Mathbox for Thierry Arnoux < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  tpr2rico Structured version   Visualization version   GIF version

Theorem tpr2rico 29286
 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 12050 . . . . . . . . . 10 (,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧𝑧 < 𝑦)})
21ixxf 12056 . . . . . . . . 9 (,):(ℝ* × ℝ*)⟶𝒫 ℝ*
3 ffn 5958 . . . . . . . . 9 ((,):(ℝ* × ℝ*)⟶𝒫 ℝ* → (,) Fn (ℝ* × ℝ*))
42, 3mp1i 13 . . . . . . . 8 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → (,) Fn (ℝ* × ℝ*))
5 elssuni 4403 . . . . . . . . . . . . . 14 (𝐴 ∈ (𝐽 ×t 𝐽) → 𝐴 (𝐽 ×t 𝐽))
6 tpr2rico.0 . . . . . . . . . . . . . . . 16 𝐽 = (topGen‘ran (,))
7 retop 22375 . . . . . . . . . . . . . . . 16 (topGen‘ran (,)) ∈ Top
86, 7eqeltri 2684 . . . . . . . . . . . . . . 15 𝐽 ∈ Top
9 uniretop 22376 . . . . . . . . . . . . . . . 16 ℝ = (topGen‘ran (,))
106unieqi 4381 . . . . . . . . . . . . . . . 16 𝐽 = (topGen‘ran (,))
119, 10eqtr4i 2635 . . . . . . . . . . . . . . 15 ℝ = 𝐽
128, 8, 11, 11txunii 21206 . . . . . . . . . . . . . 14 (ℝ × ℝ) = (𝐽 ×t 𝐽)
135, 12syl6sseqr 3615 . . . . . . . . . . . . 13 (𝐴 ∈ (𝐽 ×t 𝐽) → 𝐴 ⊆ (ℝ × ℝ))
1413ad2antrr 758 . . . . . . . . . . . 12 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → 𝐴 ⊆ (ℝ × ℝ))
15 simplr 788 . . . . . . . . . . . 12 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → 𝑋𝐴)
1614, 15sseldd 3569 . . . . . . . . . . 11 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → 𝑋 ∈ (ℝ × ℝ))
17 xp1st 7089 . . . . . . . . . . 11 (𝑋 ∈ (ℝ × ℝ) → (1st𝑋) ∈ ℝ)
1816, 17syl 17 . . . . . . . . . 10 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → (1st𝑋) ∈ ℝ)
19 simpr 476 . . . . . . . . . . . 12 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → 𝑑 ∈ ℝ+)
2019rpred 11748 . . . . . . . . . . 11 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → 𝑑 ∈ ℝ)
2120rehalfcld 11156 . . . . . . . . . 10 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → (𝑑 / 2) ∈ ℝ)
2218, 21resubcld 10337 . . . . . . . . 9 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → ((1st𝑋) − (𝑑 / 2)) ∈ ℝ)
2322rexrd 9968 . . . . . . . 8 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → ((1st𝑋) − (𝑑 / 2)) ∈ ℝ*)
2418, 21readdcld 9948 . . . . . . . . 9 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → ((1st𝑋) + (𝑑 / 2)) ∈ ℝ)
2524rexrd 9968 . . . . . . . 8 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → ((1st𝑋) + (𝑑 / 2)) ∈ ℝ*)
26 fnovrn 6707 . . . . . . . 8 (((,) Fn (ℝ* × ℝ*) ∧ ((1st𝑋) − (𝑑 / 2)) ∈ ℝ* ∧ ((1st𝑋) + (𝑑 / 2)) ∈ ℝ*) → (((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) ∈ ran (,))
274, 23, 25, 26syl3anc 1318 . . . . . . 7 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → (((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) ∈ ran (,))
28 xp2nd 7090 . . . . . . . . . . 11 (𝑋 ∈ (ℝ × ℝ) → (2nd𝑋) ∈ ℝ)
2916, 28syl 17 . . . . . . . . . 10 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → (2nd𝑋) ∈ ℝ)
3029, 21resubcld 10337 . . . . . . . . 9 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → ((2nd𝑋) − (𝑑 / 2)) ∈ ℝ)
3130rexrd 9968 . . . . . . . 8 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → ((2nd𝑋) − (𝑑 / 2)) ∈ ℝ*)
