tpr2rico - Mathbox for Thierry Arnoux

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Theorem tpr2rico 31437

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) ∧ 𝑋 ∈ 𝐴) → ∃𝑟 ∈ 𝐵 (𝑋 ∈ 𝑟 ∧ 𝑟 ⊆ 𝐴))

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.

Step	Hyp	Ref	Expression
1		df-ioo 12828	. . . . . . . . . 10 ⊢ (,) = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)})
2	1	ixxf 12834	. . . . . . . . 9 ⊢ (,):(ℝ^* × ℝ^)⟶𝒫 ℝ^
3		ffn 6505	. . . . . . . . 9 ⊢ ((,):(ℝ^* × ℝ^)⟶𝒫 ℝ^ → (,) Fn (ℝ^* × ℝ^*))
4	2, 3	mp1i 13	. . . . . . . 8 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → (,) Fn (ℝ^* × ℝ^*))
5		elssuni 4829	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ (𝐽 ×_t 𝐽) → 𝐴 ⊆ ∪ (𝐽 ×_t 𝐽))
6		tpr2rico.0	. . . . . . . . . . . . . . . 16 ⊢ 𝐽 = (topGen‘ran (,))
7		retop 23517	. . . . . . . . . . . . . . . 16 ⊢ (topGen‘ran (,)) ∈ Top
8	6, 7	eqeltri 2830	. . . . . . . . . . . . . . 15 ⊢ 𝐽 ∈ Top
9		uniretop 23518	. . . . . . . . . . . . . . . 16 ⊢ ℝ = ∪ (topGen‘ran (,))
10	6	unieqi 4810	. . . . . . . . . . . . . . . 16 ⊢ ∪ 𝐽 = ∪ (topGen‘ran (,))
11	9, 10	eqtr4i 2765	. . . . . . . . . . . . . . 15 ⊢ ℝ = ∪ 𝐽
12	8, 8, 11, 11	txunii 22347	. . . . . . . . . . . . . 14 ⊢ (ℝ × ℝ) = ∪ (𝐽 ×_t 𝐽)
13	5, 12	sseqtrrdi 3929	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ (𝐽 ×_t 𝐽) → 𝐴 ⊆ (ℝ × ℝ))
14	13	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → 𝐴 ⊆ (ℝ × ℝ))
15		simplr 769	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → 𝑋 ∈ 𝐴)
16	14, 15	sseldd 3879	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → 𝑋 ∈ (ℝ × ℝ))
17		xp1st 7749	. . . . . . . . . . 11 ⊢ (𝑋 ∈ (ℝ × ℝ) → (1^st ‘𝑋) ∈ ℝ)
18	16, 17	syl 17	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → (1^st ‘𝑋) ∈ ℝ)
19		simpr 488	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → 𝑑 ∈ ℝ⁺)
20	19	rpred 12517	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → 𝑑 ∈ ℝ)
21	20	rehalfcld 11966	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → (𝑑 / 2) ∈ ℝ)
22	18, 21	resubcld 11149	. . . . . . . . 9 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → ((1^st ‘𝑋) − (𝑑 / 2)) ∈ ℝ)
23	22	rexrd 10772	. . . . . . . 8 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → ((1^st ‘𝑋) − (𝑑 / 2)) ∈ ℝ^*)
24	18, 21	readdcld 10751	. . . . . . . . 9 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → ((1^st ‘𝑋) + (𝑑 / 2)) ∈ ℝ)
25	24	rexrd 10772	. . . . . . . 8 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → ((1^st ‘𝑋) + (𝑑 / 2)) ∈ ℝ^*)
26		fnovrn 7342	. . . . . . . 8 ⊢ (((,) Fn (ℝ^* × ℝ^) ∧ ((1^st ‘𝑋) − (𝑑 / 2)) ∈ ℝ^ ∧ ((1^st ‘𝑋) + (𝑑 / 2)) ∈ ℝ^*) → (((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) ∈ ran (,))
27	4, 23, 25, 26	syl3anc 1372	. . . . . . 7 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → (((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) ∈ ran (,))
28		xp2nd 7750	. . . . . . . . . . 11 ⊢ (𝑋 ∈ (ℝ × ℝ) → (2^nd ‘𝑋) ∈ ℝ)
29	16, 28	syl 17	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → (2^nd ‘𝑋) ∈ ℝ)
30	29, 21	resubcld 11149	. . . . . . . . 9 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → ((2^nd ‘𝑋) − (𝑑 / 2)) ∈ ℝ)
31	30	rexrd 10772	. . . . . . . 8 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → ((2^nd ‘𝑋) − (𝑑 / 2)) ∈ ℝ^*)
32	29, 21	readdcld 10751	. . . . . . . . 9 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → ((2^nd ‘𝑋) + (𝑑 / 2)) ∈ ℝ)
33	32	rexrd 10772	. . . . . . . 8 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → ((2^nd ‘𝑋) + (𝑑 / 2)) ∈ ℝ^*)
34		fnovrn 7342	. . . . . . . 8 ⊢ (((,) Fn (ℝ^* × ℝ^) ∧ ((2^nd ‘𝑋) − (𝑑 / 2)) ∈ ℝ^ ∧ ((2^nd ‘𝑋) + (𝑑 / 2)) ∈ ℝ^*) → (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2))) ∈ ran (,))
35	4, 31, 33, 34	syl3anc 1372	. . . . . . 7 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2))) ∈ ran (,))
36		eqidd 2740	. . . . . . 7 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) = ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))))
37		xpeq1 5540	. . . . . . . . 9 ⊢ (𝑥 = (((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) → (𝑥 × 𝑦) = ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × 𝑦))
38	37	eqeq2d 2750	. . . . . . . 8 ⊢ (𝑥 = (((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) → (((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) = (𝑥 × 𝑦) ↔ ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) = ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × 𝑦)))
