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Theorem tpr2rico 33187
Description: For any point of an open set of the usual topology on (ℝ Γ— 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 = (topGenβ€˜ran (,))
tpr2rico.1 𝐺 = (𝑒 ∈ ℝ, 𝑣 ∈ ℝ ↦ (𝑒 + (i Β· 𝑣)))
tpr2rico.2 𝐡 = ran (π‘₯ ∈ ran (,), 𝑦 ∈ ran (,) ↦ (π‘₯ Γ— 𝑦))
Assertion
Ref Expression
tpr2rico ((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) β†’ βˆƒπ‘Ÿ ∈ 𝐡 (𝑋 ∈ π‘Ÿ ∧ π‘Ÿ βŠ† 𝐴))
Distinct variable groups:   𝑣,𝑒,π‘₯,𝑦   π‘₯,π‘Ÿ,𝐴   𝐡,π‘Ÿ   π‘₯,𝐺   π‘₯,𝐽   π‘₯,𝑋   𝑦,π‘Ÿ,𝑋
Allowed substitution hints:   𝐴(𝑦,𝑣,𝑒)   𝐡(π‘₯,𝑦,𝑣,𝑒)   𝐺(𝑦,𝑣,𝑒,π‘Ÿ)   𝐽(𝑦,𝑣,𝑒,π‘Ÿ)   𝑋(𝑣,𝑒)

Proof of Theorem tpr2rico
Dummy variables 𝑧 π‘š 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ioo 13333 . . . . . . . . . 10 (,) = (π‘₯ ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (π‘₯ < 𝑧 ∧ 𝑧 < 𝑦)})
21ixxf 13339 . . . . . . . . 9 (,):(ℝ* Γ— ℝ*)βŸΆπ’« ℝ*
3 ffn 6718 . . . . . . . . 9 ((,):(ℝ* Γ— ℝ*)βŸΆπ’« ℝ* β†’ (,) Fn (ℝ* Γ— ℝ*))
42, 3mp1i 13 . . . . . . . 8 (((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) β†’ (,) Fn (ℝ* Γ— ℝ*))
5 elssuni 4942 . . . . . . . . . . . . . 14 (𝐴 ∈ (𝐽 Γ—t 𝐽) β†’ 𝐴 βŠ† βˆͺ (𝐽 Γ—t 𝐽))
6 tpr2rico.0 . . . . . . . . . . . . . . . 16 𝐽 = (topGenβ€˜ran (,))
7 retop 24499 . . . . . . . . . . . . . . . 16 (topGenβ€˜ran (,)) ∈ Top
86, 7eqeltri 2828 . . . . . . . . . . . . . . 15 𝐽 ∈ Top
9 uniretop 24500 . . . . . . . . . . . . . . . 16 ℝ = βˆͺ (topGenβ€˜ran (,))
106unieqi 4922 . . . . . . . . . . . . . . . 16 βˆͺ 𝐽 = βˆͺ (topGenβ€˜ran (,))
119, 10eqtr4i 2762 . . . . . . . . . . . . . . 15 ℝ = βˆͺ 𝐽
128, 8, 11, 11txunii 23318 . . . . . . . . . . . . . 14 (ℝ Γ— ℝ) = βˆͺ (𝐽 Γ—t 𝐽)
135, 12sseqtrrdi 4034 . . . . . . . . . . . . 13 (𝐴 ∈ (𝐽 Γ—t 𝐽) β†’ 𝐴 βŠ† (ℝ Γ— ℝ))
1413ad2antrr 723 . . . . . . . . . . . 12 (((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) β†’ 𝐴 βŠ† (ℝ Γ— ℝ))
15 simplr 766 . . . . . . . . . . . 12 (((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) β†’ 𝑋 ∈ 𝐴)
1614, 15sseldd 3984 . . . . . . . . . . 11 (((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) β†’ 𝑋 ∈ (ℝ Γ— ℝ))
17 xp1st 8010 . . . . . . . . . . 11 (𝑋 ∈ (ℝ Γ— ℝ) β†’ (1st β€˜π‘‹) ∈ ℝ)
1816, 17syl 17 . . . . . . . . . 10 (((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) β†’ (1st β€˜π‘‹) ∈ ℝ)
19 simpr 484 . . . . . . . . . . . 12 (((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) β†’ 𝑑 ∈ ℝ+)
2019rpred 13021 . . . . . . . . . . 11 (((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) β†’ 𝑑 ∈ ℝ)
2120rehalfcld 12464 . . . . . . . . . 10 (((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) β†’ (𝑑 / 2) ∈ ℝ)
2218, 21resubcld 11647 . . . . . . . . 9 (((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) β†’ ((1st β€˜π‘‹) βˆ’ (𝑑 / 2)) ∈ ℝ)
2322rexrd 11269 . . . . . . . 8 (((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) β†’ ((1st β€˜π‘‹) βˆ’ (𝑑 / 2)) ∈ ℝ*)
2418, 21readdcld 11248 . . . . . . . . 9 (((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) β†’ ((1st β€˜π‘‹) + (𝑑 / 2)) ∈ ℝ)
2524rexrd 11269 . . . . . . . 8 (((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) β†’ ((1st β€˜π‘‹) + (𝑑 / 2)) ∈ ℝ*)
26 fnovrn 7585 . . . . . . . 8 (((,) Fn (ℝ* Γ— ℝ*) ∧ ((1st β€˜π‘‹) βˆ’ (𝑑 / 2)) ∈ ℝ* ∧ ((1st β€˜π‘‹) + (𝑑 / 2)) ∈ ℝ*) β†’ (((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) ∈ ran (,))
274, 23, 25, 26syl3anc 1370 . . . . . . 7 (((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) β†’ (((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) ∈ ran (,))
28 xp2nd 8011 . . . . . . . . . . 11 (𝑋 ∈ (ℝ Γ— ℝ) β†’ (2nd β€˜π‘‹) ∈ ℝ)
2916, 28syl 17 . . . . . . . . . 10 (((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) β†’ (2nd β€˜π‘‹) ∈ ℝ)
3029, 21resubcld 11647 . . . . . . . . 9 (((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) β†’ ((2nd β€˜π‘‹) βˆ’ (𝑑 / 2)) ∈ ℝ)
3130rexrd 11269 . . . . . . . 8 (((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) β†’ ((2nd β€˜π‘‹) βˆ’ (𝑑 / 2)) ∈ ℝ*)
3229, 21readdcld 11248 . . . . . . . . 9 (((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) β†’ ((2nd β€˜π‘‹) + (𝑑 / 2)) ∈ ℝ)
3332rexrd 11269 . . . . . . . 8 (((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) β†’ ((2nd β€˜π‘‹) + (𝑑 / 2)) ∈ ℝ*)
34 fnovrn 7585 . . . . . . . 8 (((,) Fn (ℝ* Γ— ℝ*) ∧ ((2nd β€˜π‘‹) βˆ’ (𝑑 / 2)) ∈ ℝ* ∧ ((2nd β€˜π‘‹) + (𝑑 / 2)) ∈ ℝ*) β†’ (((2nd β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((2nd β€˜π‘‹) + (𝑑 / 2))) ∈ ran (,))
354, 31, 33, 34syl3anc 1370 . . . . . . 7 (((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) β†’ (((2nd β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((2nd β€˜π‘‹) + (𝑑 / 2))) ∈ ran (,))
36 eqidd 2732 . . . . . . 7 (((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) β†’ ((((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) Γ— (((2nd β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((2nd β€˜π‘‹) + (𝑑 / 2)))) = ((((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) Γ— (((2nd β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((2nd β€˜π‘‹) + (𝑑 / 2)))))
37 xpeq1 5691 . . . . . . . . 9 (π‘₯ = (((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) β†’ (π‘₯ Γ— 𝑦) = ((((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) Γ— 𝑦))
3837eqeq2d 2742 . . . . . . . 8 (π‘₯ = (((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) β†’ (((((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) Γ— (((2nd β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((2nd β€˜π‘‹) + (𝑑 / 2)))) = (π‘₯ Γ— 𝑦) ↔ ((((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) Γ— (((2nd β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((2nd β€˜π‘‹) + (𝑑 / 2)))) = ((((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) Γ— 𝑦)))