3229, 21readdcld 9948 . . . . . . . . 9 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → ((2nd𝑋) + (𝑑 / 2)) ∈ ℝ)
3332rexrd 9968 . . . . . . . 8 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → ((2nd𝑋) + (𝑑 / 2)) ∈ ℝ*)
34 fnovrn 6707 . . . . . . . 8 (((,) Fn (ℝ* × ℝ*) ∧ ((2nd𝑋) − (𝑑 / 2)) ∈ ℝ* ∧ ((2nd𝑋) + (𝑑 / 2)) ∈ ℝ*) → (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2))) ∈ ran (,))
354, 31, 33, 34syl3anc 1318 . . . . . . 7 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2))) ∈ ran (,))
36 eqidd 2611 . . . . . . 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 5052 . . . . . . . . 9 (𝑥 = (((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) → (𝑥 × 𝑦) = ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × 𝑦))
3837eqeq2d 2620 . . . . . . . 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 5053 . . . . . . . . 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 2620 . . . . . . . 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 3295 . . . . . . 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 1318 . . . . . 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 2610 . . . . . . 7 (𝑥 ∈ ran (,), 𝑦 ∈ ran (,) ↦ (𝑥 × 𝑦)) = (𝑥 ∈ ran (,), 𝑦 ∈ ran (,) ↦ (𝑥 × 𝑦))
44 vex 3176 . . . . . . . 8 𝑥 ∈ V
45 vex 3176 . . . . . . . 8 𝑦 ∈ V
4644, 45xpex 6860 . . . . . . 7 (𝑥 × 𝑦) ∈ V
4743, 46elrnmpt2 6671 . . . . . 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 223 . . . . 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 2699 . . . 4 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ∈ 𝐵)
5150ralrimiva 2949 . . 3 ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) → ∀𝑑 ∈ ℝ+ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ∈ 𝐵)
52 xpss 5149 . . . . . . 7 (ℝ × ℝ) ⊆ (V × V)
5352, 16sseldi 3566 . . . . . 6 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → 𝑋 ∈ (V × V))
5418rexrd 9968 . . . . . . . 8 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → (1st𝑋) ∈ ℝ*)
5519rphalfcld 11760 . . . . . . . . 9 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → (𝑑 / 2) ∈ ℝ+)
5618, 55ltsubrpd 11780 . . . . . . . 8 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → ((1st𝑋) − (𝑑 / 2)) < (1st𝑋))
5718, 55ltaddrpd 11781 . . . . . . . 8 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → (1st𝑋) < ((1st𝑋) + (𝑑 / 2)))
58 elioo1 12086 . . . . . . . . 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 691 . . . . . . . 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 1238 . . . . . . 7 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → (1st𝑋) ∈ (((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))))
6129rexrd 9968 . . . . . . . 8 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → (2nd𝑋) ∈ ℝ*)
6229, 55ltsubrpd 11780 . . . . . . . 8 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → ((2nd𝑋) − (𝑑 / 2)) < (2nd𝑋))
6329, 55ltaddrpd 11781 . . . . . . . 8 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → (2nd𝑋) < ((2nd𝑋) + (𝑑 / 2)))
64 elioo1 12086 . . . . . . . . 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 691 . . . . . . . 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 1238 . . . . . . 7 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → (2nd𝑋) ∈ (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2))))
