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Theorem tpr2rico 32961
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 13330 . . . . . . . . . 10 (,) = (π‘₯ ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (π‘₯ < 𝑧 ∧ 𝑧 < 𝑦)})
21ixxf 13336 . . . . . . . . 9 (,):(ℝ* Γ— ℝ*)βŸΆπ’« ℝ*
3 ffn 6717 . . . . . . . . 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 4941 . . . . . . . . . . . . . 14 (𝐴 ∈ (𝐽 Γ—t 𝐽) β†’ 𝐴 βŠ† βˆͺ (𝐽 Γ—t 𝐽))
6 tpr2rico.0 . . . . . . . . . . . . . . . 16 𝐽 = (topGenβ€˜ran (,))
7 retop 24285 . . . . . . . . . . . . . . . 16 (topGenβ€˜ran (,)) ∈ Top
86, 7eqeltri 2829 . . . . . . . . . . . . . . 15 𝐽 ∈ Top
9 uniretop 24286 . . . . . . . . . . . . . . . 16 ℝ = βˆͺ (topGenβ€˜ran (,))
106unieqi 4921 . . . . . . . . . . . . . . . 16 βˆͺ 𝐽 = βˆͺ (topGenβ€˜ran (,))
119, 10eqtr4i 2763 . . . . . . . . . . . . . . 15 ℝ = βˆͺ 𝐽
128, 8, 11, 11txunii 23104 . . . . . . . . . . . . . 14 (ℝ Γ— ℝ) = βˆͺ (𝐽 Γ—t 𝐽)
135, 12sseqtrrdi 4033 . . . . . . . . . . . . 13 (𝐴 ∈ (𝐽 Γ—t 𝐽) β†’ 𝐴 βŠ† (ℝ Γ— ℝ))
1413ad2antrr 724 . . . . . . . . . . . 12 (((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) β†’ 𝐴 βŠ† (ℝ Γ— ℝ))
15 simplr 767 . . . . . . . . . . . 12 (((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) β†’ 𝑋 ∈ 𝐴)
1614, 15sseldd 3983 . . . . . . . . . . 11 (((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) β†’ 𝑋 ∈ (ℝ Γ— ℝ))
17 xp1st 8009 . . . . . . . . . . 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 485 . . . . . . . . . . . 12 (((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) β†’ 𝑑 ∈ ℝ+)
2019rpred 13018 . . . . . . . . . . 11 (((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) β†’ 𝑑 ∈ ℝ)
2120rehalfcld 12461 . . . . . . . . . 10 (((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) β†’ (𝑑 / 2) ∈ ℝ)
2218, 21resubcld 11644 . . . . . . . . 9 (((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) β†’ ((1st β€˜π‘‹) βˆ’ (𝑑 / 2)) ∈ ℝ)
2322rexrd 11266 . . . . . . . 8 (((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) β†’ ((1st β€˜π‘‹) βˆ’ (𝑑 / 2)) ∈ ℝ*)
2418, 21readdcld 11245 . . . . . . . . 9 (((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) β†’ ((1st β€˜π‘‹) + (𝑑 / 2)) ∈ ℝ)
2524rexrd 11266 . . . . . . . 8 (((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) β†’ ((1st β€˜π‘‹) + (𝑑 / 2)) ∈ ℝ*)
26 fnovrn 7584 . . . . . . . 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 1371 . . . . . . 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 8010 . . . . . . . . . . 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 11644 . . . . . . . . 9 (((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) β†’ ((2nd β€˜π‘‹) βˆ’ (𝑑 / 2)) ∈ ℝ)
3130rexrd 11266 . . . . . . . 8 (((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) β†’ ((2nd β€˜π‘‹) βˆ’ (𝑑 / 2)) ∈ ℝ*)
3229, 21readdcld 11245 . . . . . . . . 9 (((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) β†’ ((2nd β€˜π‘‹) + (𝑑 / 2)) ∈ ℝ)
3332rexrd 11266 . . . . . . . 8 (((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) β†’ ((2nd β€˜π‘‹) + (𝑑 / 2)) ∈ ℝ*)
34 fnovrn 7584 . . . . . . . 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 1371 . . . . . . 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 2733 . . . . . . 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 5690 . . . . . . . . 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 2743 . . . . . . . 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 5697 . . . . . . . . 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 2743 . . . . . . . 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 3624 . . . . . . 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 1371 . . . . . 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 2732 . . . . . . 7 (π‘₯ ∈ ran (,), 𝑦 ∈ ran (,) ↦ (π‘₯ Γ— 𝑦)) = (π‘₯ ∈ ran (,), 𝑦 ∈ ran (,) ↦ (π‘₯ Γ— 𝑦))