39		xpeq2 5547	. . . . . . . . 9 ⊢ (𝑦 = (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2))) → ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × 𝑦) = ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))))
40	39	eqeq2d 2750	. . . . . . . 8 ⊢ (𝑦 = (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2))) → (((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) = ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × 𝑦) ↔ ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) = ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2))))))
41	38, 40	rspc2ev 3539	. . . . . . 7 ⊢ (((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) ∈ ran (,) ∧ (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2))) ∈ ran (,) ∧ ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) = ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2))))) → ∃𝑥 ∈ ran (,)∃𝑦 ∈ ran (,)((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) = (𝑥 × 𝑦))
42	27, 35, 36, 41	syl3anc 1372	. . . . . 6 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → ∃𝑥 ∈ ran (,)∃𝑦 ∈ ran (,)((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) = (𝑥 × 𝑦))
43		eqid 2739	. . . . . . 7 ⊢ (𝑥 ∈ ran (,), 𝑦 ∈ ran (,) ↦ (𝑥 × 𝑦)) = (𝑥 ∈ ran (,), 𝑦 ∈ ran (,) ↦ (𝑥 × 𝑦))
44		vex 3403	. . . . . . . 8 ⊢ 𝑥 ∈ V
45		vex 3403	. . . . . . . 8 ⊢ 𝑦 ∈ V
46	44, 45	xpex 7497	. . . . . . 7 ⊢ (𝑥 × 𝑦) ∈ V
47	43, 46	elrnmpo 7305	. . . . . 6 ⊢ (((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) ∈ ran (𝑥 ∈ ran (,), 𝑦 ∈ ran (,) ↦ (𝑥 × 𝑦)) ↔ ∃𝑥 ∈ ran (,)∃𝑦 ∈ ran (,)((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) = (𝑥 × 𝑦))
48	42, 47	sylibr 237	. . . . 5 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) ∈ ran (𝑥 ∈ ran (,), 𝑦 ∈ ran (,) ↦ (𝑥 × 𝑦)))
49		tpr2rico.2	. . . . 5 ⊢ 𝐵 = ran (𝑥 ∈ ran (,), 𝑦 ∈ ran (,) ↦ (𝑥 × 𝑦))
50	48, 49	eleqtrrdi 2845	. . . 4 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) ∈ 𝐵)
51	50	ralrimiva 3097	. . 3 ⊢ ((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) → ∀𝑑 ∈ ℝ⁺ ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) ∈ 𝐵)
52		xpss 5542	. . . . . . 7 ⊢ (ℝ × ℝ) ⊆ (V × V)
53	52, 16	sseldi 3876	. . . . . 6 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → 𝑋 ∈ (V × V))
54	18	rexrd 10772	. . . . . . . 8 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → (1^st ‘𝑋) ∈ ℝ^*)
55	19	rphalfcld 12529	. . . . . . . . 9 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → (𝑑 / 2) ∈ ℝ⁺)
56	18, 55	ltsubrpd 12549	. . . . . . . 8 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → ((1^st ‘𝑋) − (𝑑 / 2)) < (1^st ‘𝑋))
57	18, 55	ltaddrpd 12550	. . . . . . . 8 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → (1^st ‘𝑋) < ((1^st ‘𝑋) + (𝑑 / 2)))
58		elioo1 12864	. . . . . . . . 9 ⊢ ((((1^st ‘𝑋) − (𝑑 / 2)) ∈ ℝ^* ∧ ((1^st ‘𝑋) + (𝑑 / 2)) ∈ ℝ^) → ((1^st ‘𝑋) ∈ (((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) ↔ ((1^st ‘𝑋) ∈ ℝ^ ∧ ((1^st ‘𝑋) − (𝑑 / 2)) < (1^st ‘𝑋) ∧ (1^st ‘𝑋) < ((1^st ‘𝑋) + (𝑑 / 2)))))
59	23, 25, 58	syl2anc 587	. . . . . . . 8 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → ((1^st ‘𝑋) ∈ (((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) ↔ ((1^st ‘𝑋) ∈ ℝ^* ∧ ((1^st ‘𝑋) − (𝑑 / 2)) < (1^st ‘𝑋) ∧ (1^st ‘𝑋) < ((1^st ‘𝑋) + (𝑑 / 2)))))
60	54, 56, 57, 59	mpbir3and 1343	. . . . . . 7 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → (1^st ‘𝑋) ∈ (((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))))
61	29	rexrd 10772	. . . . . . . 8 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → (2^nd ‘𝑋) ∈ ℝ^*)
62	29, 55	ltsubrpd 12549	. . . . . . . 8 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → ((2^nd ‘𝑋) − (𝑑 / 2)) < (2^nd ‘𝑋))
63	29, 55	ltaddrpd 12550	. . . . . . . 8 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → (2^nd ‘𝑋) < ((2^nd ‘𝑋) + (𝑑 / 2)))
64		elioo1 12864	. . . . . . . . 9 ⊢ ((((2^nd ‘𝑋) − (𝑑 / 2)) ∈ ℝ^* ∧ ((2^nd ‘𝑋) + (𝑑 / 2)) ∈ ℝ^) → ((2^nd ‘𝑋) ∈ (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2))) ↔ ((2^nd ‘𝑋) ∈ ℝ^ ∧ ((2^nd ‘𝑋) − (𝑑 / 2)) < (2^nd ‘𝑋) ∧ (2^nd ‘𝑋) < ((2^nd ‘𝑋) + (𝑑 / 2)))))