6760, 66jca 553 . . . . . 6 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → ((1st𝑋) ∈ (((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) ∧ (2nd𝑋) ∈ (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))))
68 elxp7 7092 . . . . . 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 695 . . . . 5 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → 𝑋 ∈ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))))
7069ralrimiva 2949 . . . 4 ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) → ∀𝑑 ∈ ℝ+ 𝑋 ∈ ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))))
71 mnfle 11845 . . . . . . . . . . . . . . . . . 18 (((1st𝑋) − (𝑑 / 2)) ∈ ℝ* → -∞ ≤ ((1st𝑋) − (𝑑 / 2)))
7223, 71syl 17 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → -∞ ≤ ((1st𝑋) − (𝑑 / 2)))
73 pnfge 11840 . . . . . . . . . . . . . . . . . 18 (((1st𝑋) + (𝑑 / 2)) ∈ ℝ* → ((1st𝑋) + (𝑑 / 2)) ≤ +∞)
7425, 73syl 17 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → ((1st𝑋) + (𝑑 / 2)) ≤ +∞)
75 mnfxr 9975 . . . . . . . . . . . . . . . . . 18 -∞ ∈ ℝ*
76 pnfxr 9971 . . . . . . . . . . . . . . . . . 18 +∞ ∈ ℝ*
77 ioossioo 12136 . . . . . . . . . . . . . . . . . 18 (((-∞ ∈ ℝ* ∧ +∞ ∈ ℝ*) ∧ (-∞ ≤ ((1st𝑋) − (𝑑 / 2)) ∧ ((1st𝑋) + (𝑑 / 2)) ≤ +∞)) → (((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) ⊆ (-∞(,)+∞))
7875, 76, 77mpanl12 714 . . . . . . . . . . . . . . . . 17 ((-∞ ≤ ((1st𝑋) − (𝑑 / 2)) ∧ ((1st𝑋) + (𝑑 / 2)) ≤ +∞) → (((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) ⊆ (-∞(,)+∞))
7972, 74, 78syl2anc 691 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → (((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) ⊆ (-∞(,)+∞))
80 ioomax 12119 . . . . . . . . . . . . . . . 16 (-∞(,)+∞) = ℝ
8179, 80syl6sseq 3614 . . . . . . . . . . . . . . 15 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → (((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) ⊆ ℝ)
82 mnfle 11845 . . . . . . . . . . . . . . . . . 18 (((2nd𝑋) − (𝑑 / 2)) ∈ ℝ* → -∞ ≤ ((2nd𝑋) − (𝑑 / 2)))
8331, 82syl 17 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → -∞ ≤ ((2nd𝑋) − (𝑑 / 2)))
84 pnfge 11840 . . . . . . . . . . . . . . . . . 18 (((2nd𝑋) + (𝑑 / 2)) ∈ ℝ* → ((2nd𝑋) + (𝑑 / 2)) ≤ +∞)
8533, 84syl 17 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → ((2nd𝑋) + (𝑑 / 2)) ≤ +∞)
86 ioossioo 12136 . . . . . . . . . . . . . . . . . 18 (((-∞ ∈ ℝ* ∧ +∞ ∈ ℝ*) ∧ (-∞ ≤ ((2nd𝑋) − (𝑑 / 2)) ∧ ((2nd𝑋) + (𝑑 / 2)) ≤ +∞)) → (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2))) ⊆ (-∞(,)+∞))
8775, 76, 86mpanl12 714 . . . . . . . . . . . . . . . . 17 ((-∞ ≤ ((2nd𝑋) − (𝑑 / 2)) ∧ ((2nd𝑋) + (𝑑 / 2)) ≤ +∞) → (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2))) ⊆ (-∞(,)+∞))
8883, 85, 87syl2anc 691 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2))) ⊆ (-∞(,)+∞))
8988, 80syl6sseq 3614 . . . . . . . . . . . . . . 15 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2))) ⊆ ℝ)
90 xpss12 5148 . . . . . . . . . . . . . . 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 691 . . . . . . . . . . . . . 14 (((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) → ((((1st𝑋) − (𝑑 / 2))(,)((1st𝑋) + (𝑑 / 2))) × (((2nd𝑋) − (𝑑 / 2))(,)((2nd𝑋) + (𝑑 / 2)))) ⊆ (ℝ × ℝ))