39 xpeq2 5698 . . . . . . . . 9 (𝑦 = (((2nd β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((2nd β€˜π‘‹) + (𝑑 / 2))) β†’ ((((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) Γ— 𝑦) = ((((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) Γ— (((2nd β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((2nd β€˜π‘‹) + (𝑑 / 2)))))
4039eqeq2d 2742 . . . . . . . 8 (𝑦 = (((2nd β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((2nd β€˜π‘‹) + (𝑑 / 2))) β†’ (((((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) Γ— (((2nd β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((2nd β€˜π‘‹) + (𝑑 / 2)))) = ((((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) Γ— 𝑦) ↔ ((((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) Γ— (((2nd β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((2nd β€˜π‘‹) + (𝑑 / 2)))) = ((((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) Γ— (((2nd β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((2nd β€˜π‘‹) + (𝑑 / 2))))))
4138, 40rspc2ev 3625 . . . . . . 7 (((((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) ∈ ran (,) ∧ (((2nd β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((2nd β€˜π‘‹) + (𝑑 / 2))) ∈ ran (,) ∧ ((((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) Γ— (((2nd β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((2nd β€˜π‘‹) + (𝑑 / 2)))) = ((((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) Γ— (((2nd β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((2nd β€˜π‘‹) + (𝑑 / 2))))) β†’ βˆƒπ‘₯ ∈ ran (,)βˆƒπ‘¦ ∈ ran (,)((((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) Γ— (((2nd β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((2nd β€˜π‘‹) + (𝑑 / 2)))) = (π‘₯ Γ— 𝑦))
4227, 35, 36, 41syl3anc 1370 . . . . . 6 (((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) β†’ βˆƒπ‘₯ ∈ ran (,)βˆƒπ‘¦ ∈ ran (,)((((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) Γ— (((2nd β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((2nd β€˜π‘‹) + (𝑑 / 2)))) = (π‘₯ Γ— 𝑦))
43 eqid 2731 . . . . . . 7 (π‘₯ ∈ ran (,), 𝑦 ∈ ran (,) ↦ (π‘₯ Γ— 𝑦)) = (π‘₯ ∈ ran (,), 𝑦 ∈ ran (,) ↦ (π‘₯ Γ— 𝑦))
44 vex 3477 . . . . . . . 8 π‘₯ ∈ V
45 vex 3477 . . . . . . . 8 𝑦 ∈ V
4644, 45xpex 7743 . . . . . . 7 (π‘₯ Γ— 𝑦) ∈ V
4743, 46elrnmpo 7548 . . . . . 6 (((((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) Γ— (((2nd β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((2nd β€˜π‘‹) + (𝑑 / 2)))) ∈ ran (π‘₯ ∈ ran (,), 𝑦 ∈ ran (,) ↦ (π‘₯ Γ— 𝑦)) ↔ βˆƒπ‘₯ ∈ ran (,)βˆƒπ‘¦ ∈ ran (,)((((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) Γ— (((2nd β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((2nd β€˜π‘‹) + (𝑑 / 2)))) = (π‘₯ Γ— 𝑦))
4842, 47sylibr 233 . . . . 5 (((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) β†’ ((((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) Γ— (((2nd β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((2nd β€˜π‘‹) + (𝑑 / 2)))) ∈ ran (π‘₯ ∈ ran (,), 𝑦 ∈ ran (,) ↦ (π‘₯ Γ— 𝑦)))
49 tpr2rico.2 . . . . 5 𝐡 = ran (π‘₯ ∈ ran (,), 𝑦 ∈ ran (,) ↦ (π‘₯ Γ— 𝑦))
5048, 49eleqtrrdi 2843 . . . 4 (((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) β†’ ((((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) Γ— (((2nd β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((2nd β€˜π‘‹) + (𝑑 / 2)))) ∈ 𝐡)
5150ralrimiva 3145 . . 3 ((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) β†’ βˆ€π‘‘ ∈ ℝ+ ((((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) Γ— (((2nd β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((2nd β€˜π‘‹) + (𝑑 / 2)))) ∈ 𝐡)
52 xpss 5693 . . . . . . 7 (ℝ Γ— ℝ) βŠ† (V Γ— V)
5352, 16sselid 3981 . . . . . 6 (((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) β†’ 𝑋 ∈ (V Γ— V))
5418rexrd 11269 . . . . . . . 8 (((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) β†’ (1st β€˜π‘‹) ∈ ℝ*)
5519rphalfcld 13033 . . . . . . . . 9 (((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) β†’ (𝑑 / 2) ∈ ℝ+)
5618, 55ltsubrpd 13053 . . . . . . . 8 (((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) β†’ ((1st β€˜π‘‹) βˆ’ (𝑑 / 2)) < (1st β€˜π‘‹))
5718, 55ltaddrpd 13054 . . . . . . . 8 (((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) β†’ (1st β€˜π‘‹) < ((1st β€˜π‘‹) + (𝑑 / 2)))
58 elioo1 13369 . . . . . . . . 9 ((((1st β€˜π‘‹) βˆ’ (𝑑 / 2)) ∈ ℝ* ∧ ((1st β€˜π‘‹) + (𝑑 / 2)) ∈ ℝ*) β†’ ((1st β€˜π‘‹) ∈ (((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) ↔ ((1st β€˜π‘‹) ∈ ℝ* ∧ ((1st β€˜π‘‹) βˆ’ (𝑑 / 2)) < (1st β€˜π‘‹) ∧ (1st β€˜π‘‹) < ((1st β€˜π‘‹) + (𝑑 / 2)))))
5923, 25, 58syl2anc 583 . . . . . . . 8 (((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) β†’ ((1st β€˜π‘‹) ∈ (((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) ↔ ((1st β€˜π‘‹) ∈ ℝ* ∧ ((1st β€˜π‘‹) βˆ’ (𝑑 / 2)) < (1st β€˜π‘‹) ∧ (1st β€˜π‘‹) < ((1st β€˜π‘‹) + (𝑑 / 2)))))
6054, 56, 57, 59mpbir3and 1341 . . . . . . 7 (((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) β†’ (1st β€˜π‘‹) ∈ (((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))))
6129rexrd 11269 . . . . . . . 8 (((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) β†’ (2nd β€˜π‘‹) ∈ ℝ*)
6229, 55ltsubrpd 13053 . . . . . . . 8 (((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) β†’ ((2nd β€˜π‘‹) βˆ’ (𝑑 / 2)) < (2nd β€˜π‘‹))
6329, 55ltaddrpd 13054 . . . . . . . 8 (((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) β†’ (2nd β€˜π‘‹) < ((2nd β€˜π‘‹) + (𝑑 / 2)))
