ℝ)
2524rexrd 10347 . . . . . . . 8 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → ((1st𝑋) + (𝑑 / 2)) ∈ ℝ*)
26 fnovrn 7011 . . . . . . . 8 (((,) Fn (ℝ* × ℝ*) ∧ ((1st𝑋) − (𝑑 / 2)) ∈ ℝ* ∧ ((1st𝑋) + (𝑑 / 2)) ∈ ℝ*) → (((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) ∈ ran (,))
274, 23, 25, 26syl3anc 1490 . . . . . . 7 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → (((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) ∈ ran (,))
28 xp2nd 7403 . . . . . . . . . . 11 (𝑋 ∈ (ℝ × ℝ) → (2nd𝑋) ∈ ℝ)
2916, 28syl 17 . . . . . . . . . 10 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → (2nd𝑋) ∈ ℝ)
3029, 21resubcld 10716 . . . . . . . . 9 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → ((2nd𝑋) − (𝑑 / 2)) ∈ ℝ)
3130rexrd 10347 . . . . . . . 8 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → ((2nd𝑋) − (𝑑 / 2)) ∈ ℝ*)
3229, 21readdcld 10327 . . . . . . . . 9 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → ((2nd𝑋) + (𝑑 / 2)) ∈ ℝ)
3332rexrd 10347 . . . . . . . 8 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → ((2nd𝑋) + (𝑑 / 2)) ∈ ℝ*)
34 fnovrn 7011 . . . . . . . 8 (((,) Fn (ℝ* × ℝ*) ∧ ((2nd𝑋) − (𝑑 / 2)) ∈ ℝ* ∧ ((2nd𝑋) + (𝑑 / 2)) ∈ ℝ*) → (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2))) ∈ ran (,))
354, 31, 33, 34syl3anc 1490 . . . . . . 7 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2))) ∈ ran (,))
36 eqidd 2766 . . . . . . 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 5293 . . . . . . . . 9 (𝑥 = (((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) → (𝑥 × 𝑦) = ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × 𝑦))
3837eqeq2d 2775 . . . . . . . 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 5300 . . . . . . . . 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 2775 . . . . . . . 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 3477 . . . . . . 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 1490 . . . . . 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 2765 . . . . . . 7 (𝑥 ∈ ran (,), 𝑦 ∈ ran (,) ↦ (𝑥 × 𝑦)) = (𝑥 ∈ ran (,), 𝑦 ∈ ran (,) ↦ (𝑥 × 𝑦))
44 vex 3353 . . . . . . . 8 𝑥 ∈ V
45 vex 3353 . . . . . . . 8 𝑦 ∈ V
4644, 45xpex 7164 . . . . . . 7 (𝑥 × 𝑦) ∈ V
4743, 46elrnmpt2 6975 . . . . . 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 225 . . . . 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, 49syl6eleqr 2855 . . . 4 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ∈ 𝐵)
5150ralrimiva 3113 . . 3 ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) → ∀𝑑 ∈ ℝ+ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ∈ 𝐵)
52 xpss 5295 . . . . . . 7 (ℝ × ℝ) ⊆ (V × V)
5352, 16sseldi 3761 . . . . . 6 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → 𝑋 ∈ (V × V))
5418rexrd 10347 . . . . . . . 8 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → (1st𝑋) ∈ ℝ*)
5519rphalfcld 12087 . . . . . . . . 9 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → (𝑑 / 2) ∈ ℝ+)
5618, 55ltsubrpd 12107 . . . . . . . 8 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → ((1st𝑋) − (𝑑 / 2)) < (1st𝑋))
5718, 55ltaddrpd 12108 . . . . . . . 8 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → (1st𝑋) < ((1st𝑋) + (𝑑 / 2)))
58 elioo1 12422 . . . . . . . . 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 579 . . . . . . . 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 1442 . . . . . . 7 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → (1st𝑋) ∈ (((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))))
6129rexrd 10347 . . . . . . . 8 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → (2nd𝑋) ∈ ℝ*)