65	31, 33, 64	syl2anc 587	. . . . . . . 8 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → ((2^nd ‘𝑋) ∈ (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2))) ↔ ((2^nd ‘𝑋) ∈ ℝ^* ∧ ((2^nd ‘𝑋) − (𝑑 / 2)) < (2^nd ‘𝑋) ∧ (2^nd ‘𝑋) < ((2^nd ‘𝑋) + (𝑑 / 2)))))
66	61, 62, 63, 65	mpbir3and 1343	. . . . . . 7 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → (2^nd ‘𝑋) ∈ (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2))))
67	60, 66	jca 515	. . . . . 6 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → ((1^st ‘𝑋) ∈ (((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) ∧ (2^nd ‘𝑋) ∈ (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))))
68		elxp7 7752	. . . . . 6 ⊢ (𝑋 ∈ ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) ↔ (𝑋 ∈ (V × V) ∧ ((1^st ‘𝑋) ∈ (((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) ∧ (2^nd ‘𝑋) ∈ (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2))))))
69	53, 67, 68	sylanbrc 586	. . . . 5 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → 𝑋 ∈ ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))))
70	69	ralrimiva 3097	. . . 4 ⊢ ((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) → ∀𝑑 ∈ ℝ⁺ 𝑋 ∈ ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))))
71		mnfle 12615	. . . . . . . . . . . . . . . . . 18 ⊢ (((1^st ‘𝑋) − (𝑑 / 2)) ∈ ℝ^* → -∞ ≤ ((1^st ‘𝑋) − (𝑑 / 2)))
72	23, 71	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → -∞ ≤ ((1^st ‘𝑋) − (𝑑 / 2)))
73		pnfge 12611	. . . . . . . . . . . . . . . . . 18 ⊢ (((1^st ‘𝑋) + (𝑑 / 2)) ∈ ℝ^* → ((1^st ‘𝑋) + (𝑑 / 2)) ≤ +∞)
74	25, 73	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → ((1^st ‘𝑋) + (𝑑 / 2)) ≤ +∞)
75		mnfxr 10779	. . . . . . . . . . . . . . . . . 18 ⊢ -∞ ∈ ℝ^*
76		pnfxr 10776	. . . . . . . . . . . . . . . . . 18 ⊢ +∞ ∈ ℝ^*
77		ioossioo 12918	. . . . . . . . . . . . . . . . . 18 ⊢ (((-∞ ∈ ℝ^* ∧ +∞ ∈ ℝ^*) ∧ (-∞ ≤ ((1^st ‘𝑋) − (𝑑 / 2)) ∧ ((1^st ‘𝑋) + (𝑑 / 2)) ≤ +∞)) → (((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) ⊆ (-∞(,)+∞))
78	75, 76, 77	mpanl12 702	. . . . . . . . . . . . . . . . 17 ⊢ ((-∞ ≤ ((1^st ‘𝑋) − (𝑑 / 2)) ∧ ((1^st ‘𝑋) + (𝑑 / 2)) ≤ +∞) → (((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) ⊆ (-∞(,)+∞))
79	72, 74, 78	syl2anc 587	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → (((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) ⊆ (-∞(,)+∞))
80		ioomax 12899	. . . . . . . . . . . . . . . 16 ⊢ (-∞(,)+∞) = ℝ
81	79, 80	sseqtrdi 3928	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → (((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) ⊆ ℝ)
82		mnfle 12615	. . . . . . . . . . . . . . . . . 18 ⊢ (((2^nd ‘𝑋) − (𝑑 / 2)) ∈ ℝ^* → -∞ ≤ ((2^nd ‘𝑋) − (𝑑 / 2)))
83	31, 82	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → -∞ ≤ ((2^nd ‘𝑋) − (𝑑 / 2)))
84		pnfge 12611	. . . . . . . . . . . . . . . . . 18 ⊢ (((2^nd ‘𝑋) + (𝑑 / 2)) ∈ ℝ^* → ((2^nd ‘𝑋) + (𝑑 / 2)) ≤ +∞)
85	33, 84	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → ((2^nd ‘𝑋) + (𝑑 / 2)) ≤ +∞)
86		ioossioo 12918	. . . . . . . . . . . . . . . . . 18 ⊢ (((-∞ ∈ ℝ^* ∧ +∞ ∈ ℝ^*) ∧ (-∞ ≤ ((2^nd ‘𝑋) − (𝑑 / 2)) ∧ ((2^nd ‘𝑋) + (𝑑 / 2)) ≤ +∞)) → (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2))) ⊆ (-∞(,)+∞))
87	75, 76, 86	mpanl12 702	. . . . . . . . . . . . . . . . 17 ⊢ ((-∞ ≤ ((2^nd ‘𝑋) − (𝑑 / 2)) ∧ ((2^nd ‘𝑋) + (𝑑 / 2)) ≤ +∞) → (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2))) ⊆ (-∞(,)+∞))
88	83, 85, 87	syl2anc 587	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2))) ⊆ (-∞(,)+∞))
89	88, 80	sseqtrdi 3928	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2))) ⊆ ℝ)
90		xpss12 5541	. . . . . . . . . . . . . . 15 ⊢ (((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) ⊆ ℝ ∧ (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2))) ⊆ ℝ) → ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) ⊆ (ℝ × ℝ))
91	81, 89, 90	syl2anc 587	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) ⊆ (ℝ × ℝ))
92	91	sselda 3878	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑥 ∈ ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2))))) → 𝑥 ∈ (ℝ × ℝ))
93	92	expcom 417	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) → (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → 𝑥 ∈ (ℝ × ℝ)))
94	93	ancld 554	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) → (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑥 ∈ (ℝ × ℝ))))