64 elioo1 13369 . . . . . . . . 9 ((((2nd β€˜π‘‹) βˆ’ (𝑑 / 2)) ∈ ℝ* ∧ ((2nd β€˜π‘‹) + (𝑑 / 2)) ∈ ℝ*) β†’ ((2nd β€˜π‘‹) ∈ (((2nd β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((2nd β€˜π‘‹) + (𝑑 / 2))) ↔ ((2nd β€˜π‘‹) ∈ ℝ* ∧ ((2nd β€˜π‘‹) βˆ’ (𝑑 / 2)) < (2nd β€˜π‘‹) ∧ (2nd β€˜π‘‹) < ((2nd β€˜π‘‹) + (𝑑 / 2)))))
6531, 33, 64syl2anc 583 . . . . . . . 8 (((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) β†’ ((2nd β€˜π‘‹) ∈ (((2nd β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((2nd β€˜π‘‹) + (𝑑 / 2))) ↔ ((2nd β€˜π‘‹) ∈ ℝ* ∧ ((2nd β€˜π‘‹) βˆ’ (𝑑 / 2)) < (2nd β€˜π‘‹) ∧ (2nd β€˜π‘‹) < ((2nd β€˜π‘‹) + (𝑑 / 2)))))
6661, 62, 63, 65mpbir3and 1341 . . . . . . 7 (((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) β†’ (2nd β€˜π‘‹) ∈ (((2nd β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((2nd β€˜π‘‹) + (𝑑 / 2))))
6760, 66jca 511 . . . . . 6 (((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) β†’ ((1st β€˜π‘‹) ∈ (((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) ∧ (2nd β€˜π‘‹) ∈ (((2nd β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((2nd β€˜π‘‹) + (𝑑 / 2)))))
68 elxp7 8013 . . . . . 6 (𝑋 ∈ ((((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) Γ— (((2nd β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((2nd β€˜π‘‹) + (𝑑 / 2)))) ↔ (𝑋 ∈ (V Γ— V) ∧ ((1st β€˜π‘‹) ∈ (((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) ∧ (2nd β€˜π‘‹) ∈ (((2nd β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((2nd β€˜π‘‹) + (𝑑 / 2))))))
6953, 67, 68sylanbrc 582 . . . . 5 (((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) β†’ 𝑋 ∈ ((((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) Γ— (((2nd β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((2nd β€˜π‘‹) + (𝑑 / 2)))))
7069ralrimiva 3145 . . . 4 ((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) β†’ βˆ€π‘‘ ∈ ℝ+ 𝑋 ∈ ((((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) Γ— (((2nd β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((2nd β€˜π‘‹) + (𝑑 / 2)))))
71 mnfle 13119 . . . . . . . . . . . . . . . . . 18 (((1st β€˜π‘‹) βˆ’ (𝑑 / 2)) ∈ ℝ* β†’ -∞ ≀ ((1st β€˜π‘‹) βˆ’ (𝑑 / 2)))
7223, 71syl 17 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) β†’ -∞ ≀ ((1st β€˜π‘‹) βˆ’ (𝑑 / 2)))
73 pnfge 13115 . . . . . . . . . . . . . . . . . 18 (((1st β€˜π‘‹) + (𝑑 / 2)) ∈ ℝ* β†’ ((1st β€˜π‘‹) + (𝑑 / 2)) ≀ +∞)
7425, 73syl 17 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) β†’ ((1st β€˜π‘‹) + (𝑑 / 2)) ≀ +∞)
75 mnfxr 11276 . . . . . . . . . . . . . . . . . 18 -∞ ∈ ℝ*
76 pnfxr 11273 . . . . . . . . . . . . . . . . . 18 +∞ ∈ ℝ*
77 ioossioo 13423 . . . . . . . . . . . . . . . . . 18 (((-∞ ∈ ℝ* ∧ +∞ ∈ ℝ*) ∧ (-∞ ≀ ((1st β€˜π‘‹) βˆ’ (𝑑 / 2)) ∧ ((1st β€˜π‘‹) + (𝑑 / 2)) ≀ +∞)) β†’ (((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) βŠ† (-∞(,)+∞))
7875, 76, 77mpanl12 699 . . . . . . . . . . . . . . . . 17 ((-∞ ≀ ((1st β€˜π‘‹) βˆ’ (𝑑 / 2)) ∧ ((1st β€˜π‘‹) + (𝑑 / 2)) ≀ +∞) β†’ (((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) βŠ† (-∞(,)+∞))
7972, 74, 78syl2anc 583 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) β†’ (((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) βŠ† (-∞(,)+∞))
80 ioomax 13404 . . . . . . . . . . . . . . . 16 (-∞(,)+∞) = ℝ
8179, 80sseqtrdi 4033 . . . . . . . . . . . . . . 15 (((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) β†’ (((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) βŠ† ℝ)
82 mnfle 13119 . . . . . . . . . . . . . . . . . 18 (((2nd β€˜π‘‹) βˆ’ (𝑑 / 2)) ∈ ℝ* β†’ -∞ ≀ ((2nd β€˜π‘‹) βˆ’ (𝑑 / 2)))
8331, 82syl 17 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) β†’ -∞ ≀ ((2nd β€˜π‘‹) βˆ’ (𝑑 / 2)))
84 pnfge 13115 . . . . . . . . . . . . . . . . . 18 (((2nd β€˜π‘‹) + (𝑑 / 2)) ∈ ℝ* β†’ ((2nd β€˜π‘‹) + (𝑑 / 2)) ≀ +∞)
8533, 84syl 17 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) β†’ ((2nd β€˜π‘‹) + (𝑑 / 2)) ≀ +∞)
86 ioossioo 13423 . . . . . . . . . . . . . . . . . 18 (((-∞ ∈ ℝ* ∧ +∞ ∈ ℝ*) ∧ (-∞ ≀ ((2nd β€˜π‘‹) βˆ’ (𝑑 / 2)) ∧ ((2nd β€˜π‘‹) + (𝑑 / 2)) ≀ +∞)) β†’ (((2nd β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((2nd β€˜π‘‹) + (𝑑 / 2))) βŠ† (-∞(,)+∞))
8775, 76, 86mpanl12 699 . . . . . . . . . . . . . . . . 17 ((-∞ ≀ ((2nd β€˜π‘‹) βˆ’ (𝑑 / 2)) ∧ ((2nd β€˜π‘‹) + (𝑑 / 2)) ≀ +∞) β†’ (((2nd β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((2nd β€˜π‘‹) + (𝑑 / 2))) βŠ† (-∞(,)+∞))
8883, 85, 87syl2anc 583 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) β†’ (((2nd β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((2nd β€˜π‘‹) + (𝑑 / 2))) βŠ† (-∞(,)+∞))
8988, 80sseqtrdi 4033 . . . . . . . . . . . . . . 15 (((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) β†’ (((2nd β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((2nd β€˜π‘‹) + (𝑑 / 2))) βŠ† ℝ)
90 xpss12 5692 . . . . . . . . . . . . . . 15 (((((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) βŠ† ℝ ∧ (((2nd β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((2nd β€˜π‘‹) + (𝑑 / 2))) βŠ† ℝ) β†’ ((((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) Γ— (((2nd β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((2nd β€˜π‘‹) + (𝑑 / 2)))) βŠ† (ℝ Γ— ℝ))
9181, 89, 90syl2anc 583 . . . . . . . . . . . . . 14 (((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) β†’ ((((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) Γ— (((2nd β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((2nd β€˜π‘‹) + (𝑑 / 2)))) βŠ† (ℝ Γ— ℝ))
9291sselda 3983 . . . . . . . . . . . . 13 ((((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) ∧ π‘₯ ∈ ((((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) Γ— (((2nd β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((2nd β€˜π‘‹) + (𝑑 / 2))))) β†’ π‘₯ ∈ (ℝ Γ— ℝ))