6229, 55ltsubrpd 12107 . . . . . . . 8 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → ((2nd𝑋) − (𝑑 / 2)) < (2nd𝑋))
6329, 55ltaddrpd 12108 . . . . . . . 8 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → (2nd𝑋) < ((2nd𝑋) + (𝑑 / 2)))
64 elioo1 12422 . . . . . . . . 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 579 . . . . . . . 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 1442 . . . . . . 7 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → (2nd𝑋) ∈ (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2))))
6760, 66jca 507 . . . . . 6 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → ((1st𝑋) ∈ (((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) ∧ (2nd𝑋) ∈ (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))))
68 elxp7 7405 . . . . . 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 578 . . . . 5 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → 𝑋 ∈ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))))
7069ralrimiva 3113 . . . 4 ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) → ∀𝑑 ∈ ℝ+ 𝑋 ∈ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))))
71 mnfle 12174 . . . . . . . . . . . . . . . . . 18 (((1st𝑋) − (𝑑 / 2)) ∈ ℝ* → -∞ ≤ ((1st𝑋) − (𝑑 / 2)))
7223, 71syl 17 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → -∞ ≤ ((1st𝑋) − (𝑑 / 2)))
73 pnfge 12169 . . . . . . . . . . . . . . . . . 18 (((1st𝑋) + (𝑑 / 2)) ∈ ℝ* → ((1st𝑋) + (𝑑 / 2)) ≤ +∞)
7425, 73syl 17 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → ((1st𝑋) + (𝑑 / 2)) ≤ +∞)
75 mnfxr 10354 . . . . . . . . . . . . . . . . . 18 -∞ ∈ ℝ*
76 pnfxr 10350 . . . . . . . . . . . . . . . . . 18 +∞ ∈ ℝ*
77 ioossioo 12473 . . . . . . . . . . . . . . . . . 18 (((-∞ ∈ ℝ* ∧ +∞ ∈ ℝ*) ∧ (-∞ ≤ ((1st𝑋) − (𝑑 / 2)) ∧ ((1st𝑋) + (𝑑 / 2)) ≤ +∞)) → (((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) ⊆ (-∞(,)+∞))
7875, 76, 77mpanl12 693 . . . . . . . . . . . . . . . . 17 ((-∞ ≤ ((1st𝑋) − (𝑑 / 2)) ∧ ((1st𝑋) + (𝑑 / 2)) ≤ +∞) → (((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) ⊆ (-∞(,)+∞))
7972, 74, 78syl2anc 579 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → (((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) ⊆ (-∞(,)+∞))
80 ioomax 12455 . . . . . . . . . . . . . . . 16 (-∞(,)+∞) = ℝ
8179, 80syl6sseq 3813 . . . . . . . . . . . . . . 15 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → (((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) ⊆ ℝ)
82 mnfle 12174 . . . . . . . . . . . . . . . . . 18 (((2nd𝑋) − (𝑑 / 2)) ∈ ℝ* → -∞ ≤ ((2nd𝑋) − (𝑑 / 2)))
8331, 82syl 17 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → -∞ ≤ ((2nd𝑋) − (𝑑 / 2)))
84 pnfge 12169 . . . . . . . . . . . . . . . . . 18 (((2nd𝑋) + (𝑑 / 2)) ∈ ℝ* → ((2nd𝑋) + (𝑑 / 2)) ≤ +∞)
8533, 84syl 17 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → ((2nd𝑋) + (𝑑 / 2)) ≤ +∞)
86 ioossioo 12473 . . . . . . . . . . . . . . . . . 18 (((-∞ ∈ ℝ* ∧ +∞ ∈ ℝ*) ∧ (-∞ ≤ ((2nd𝑋) − (𝑑 / 2)) ∧ ((2nd𝑋) + (𝑑 / 2)) ≤ +∞)) → (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2))) ⊆ (-∞(,)+∞))
8775, 76, 86mpanl12 693 . . . . . . . . . . . . . . . . 17 ((-∞ ≤ ((2nd𝑋) − (𝑑 / 2)) ∧ ((2nd𝑋) + (𝑑 / 2)) ≤ +∞) → (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2))) ⊆ (-∞(,)+∞))
8883, 85, 87syl2anc 579 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2))) ⊆ (-∞(,)+∞))
8988, 80syl6sseq 3813 . . . . . . . . . . . . . . 15 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2))) ⊆ ℝ)
90 xpss12 5294 . . . . . . . . . . . . . . 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 579 . . . . . . . . . . . . . 14 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ⊆ (ℝ × ℝ))