95	94	imdistanri 573	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑥 ∈ ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2))))) → ((((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑥 ∈ (ℝ × ℝ)) ∧ 𝑥 ∈ ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2))))))
96	13	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ (𝑋 ∈ 𝐴 ∧ 𝑑 ∈ ℝ⁺ ∧ 𝑥 ∈ (ℝ × ℝ))) → 𝐴 ⊆ (ℝ × ℝ))
97		simpr1 1195	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ (𝑋 ∈ 𝐴 ∧ 𝑑 ∈ ℝ⁺ ∧ 𝑥 ∈ (ℝ × ℝ))) → 𝑋 ∈ 𝐴)
98	96, 97	sseldd 3879	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ (𝑋 ∈ 𝐴 ∧ 𝑑 ∈ ℝ⁺ ∧ 𝑥 ∈ (ℝ × ℝ))) → 𝑋 ∈ (ℝ × ℝ))
99	98	3anassrs 1361	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑥 ∈ (ℝ × ℝ)) → 𝑋 ∈ (ℝ × ℝ))
100		simpr 488	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑥 ∈ (ℝ × ℝ)) → 𝑥 ∈ (ℝ × ℝ))
101		simplr 769	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑥 ∈ (ℝ × ℝ)) → 𝑑 ∈ ℝ⁺)
102	101	rphalfcld 12529	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑥 ∈ (ℝ × ℝ)) → (𝑑 / 2) ∈ ℝ⁺)
103		tpr2rico.1	. . . . . . . . . . . . . . 15 ⊢ 𝐺 = (𝑢 ∈ ℝ, 𝑣 ∈ ℝ ↦ (𝑢 + (i · 𝑣)))
104	103	cnre2csqima 31436	. . . . . . . . . . . . . 14 ⊢ ((𝑋 ∈ (ℝ × ℝ) ∧ 𝑥 ∈ (ℝ × ℝ) ∧ (𝑑 / 2) ∈ ℝ⁺) → (𝑥 ∈ ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) → ((abs‘(ℜ‘((𝐺‘𝑥) − (𝐺‘𝑋)))) < (𝑑 / 2) ∧ (abs‘(ℑ‘((𝐺‘𝑥) − (𝐺‘𝑋)))) < (𝑑 / 2))))
105	99, 100, 102, 104	syl3anc 1372	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑥 ∈ (ℝ × ℝ)) → (𝑥 ∈ ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) → ((abs‘(ℜ‘((𝐺‘𝑥) − (𝐺‘𝑋)))) < (𝑑 / 2) ∧ (abs‘(ℑ‘((𝐺‘𝑥) − (𝐺‘𝑋)))) < (𝑑 / 2))))
106		eqid 2739	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
107	103, 6, 106	cnrehmeo 23708	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝐺 ∈ ((𝐽 ×_t 𝐽)Homeo(TopOpen‘ℂ_fld))
108	106	cnfldtopon 23538	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
109	108	toponunii 21670	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ℂ = ∪ (TopOpen‘ℂ_fld)
110	12, 109	hmeof1o 22518	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐺 ∈ ((𝐽 ×_t 𝐽)Homeo(TopOpen‘ℂ_fld)) → 𝐺:(ℝ × ℝ)–1-1-onto→ℂ)
111		f1of 6621	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐺:(ℝ × ℝ)–1-1-onto→ℂ → 𝐺:(ℝ × ℝ)⟶ℂ)
112	107, 110, 111	mp2b 10	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐺:(ℝ × ℝ)⟶ℂ
113	112	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑥 ∈ (ℝ × ℝ)) → 𝐺:(ℝ × ℝ)⟶ℂ)
114	113, 99	ffvelrnd 6865	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑥 ∈ (ℝ × ℝ)) → (𝐺‘𝑋) ∈ ℂ)
115	112	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → 𝐺:(ℝ × ℝ)⟶ℂ)
116	115	ffvelrnda 6864	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑥 ∈ (ℝ × ℝ)) → (𝐺‘𝑥) ∈ ℂ)
117		sqsscirc2 31434	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐺‘𝑋) ∈ ℂ ∧ (𝐺‘𝑥) ∈ ℂ) ∧ 𝑑 ∈ ℝ⁺) → (((abs‘(ℜ‘((𝐺‘𝑥) − (𝐺‘𝑋)))) < (𝑑 / 2) ∧ (abs‘(ℑ‘((𝐺‘𝑥) − (𝐺‘𝑋)))) < (𝑑 / 2)) → (abs‘((𝐺‘𝑥) − (𝐺‘𝑋))) < 𝑑))
118	114, 116, 101, 117	syl21anc 837	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑥 ∈ (ℝ × ℝ)) → (((abs‘(ℜ‘((𝐺‘𝑥) − (𝐺‘𝑋)))) < (𝑑 / 2) ∧ (abs‘(ℑ‘((𝐺‘𝑥) − (𝐺‘𝑋)))) < (𝑑 / 2)) → (abs‘((𝐺‘𝑥) − (𝐺‘𝑋))) < 𝑑))
119	118	imp 410	. . . . . . . . . . . . . . 15 ⊢ (((((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑥 ∈ (ℝ × ℝ)) ∧ ((abs‘(ℜ‘((𝐺‘𝑥) − (𝐺‘𝑋)))) < (𝑑 / 2) ∧ (abs‘(ℑ‘((𝐺‘𝑥) − (𝐺‘𝑋)))) < (𝑑 / 2))) → (abs‘((𝐺‘𝑥) − (𝐺‘𝑋))) < 𝑑)
120	101	rpxrd 12518	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑥 ∈ (ℝ × ℝ)) → 𝑑 ∈ ℝ^*)
121	120	adantr 484	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑥 ∈ (ℝ × ℝ)) ∧ (abs‘((𝐺‘𝑥) − (𝐺‘𝑋))) < 𝑑) → 𝑑 ∈ ℝ^*)
122		cnxmet 23528	. . . . . . . . . . . . . . . . 17 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
123	121, 122	jctil 523	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑥 ∈ (ℝ × ℝ)) ∧ (abs‘((𝐺‘𝑥) − (𝐺‘𝑋))) < 𝑑) → ((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑑 ∈ ℝ^*))
124	114	adantr 484	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑥 ∈ (ℝ × ℝ)) ∧ (abs‘((𝐺‘𝑥) − (𝐺‘𝑋))) < 𝑑) → (𝐺‘𝑋) ∈ ℂ)
125	116	adantr 484	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑥 ∈ (ℝ × ℝ)) ∧ (abs‘((𝐺‘𝑥) − (𝐺‘𝑋))) < 𝑑) → (𝐺‘𝑥) ∈ ℂ)
126	124, 125	jca 515	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑥 ∈ (ℝ × ℝ)) ∧ (abs‘((𝐺‘𝑥) − (𝐺‘𝑋))) < 𝑑) → ((𝐺‘𝑋) ∈ ℂ ∧ (𝐺‘𝑥) ∈ ℂ))
127		eqid 2739	. . . . . . . . . . . . . . . . . . 19 ⊢ (abs ∘ − ) = (abs ∘ − )