9392expcom 413 . . . . . . . . . . . 12 (π‘₯ ∈ ((((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) Γ— (((2nd β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((2nd β€˜π‘‹) + (𝑑 / 2)))) β†’ (((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) β†’ π‘₯ ∈ (ℝ Γ— ℝ)))
9493ancld 550 . . . . . . . . . . 11 (π‘₯ ∈ ((((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) Γ— (((2nd β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((2nd β€˜π‘‹) + (𝑑 / 2)))) β†’ (((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) β†’ (((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) ∧ π‘₯ ∈ (ℝ Γ— ℝ))))
9594imdistanri 569 . . . . . . . . . 10 ((((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) ∧ π‘₯ ∈ ((((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) Γ— (((2nd β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((2nd β€˜π‘‹) + (𝑑 / 2))))) β†’ ((((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) ∧ π‘₯ ∈ (ℝ Γ— ℝ)) ∧ π‘₯ ∈ ((((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) Γ— (((2nd β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((2nd β€˜π‘‹) + (𝑑 / 2))))))
9613adantr 480 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ (𝑋 ∈ 𝐴 ∧ 𝑑 ∈ ℝ+ ∧ π‘₯ ∈ (ℝ Γ— ℝ))) β†’ 𝐴 βŠ† (ℝ Γ— ℝ))
97 simpr1 1193 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ (𝑋 ∈ 𝐴 ∧ 𝑑 ∈ ℝ+ ∧ π‘₯ ∈ (ℝ Γ— ℝ))) β†’ 𝑋 ∈ 𝐴)
9896, 97sseldd 3984 . . . . . . . . . . . . . . 15 ((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ (𝑋 ∈ 𝐴 ∧ 𝑑 ∈ ℝ+ ∧ π‘₯ ∈ (ℝ Γ— ℝ))) β†’ 𝑋 ∈ (ℝ Γ— ℝ))
99983anassrs 1359 . . . . . . . . . . . . . 14 ((((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) ∧ π‘₯ ∈ (ℝ Γ— ℝ)) β†’ 𝑋 ∈ (ℝ Γ— ℝ))
100 simpr 484 . . . . . . . . . . . . . 14 ((((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) ∧ π‘₯ ∈ (ℝ Γ— ℝ)) β†’ π‘₯ ∈ (ℝ Γ— ℝ))
101 simplr 766 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) ∧ π‘₯ ∈ (ℝ Γ— ℝ)) β†’ 𝑑 ∈ ℝ+)
102101rphalfcld 13033 . . . . . . . . . . . . . 14 ((((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) ∧ π‘₯ ∈ (ℝ Γ— ℝ)) β†’ (𝑑 / 2) ∈ ℝ+)
103 tpr2rico.1 . . . . . . . . . . . . . . 15 𝐺 = (𝑒 ∈ ℝ, 𝑣 ∈ ℝ ↦ (𝑒 + (i Β· 𝑣)))
104103cnre2csqima 33186 . . . . . . . . . . . . . 14 ((𝑋 ∈ (ℝ Γ— ℝ) ∧ π‘₯ ∈ (ℝ Γ— ℝ) ∧ (𝑑 / 2) ∈ ℝ+) β†’ (π‘₯ ∈ ((((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) Γ— (((2nd β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((2nd β€˜π‘‹) + (𝑑 / 2)))) β†’ ((absβ€˜(β„œβ€˜((πΊβ€˜π‘₯) βˆ’ (πΊβ€˜π‘‹)))) < (𝑑 / 2) ∧ (absβ€˜(β„‘β€˜((πΊβ€˜π‘₯) βˆ’ (πΊβ€˜π‘‹)))) < (𝑑 / 2))))
10599, 100, 102, 104syl3anc 1370 . . . . . . . . . . . . 13 ((((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) ∧ π‘₯ ∈ (ℝ Γ— ℝ)) β†’ (π‘₯ ∈ ((((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) Γ— (((2nd β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((2nd β€˜π‘‹) + (𝑑 / 2)))) β†’ ((absβ€˜(β„œβ€˜((πΊβ€˜π‘₯) βˆ’ (πΊβ€˜π‘‹)))) < (𝑑 / 2) ∧ (absβ€˜(β„‘β€˜((πΊβ€˜π‘₯) βˆ’ (πΊβ€˜π‘‹)))) < (𝑑 / 2))))
106 eqid 2731 . . . . . . . . . . . . . . . . . . . . 21 (TopOpenβ€˜β„‚fld) = (TopOpenβ€˜β„‚fld)
107103, 6, 106cnrehmeo 24699 . . . . . . . . . . . . . . . . . . . 20 𝐺 ∈ ((𝐽 Γ—t 𝐽)Homeo(TopOpenβ€˜β„‚fld))
108106cnfldtopon 24520 . . . . . . . . . . . . . . . . . . . . . 22 (TopOpenβ€˜β„‚fld) ∈ (TopOnβ€˜β„‚)
109108toponunii 22639 . . . . . . . . . . . . . . . . . . . . 21 β„‚ = βˆͺ (TopOpenβ€˜β„‚fld)
11012, 109hmeof1o 23489 . . . . . . . . . . . . . . . . . . . 20 (𝐺 ∈ ((𝐽 Γ—t 𝐽)Homeo(TopOpenβ€˜β„‚fld)) β†’ 𝐺:(ℝ Γ— ℝ)–1-1-ontoβ†’β„‚)
111 f1of 6834 . . . . . . . . . . . . . . . . . . . 20 (𝐺:(ℝ Γ— ℝ)–1-1-ontoβ†’β„‚ β†’ 𝐺:(ℝ Γ— ℝ)βŸΆβ„‚)
112107, 110, 111mp2b 10 . . . . . . . . . . . . . . . . . . 19 𝐺:(ℝ Γ— ℝ)βŸΆβ„‚
113112a1i 11 . . . . . . . . . . . . . . . . . 18 ((((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) ∧ π‘₯ ∈ (ℝ Γ— ℝ)) β†’ 𝐺:(ℝ Γ— ℝ)βŸΆβ„‚)
114113, 99ffvelcdmd 7088 . . . . . . . . . . . . . . . . 17 ((((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) ∧ π‘₯ ∈ (ℝ Γ— ℝ)) β†’ (πΊβ€˜π‘‹) ∈ β„‚)
115112a1i 11 . . . . . . . . . . . . . . . . . 18 (((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) β†’ 𝐺:(ℝ Γ— ℝ)βŸΆβ„‚)
116115ffvelcdmda 7087 . . . . . . . . . . . . . . . . 17 ((((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) ∧ π‘₯ ∈ (ℝ Γ— ℝ)) β†’ (πΊβ€˜π‘₯) ∈ β„‚)
117 sqsscirc2 33184 . . . . . . . . . . . . . . . . 17 ((((πΊβ€˜π‘‹) ∈ β„‚ ∧ (πΊβ€˜π‘₯) ∈ β„‚) ∧ 𝑑 ∈ ℝ+) β†’ (((absβ€˜(β„œβ€˜((πΊβ€˜π‘₯) βˆ’ (πΊβ€˜π‘‹)))) < (𝑑 / 2) ∧ (absβ€˜(β„‘β€˜((πΊβ€˜π‘₯) βˆ’ (πΊβ€˜π‘‹)))) < (𝑑 / 2)) β†’ (absβ€˜((πΊβ€˜π‘₯) βˆ’ (πΊβ€˜π‘‹))) < 𝑑))
118114, 116, 101, 117syl21anc 835 . . . . . . . . . . . . . . . 16 ((((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) ∧ π‘₯ ∈ (ℝ Γ— ℝ)) β†’ (((absβ€˜(β„œβ€˜((πΊβ€˜π‘₯) βˆ’ (πΊβ€˜π‘‹)))) < (𝑑 / 2) ∧ (absβ€˜(β„‘β€˜((πΊβ€˜π‘₯) βˆ’ (πΊβ€˜π‘‹)))) < (𝑑 / 2)) β†’ (absβ€˜((πΊβ€˜π‘₯) βˆ’ (πΊβ€˜π‘‹))) < 𝑑))
119118imp 406 . . . . . . . . . . . . . . 15 (((((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) ∧ π‘₯ ∈ (ℝ Γ— ℝ)) ∧ ((absβ€˜(β„œβ€˜((πΊβ€˜π‘₯) βˆ’ (πΊβ€˜π‘‹)))) < (𝑑 / 2) ∧ (absβ€˜(β„‘β€˜((πΊβ€˜π‘₯) βˆ’ (πΊβ€˜π‘‹)))) < (𝑑 / 2))) β†’ (absβ€˜((πΊβ€˜π‘₯) βˆ’ (πΊβ€˜π‘‹))) < 𝑑)
120101rpxrd 13022 . . . . . . . . . . . . . . . . . 18 ((((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) ∧ π‘₯ ∈ (ℝ Γ— ℝ)) β†’ 𝑑 ∈ ℝ*)
121120adantr 480 . . . . . . . . . . . . . . . . 17 (((((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) ∧ π‘₯ ∈ (ℝ Γ— ℝ)) ∧ (absβ€˜((πΊβ€˜π‘₯) βˆ’ (πΊβ€˜π‘‹))) < 𝑑) β†’ 𝑑 ∈ ℝ*)