9291sselda 3568 . . . . . . . . . . . . 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 450 . . . . . . . . . . . 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 574 . . . . . . . . . . 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 723 . . . . . . . . . 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 480 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ (𝑋𝐴𝑑 ∈ ℝ+𝑥 ∈ (ℝ × ℝ))) → 𝐴 ⊆ (ℝ × ℝ))
97 simpr1 1060 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ (𝑋𝐴𝑑 ∈ ℝ+𝑥 ∈ (ℝ × ℝ))) → 𝑋𝐴)
9896, 97sseldd 3569 . . . . . . . . . . . . . . 15 ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ (𝑋𝐴𝑑 ∈ ℝ+𝑥 ∈ (ℝ × ℝ))) → 𝑋 ∈ (ℝ × ℝ))
99983anassrs 1282 . . . . . . . . . . . . . 14 ((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) → 𝑋 ∈ (ℝ × ℝ))
100 simpr 476 . . . . . . . . . . . . . 14 ((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) → 𝑥 ∈ (ℝ × ℝ))
101 simplr 788 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) → 𝑑 ∈ ℝ+)
102101rphalfcld 11760 . . . . . . . . . . . . . 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 29285 . . . . . . . . . . . . . 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 1318 . . . . . . . . . . . . 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 2610 . . . . . . . . . . . . . . . . . . . . 21 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
107103, 6, 106cnrehmeo 22560 . . . . . . . . . . . . . . . . . . . 20 𝐺 ∈ ((𝐽 ×t 𝐽)Homeo(TopOpen‘ℂfld))
108106cnfldtopon 22396 . . . . . . . . . . . . . . . . . . . . . 22 (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
109108toponunii 20547 . . . . . . . . . . . . . . . . . . . . 21 ℂ = (TopOpen‘ℂfld)
11012, 109hmeof1o 21377 . . . . . . . . . . . . . . . . . . . 20 (𝐺 ∈ ((𝐽 ×t 𝐽)Homeo(TopOpen‘ℂfld)) → 𝐺:(ℝ × ℝ)–1-1-onto→ℂ)
111 f1of 6050 . . . . . . . . . . . . . . . . . . . 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 6268 . . . . . . . . . . . . . . . . 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 6267 . . . . . . . . . . . . . . . . 17 ((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) → (𝐺𝑥) ∈ ℂ)
117 sqsscirc2 29283 . . . . . . . . . . . . . . . . 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 1317 . . . . . . . . . . . . . . . 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 444 . . . . . . . . . . . . . . 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 11749 . . . . . . . . . . . . . . . . . 18 ((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) → 𝑑 ∈ ℝ*)
121120adantr 480 . . . . . . . . . . . . . . . . 17 (((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) ∧ (abs‘((𝐺𝑥) − (𝐺𝑋))) < 𝑑) → 𝑑 ∈ ℝ*)
122 cnxmet 22386 . . . . . . . . . . . . . . . . 17 (abs ∘ − ) ∈ (∞Met‘ℂ)
123121, 122jctil 558 . . . . . . . . . . . . . . . 16 (((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) ∧ (abs‘((𝐺𝑥) − (𝐺𝑋))) < 𝑑) → ((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑑 ∈ ℝ*))
124114adantr 480 . . . . . . . . . . . . . . . . 17 (((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) ∧ (abs‘((𝐺𝑥) − (𝐺𝑋))) < 𝑑) → (𝐺𝑋) ∈ ℂ)
125116adantr 480 . . . . . . . . . . . . . . . . 17 (((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) ∧ (abs‘((𝐺𝑥) − (𝐺𝑋))) < 𝑑) → (𝐺𝑥) ∈ ℂ)