128	127	cnmetdval 23526	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐺‘𝑥) ∈ ℂ ∧ (𝐺‘𝑋) ∈ ℂ) → ((𝐺‘𝑥)(abs ∘ − )(𝐺‘𝑋)) = (abs‘((𝐺‘𝑥) − (𝐺‘𝑋))))
129	125, 124, 128	syl2anc 587	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑥 ∈ (ℝ × ℝ)) ∧ (abs‘((𝐺‘𝑥) − (𝐺‘𝑋))) < 𝑑) → ((𝐺‘𝑥)(abs ∘ − )(𝐺‘𝑋)) = (abs‘((𝐺‘𝑥) − (𝐺‘𝑋))))
130		simpr 488	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑥 ∈ (ℝ × ℝ)) ∧ (abs‘((𝐺‘𝑥) − (𝐺‘𝑋))) < 𝑑) → (abs‘((𝐺‘𝑥) − (𝐺‘𝑋))) < 𝑑)
131	129, 130	eqbrtrd 5053	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑥 ∈ (ℝ × ℝ)) ∧ (abs‘((𝐺‘𝑥) − (𝐺‘𝑋))) < 𝑑) → ((𝐺‘𝑥)(abs ∘ − )(𝐺‘𝑋)) < 𝑑)
132		elbl3 23148	. . . . . . . . . . . . . . . . 17 ⊢ ((((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑑 ∈ ℝ^*) ∧ ((𝐺‘𝑋) ∈ ℂ ∧ (𝐺‘𝑥) ∈ ℂ)) → ((𝐺‘𝑥) ∈ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑) ↔ ((𝐺‘𝑥)(abs ∘ − )(𝐺‘𝑋)) < 𝑑))
133	132	biimpar 481	. . . . . . . . . . . . . . . 16 ⊢ (((((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑑 ∈ ℝ^*) ∧ ((𝐺‘𝑋) ∈ ℂ ∧ (𝐺‘𝑥) ∈ ℂ)) ∧ ((𝐺‘𝑥)(abs ∘ − )(𝐺‘𝑋)) < 𝑑) → (𝐺‘𝑥) ∈ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑))
134	123, 126, 131, 133	syl21anc 837	. . . . . . . . . . . . . . 15 ⊢ (((((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑥 ∈ (ℝ × ℝ)) ∧ (abs‘((𝐺‘𝑥) − (𝐺‘𝑋))) < 𝑑) → (𝐺‘𝑥) ∈ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑))
135	119, 134	syldan 594	. . . . . . . . . . . . . 14 ⊢ (((((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑥 ∈ (ℝ × ℝ)) ∧ ((abs‘(ℜ‘((𝐺‘𝑥) − (𝐺‘𝑋)))) < (𝑑 / 2) ∧ (abs‘(ℑ‘((𝐺‘𝑥) − (𝐺‘𝑋)))) < (𝑑 / 2))) → (𝐺‘𝑥) ∈ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑))
136	135	ex 416	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑥 ∈ (ℝ × ℝ)) → (((abs‘(ℜ‘((𝐺‘𝑥) − (𝐺‘𝑋)))) < (𝑑 / 2) ∧ (abs‘(ℑ‘((𝐺‘𝑥) − (𝐺‘𝑋)))) < (𝑑 / 2)) → (𝐺‘𝑥) ∈ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑)))
137	105, 136	syld 47	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑥 ∈ (ℝ × ℝ)) → (𝑥 ∈ ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) → (𝐺‘𝑥) ∈ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑)))
138		f1ocnv 6633	. . . . . . . . . . . . . . 15 ⊢ (𝐺:(ℝ × ℝ)–1-1-onto→ℂ → ^◡𝐺:ℂ–1-1-onto→(ℝ × ℝ))
139	107, 110, 138	mp2b 10	. . . . . . . . . . . . . 14 ⊢ ^◡𝐺:ℂ–1-1-onto→(ℝ × ℝ)
140		f1ofun 6623	. . . . . . . . . . . . . 14 ⊢ (^◡𝐺:ℂ–1-1-onto→(ℝ × ℝ) → Fun ^◡𝐺)
141	139, 140	ax-mp 5	. . . . . . . . . . . . 13 ⊢ Fun ^◡𝐺
142		f1odm 6625	. . . . . . . . . . . . . . 15 ⊢ (^◡𝐺:ℂ–1-1-onto→(ℝ × ℝ) → dom ^◡𝐺 = ℂ)
143	139, 142	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ dom ^◡𝐺 = ℂ
144	116, 143	eleqtrrdi 2845	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑥 ∈ (ℝ × ℝ)) → (𝐺‘𝑥) ∈ dom ^◡𝐺)
145		funfvima 7006	. . . . . . . . . . . . 13 ⊢ ((Fun ^◡𝐺 ∧ (𝐺‘𝑥) ∈ dom ^◡𝐺) → ((𝐺‘𝑥) ∈ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑) → (^◡𝐺‘(𝐺‘𝑥)) ∈ (^◡𝐺 “ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑))))
146	141, 144, 145	sylancr 590	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑥 ∈ (ℝ × ℝ)) → ((𝐺‘𝑥) ∈ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑) → (^◡𝐺‘(𝐺‘𝑥)) ∈ (^◡𝐺 “ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑))))
147	107, 110	mp1i 13	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑥 ∈ (ℝ × ℝ)) → 𝐺:(ℝ × ℝ)–1-1-onto→ℂ)
148		f1ocnvfv1 7047	. . . . . . . . . . . . . . 15 ⊢ ((𝐺:(ℝ × ℝ)–1-1-onto→ℂ ∧ 𝑥 ∈ (ℝ × ℝ)) → (^◡𝐺‘(𝐺‘𝑥)) = 𝑥)
149	147, 100, 148	syl2anc 587	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑥 ∈ (ℝ × ℝ)) → (^◡𝐺‘(𝐺‘𝑥)) = 𝑥)
150	149	eleq1d 2818	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑥 ∈ (ℝ × ℝ)) → ((^◡𝐺‘(𝐺‘𝑥)) ∈ (^◡𝐺 “ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑)) ↔ 𝑥 ∈ (^◡𝐺 “ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑))))
151	150	biimpd 232	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑥 ∈ (ℝ × ℝ)) → ((^◡𝐺‘(𝐺‘𝑥)) ∈ (^◡𝐺 “ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑)) → 𝑥 ∈ (^◡𝐺 “ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑))))
152	137, 146, 151	3syld 60	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑥 ∈ (ℝ × ℝ)) → (𝑥 ∈ ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) → 𝑥 ∈ (^◡𝐺 “ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑))))