122 cnxmet 24510 . . . . . . . . . . . . . . . . 17 (abs ∘ βˆ’ ) ∈ (∞Metβ€˜β„‚)
123121, 122jctil 519 . . . . . . . . . . . . . . . 16 (((((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) ∧ π‘₯ ∈ (ℝ Γ— ℝ)) ∧ (absβ€˜((πΊβ€˜π‘₯) βˆ’ (πΊβ€˜π‘‹))) < 𝑑) β†’ ((abs ∘ βˆ’ ) ∈ (∞Metβ€˜β„‚) ∧ 𝑑 ∈ ℝ*))
124114adantr 480 . . . . . . . . . . . . . . . . 17 (((((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) ∧ π‘₯ ∈ (ℝ Γ— ℝ)) ∧ (absβ€˜((πΊβ€˜π‘₯) βˆ’ (πΊβ€˜π‘‹))) < 𝑑) β†’ (πΊβ€˜π‘‹) ∈ β„‚)
125116adantr 480 . . . . . . . . . . . . . . . . 17 (((((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) ∧ π‘₯ ∈ (ℝ Γ— ℝ)) ∧ (absβ€˜((πΊβ€˜π‘₯) βˆ’ (πΊβ€˜π‘‹))) < 𝑑) β†’ (πΊβ€˜π‘₯) ∈ β„‚)
126124, 125jca 511 . . . . . . . . . . . . . . . 16 (((((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) ∧ π‘₯ ∈ (ℝ Γ— ℝ)) ∧ (absβ€˜((πΊβ€˜π‘₯) βˆ’ (πΊβ€˜π‘‹))) < 𝑑) β†’ ((πΊβ€˜π‘‹) ∈ β„‚ ∧ (πΊβ€˜π‘₯) ∈ β„‚))
127 eqid 2731 . . . . . . . . . . . . . . . . . . 19 (abs ∘ βˆ’ ) = (abs ∘ βˆ’ )
128127cnmetdval 24508 . . . . . . . . . . . . . . . . . 18 (((πΊβ€˜π‘₯) ∈ β„‚ ∧ (πΊβ€˜π‘‹) ∈ β„‚) β†’ ((πΊβ€˜π‘₯)(abs ∘ βˆ’ )(πΊβ€˜π‘‹)) = (absβ€˜((πΊβ€˜π‘₯) βˆ’ (πΊβ€˜π‘‹))))
129125, 124, 128syl2anc 583 . . . . . . . . . . . . . . . . 17 (((((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) ∧ π‘₯ ∈ (ℝ Γ— ℝ)) ∧ (absβ€˜((πΊβ€˜π‘₯) βˆ’ (πΊβ€˜π‘‹))) < 𝑑) β†’ ((πΊβ€˜π‘₯)(abs ∘ βˆ’ )(πΊβ€˜π‘‹)) = (absβ€˜((πΊβ€˜π‘₯) βˆ’ (πΊβ€˜π‘‹))))
130 simpr 484 . . . . . . . . . . . . . . . . 17 (((((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) ∧ π‘₯ ∈ (ℝ Γ— ℝ)) ∧ (absβ€˜((πΊβ€˜π‘₯) βˆ’ (πΊβ€˜π‘‹))) < 𝑑) β†’ (absβ€˜((πΊβ€˜π‘₯) βˆ’ (πΊβ€˜π‘‹))) < 𝑑)
131129, 130eqbrtrd 5171 . . . . . . . . . . . . . . . 16 (((((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) ∧ π‘₯ ∈ (ℝ Γ— ℝ)) ∧ (absβ€˜((πΊβ€˜π‘₯) βˆ’ (πΊβ€˜π‘‹))) < 𝑑) β†’ ((πΊβ€˜π‘₯)(abs ∘ βˆ’ )(πΊβ€˜π‘‹)) < 𝑑)
132 elbl3 24119 . . . . . . . . . . . . . . . . 17 ((((abs ∘ βˆ’ ) ∈ (∞Metβ€˜β„‚) ∧ 𝑑 ∈ ℝ*) ∧ ((πΊβ€˜π‘‹) ∈ β„‚ ∧ (πΊβ€˜π‘₯) ∈ β„‚)) β†’ ((πΊβ€˜π‘₯) ∈ ((πΊβ€˜π‘‹)(ballβ€˜(abs ∘ βˆ’ ))𝑑) ↔ ((πΊβ€˜π‘₯)(abs ∘ βˆ’ )(πΊβ€˜π‘‹)) < 𝑑))
133132biimpar 477 . . . . . . . . . . . . . . . 16 (((((abs ∘ βˆ’ ) ∈ (∞Metβ€˜β„‚) ∧ 𝑑 ∈ ℝ*) ∧ ((πΊβ€˜π‘‹) ∈ β„‚ ∧ (πΊβ€˜π‘₯) ∈ β„‚)) ∧ ((πΊβ€˜π‘₯)(abs ∘ βˆ’ )(πΊβ€˜π‘‹)) < 𝑑) β†’ (πΊβ€˜π‘₯) ∈ ((πΊβ€˜π‘‹)(ballβ€˜(abs ∘ βˆ’ ))𝑑))
134123, 126, 131, 133syl21anc 835 . . . . . . . . . . . . . . 15 (((((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) ∧ π‘₯ ∈ (ℝ Γ— ℝ)) ∧ (absβ€˜((πΊβ€˜π‘₯) βˆ’ (πΊβ€˜π‘‹))) < 𝑑) β†’ (πΊβ€˜π‘₯) ∈ ((πΊβ€˜π‘‹)(ballβ€˜(abs ∘ βˆ’ ))𝑑))
135119, 134syldan 590 . . . . . . . . . . . . . 14 (((((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) ∧ π‘₯ ∈ (ℝ Γ— ℝ)) ∧ ((absβ€˜(β„œβ€˜((πΊβ€˜π‘₯) βˆ’ (πΊβ€˜π‘‹)))) < (𝑑 / 2) ∧ (absβ€˜(β„‘β€˜((πΊβ€˜π‘₯) βˆ’ (πΊβ€˜π‘‹)))) < (𝑑 / 2))) β†’ (πΊβ€˜π‘₯) ∈ ((πΊβ€˜π‘‹)(ballβ€˜(abs ∘ βˆ’ ))𝑑))
136135ex 412 . . . . . . . . . . . . 13 ((((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) ∧ π‘₯ ∈ (ℝ Γ— ℝ)) β†’ (((absβ€˜(β„œβ€˜((πΊβ€˜π‘₯) βˆ’ (πΊβ€˜π‘‹)))) < (𝑑 / 2) ∧ (absβ€˜(β„‘β€˜((πΊβ€˜π‘₯) βˆ’ (πΊβ€˜π‘‹)))) < (𝑑 / 2)) β†’ (πΊβ€˜π‘₯) ∈ ((πΊβ€˜π‘‹)(ballβ€˜(abs ∘ βˆ’ ))𝑑)))
137105, 136syld 47 . . . . . . . . . . . 12 ((((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) ∧ π‘₯ ∈ (ℝ Γ— ℝ)) β†’ (π‘₯ ∈ ((((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) Γ— (((2nd β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((2nd β€˜π‘‹) + (𝑑 / 2)))) β†’ (πΊβ€˜π‘₯) ∈ ((πΊβ€˜π‘‹)(ballβ€˜(abs ∘ βˆ’ ))𝑑)))
138 f1ocnv 6846 . . . . . . . . . . . . . . 15 (𝐺:(ℝ Γ— ℝ)–1-1-ontoβ†’β„‚ β†’ ◑𝐺:ℂ–1-1-ontoβ†’(ℝ Γ— ℝ))
139107, 110, 138mp2b 10 . . . . . . . . . . . . . 14 ◑𝐺:ℂ–1-1-ontoβ†’(ℝ Γ— ℝ)
140 f1ofun 6836 . . . . . . . . . . . . . 14 (◑𝐺:ℂ–1-1-ontoβ†’(ℝ Γ— ℝ) β†’ Fun ◑𝐺)
141139, 140ax-mp 5 . . . . . . . . . . . . 13 Fun ◑𝐺
142 f1odm 6838 . . . . . . . . . . . . . . 15 (◑𝐺:ℂ–1-1-ontoβ†’(ℝ Γ— ℝ) β†’ dom ◑𝐺 = β„‚)
143139, 142ax-mp 5 . . . . . . . . . . . . . 14 dom ◑𝐺 = β„‚
144116, 143eleqtrrdi 2843 . . . . . . . . . . . . 13 ((((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) ∧ π‘₯ ∈ (ℝ Γ— ℝ)) β†’ (πΊβ€˜π‘₯) ∈ dom ◑𝐺)
145 funfvima 7235 . . . . . . . . . . . . 13 ((Fun ◑𝐺 ∧ (πΊβ€˜π‘₯) ∈ dom ◑𝐺) β†’ ((πΊβ€˜π‘₯) ∈ ((πΊβ€˜π‘‹)(ballβ€˜(abs ∘ βˆ’ ))𝑑) β†’ (β—‘πΊβ€˜(πΊβ€˜π‘₯)) ∈ (◑𝐺 β€œ ((πΊβ€˜π‘‹)(ballβ€˜(abs ∘ βˆ’ ))𝑑))))
146141, 144, 145sylancr 586 . . . . . . . . . . . 12 ((((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) ∧ π‘₯ ∈ (ℝ Γ— ℝ)) β†’ ((πΊβ€˜π‘₯) ∈ ((πΊβ€˜π‘‹)(ballβ€˜(abs ∘ βˆ’ ))𝑑) β†’ (β—‘πΊβ€˜(πΊβ€˜π‘₯)) ∈ (◑𝐺 β€œ ((πΊβ€˜π‘‹)(ballβ€˜(abs ∘ βˆ’ ))𝑑))))
147107, 110mp1i 13 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) ∧ π‘₯ ∈ (ℝ Γ— ℝ)) β†’ 𝐺:(ℝ Γ— ℝ)–1-1-ontoβ†’β„‚)
148 f1ocnvfv1 7277 . . . . . . . . . . . . . . 15 ((𝐺:(ℝ Γ— ℝ)–1-1-ontoβ†’β„‚ ∧ π‘₯ ∈ (ℝ Γ— ℝ)) β†’ (β—‘πΊβ€˜(πΊβ€˜π‘₯)) = π‘₯)
149147, 100, 148syl2anc 583 . . . . . . . . . . . . . 14 ((((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) ∧ π‘₯ ∈ (ℝ Γ— ℝ)) β†’ (β—‘πΊβ€˜(πΊβ€˜π‘₯)) = π‘₯)
150149eleq1d 2817 . . . . . . . . . . . . 13 ((((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) ∧ π‘₯ ∈ (ℝ Γ— ℝ)) β†’ ((β—‘πΊβ€˜(πΊβ€˜π‘₯)) ∈ (◑𝐺 β€œ ((πΊβ€˜π‘‹)(ballβ€˜(abs ∘ βˆ’ ))𝑑)) ↔ π‘₯ ∈ (◑𝐺 β€œ ((πΊβ€˜π‘‹)(ballβ€˜(abs ∘ βˆ’ ))𝑑))))
151150biimpd 228 . . . . . . . . . . . 12 ((((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) ∧ π‘₯ ∈ (ℝ Γ— ℝ)) β†’ ((β—‘πΊβ€˜(πΊβ€˜π‘₯)) ∈ (◑𝐺 β€œ ((πΊβ€˜π‘‹)(ballβ€˜(abs ∘ βˆ’ ))𝑑)) β†’ π‘₯ ∈ (◑𝐺 β€œ ((πΊβ€˜π‘‹)(ballβ€˜(abs ∘ βˆ’ ))𝑑))))