126124, 125jca 553 . . . . . . . . . . . . . . . 16 (((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) ∧ (abs‘((𝐺𝑥) − (𝐺𝑋))) < 𝑑) → ((𝐺𝑋) ∈ ℂ ∧ (𝐺𝑥) ∈ ℂ))
127 eqid 2610 . . . . . . . . . . . . . . . . . . 19 (abs ∘ − ) = (abs ∘ − )
128127cnmetdval 22384 . . . . . . . . . . . . . . . . . 18 (((𝐺𝑥) ∈ ℂ ∧ (𝐺𝑋) ∈ ℂ) → ((𝐺𝑥)(abs ∘ − )(𝐺𝑋)) = (abs‘((𝐺𝑥) − (𝐺𝑋))))
129125, 124, 128syl2anc 691 . . . . . . . . . . . . . . . . 17 (((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) ∧ (abs‘((𝐺𝑥) − (𝐺𝑋))) < 𝑑) → ((𝐺𝑥)(abs ∘ − )(𝐺𝑋)) = (abs‘((𝐺𝑥) − (𝐺𝑋))))
130 simpr 476 . . . . . . . . . . . . . . . . 17 (((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) ∧ (abs‘((𝐺𝑥) − (𝐺𝑋))) < 𝑑) → (abs‘((𝐺𝑥) − (𝐺𝑋))) < 𝑑)
131129, 130eqbrtrd 4605 . . . . . . . . . . . . . . . 16 (((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) ∧ (abs‘((𝐺𝑥) − (𝐺𝑋))) < 𝑑) → ((𝐺𝑥)(abs ∘ − )(𝐺𝑋)) < 𝑑)
132 elbl3 22007 . . . . . . . . . . . . . . . . 17 ((((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑑 ∈ ℝ*) ∧ ((𝐺𝑋) ∈ ℂ ∧ (𝐺𝑥) ∈ ℂ)) → ((𝐺𝑥) ∈ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑) ↔ ((𝐺𝑥)(abs ∘ − )(𝐺𝑋)) < 𝑑))
133132biimpar 501 . . . . . . . . . . . . . . . 16 (((((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑑 ∈ ℝ*) ∧ ((𝐺𝑋) ∈ ℂ ∧ (𝐺𝑥) ∈ ℂ)) ∧ ((𝐺𝑥)(abs ∘ − )(𝐺𝑋)) < 𝑑) → (𝐺𝑥) ∈ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑))
134123, 126, 131, 133syl21anc 1317 . . . . . . . . . . . . . . 15 (((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) ∧ (abs‘((𝐺𝑥) − (𝐺𝑋))) < 𝑑) → (𝐺𝑥) ∈ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑))
135119, 134syldan 486 . . . . . . . . . . . . . 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 449 . . . . . . . . . . . . 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 46 . . . . . . . . . . . 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 6062 . . . . . . . . . . . . . . 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 6052 . . . . . . . . . . . . . 14 (𝐺:ℂ–1-1-onto→(ℝ × ℝ) → Fun 𝐺)
141139, 140ax-mp 5 . . . . . . . . . . . . 13 Fun 𝐺
142 f1odm 6054 . . . . . . . . . . . . . . 15 (𝐺:ℂ–1-1-onto→(ℝ × ℝ) → dom 𝐺 = ℂ)
143139, 142ax-mp 5 . . . . . . . . . . . . . 14 dom 𝐺 = ℂ
144116, 143syl6eleqr 2699 . . . . . . . . . . . . 13 ((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) → (𝐺𝑥) ∈ dom 𝐺)
145 funfvima 6396 . . . . . . . . . . . . 13 ((Fun 𝐺 ∧ (𝐺𝑥) ∈ dom 𝐺) → ((𝐺𝑥) ∈ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑) → (𝐺‘(𝐺𝑥)) ∈ (𝐺 “ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑))))
146141, 144, 145sylancr 694 . . . . . . . . . . . 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 6432 . . . . . . . . . . . . . . 15 ((𝐺:(ℝ × ℝ)–1-1-onto→ℂ ∧ 𝑥 ∈ (ℝ × ℝ)) → (𝐺‘(𝐺𝑥)) = 𝑥)
149147, 100, 148syl2anc 691 . . . . . . . . . . . . . 14 ((((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ (ℝ × ℝ)) → (𝐺‘(𝐺𝑥)) = 𝑥)
150149eleq1d 2672 . . . . . . . . . . . . 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 218 . . . . . . . . . . . 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 58 . . . . . . . . . . 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 444 . . . . . . . . . 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 449 . . . . . . . 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 3574 . . . . . . 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 2949 . . . . . 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 6660 . . . . . . . . . 10 Fun 𝐺