153	152	imp 410	. . . . . . . . . 10 ⊢ (((((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑥 ∈ (ℝ × ℝ)) ∧ 𝑥 ∈ ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2))))) → 𝑥 ∈ (^◡𝐺 “ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑)))
154	95, 153	syl 17	. . . . . . . . 9 ⊢ ((((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑥 ∈ ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2))))) → 𝑥 ∈ (^◡𝐺 “ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑)))
155	154	ex 416	. . . . . . . 8 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → (𝑥 ∈ ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) → 𝑥 ∈ (^◡𝐺 “ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑))))
156	155	ssrdv 3884	. . . . . . 7 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) ⊆ (^◡𝐺 “ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑)))
157	156	ralrimiva 3097	. . . . . 6 ⊢ ((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) → ∀𝑑 ∈ ℝ⁺ ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) ⊆ (^◡𝐺 “ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑)))
158	103	mpofun 7293	. . . . . . . . . 10 ⊢ Fun 𝐺
159	158	a1i 11	. . . . . . . . 9 ⊢ ((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) → Fun 𝐺)
160	13	sselda 3878	. . . . . . . . . 10 ⊢ ((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) → 𝑋 ∈ (ℝ × ℝ))
161		f1odm 6625	. . . . . . . . . . 11 ⊢ (𝐺:(ℝ × ℝ)–1-1-onto→ℂ → dom 𝐺 = (ℝ × ℝ))
162	107, 110, 161	mp2b 10	. . . . . . . . . 10 ⊢ dom 𝐺 = (ℝ × ℝ)
163	160, 162	eleqtrrdi 2845	. . . . . . . . 9 ⊢ ((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) → 𝑋 ∈ dom 𝐺)
164		simpr 488	. . . . . . . . 9 ⊢ ((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) → 𝑋 ∈ 𝐴)
165		funfvima 7006	. . . . . . . . . 10 ⊢ ((Fun 𝐺 ∧ 𝑋 ∈ dom 𝐺) → (𝑋 ∈ 𝐴 → (𝐺‘𝑋) ∈ (𝐺 “ 𝐴)))
166	165	imp 410	. . . . . . . . 9 ⊢ (((Fun 𝐺 ∧ 𝑋 ∈ dom 𝐺) ∧ 𝑋 ∈ 𝐴) → (𝐺‘𝑋) ∈ (𝐺 “ 𝐴))
167	159, 163, 164, 166	syl21anc 837	. . . . . . . 8 ⊢ ((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) → (𝐺‘𝑋) ∈ (𝐺 “ 𝐴))
168		hmeoima 22519	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ ((𝐽 ×_t 𝐽)Homeo(TopOpen‘ℂ_fld)) ∧ 𝐴 ∈ (𝐽 ×_t 𝐽)) → (𝐺 “ 𝐴) ∈ (TopOpen‘ℂ_fld))
169	107, 168	mpan 690	. . . . . . . . . 10 ⊢ (𝐴 ∈ (𝐽 ×_t 𝐽) → (𝐺 “ 𝐴) ∈ (TopOpen‘ℂ_fld))
170	106	cnfldtopn 23537	. . . . . . . . . . . . 13 ⊢ (TopOpen‘ℂ_fld) = (MetOpen‘(abs ∘ − ))
171	170	elmopn2 23201	. . . . . . . . . . . 12 ⊢ ((abs ∘ − ) ∈ (∞Met‘ℂ) → ((𝐺 “ 𝐴) ∈ (TopOpen‘ℂ_fld) ↔ ((𝐺 “ 𝐴) ⊆ ℂ ∧ ∀𝑚 ∈ (𝐺 “ 𝐴)∃𝑑 ∈ ℝ⁺ (𝑚(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺 “ 𝐴))))
172	122, 171	ax-mp 5	. . . . . . . . . . 11 ⊢ ((𝐺 “ 𝐴) ∈ (TopOpen‘ℂ_fld) ↔ ((𝐺 “ 𝐴) ⊆ ℂ ∧ ∀𝑚 ∈ (𝐺 “ 𝐴)∃𝑑 ∈ ℝ⁺ (𝑚(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺 “ 𝐴)))
173	172	simprbi 500	. . . . . . . . . 10 ⊢ ((𝐺 “ 𝐴) ∈ (TopOpen‘ℂ_fld) → ∀𝑚 ∈ (𝐺 “ 𝐴)∃𝑑 ∈ ℝ⁺ (𝑚(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺 “ 𝐴))
174	169, 173	syl 17	. . . . . . . . 9 ⊢ (𝐴 ∈ (𝐽 ×_t 𝐽) → ∀𝑚 ∈ (𝐺 “ 𝐴)∃𝑑 ∈ ℝ⁺ (𝑚(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺 “ 𝐴))
175	174	adantr 484	. . . . . . . 8 ⊢ ((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) → ∀𝑚 ∈ (𝐺 “ 𝐴)∃𝑑 ∈ ℝ⁺ (𝑚(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺 “ 𝐴))
176		oveq1 7180	. . . . . . . . . . 11 ⊢ (𝑚 = (𝐺‘𝑋) → (𝑚(ball‘(abs ∘ − ))𝑑) = ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑))
177	176	sseq1d 3909	. . . . . . . . . 10 ⊢ (𝑚 = (𝐺‘𝑋) → ((𝑚(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺 “ 𝐴) ↔ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺 “ 𝐴)))
178	177	rexbidv 3208	. . . . . . . . 9 ⊢ (𝑚 = (𝐺‘𝑋) → (∃𝑑 ∈ ℝ⁺ (𝑚(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺 “ 𝐴) ↔ ∃𝑑 ∈ ℝ⁺ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺 “ 𝐴)))
179	178	rspcva 3525	. . . . . . . 8 ⊢ (((𝐺‘𝑋) ∈ (𝐺 “ 𝐴) ∧ ∀𝑚 ∈ (𝐺 “ 𝐴)∃𝑑 ∈ ℝ⁺ (𝑚(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺 “ 𝐴)) → ∃𝑑 ∈ ℝ⁺ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺 “ 𝐴))
180	167, 175, 179	syl2anc 587	. . . . . . 7 ⊢ ((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) → ∃𝑑 ∈ ℝ⁺ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺 “ 𝐴))
181		imass2 5940	. . . . . . . . . 10 ⊢ (((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺 “ 𝐴) → (^◡𝐺 “ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑)) ⊆ (^◡𝐺 “ (𝐺 “ 𝐴)))