152137, 146, 1513syld 60 . . . . . . . . . . 11 ((((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) ∧ π‘₯ ∈ (ℝ Γ— ℝ)) β†’ (π‘₯ ∈ ((((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) Γ— (((2nd β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((2nd β€˜π‘‹) + (𝑑 / 2)))) β†’ π‘₯ ∈ (◑𝐺 β€œ ((πΊβ€˜π‘‹)(ballβ€˜(abs ∘ βˆ’ ))𝑑))))
153152imp 406 . . . . . . . . . 10 (((((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) ∧ π‘₯ ∈ (ℝ Γ— ℝ)) ∧ π‘₯ ∈ ((((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) Γ— (((2nd β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((2nd β€˜π‘‹) + (𝑑 / 2))))) β†’ π‘₯ ∈ (◑𝐺 β€œ ((πΊβ€˜π‘‹)(ballβ€˜(abs ∘ βˆ’ ))𝑑)))
15495, 153syl 17 . . . . . . . . 9 ((((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) ∧ π‘₯ ∈ ((((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) Γ— (((2nd β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((2nd β€˜π‘‹) + (𝑑 / 2))))) β†’ π‘₯ ∈ (◑𝐺 β€œ ((πΊβ€˜π‘‹)(ballβ€˜(abs ∘ βˆ’ ))𝑑)))
155154ex 412 . . . . . . . 8 (((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) β†’ (π‘₯ ∈ ((((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) Γ— (((2nd β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((2nd β€˜π‘‹) + (𝑑 / 2)))) β†’ π‘₯ ∈ (◑𝐺 β€œ ((πΊβ€˜π‘‹)(ballβ€˜(abs ∘ βˆ’ ))𝑑))))
156155ssrdv 3989 . . . . . . 7 (((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) β†’ ((((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) Γ— (((2nd β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((2nd β€˜π‘‹) + (𝑑 / 2)))) βŠ† (◑𝐺 β€œ ((πΊβ€˜π‘‹)(ballβ€˜(abs ∘ βˆ’ ))𝑑)))
157156ralrimiva 3145 . . . . . 6 ((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) β†’ βˆ€π‘‘ ∈ ℝ+ ((((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) Γ— (((2nd β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((2nd β€˜π‘‹) + (𝑑 / 2)))) βŠ† (◑𝐺 β€œ ((πΊβ€˜π‘‹)(ballβ€˜(abs ∘ βˆ’ ))𝑑)))
158103mpofun 7535 . . . . . . . . . 10 Fun 𝐺
159158a1i 11 . . . . . . . . 9 ((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) β†’ Fun 𝐺)
16013sselda 3983 . . . . . . . . . 10 ((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) β†’ 𝑋 ∈ (ℝ Γ— ℝ))
161 f1odm 6838 . . . . . . . . . . 11 (𝐺:(ℝ Γ— ℝ)–1-1-ontoβ†’β„‚ β†’ dom 𝐺 = (ℝ Γ— ℝ))
162107, 110, 161mp2b 10 . . . . . . . . . 10 dom 𝐺 = (ℝ Γ— ℝ)
163160, 162eleqtrrdi 2843 . . . . . . . . 9 ((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) β†’ 𝑋 ∈ dom 𝐺)
164 simpr 484 . . . . . . . . 9 ((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) β†’ 𝑋 ∈ 𝐴)
165 funfvima 7235 . . . . . . . . . 10 ((Fun 𝐺 ∧ 𝑋 ∈ dom 𝐺) β†’ (𝑋 ∈ 𝐴 β†’ (πΊβ€˜π‘‹) ∈ (𝐺 β€œ 𝐴)))
166165imp 406 . . . . . . . . 9 (((Fun 𝐺 ∧ 𝑋 ∈ dom 𝐺) ∧ 𝑋 ∈ 𝐴) β†’ (πΊβ€˜π‘‹) ∈ (𝐺 β€œ 𝐴))
167159, 163, 164, 166syl21anc 835 . . . . . . . 8 ((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) β†’ (πΊβ€˜π‘‹) ∈ (𝐺 β€œ 𝐴))
168 hmeoima 23490 . . . . . . . . . . 11 ((𝐺 ∈ ((𝐽 Γ—t 𝐽)Homeo(TopOpenβ€˜β„‚fld)) ∧ 𝐴 ∈ (𝐽 Γ—t 𝐽)) β†’ (𝐺 β€œ 𝐴) ∈ (TopOpenβ€˜β„‚fld))
169107, 168mpan 687 . . . . . . . . . 10 (𝐴 ∈ (𝐽 Γ—t 𝐽) β†’ (𝐺 β€œ 𝐴) ∈ (TopOpenβ€˜β„‚fld))
170106cnfldtopn 24519 . . . . . . . . . . . . 13 (TopOpenβ€˜β„‚fld) = (MetOpenβ€˜(abs ∘ βˆ’ ))
171170elmopn2 24172 . . . . . . . . . . . 12 ((abs ∘ βˆ’ ) ∈ (∞Metβ€˜β„‚) β†’ ((𝐺 β€œ 𝐴) ∈ (TopOpenβ€˜β„‚fld) ↔ ((𝐺 β€œ 𝐴) βŠ† β„‚ ∧ βˆ€π‘š ∈ (𝐺 β€œ 𝐴)βˆƒπ‘‘ ∈ ℝ+ (π‘š(ballβ€˜(abs ∘ βˆ’ ))𝑑) βŠ† (𝐺 β€œ 𝐴))))
172122, 171ax-mp 5 . . . . . . . . . . 11 ((𝐺 β€œ 𝐴) ∈ (TopOpenβ€˜β„‚fld) ↔ ((𝐺 β€œ 𝐴) βŠ† β„‚ ∧ βˆ€π‘š ∈ (𝐺 β€œ 𝐴)βˆƒπ‘‘ ∈ ℝ+ (π‘š(ballβ€˜(abs ∘ βˆ’ ))𝑑) βŠ† (𝐺 β€œ 𝐴)))
173172simprbi 496 . . . . . . . . . 10 ((𝐺 β€œ 𝐴) ∈ (TopOpenβ€˜β„‚fld) β†’ βˆ€π‘š ∈ (𝐺 β€œ 𝐴)βˆƒπ‘‘ ∈ ℝ+ (π‘š(ballβ€˜(abs ∘ βˆ’ ))𝑑) βŠ† (𝐺 β€œ 𝐴))
174169, 173syl 17 . . . . . . . . 9 (𝐴 ∈ (𝐽 Γ—t 𝐽) β†’ βˆ€π‘š ∈ (𝐺 β€œ 𝐴)βˆƒπ‘‘ ∈ ℝ+ (π‘š(ballβ€˜(abs ∘ βˆ’ ))𝑑) βŠ† (𝐺 β€œ 𝐴))
175174adantr 480 . . . . . . . 8 ((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) β†’ βˆ€π‘š ∈ (𝐺 β€œ 𝐴)βˆƒπ‘‘ ∈ ℝ+ (π‘š(ballβ€˜(abs ∘ βˆ’ ))𝑑) βŠ† (𝐺 β€œ 𝐴))
176 oveq1 7419 . . . . . . . . . . 11 (π‘š = (πΊβ€˜π‘‹) β†’ (π‘š(ballβ€˜(abs ∘ βˆ’ ))𝑑) = ((πΊβ€˜π‘‹)(ballβ€˜(abs ∘ βˆ’ ))𝑑))
177176sseq1d 4014 . . . . . . . . . 10 (π‘š = (πΊβ€˜π‘‹) β†’ ((π‘š(ballβ€˜(abs ∘ βˆ’ ))𝑑) βŠ† (𝐺 β€œ 𝐴) ↔ ((πΊβ€˜π‘‹)(ballβ€˜(abs ∘ βˆ’ ))𝑑) βŠ† (𝐺 β€œ 𝐴)))
178177rexbidv 3177 . . . . . . . . 9 (π‘š = (πΊβ€˜π‘‹) β†’ (βˆƒπ‘‘ ∈ ℝ+ (π‘š(ballβ€˜(abs ∘ βˆ’ ))𝑑) βŠ† (𝐺 β€œ 𝐴) ↔ βˆƒπ‘‘ ∈ ℝ+ ((πΊβ€˜π‘‹)(ballβ€˜(abs ∘ βˆ’ ))𝑑) βŠ† (𝐺 β€œ 𝐴)))
179178rspcva 3611 . . . . . . . 8 (((πΊβ€˜π‘‹) ∈ (𝐺 β€œ 𝐴) ∧ βˆ€π‘š ∈ (𝐺 β€œ 𝐴)βˆƒπ‘‘ ∈ ℝ+ (π‘š(ballβ€˜(abs ∘ βˆ’ ))𝑑) βŠ† (𝐺 β€œ 𝐴)) β†’ βˆƒπ‘‘ ∈ ℝ+ ((πΊβ€˜π‘‹)(ballβ€˜(abs ∘ βˆ’ ))𝑑) βŠ† (𝐺 β€œ 𝐴))
180167, 175, 179syl2anc 583 . . . . . . 7 ((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) β†’ βˆƒπ‘‘ ∈ ℝ+ ((πΊβ€˜π‘‹)(ballβ€˜(abs ∘ βˆ’ ))𝑑) βŠ† (𝐺 β€œ 𝐴))
181 imass2 6102 . . . . . . . . . 10 (((πΊβ€˜π‘‹)(ballβ€˜(abs ∘ βˆ’ ))𝑑) βŠ† (𝐺 β€œ 𝐴) β†’ (◑𝐺 β€œ ((πΊβ€˜π‘‹)(ballβ€˜(abs ∘ βˆ’ ))𝑑)) βŠ† (◑𝐺 β€œ (𝐺 β€œ 𝐴)))
182 f1of1 6833 . . . . . . . . . . . . 13 (𝐺:(ℝ Γ— ℝ)–1-1-ontoβ†’β„‚ β†’ 𝐺:(ℝ Γ— ℝ)–1-1β†’β„‚)
183107, 110, 182mp2b 10 . . . . . . . . . . . 12 𝐺:(ℝ Γ— ℝ)–1-1β†’β„‚
184 f1imacnv 6850 . . . . . . . . . . . 12 ((𝐺:(ℝ Γ— ℝ)–1-1β†’β„‚ ∧ 𝐴 βŠ† (ℝ Γ— ℝ)) β†’ (◑𝐺 β€œ (𝐺 β€œ 𝐴)) = 𝐴)
185183, 13, 184sylancr 586 . . . . . . . . . . 11 (𝐴 ∈ (𝐽 Γ—t 𝐽) β†’ (◑𝐺 β€œ (𝐺 β€œ 𝐴)) = 𝐴)
186185sseq2d 4015 . . . . . . . . . 10 (𝐴 ∈ (𝐽 Γ—t 𝐽) β†’ ((◑𝐺 β€œ ((πΊβ€˜π‘‹)(ballβ€˜(abs ∘ βˆ’ ))𝑑)) βŠ† (◑𝐺 β€œ (𝐺 β€œ 𝐴)) ↔ (◑𝐺 β€œ ((πΊβ€˜π‘‹)(ballβ€˜(abs ∘ βˆ’ ))𝑑)) βŠ† 𝐴))
187181, 186imbitrid 243 . . . . . . . . 9 (𝐴 ∈ (𝐽 Γ—t 𝐽) β†’ (((πΊβ€˜π‘‹)(ballβ€˜(abs ∘ βˆ’ ))𝑑) βŠ† (𝐺 β€œ 𝐴) β†’ (◑𝐺 β€œ ((πΊβ€˜π‘‹)(ballβ€˜(abs ∘ βˆ’ ))𝑑)) βŠ† 𝐴))