44 vex 3478 . . . . . . . 8 π‘₯ ∈ V
45 vex 3478 . . . . . . . 8 𝑦 ∈ V
4644, 45xpex 7742 . . . . . . 7 (π‘₯ Γ— 𝑦) ∈ V
4743, 46elrnmpo 7547 . . . . . 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 2844 . . . 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 3146 . . 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 5692 . . . . . . 7 (ℝ Γ— ℝ) βŠ† (V Γ— V)
5352, 16sselid 3980 . . . . . 6 (((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) β†’ 𝑋 ∈ (V Γ— V))
5418rexrd 11266 . . . . . . . 8 (((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) β†’ (1st β€˜π‘‹) ∈ ℝ*)
5519rphalfcld 13030 . . . . . . . . 9 (((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) β†’ (𝑑 / 2) ∈ ℝ+)
5618, 55ltsubrpd 13050 . . . . . . . 8 (((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) β†’ ((1st β€˜π‘‹) βˆ’ (𝑑 / 2)) < (1st β€˜π‘‹))
5718, 55ltaddrpd 13051 . . . . . . . 8 (((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) β†’ (1st β€˜π‘‹) < ((1st β€˜π‘‹) + (𝑑 / 2)))
58 elioo1 13366 . . . . . . . . 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 584 . . . . . . . 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 1342 . . . . . . 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 11266 . . . . . . . 8 (((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) β†’ (2nd β€˜π‘‹) ∈ ℝ*)
6229, 55ltsubrpd 13050 . . . . . . . 8 (((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) β†’ ((2nd β€˜π‘‹) βˆ’ (𝑑 / 2)) < (2nd β€˜π‘‹))
6329, 55ltaddrpd 13051 . . . . . . . 8 (((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) β†’ (2nd β€˜π‘‹) < ((2nd β€˜π‘‹) + (𝑑 / 2)))
64 elioo1 13366 . . . . . . . . 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 584 . . . . . . . 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 1342 . . . . . . 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 512 . . . . . 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 8012 . . . . . 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 583 . . . . 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 3146 . . . 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 13116 . . . . . . . . . . . . . . . . . 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 13112 . . . . . . . . . . . . . . . . . 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 11273 . . . . . . . . . . . . . . . . . 18 -∞ ∈ ℝ*
76 pnfxr 11270 . . . . . . . . . . . . . . . . . 18 +∞ ∈ ℝ*
77 ioossioo 13420 . . . . . . . . . . . . . . . . . 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 700 . . . . . . . . . . . . . . . . 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 584 . . . . . . . . . . . . . . . 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 13401 . . . . . . . . . . . . . . . 16 (-∞(,)+∞) = ℝ
8179, 80sseqtrdi 4032 . . . . . . . . . . . . . . 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 13116 . . . . . . . . . . . . . . . . . 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 13112 . . . . . . . . . . . . . . . . . 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 13420 . . . . . . . . . . . . . . . . . 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 700 . . . . . . . . . . . . . . . . 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 584 . . . . . . . . . . . . . . . 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 4032 . . . . . . . . . . . . . . 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 5691 . . . . . . . . . . . . . . 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 584 . . . . . . . . . . . . . 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 3982 . . . . . . . . . . . . 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 414 . . . . . . . . . . . 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 551 . . . . . . . . . . 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 570 . . . . . . . . . 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 481 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ (𝑋 ∈ 𝐴 ∧ 𝑑 ∈ ℝ+ ∧ π‘₯ ∈ (ℝ Γ— ℝ))) β†’ 𝐴 βŠ† (ℝ Γ— ℝ))