159158a1i 11 . . . . . . . . 9 ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) → Fun 𝐺)
16013sselda 3568 . . . . . . . . . 10 ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) → 𝑋 ∈ (ℝ × ℝ))
161 f1odm 6054 . . . . . . . . . . 11 (𝐺:(ℝ × ℝ)–1-1-onto→ℂ → dom 𝐺 = (ℝ × ℝ))
162107, 110, 161mp2b 10 . . . . . . . . . 10 dom 𝐺 = (ℝ × ℝ)
163160, 162syl6eleqr 2699 . . . . . . . . 9 ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) → 𝑋 ∈ dom 𝐺)
164 simpr 476 . . . . . . . . 9 ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) → 𝑋𝐴)
165 funfvima 6396 . . . . . . . . . 10 ((Fun 𝐺𝑋 ∈ dom 𝐺) → (𝑋𝐴 → (𝐺𝑋) ∈ (𝐺𝐴)))
166165imp 444 . . . . . . . . 9 (((Fun 𝐺𝑋 ∈ dom 𝐺) ∧ 𝑋𝐴) → (𝐺𝑋) ∈ (𝐺𝐴))
167159, 163, 164, 166syl21anc 1317 . . . . . . . 8 ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) → (𝐺𝑋) ∈ (𝐺𝐴))
168 hmeoima 21378 . . . . . . . . . . 11 ((𝐺 ∈ ((𝐽 ×t 𝐽)Homeo(TopOpen‘ℂfld)) ∧ 𝐴 ∈ (𝐽 ×t 𝐽)) → (𝐺𝐴) ∈ (TopOpen‘ℂfld))
169107, 168mpan 702 . . . . . . . . . 10 (𝐴 ∈ (𝐽 ×t 𝐽) → (𝐺𝐴) ∈ (TopOpen‘ℂfld))
170106cnfldtopn 22395 . . . . . . . . . . . . 13 (TopOpen‘ℂfld) = (MetOpen‘(abs ∘ − ))
171170elmopn2 22060 . . . . . . . . . . . 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 479 . . . . . . . . . 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 480 . . . . . . . 8 ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) → ∀𝑚 ∈ (𝐺𝐴)∃𝑑 ∈ ℝ+ (𝑚(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺𝐴))
176 oveq1 6556 . . . . . . . . . . 11 (𝑚 = (𝐺𝑋) → (𝑚(ball‘(abs ∘ − ))𝑑) = ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑))
177176sseq1d 3595 . . . . . . . . . 10 (𝑚 = (𝐺𝑋) → ((𝑚(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺𝐴) ↔ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺𝐴)))
178177rexbidv 3034 . . . . . . . . 9 (𝑚 = (𝐺𝑋) → (∃𝑑 ∈ ℝ+ (𝑚(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺𝐴) ↔ ∃𝑑 ∈ ℝ+ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺𝐴)))
179178rspcva 3280 . . . . . . . 8 (((𝐺𝑋) ∈ (𝐺𝐴) ∧ ∀𝑚 ∈ (𝐺𝐴)∃𝑑 ∈ ℝ+ (𝑚(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺𝐴)) → ∃𝑑 ∈ ℝ+ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺𝐴))
180167, 175, 179syl2anc 691 . . . . . . 7 ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) → ∃𝑑 ∈ ℝ+ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺𝐴))
181 imass2 5420 . . . . . . . . . 10 (((𝐺𝑋)(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺𝐴) → (𝐺 “ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑)) ⊆ (𝐺 “ (𝐺𝐴)))
182 f1of1 6049 . . . . . . . . . . . . 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 6066 . . . . . . . . . . . 12 ((𝐺:(ℝ × ℝ)–1-1→ℂ ∧ 𝐴 ⊆ (ℝ × ℝ)) → (𝐺 “ (𝐺𝐴)) = 𝐴)
185183, 13, 184sylancr 694 . . . . . . . . . . 11 (𝐴 ∈ (𝐽 ×t 𝐽) → (𝐺 “ (𝐺𝐴)) = 𝐴)
186185sseq2d 3596 . . . . . . . . . 10 (𝐴 ∈ (𝐽 ×t 𝐽) → ((𝐺 “ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑)) ⊆ (𝐺 “ (𝐺𝐴)) ↔ (𝐺 “ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑)) ⊆ 𝐴))