182		f1of1 6620	. . . . . . . . . . . . 13 ⊢ (𝐺:(ℝ × ℝ)–1-1-onto→ℂ → 𝐺:(ℝ × ℝ)–1-1→ℂ)
183	107, 110, 182	mp2b 10	. . . . . . . . . . . 12 ⊢ 𝐺:(ℝ × ℝ)–1-1→ℂ
184		f1imacnv 6637	. . . . . . . . . . . 12 ⊢ ((𝐺:(ℝ × ℝ)–1-1→ℂ ∧ 𝐴 ⊆ (ℝ × ℝ)) → (^◡𝐺 “ (𝐺 “ 𝐴)) = 𝐴)
185	183, 13, 184	sylancr 590	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (𝐽 ×_t 𝐽) → (^◡𝐺 “ (𝐺 “ 𝐴)) = 𝐴)
186	185	sseq2d 3910	. . . . . . . . . 10 ⊢ (𝐴 ∈ (𝐽 ×_t 𝐽) → ((^◡𝐺 “ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑)) ⊆ (^◡𝐺 “ (𝐺 “ 𝐴)) ↔ (^◡𝐺 “ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑)) ⊆ 𝐴))
187	181, 186	syl5ib 247	. . . . . . . . 9 ⊢ (𝐴 ∈ (𝐽 ×_t 𝐽) → (((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺 “ 𝐴) → (^◡𝐺 “ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑)) ⊆ 𝐴))
188	187	reximdv 3184	. . . . . . . 8 ⊢ (𝐴 ∈ (𝐽 ×_t 𝐽) → (∃𝑑 ∈ ℝ⁺ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺 “ 𝐴) → ∃𝑑 ∈ ℝ⁺ (^◡𝐺 “ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑)) ⊆ 𝐴))
189	188	adantr 484	. . . . . . 7 ⊢ ((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) → (∃𝑑 ∈ ℝ⁺ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺 “ 𝐴) → ∃𝑑 ∈ ℝ⁺ (^◡𝐺 “ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑)) ⊆ 𝐴))
190	180, 189	mpd 15	. . . . . 6 ⊢ ((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) → ∃𝑑 ∈ ℝ⁺ (^◡𝐺 “ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑)) ⊆ 𝐴)
191		r19.29 3168	. . . . . 6 ⊢ ((∀𝑑 ∈ ℝ⁺ ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) ⊆ (^◡𝐺 “ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑)) ∧ ∃𝑑 ∈ ℝ⁺ (^◡𝐺 “ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑)) ⊆ 𝐴) → ∃𝑑 ∈ ℝ⁺ (((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) ⊆ (^◡𝐺 “ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑)) ∧ (^◡𝐺 “ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑)) ⊆ 𝐴))
192	157, 190, 191	syl2anc 587	. . . . 5 ⊢ ((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) → ∃𝑑 ∈ ℝ⁺ (((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) ⊆ (^◡𝐺 “ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑)) ∧ (^◡𝐺 “ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑)) ⊆ 𝐴))
193		sstr 3886	. . . . . 6 ⊢ ((((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) ⊆ (^◡𝐺 “ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑)) ∧ (^◡𝐺 “ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑)) ⊆ 𝐴) → ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) ⊆ 𝐴)
194	193	reximi 3158	. . . . 5 ⊢ (∃𝑑 ∈ ℝ⁺ (((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) ⊆ (^◡𝐺 “ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑)) ∧ (^◡𝐺 “ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑)) ⊆ 𝐴) → ∃𝑑 ∈ ℝ⁺ ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) ⊆ 𝐴)
195	192, 194	syl 17	. . . 4 ⊢ ((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) → ∃𝑑 ∈ ℝ⁺ ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) ⊆ 𝐴)
196		r19.29 3168	. . . 4 ⊢ ((∀𝑑 ∈ ℝ⁺ 𝑋 ∈ ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) ∧ ∃𝑑 ∈ ℝ⁺ ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) ⊆ 𝐴) → ∃𝑑 ∈ ℝ⁺ (𝑋 ∈ ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) ∧ ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) ⊆ 𝐴))
197	70, 195, 196	syl2anc 587	. . 3 ⊢ ((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) → ∃𝑑 ∈ ℝ⁺ (𝑋 ∈ ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) ∧ ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) ⊆ 𝐴))
198		r19.29 3168	. . 3 ⊢ ((∀𝑑 ∈ ℝ⁺ ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) ∈ 𝐵 ∧ ∃𝑑 ∈ ℝ⁺ (𝑋 ∈ ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) ∧ ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) ⊆ 𝐴)) → ∃𝑑 ∈ ℝ⁺ (((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) ∈ 𝐵 ∧ (𝑋 ∈ ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) ∧ ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) ⊆ 𝐴)))
199	51, 197, 198	syl2anc 587	. 2 ⊢ ((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) → ∃𝑑 ∈ ℝ⁺ (((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) ∈ 𝐵 ∧ (𝑋 ∈ ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) ∧ ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) ⊆ 𝐴)))
200		eleq2 2822	. . . . 5 ⊢ (𝑟 = ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) → (𝑋 ∈ 𝑟 ↔ 𝑋 ∈ ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2))))))