97 simpr1 1194 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ (𝑋 ∈ 𝐴 ∧ 𝑑 ∈ ℝ+ ∧ π‘₯ ∈ (ℝ Γ— ℝ))) β†’ 𝑋 ∈ 𝐴)
9896, 97sseldd 3983 . . . . . . . . . . . . . . 15 ((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ (𝑋 ∈ 𝐴 ∧ 𝑑 ∈ ℝ+ ∧ π‘₯ ∈ (ℝ Γ— ℝ))) β†’ 𝑋 ∈ (ℝ Γ— ℝ))
99983anassrs 1360 . . . . . . . . . . . . . 14 ((((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) ∧ π‘₯ ∈ (ℝ Γ— ℝ)) β†’ 𝑋 ∈ (ℝ Γ— ℝ))
100 simpr 485 . . . . . . . . . . . . . 14 ((((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) ∧ π‘₯ ∈ (ℝ Γ— ℝ)) β†’ π‘₯ ∈ (ℝ Γ— ℝ))
101 simplr 767 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) ∧ π‘₯ ∈ (ℝ Γ— ℝ)) β†’ 𝑑 ∈ ℝ+)
102101rphalfcld 13030 . . . . . . . . . . . . . 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 32960 . . . . . . . . . . . . . 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 1371 . . . . . . . . . . . . 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 2732 . . . . . . . . . . . . . . . . . . . . 21 (TopOpenβ€˜β„‚fld) = (TopOpenβ€˜β„‚fld)
107103, 6, 106cnrehmeo 24476 . . . . . . . . . . . . . . . . . . . 20 𝐺 ∈ ((𝐽 Γ—t 𝐽)Homeo(TopOpenβ€˜β„‚fld))
108106cnfldtopon 24306 . . . . . . . . . . . . . . . . . . . . . 22 (TopOpenβ€˜β„‚fld) ∈ (TopOnβ€˜β„‚)
109108toponunii 22425 . . . . . . . . . . . . . . . . . . . . 21 β„‚ = βˆͺ (TopOpenβ€˜β„‚fld)
11012, 109hmeof1o 23275 . . . . . . . . . . . . . . . . . . . 20 (𝐺 ∈ ((𝐽 Γ—t 𝐽)Homeo(TopOpenβ€˜β„‚fld)) β†’ 𝐺:(ℝ Γ— ℝ)–1-1-ontoβ†’β„‚)
111 f1of 6833 . . . . . . . . . . . . . . . . . . . 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 7087 . . . . . . . . . . . . . . . . 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 7086 . . . . . . . . . . . . . . . . 17 ((((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) ∧ π‘₯ ∈ (ℝ Γ— ℝ)) β†’ (πΊβ€˜π‘₯) ∈ β„‚)
117 sqsscirc2 32958 . . . . . . . . . . . . . . . . 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 836 . . . . . . . . . . . . . . . 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 407 . . . . . . . . . . . . . . 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 13019 . . . . . . . . . . . . . . . . . 18 ((((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) ∧ π‘₯ ∈ (ℝ Γ— ℝ)) β†’ 𝑑 ∈ ℝ*)
121120adantr 481 . . . . . . . . . . . . . . . . 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 24296 . . . . . . . . . . . . . . . . 17 (abs ∘ βˆ’ ) ∈ (∞Metβ€˜β„‚)
123121, 122jctil 520 . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . 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 512 . . . . . . . . . . . . . . . 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 2732 . . . . . . . . . . . . . . . . . . 19 (abs ∘ βˆ’ ) = (abs ∘ βˆ’ )
128127cnmetdval 24294 . . . . . . . . . . . . . . . . . 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 584 . . . . . . . . . . . . . . . . 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 485 . . . . . . . . . . . . . . . . 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 5170 . . . . . . . . . . . . . . . 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 23905 . . . . . . . . . . . . . . . . 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 478 . . . . . . . . . . . . . . . 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 836 . . . . . . . . . . . . . . 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 591 . . . . . . . . . . . . . 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 413 . . . . . . . . . . . . 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 6845 . . . . . . . . . . . . . . 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 6835 . . . . . . . . . . . . . 14 (◑𝐺:ℂ–1-1-ontoβ†’(ℝ Γ— ℝ) β†’ Fun ◑𝐺)