9291sselda 3763 . . . . . . . . . . . . 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 402 . . . . . . . . . . . 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 546 . . . . . . . . . . 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 565 . . . . . . . . . 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 472 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ (𝑋𝐴𝑑 ∈ ℝ+𝑥 ∈ (ℝ × ℝ))) → 𝐴 ⊆ (ℝ × ℝ))
97 simpr1 1248 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ (𝑋𝐴𝑑 ∈ ℝ+𝑥 ∈ (ℝ × ℝ))) → 𝑋𝐴)
9896, 97sseldd 3764 . . . . . . . . . . . . . . 15 ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ (𝑋𝐴𝑑 ∈ ℝ+𝑥 ∈ (ℝ × ℝ))) → 𝑋 ∈ (ℝ × ℝ))
99983anassrs 1469 . . . . . . . . . . . . . 14 ((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) → 𝑋 ∈ (ℝ × ℝ))
100 simpr 477 . . . . . . . . . . . . . 14 ((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) → 𝑥 ∈ (ℝ × ℝ))
101 simplr 785 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) → 𝑑 ∈ ℝ+)
102101rphalfcld 12087 . . . . . . . . . . . . . 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 30425 . . . . . . . . . . . . . 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 1490 . . . . . . . . . . . . 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 2765 . . . . . . . . . . . . . . . . . . . . 21 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
107103, 6, 106cnrehmeo 23045 . . . . . . . . . . . . . . . . . . . 20 𝐺 ∈ ((𝐽 ×t 𝐽)Homeo(TopOpen‘ℂfld))
108106cnfldtopon 22879 . . . . . . . . . . . . . . . . . . . . . 22 (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
109108toponunii 21014 . . . . . . . . . . . . . . . . . . . . 21 ℂ = (TopOpen‘ℂfld)
11012, 109hmeof1o 21861 . . . . . . . . . . . . . . . . . . . 20 (𝐺 ∈ ((𝐽 ×t 𝐽)Homeo(TopOpen‘ℂfld)) → 𝐺:(ℝ × ℝ)–1-1-onto→ℂ)
111 f1of 6324 . . . . . . . . . . . . . . . . . . . 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 6554 . . . . . . . . . . . . . . . . 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 6553 . . . . . . . . . . . . . . . . 17 ((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) → (𝐺𝑥) ∈ ℂ)
117 sqsscirc2 30423 . . . . . . . . . . . . . . . . 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 866 . . . . . . . . . . . . . . . 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 395 . . . . . . . . . . . . . . 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 12076 . . . . . . . . . . . . . . . . . 18 ((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) → 𝑑 ∈ ℝ*)
121120adantr 472 . . . . . . . . . . . . . . . . 17 (((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) ∧ (abs‘((𝐺𝑥) − (𝐺𝑋))) < 𝑑) → 𝑑 ∈ ℝ*)
122 cnxmet 22869 . . . . . . . . . . . . . . . . 17 (abs ∘ − ) ∈ (∞Met‘ℂ)
123121, 122jctil 515 . . . . . . . . . . . . . . . 16 (((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) ∧ (abs‘((𝐺𝑥) − (𝐺𝑋))) < 𝑑) → ((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑑 ∈ ℝ*))
124114adantr 472 . . . . . . . . . . . . . . . . 17 (((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) ∧ (abs‘((𝐺𝑥) − (𝐺𝑋))) < 𝑑) → (𝐺𝑋) ∈ ℂ)
125116adantr 472 . . . . . . . . . . . . . . . . 17 (((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) ∧ (abs‘((𝐺𝑥) − (𝐺𝑋))) < 𝑑) → (𝐺𝑥) ∈ ℂ)
126124, 125jca 507 . . . . . . . . . . . . . . . 16 (((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) ∧ (abs‘((𝐺𝑥) − (𝐺𝑋))) < 𝑑) → ((𝐺𝑋) ∈ ℂ ∧ (𝐺𝑥) ∈ ℂ))
127 eqid 2765 . . . . . . . . . . . . . . . . . . 19 (abs ∘ − ) = (abs ∘ − )
128127cnmetdval 22867 . . . . . . . . . . . . . . . . . 18 (((𝐺𝑥) ∈ ℂ ∧ (𝐺𝑋) ∈ ℂ) → ((𝐺𝑥)(abs ∘ − )(𝐺𝑋)) = (abs‘((𝐺𝑥) − (𝐺𝑋))))