188187reximdv 3169 . . . . . . . 8 (𝐴 ∈ (𝐽 Γ—t 𝐽) β†’ (βˆƒπ‘‘ ∈ ℝ+ ((πΊβ€˜π‘‹)(ballβ€˜(abs ∘ βˆ’ ))𝑑) βŠ† (𝐺 β€œ 𝐴) β†’ βˆƒπ‘‘ ∈ ℝ+ (◑𝐺 β€œ ((πΊβ€˜π‘‹)(ballβ€˜(abs ∘ βˆ’ ))𝑑)) βŠ† 𝐴))
189188adantr 480 . . . . . . 7 ((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) β†’ (βˆƒπ‘‘ ∈ ℝ+ ((πΊβ€˜π‘‹)(ballβ€˜(abs ∘ βˆ’ ))𝑑) βŠ† (𝐺 β€œ 𝐴) β†’ βˆƒπ‘‘ ∈ ℝ+ (◑𝐺 β€œ ((πΊβ€˜π‘‹)(ballβ€˜(abs ∘ βˆ’ ))𝑑)) βŠ† 𝐴))
190180, 189mpd 15 . . . . . 6 ((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) β†’ βˆƒπ‘‘ ∈ ℝ+ (◑𝐺 β€œ ((πΊβ€˜π‘‹)(ballβ€˜(abs ∘ βˆ’ ))𝑑)) βŠ† 𝐴)
191 r19.29 3113 . . . . . 6 ((βˆ€π‘‘ ∈ ℝ+ ((((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) Γ— (((2nd β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((2nd β€˜π‘‹) + (𝑑 / 2)))) βŠ† (◑𝐺 β€œ ((πΊβ€˜π‘‹)(ballβ€˜(abs ∘ βˆ’ ))𝑑)) ∧ βˆƒπ‘‘ ∈ ℝ+ (◑𝐺 β€œ ((πΊβ€˜π‘‹)(ballβ€˜(abs ∘ βˆ’ ))𝑑)) βŠ† 𝐴) β†’ βˆƒπ‘‘ ∈ ℝ+ (((((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) Γ— (((2nd β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((2nd β€˜π‘‹) + (𝑑 / 2)))) βŠ† (◑𝐺 β€œ ((πΊβ€˜π‘‹)(ballβ€˜(abs ∘ βˆ’ ))𝑑)) ∧ (◑𝐺 β€œ ((πΊβ€˜π‘‹)(ballβ€˜(abs ∘ βˆ’ ))𝑑)) βŠ† 𝐴))
192157, 190, 191syl2anc 583 . . . . 5 ((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) β†’ βˆƒπ‘‘ ∈ ℝ+ (((((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) Γ— (((2nd β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((2nd β€˜π‘‹) + (𝑑 / 2)))) βŠ† (◑𝐺 β€œ ((πΊβ€˜π‘‹)(ballβ€˜(abs ∘ βˆ’ ))𝑑)) ∧ (◑𝐺 β€œ ((πΊβ€˜π‘‹)(ballβ€˜(abs ∘ βˆ’ ))𝑑)) βŠ† 𝐴))
193 sstr 3991 . . . . . 6 ((((((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) Γ— (((2nd β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((2nd β€˜π‘‹) + (𝑑 / 2)))) βŠ† (◑𝐺 β€œ ((πΊβ€˜π‘‹)(ballβ€˜(abs ∘ βˆ’ ))𝑑)) ∧ (◑𝐺 β€œ ((πΊβ€˜π‘‹)(ballβ€˜(abs ∘ βˆ’ ))𝑑)) βŠ† 𝐴) β†’ ((((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) Γ— (((2nd β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((2nd β€˜π‘‹) + (𝑑 / 2)))) βŠ† 𝐴)
194193reximi 3083 . . . . 5 (βˆƒπ‘‘ ∈ ℝ+ (((((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) Γ— (((2nd β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((2nd β€˜π‘‹) + (𝑑 / 2)))) βŠ† (◑𝐺 β€œ ((πΊβ€˜π‘‹)(ballβ€˜(abs ∘ βˆ’ ))𝑑)) ∧ (◑𝐺 β€œ ((πΊβ€˜π‘‹)(ballβ€˜(abs ∘ βˆ’ ))𝑑)) βŠ† 𝐴) β†’ βˆƒπ‘‘ ∈ ℝ+ ((((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) Γ— (((2nd β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((2nd β€˜π‘‹) + (𝑑 / 2)))) βŠ† 𝐴)
195192, 194syl 17 . . . 4 ((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) β†’ βˆƒπ‘‘ ∈ ℝ+ ((((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) Γ— (((2nd β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((2nd β€˜π‘‹) + (𝑑 / 2)))) βŠ† 𝐴)
196 r19.29 3113 . . . 4 ((βˆ€π‘‘ ∈ ℝ+ 𝑋 ∈ ((((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) Γ— (((2nd β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((2nd β€˜π‘‹) + (𝑑 / 2)))) ∧ βˆƒπ‘‘ ∈ ℝ+ ((((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) Γ— (((2nd β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((2nd β€˜π‘‹) + (𝑑 / 2)))) βŠ† 𝐴) β†’ βˆƒπ‘‘ ∈ ℝ+ (𝑋 ∈ ((((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) Γ— (((2nd β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((2nd β€˜π‘‹) + (𝑑 / 2)))) ∧ ((((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) Γ— (((2nd β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((2nd β€˜π‘‹) + (𝑑 / 2)))) βŠ† 𝐴))
19770, 195, 196syl2anc 583 . . 3 ((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) β†’ βˆƒπ‘‘ ∈ ℝ+ (𝑋 ∈ ((((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) Γ— (((2nd β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((2nd β€˜π‘‹) + (𝑑 / 2)))) ∧ ((((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) Γ— (((2nd β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((2nd β€˜π‘‹) + (𝑑 / 2)))) βŠ† 𝐴))
198 r19.29 3113 . . 3 ((βˆ€π‘‘ ∈ ℝ+ ((((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) Γ— (((2nd β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((2nd β€˜π‘‹) + (𝑑 / 2)))) ∈ 𝐡 ∧ βˆƒπ‘‘ ∈ ℝ+ (𝑋 ∈ ((((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) Γ— (((2nd β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((2nd β€˜π‘‹) + (𝑑 / 2)))) ∧ ((((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) Γ— (((2nd β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((2nd β€˜π‘‹) + (𝑑 / 2)))) βŠ† 𝐴)) β†’ βˆƒπ‘‘ ∈ ℝ+ (((((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) Γ— (((2nd β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((2nd β€˜π‘‹) + (𝑑 / 2)))) ∈ 𝐡 ∧ (𝑋 ∈ ((((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) Γ— (((2nd β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((2nd β€˜π‘‹) + (𝑑 / 2)))) ∧ ((((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) Γ— (((2nd β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((2nd β€˜π‘‹) + (𝑑 / 2)))) βŠ† 𝐴)))
19951, 197, 198syl2anc 583 . 2 ((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) β†’ βˆƒπ‘‘ ∈ ℝ+ (((((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) Γ— (((2nd β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((2nd β€˜π‘‹) + (𝑑 / 2)))) ∈ 𝐡 ∧ (𝑋 ∈ ((((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) Γ— (((2nd β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((2nd β€˜π‘‹) + (𝑑 / 2)))) ∧ ((((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) Γ— (((2nd β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((2nd β€˜π‘‹) + (𝑑 / 2)))) βŠ† 𝐴)))