141139, 140ax-mp 5 . . . . . . . . . . . . 13 Fun ◑𝐺
142 f1odm 6837 . . . . . . . . . . . . . . 15 (◑𝐺:ℂ–1-1-ontoβ†’(ℝ Γ— ℝ) β†’ dom ◑𝐺 = β„‚)
143139, 142ax-mp 5 . . . . . . . . . . . . . 14 dom ◑𝐺 = β„‚
144116, 143eleqtrrdi 2844 . . . . . . . . . . . . 13 ((((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) ∧ π‘₯ ∈ (ℝ Γ— ℝ)) β†’ (πΊβ€˜π‘₯) ∈ dom ◑𝐺)
145 funfvima 7234 . . . . . . . . . . . . 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 587 . . . . . . . . . . . 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 7276 . . . . . . . . . . . . . . 15 ((𝐺:(ℝ Γ— ℝ)–1-1-ontoβ†’β„‚ ∧ π‘₯ ∈ (ℝ Γ— ℝ)) β†’ (β—‘πΊβ€˜(πΊβ€˜π‘₯)) = π‘₯)
149147, 100, 148syl2anc 584 . . . . . . . . . . . . . 14 ((((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ+) ∧ π‘₯ ∈ (ℝ Γ— ℝ)) β†’ (β—‘πΊβ€˜(πΊβ€˜π‘₯)) = π‘₯)
150149eleq1d 2818 . . . . . . . . . . . . 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 407 . . . . . . . . . 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 413 . . . . . . . 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 3988 . . . . . . 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 3146 . . . . . 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 7534 . . . . . . . . . 10 Fun 𝐺
159158a1i 11 . . . . . . . . 9 ((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) β†’ Fun 𝐺)
16013sselda 3982 . . . . . . . . . 10 ((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) β†’ 𝑋 ∈ (ℝ Γ— ℝ))
161 f1odm 6837 . . . . . . . . . . 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 2844 . . . . . . . . 9 ((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) β†’ 𝑋 ∈ dom 𝐺)
164 simpr 485 . . . . . . . . 9 ((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) β†’ 𝑋 ∈ 𝐴)
165 funfvima 7234 . . . . . . . . . 10 ((Fun 𝐺 ∧ 𝑋 ∈ dom 𝐺) β†’ (𝑋 ∈ 𝐴 β†’ (πΊβ€˜π‘‹) ∈ (𝐺 β€œ 𝐴)))
166165imp 407 . . . . . . . . 9 (((Fun 𝐺 ∧ 𝑋 ∈ dom 𝐺) ∧ 𝑋 ∈ 𝐴) β†’ (πΊβ€˜π‘‹) ∈ (𝐺 β€œ 𝐴))
167159, 163, 164, 166syl21anc 836 . . . . . . . 8 ((𝐴 ∈ (𝐽 Γ—t 𝐽) ∧ 𝑋 ∈ 𝐴) β†’ (πΊβ€˜π‘‹) ∈ (𝐺 β€œ 𝐴))
168 hmeoima 23276 . . . . . . . . . . 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 688 . . . . . . . . . 10 (𝐴 ∈ (𝐽 Γ—t 𝐽) β†’ (𝐺 β€œ 𝐴) ∈ (TopOpenβ€˜β„‚fld))
170106cnfldtopn 24305 . . . . . . . . . . . . 13 (TopOpenβ€˜β„‚fld) = (MetOpenβ€˜(abs ∘ βˆ’ ))
171170elmopn2 23958 . . . . . . . . . . . 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 497 . . . . . . . . . 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 481 . . . . . . . 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 7418 . . . . . . . . . . 11 (π‘š = (πΊβ€˜π‘‹) β†’ (π‘š(ballβ€˜(abs ∘ βˆ’ ))𝑑) = ((πΊβ€˜π‘‹)(ballβ€˜(abs ∘ βˆ’ ))𝑑))
177176sseq1d 4013 . . . . . . . . . 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 3178 . . . . . . . . 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 3610 . . . . . . . 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 584 . . . . . . 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 6101 . . . . . . . . . 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 6832 . . . . . . . . . . . . 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 6849 . . . . . . . . . . . 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 587 . . . . . . . . . . 11 (𝐴 ∈ (𝐽 Γ—t 𝐽) β†’ (◑𝐺 β€œ (𝐺 β€œ 𝐴)) = 𝐴)