187181, 186syl5ib 233 . . . . . . . . 9 (𝐴 ∈ (𝐽 ×t 𝐽) → (((𝐺𝑋)(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺𝐴) → (𝐺 “ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑)) ⊆ 𝐴))
188187reximdv 2999 . . . . . . . 8 (𝐴 ∈ (𝐽 ×t 𝐽) → (∃𝑑 ∈ ℝ+ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺𝐴) → ∃𝑑 ∈ ℝ+ (𝐺 “ ((𝐺𝑋)(ball‘(abs ∘ − ))𝑑)) ⊆ 𝐴))
189188adantr 480 . . . . . . 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 3054 . . . . . 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 691 . . . . 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 3576 . . . . . 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 2994 . . . . 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 3054 . . . 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 691 . . 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 3054 . . 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 691 . 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 2677 . . . . 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 3589 . . . . 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 743 . . . 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 3282 . . 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 3011 . 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 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ⊆ wss 3540  𝒫 cpw 4108  ∪ cuni 4372   class class class wbr 4583   × cxp 5036  ◡ccnv 5037  dom cdm 5038  ran crn 5039   “ cima 5041   ∘ ccom 5042  Fun wfun 5798   Fn wfn 5799  ⟶wf 5800  –1-1→wf1 5801  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  1st c1st 7057  2nd c2nd 7058  ℂcc 9813  ℝcr 9814  ici 9817   + caddc 9818   · cmul 9820  +∞cpnf 9950  -∞cmnf 9951  ℝ*cxr 9952   < clt 9953   ≤ cle 9954   − cmin 10145   / cdiv 10563  2c2 10947  ℝ+crp 11708  (,)cioo 12046  ℜcre 13685  ℑcim 13686  abscabs 13822  TopOpenctopn 15905  topGenctg 15921  ∞Metcxmt 19552  ballcbl 19554  ℂfldccnfld 19567  Topctop 20517   ×t ctx 21173  Homeochmeo 21366 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893  ax-addf 9894  ax-mulf 9895 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-fi 8200  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-q 11665  df-rp 11709  df-xneg 11822  df-xadd 11823  df-xmul 11824  df-ioo 12050  df-icc 12053  df-fz 12198  df-fzo 12335  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-starv 15783  df-sca 15784  df-vsca 15785  df-ip 15786  df-tset 15787  df-ple 15788  df-ds 15791  df-unif 15792  df-hom 15793  df-cco 15794  df-rest 15906  df-topn 15907  df-0g 15925  df-gsum 15926  df-topgen 15927  df-pt 15928  df-prds 15931  df-xrs 15985  df-qtop 15990  df-imas 15991  df-xps 15993  df-mre 16069  df-mrc 16070  df-acs 16072  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-submnd 17159  df-mulg 17364  df-cntz 17573  df-cmn 18018  df-psmet 19559  df-xmet 19560  df-met 19561  df-bl 19562  df-mopn 19563  df-cnfld 19568  df-top 20521  df-bases 20522  df-topon 20523  df-topsp 20524  df-cn 20841  df-cnp 20842  df-tx 21175  df-hmeo 21368  df-xms 21935  df-ms 21936  df-tms 21937  df-cncf 22489 This theorem is referenced by:  dya2iocnei  29671
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