129125, 124, 128syl2anc 579 . . . . . . . . . . . . . . . . 17 (((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) ∧ (abs‘((𝐺𝑥) − (𝐺𝑋))) < 𝑑) → ((𝐺𝑥)(abs ∘ − )(𝐺𝑋)) = (abs‘((𝐺𝑥) − (𝐺𝑋))))
130 simpr 477 . . . . . . . . . . . . . . . . 17 (((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) ∧ (abs‘((𝐺𝑥) − (𝐺𝑋))) < 𝑑) → (abs‘((𝐺𝑥) − (𝐺𝑋))) < 𝑑)
131129, 130eqbrtrd 4833 . . . . . . . . . . . . . . . 16 (((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) ∧ (abs‘((𝐺𝑥) − (𝐺𝑋))) < 𝑑) → ((𝐺𝑥)(abs ∘ − )(𝐺𝑋)) < 𝑑)
132 elbl3 22490 . . . . . . . . . . . . . . . . 17 ((((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑑 ∈ ℝ*) ∧ ((𝐺𝑋) ∈ ℂ ∧ (𝐺𝑥) ∈ ℂ)) → ((𝐺𝑥) ∈ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑) ↔ ((𝐺𝑥)(abs ∘ − )(𝐺𝑋)) < 𝑑))
133132biimpar 469 . . . . . . . . . . . . . . . 16 (((((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑑 ∈ ℝ*) ∧ ((𝐺𝑋) ∈ ℂ ∧ (𝐺𝑥) ∈ ℂ)) ∧ ((𝐺𝑥)(abs ∘ − )(𝐺𝑋)) < 𝑑) → (𝐺𝑥) ∈ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑))
134123, 126, 131, 133syl21anc 866 . . . . . . . . . . . . . . 15 (((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) ∧ (abs‘((𝐺𝑥) − (𝐺𝑋))) < 𝑑) → (𝐺𝑥) ∈ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑))
135119, 134syldan 585 . . . . . . . . . . . . . 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 401 . . . . . . . . . . . . 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 6336 . . . . . . . . . . . . . . 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 6326 . . . . . . . . . . . . . 14 (𝐺:ℂ–1-1-onto→(ℝ × ℝ) → Fun 𝐺)
141139, 140ax-mp 5 . . . . . . . . . . . . 13 Fun 𝐺
142 f1odm 6328 . . . . . . . . . . . . . . 15 (𝐺:ℂ–1-1-onto→(ℝ × ℝ) → dom 𝐺 = ℂ)
143139, 142ax-mp 5 . . . . . . . . . . . . . 14 dom 𝐺 = ℂ
144116, 143syl6eleqr 2855 . . . . . . . . . . . . 13 ((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) → (𝐺𝑥) ∈ dom 𝐺)
145 funfvima 6689 . . . . . . . . . . . . 13 ((Fun 𝐺 ∧ (𝐺𝑥) ∈ dom 𝐺) → ((𝐺𝑥) ∈ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑) → (𝐺‘(𝐺𝑥)) ∈ (𝐺 “ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑))))
146141, 144, 145sylancr 581 . . . . . . . . . . . 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 6728 . . . . . . . . . . . . . . 15 ((𝐺:(ℝ × ℝ)–1-1-onto→ℂ ∧ 𝑥 ∈ (ℝ × ℝ)) → (𝐺‘(𝐺𝑥)) = 𝑥)
149147, 100, 148syl2anc 579 . . . . . . . . . . . . . 14 ((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) → (𝐺‘(𝐺𝑥)) = 𝑥)
150149eleq1d 2829 . . . . . . . . . . . . 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 220 . . . . . . . . . . . 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 395 . . . . . . . . . 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 401 . . . . . . . 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 3769 . . . . . . 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 3113 . . . . . 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)))
158103mpt2fun 6964 . . . . . . . . . 10 Fun 𝐺
159158a1i 11 . . . . . . . . 9 ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) → Fun 𝐺)
16013sselda 3763 . . . . . . . . . 10 ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) → 𝑋 ∈ (ℝ × ℝ))
161 f1odm 6328 . . . . . . . . . . 11 (𝐺:(ℝ × ℝ)–1-1-onto→ℂ → dom 𝐺 = (ℝ × ℝ))
162107, 110, 161mp2b 10 . . . . . . . . . 10 dom 𝐺 = (ℝ × ℝ)
163160, 162syl6eleqr 2855 . . . . . . . . 9 ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) → 𝑋 ∈ dom 𝐺)
164 simpr 477 . . . . . . . . 9 ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) → 𝑋𝐴)
165 funfvima 6689 . . . . . . . . . 10 ((Fun 𝐺𝑋 ∈ dom 𝐺) → (𝑋𝐴 → (𝐺𝑋) ∈ (𝐺𝐴)))
166165imp 395 . . . . . . . . 9 (((Fun 𝐺𝑋 ∈ dom 𝐺) ∧ 𝑋𝐴) → (𝐺𝑋) ∈ (𝐺𝐴))