186185sseq2d 4014 . . . . . . . . . 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 3170 . . . . . . . 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 481 . . . . . . 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 3114 . . . . . 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 584 . . . . 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 3990 . . . . . 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 3084 . . . . 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 3114 . . . 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 584 . . 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 3114 . . 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 584 . 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 2822 . . . . 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 4007 . . . . 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 631 . . . 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 3612 . . 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 3151 . 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 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  βˆƒwrex 3070  Vcvv 3474   βŠ† wss 3948  π’« cpw 4602  βˆͺ cuni 4908   class class class wbr 5148   Γ— cxp 5674  β—‘ccnv 5675  dom cdm 5676  ran crn 5677   β€œ cima 5679   ∘ ccom 5680  Fun wfun 6537   Fn wfn 6538  βŸΆwf 6539  β€“1-1β†’wf1 6540  β€“1-1-ontoβ†’wf1o 6542  β€˜cfv 6543  (class class class)co 7411   ∈ cmpo 7413  1st c1st 7975  2nd c2nd 7976  β„‚cc 11110  β„cr 11111  ici 11114   + caddc 11115   Β· cmul 11117  +∞cpnf 11247  -∞cmnf 11248  β„*cxr 11249   < clt 11250   ≀ cle 11251   βˆ’ cmin 11446   / cdiv 11873  2c2 12269  β„+crp 12976  (,)cioo 13326  β„œcre 15046  β„‘cim 15047  abscabs 15183  TopOpenctopn 17369  topGenctg 17385  βˆžMetcxmet 20935  ballcbl 20937  β„‚fldccnfld 20950  Topctop 22402   Γ—t ctx 23071  Homeochmeo 23264
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189  ax-pre-sup 11190  ax-addf 11191  ax-mulf 11192
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-of 7672  df-om 7858  df-1st 7977  df-2nd 7978  df-supp 8149  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-2o 8469  df-er 8705  df-map 8824  df-ixp 8894  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-fsupp 9364  df-fi 9408  df-sup 9439  df-inf 9440  df-oi 9507  df-card 9936  df-pnf 11252  df-mnf 11253  df-xr 11254  df-ltxr 11255  df-le 11256  df-sub 11448  df-neg 11449  df-div 11874  df-nn 12215  df-2 12277  df-3 12278  df-4 12279  df-5 12280  df-6 12281  df-7 12282  df-8 12283  df-9 12284  df-n0 12475  df-z 12561  df-dec 12680  df-uz 12825  df-q 12935  df-rp 12977  df-xneg 13094  df-xadd 13095  df-xmul 13096  df-ioo 13330  df-icc 13333  df-fz 13487  df-fzo 13630  df-seq 13969  df-exp 14030  df-hash 14293  df-cj 15048  df-re 15049  df-im 15050  df-sqrt 15184  df-abs 15185  df-struct 17082  df-sets 17099  df-slot 17117  df-ndx 17129  df-base 17147  df-ress 17176  df-plusg 17212  df-mulr 17213  df-starv 17214  df-sca 17215  df-vsca 17216  df-ip 17217  df-tset 17218  df-ple 17219  df-ds 17221  df-unif 17222  df-hom 17223  df-cco 17224  df-rest 17370  df-topn 17371  df-0g 17389  df-gsum 17390  df-topgen 17391  df-pt 17392  df-prds 17395  df-xrs 17450  df-qtop 17455  df-imas 17456  df-xps 17458  df-mre 17532  df-mrc 17533  df-acs 17535  df-mgm 18563  df-sgrp 18612  df-mnd 18628  df-submnd 18674  df-mulg 18953  df-cntz 19183  df-cmn 19652  df-psmet 20942  df-xmet 20943  df-met 20944  df-bl 20945  df-mopn 20946  df-cnfld 20951  df-top 22403  df-topon 22420  df-topsp 22442  df-bases 22456  df-cn 22738  df-cnp 22739  df-tx 23073  df-hmeo 23266  df-xms 23833  df-ms 23834  df-tms 23835  df-cncf 24401
This theorem is referenced by:  dya2iocnei  33350
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