200 eleq2 2821 . . . . 5 (π‘Ÿ = ((((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) Γ— (((2nd β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((2nd β€˜π‘‹) + (𝑑 / 2)))) β†’ (𝑋 ∈ π‘Ÿ ↔ 𝑋 ∈ ((((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) Γ— (((2nd β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((2nd β€˜π‘‹) + (𝑑 / 2))))))
201 sseq1 4008 . . . . 5 (π‘Ÿ = ((((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) Γ— (((2nd β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((2nd β€˜π‘‹) + (𝑑 / 2)))) β†’ (π‘Ÿ βŠ† 𝐴 ↔ ((((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) Γ— (((2nd β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((2nd β€˜π‘‹) + (𝑑 / 2)))) βŠ† 𝐴))
202200, 201anbi12d 630 . . . 4 (π‘Ÿ = ((((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) Γ— (((2nd β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((2nd β€˜π‘‹) + (𝑑 / 2)))) β†’ ((𝑋 ∈ π‘Ÿ ∧ π‘Ÿ βŠ† 𝐴) ↔ (𝑋 ∈ ((((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) Γ— (((2nd β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((2nd β€˜π‘‹) + (𝑑 / 2)))) ∧ ((((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) Γ— (((2nd β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((2nd β€˜π‘‹) + (𝑑 / 2)))) βŠ† 𝐴)))
203202rspcev 3613 . . 3 ((((((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) Γ— (((2nd β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((2nd β€˜π‘‹) + (𝑑 / 2)))) ∈ 𝐡 ∧ (𝑋 ∈ ((((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) Γ— (((2nd β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((2nd β€˜π‘‹) + (𝑑 / 2)))) ∧ ((((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) Γ— (((2nd β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((2nd β€˜π‘‹) + (𝑑 / 2)))) βŠ† 𝐴)) β†’ βˆƒπ‘Ÿ ∈ 𝐡 (𝑋 ∈ π‘Ÿ ∧ π‘Ÿ βŠ† 𝐴))
204203rexlimivw 3150 . 2 (βˆƒπ‘‘ ∈ ℝ+ (((((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) Γ— (((2nd β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((2nd β€˜π‘‹) + (𝑑 / 2)))) ∈ 𝐡 ∧ (𝑋 ∈ ((((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) Γ— (((2nd β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((2nd β€˜π‘‹) + (𝑑 / 2)))) ∧ ((((1st β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((1st β€˜π‘‹) + (𝑑 / 2))) Γ— (((2nd β€˜π‘‹) βˆ’ (𝑑 / 2))(,)((2nd β€˜π‘‹) + (𝑑 / 2)))) βŠ† 𝐴)) β†’ βˆƒπ‘Ÿ ∈ 𝐡 (𝑋 ∈ π‘Ÿ ∧ π‘Ÿ βŠ† 𝐴))
205199, 204syl 17 1 ((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) β†’ βˆƒπ‘Ÿ ∈ 𝐡 (𝑋 ∈ π‘Ÿ ∧ π‘Ÿ βŠ† 𝐴))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105  βˆ€wral 3060  βˆƒwrex 3069  Vcvv 3473   βŠ† wss 3949  π’« cpw 4603  βˆͺ cuni 4909   class class class wbr 5149   Γ— cxp 5675  β—‘ccnv 5676  dom cdm 5677  ran crn 5678   β€œ cima 5680   ∘ ccom 5681  Fun wfun 6538   Fn wfn 6539  βŸΆwf 6540  β€“1-1β†’wf1 6541  β€“1-1-ontoβ†’wf1o 6543  β€˜cfv 6544  (class class class)co 7412   ∈ cmpo 7414  1st c1st 7976  2nd c2nd 7977  β„‚cc 11111  β„cr 11112  ici 11115   + caddc 11116   Β· cmul 11118  +∞cpnf 11250  -∞cmnf 11251  β„*cxr 11252   < clt 11253   ≀ cle 11254   βˆ’ cmin 11449   / cdiv 11876  2c2 12272  β„+crp 12979  (,)cioo 13329  β„œcre 15049  β„‘cim 15050  abscabs 15186  TopOpenctopn 17372  topGenctg 17388  βˆžMetcxmet 21130  ballcbl 21132  β„‚fldccnfld 21145  Topctop 22616   Γ—t ctx 23285  Homeochmeo 23478
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7728  ax-cnex 11169  ax-resscn 11170  ax-1cn 11171  ax-icn 11172  ax-addcl 11173  ax-addrcl 11174  ax-mulcl 11175  ax-mulrcl 11176  ax-mulcom 11177  ax-addass 11178  ax-mulass 11179  ax-distr 11180  ax-i2m1 11181  ax-1ne0 11182  ax-1rid 11183  ax-rnegex 11184  ax-rrecex 11185  ax-cnre 11186  ax-pre-lttri 11187  ax-pre-lttrn 11188  ax-pre-ltadd 11189  ax-pre-mulgt0 11190  ax-pre-sup 11191  ax-addf 11192
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-iin 5001  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-se 5633  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-of 7673  df-om 7859  df-1st 7978  df-2nd 7979  df-supp 8150  df-frecs 8269  df-wrecs 8300  df-recs 8374  df-rdg 8413  df-1o 8469  df-2o 8470  df-er 8706  df-map 8825  df-ixp 8895  df-en 8943  df-dom 8944  df-sdom 8945  df-fin 8946  df-fsupp 9365  df-fi 9409  df-sup 9440  df-inf 9441  df-oi 9508  df-card 9937  df-pnf 11255  df-mnf 11256  df-xr 11257  df-ltxr 11258  df-le 11259  df-sub 11451  df-neg 11452  df-div 11877  df-nn 12218  df-2 12280  df-3 12281  df-4 12282  df-5 12283  df-6 12284  df-7 12285  df-8 12286  df-9 12287  df-n0 12478  df-z 12564  df-dec 12683  df-uz 12828  df-q 12938  df-rp 12980  df-xneg 13097  df-xadd 13098  df-xmul 13099  df-ioo 13333  df-icc 13336  df-fz 13490  df-fzo 13633  df-seq 13972  df-exp 14033  df-hash 14296  df-cj 15051  df-re 15052  df-im 15053  df-sqrt 15187  df-abs 15188  df-struct 17085  df-sets 17102  df-slot 17120  df-ndx 17132  df-base 17150  df-ress 17179  df-plusg 17215  df-mulr 17216  df-starv 17217  df-sca 17218  df-vsca 17219  df-ip 17220  df-tset 17221  df-ple 17222  df-ds 17224  df-unif 17225  df-hom 17226  df-cco 17227  df-rest 17373  df-topn 17374  df-0g 17392  df-gsum 17393  df-topgen 17394  df-pt 17395  df-prds 17398  df-xrs 17453  df-qtop 17458  df-imas 17459  df-xps 17461  df-mre 17535  df-mrc 17536  df-acs 17538  df-mgm 18566  df-sgrp 18645  df-mnd 18661  df-submnd 18707  df-mulg 18988  df-cntz 19223  df-cmn 19692  df-psmet 21137  df-xmet 21138  df-met 21139  df-bl 21140  df-mopn 21141  df-cnfld 21146  df-top 22617  df-topon 22634  df-topsp 22656  df-bases 22670  df-cn 22952  df-cnp 22953  df-tx 23287  df-hmeo 23480  df-xms 24047  df-ms 24048  df-tms 24049  df-cncf 24619
This theorem is referenced by:  dya2iocnei  33576
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