167159, 163, 164, 166syl21anc 866 . . . . . . . 8 ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) → (𝐺𝑋) ∈ (𝐺𝐴))
168 hmeoima 21862 . . . . . . . . . . 11 ((𝐺 ∈ ((𝐽 ×t 𝐽)Homeo(TopOpen‘ℂfld)) ∧ 𝐴 ∈ (𝐽 ×t 𝐽)) → (𝐺𝐴) ∈ (TopOpen‘ℂfld))
169107, 168mpan 681 . . . . . . . . . 10 (𝐴 ∈ (𝐽 ×t 𝐽) → (𝐺𝐴) ∈ (TopOpen‘ℂfld))
170106cnfldtopn 22878 . . . . . . . . . . . . 13 (TopOpen‘ℂfld) = (MetOpen‘(abs ∘ − ))
171170elmopn2 22543 . . . . . . . . . . . 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 490 . . . . . . . . . 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 472 . . . . . . . 8 ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) → ∀𝑚 ∈ (𝐺𝐴)∃𝑑 ∈ ℝ+ (𝑚(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺𝐴))
176 oveq1 6853 . . . . . . . . . . 11 (𝑚 = (𝐺𝑋) → (𝑚(ball‘(abs ∘ − ))𝑑) = ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑))
177176sseq1d 3794 . . . . . . . . . 10 (𝑚 = (𝐺𝑋) → ((𝑚(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺𝐴) ↔ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺𝐴)))
178177rexbidv 3199 . . . . . . . . 9 (𝑚 = (𝐺𝑋) → (∃𝑑 ∈ ℝ+ (𝑚(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺𝐴) ↔ ∃𝑑 ∈ ℝ+ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺𝐴)))
179178rspcva 3460 . . . . . . . 8 (((𝐺𝑋) ∈ (𝐺𝐴) ∧ ∀𝑚 ∈ (𝐺𝐴)∃𝑑 ∈ ℝ+ (𝑚(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺𝐴)) → ∃𝑑 ∈ ℝ+ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺𝐴))
180167, 175, 179syl2anc 579 . . . . . . 7 ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) → ∃𝑑 ∈ ℝ+ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺𝐴))
181 imass2 5685 . . . . . . . . . 10 (((𝐺𝑋)(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺𝐴) → (𝐺 “ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑)) ⊆ (𝐺 “ (𝐺𝐴)))
182 f1of1 6323 . . . . . . . . . . . . 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 6340 . . . . . . . . . . . 12 ((𝐺:(ℝ × ℝ)–1-1→ℂ ∧ 𝐴 ⊆ (ℝ × ℝ)) → (𝐺 “ (𝐺𝐴)) = 𝐴)
185183, 13, 184sylancr 581 . . . . . . . . . . 11 (𝐴 ∈ (𝐽 ×t 𝐽) → (𝐺 “ (𝐺𝐴)) = 𝐴)
186185sseq2d 3795 . . . . . . . . . 10 (𝐴 ∈ (𝐽 ×t 𝐽) → ((𝐺 “ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑)) ⊆ (𝐺 “ (𝐺𝐴)) ↔ (𝐺 “ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑)) ⊆ 𝐴))
187181, 186syl5ib 235 . . . . . . . . 9 (𝐴 ∈ (𝐽 ×t 𝐽) → (((𝐺𝑋)(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺𝐴) → (𝐺 “ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑)) ⊆ 𝐴))
188187reximdv 3162 . . . . . . . 8 (𝐴 ∈ (𝐽 ×t 𝐽) → (∃𝑑 ∈ ℝ+ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺𝐴) → ∃𝑑 ∈ ℝ+ (𝐺 “ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑)) ⊆ 𝐴))
189188adantr 472 . . . . . . 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 3219 . . . . . 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 579 . . . . 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 3771 . . . . . 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 3157 . . . . 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 3219 . . . 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 579 . . 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 3219 . . 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 579 . 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 2833 . . . . 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 3788 . . . . 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 624 . . . 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 3462 . . 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 3176 . 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 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  wral 3055  wrex 3056  Vcvv 3350  wss 3734  𝒫 cpw 4317   cuni 4596   class class class wbr 4811   × cxp 5277  ccnv 5278  dom cdm 5279  ran crn 5280  cima 5282  ccom 5283  Fun wfun 6064   Fn wfn 6065  wf 6066  1-1wf1 6067  1-1-ontowf1o 6069  cfv 6070  (class class class)co 6846  cmpt2 6848  1st c1st 7368  2nd c2nd 7369  cc 10191  cr 10192  ici 10195   + caddc 10196   · cmul 10198  +∞cpnf 10329  -∞cmnf 10330  *cxr 10331   < clt 10332  cle 10333  cmin 10524   / cdiv 10942  2c2 11331  +crp 12033  (,)cioo 12382  cre 14136  cim 14137  abscabs 14273  TopOpenctopn 16362  topGenctg 16378  ∞Metcxmet 20018  ballcbl 20020  fldccnfld 20033  Topctop 20991   ×t ctx 21657  Homeochmeo 21850
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4932  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7151  ax-inf2 8757  ax-cnex 10249  ax-resscn 10250  ax-1cn 10251  ax-icn 10252  ax-addcl 10253  ax-addrcl 10254  ax-mulcl 10255  ax-mulrcl 10256  ax-mulcom 10257  ax-addass 10258  ax-mulass 10259  ax-distr 10260  ax-i2m1 10261  ax-1ne0 10262  ax-1rid 10263  ax-rnegex 10264  ax-rrecex 10265  ax-cnre 10266  ax-pre-lttri 10267  ax-pre-lttrn 10268  ax-pre-ltadd 10269  ax-pre-mulgt0 10270  ax-pre-sup 10271  ax-addf 10272  ax-mulf 10273
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-pss 3750  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-tp 4341  df-op 4343  df-uni 4597  df-int 4636  df-iun 4680  df-iin 4681  df-br 4812  df-opab 4874  df-mpt 4891  df-tr 4914  df-id 5187  df-eprel 5192  df-po 5200  df-so 5201  df-fr 5238  df-se 5239  df-we 5240  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-pred 5867  df-ord 5913  df-on 5914  df-lim 5915  df-suc 5916  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-isom 6079  df-riota 6807  df-ov 6849  df-oprab 6850  df-mpt2 6851  df-of 7099  df-om 7268  df-1st 7370  df-2nd 7371  df-supp 7502  df-wrecs 7614  df-recs 7676  df-rdg 7714  df-1o 7768  df-2o 7769  df-oadd 7772  df-er 7951  df-map 8066  df-ixp 8118  df-en 8165  df-dom 8166  df-sdom 8167  df-fin 8168  df-fsupp 8487  df-fi 8528  df-sup 8559  df-inf 8560  df-oi 8626  df-card 9020  df-cda 9247  df-pnf 10334  df-mnf 10335  df-xr 10336  df-ltxr 10337  df-le 10338  df-sub 10526  df-neg 10527  df-div 10943  df-nn 11279  df-2 11339  df-3 11340  df-4 11341  df-5 11342  df-6 11343  df-7 11344  df-8 11345  df-9 11346  df-n0 11543  df-z 11629  df-dec 11746  df-uz 11892  df-q 11995  df-rp 12034  df-xneg 12151  df-xadd 12152  df-xmul 12153  df-ioo 12386  df-icc 12389  df-fz 12539  df-fzo 12679  df-seq 13014  df-exp 13073  df-hash 13327  df-cj 14138  df-re 14139  df-im 14140  df-sqrt 14274  df-abs 14275  df-struct 16146  df-ndx 16147  df-slot 16148  df-base 16150  df-sets 16151  df-ress 16152  df-plusg 16241  df-mulr 16242  df-starv 16243  df-sca 16244  df-vsca 16245  df-ip 16246  df-tset 16247  df-ple 16248  df-ds 16250  df-unif 16251  df-hom 16252  df-cco 16253  df-rest 16363  df-topn 16364  df-0g 16382  df-gsum 16383  df-topgen 16384  df-pt 16385  df-prds 16388  df-xrs 16442  df-qtop 16447  df-imas 16448  df-xps 16450  df-mre 16526  df-mrc 16527  df-acs 16529  df-mgm 17522  df-sgrp 17564  df-mnd 17575  df-submnd 17616  df-mulg 17822  df-cntz 18027  df-cmn 18475  df-psmet 20025  df-xmet 20026  df-met 20027  df-bl 20028  df-mopn 20029  df-cnfld 20034  df-top 20992  df-topon 21009  df-topsp 21031  df-bases 21044  df-cn 21325  df-cnp 21326  df-tx 21659  df-hmeo 21852  df-xms 22418  df-ms 22419  df-tms 22420  df-cncf 22974
This theorem is referenced by:  dya2iocnei  30812
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