201		sseq1 3903	. . . . 5 ⊢ (𝑟 = ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) → (𝑟 ⊆ 𝐴 ↔ ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) ⊆ 𝐴))
202	200, 201	anbi12d 634	. . . 4 ⊢ (𝑟 = ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) → ((𝑋 ∈ 𝑟 ∧ 𝑟 ⊆ 𝐴) ↔ (𝑋 ∈ ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) ∧ ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) ⊆ 𝐴)))
203	202	rspcev 3527	. . 3 ⊢ ((((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) ∈ 𝐵 ∧ (𝑋 ∈ ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) ∧ ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) ⊆ 𝐴)) → ∃𝑟 ∈ 𝐵 (𝑋 ∈ 𝑟 ∧ 𝑟 ⊆ 𝐴))
204	203	rexlimivw 3193	. 2 ⊢ (∃𝑑 ∈ ℝ⁺ (((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) ∈ 𝐵 ∧ (𝑋 ∈ ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) ∧ ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) ⊆ 𝐴)) → ∃𝑟 ∈ 𝐵 (𝑋 ∈ 𝑟 ∧ 𝑟 ⊆ 𝐴))
205	199, 204	syl 17	1 ⊢ ((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) → ∃𝑟 ∈ 𝐵 (𝑋 ∈ 𝑟 ∧ 𝑟 ⊆ 𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1088 = wceq 1542 ∈ wcel 2114 ∀wral 3054 ∃wrex 3055 Vcvv 3399 ⊆ wss 3844 𝒫 cpw 4489 ∪ cuni 4797 class class class wbr 5031 × cxp 5524 ^◡ccnv 5525 dom cdm 5526 ran crn 5527 “ cima 5529 ∘ ccom 5530 Fun wfun 6334 Fn wfn 6335 ⟶wf 6336 –1-1→wf1 6337 –1-1-onto→wf1o 6339 ‘cfv 6340 (class class class)co 7173 ∈ cmpo 7175 1^st c1st 7715 2^nd c2nd 7716 ℂcc 10616 ℝcr 10617 ici 10620 + caddc 10621 · cmul 10623 +∞cpnf 10753 -∞cmnf 10754 ℝ^*cxr 10755 < clt 10756 ≤ cle 10757 − cmin 10951 / cdiv 11378 2c2 11774 ℝ⁺crp 12475 (,)cioo 12824 ℜcre 14549 ℑcim 14550 abscabs 14686 TopOpenctopn 16801 topGenctg 16817 ∞Metcxmet 20205 ballcbl 20207 ℂ_fldccnfld 20220 Topctop 21647 ×_t ctx 22314 Homeochmeo 22507

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5155 ax-sep 5168 ax-nul 5175 ax-pow 5233 ax-pr 5297 ax-un 7482 ax-cnex 10674 ax-resscn 10675 ax-1cn 10676 ax-icn 10677 ax-addcl 10678 ax-addrcl 10679 ax-mulcl 10680 ax-mulrcl 10681 ax-mulcom 10682 ax-addass 10683 ax-mulass 10684 ax-distr 10685 ax-i2m1 10686 ax-1ne0 10687 ax-1rid 10688 ax-rnegex 10689 ax-rrecex 10690 ax-cnre 10691 ax-pre-lttri 10692 ax-pre-lttrn 10693 ax-pre-ltadd 10694 ax-pre-mulgt0 10695 ax-pre-sup 10696 ax-addf 10697 ax-mulf 10698

This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3401 df-sbc 3682 df-csb 3792 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-pss 3863 df-nul 4213 df-if 4416 df-pw 4491 df-sn 4518 df-pr 4520 df-tp 4522 df-op 4524 df-uni 4798 df-int 4838 df-iun 4884 df-iin 4885 df-br 5032 df-opab 5094 df-mpt 5112 df-tr 5138 df-id 5430 df-eprel 5435 df-po 5443 df-so 5444 df-fr 5484 df-se 5485 df-we 5486 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-rn 5537 df-res 5538 df-ima 5539 df-pred 6130 df-ord 6176 df-on 6177 df-lim 6178 df-suc 6179 df-iota 6298 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-isom 6349 df-riota 7130 df-ov 7176 df-oprab 7177 df-mpo 7178 df-of 7428 df-om 7603 df-1st 7717 df-2nd 7718 df-supp 7860 df-wrecs 7979 df-recs 8040 df-rdg 8078 df-1o 8134 df-2o 8135 df-er 8323 df-map 8442 df-ixp 8511 df-en 8559 df-dom 8560 df-sdom 8561 df-fin 8562 df-fsupp 8910 df-fi 8951 df-sup 8982 df-inf 8983 df-oi 9050 df-card 9444 df-pnf 10758 df-mnf 10759 df-xr 10760 df-ltxr 10761 df-le 10762 df-sub 10953 df-neg 10954 df-div 11379 df-nn 11720 df-2 11782 df-3 11783 df-4 11784 df-5 11785 df-6 11786 df-7 11787 df-8 11788 df-9 11789 df-n0 11980 df-z 12066 df-dec 12183 df-uz 12328 df-q 12434 df-rp 12476 df-xneg 12593 df-xadd 12594 df-xmul 12595 df-ioo 12828 df-icc 12831 df-fz 12985 df-fzo 13128 df-seq 13464 df-exp 13525 df-hash 13786 df-cj 14551 df-re 14552 df-im 14553 df-sqrt 14687 df-abs 14688 df-struct 16591 df-ndx 16592 df-slot 16593 df-base 16595 df-sets 16596 df-ress 16597 df-plusg 16684 df-mulr 16685 df-starv 16686 df-sca 16687 df-vsca 16688 df-ip 16689 df-tset 16690 df-ple 16691 df-ds 16693 df-unif 16694 df-hom 16695 df-cco 16696 df-rest 16802 df-topn 16803 df-0g 16821 df-gsum 16822 df-topgen 16823 df-pt 16824 df-prds 16827 df-xrs 16881 df-qtop 16886 df-imas 16887 df-xps 16889 df-mre 16963 df-mrc 16964 df-acs 16966 df-mgm 17971 df-sgrp 18020 df-mnd 18031 df-submnd 18076 df-mulg 18346 df-cntz 18568 df-cmn 19029 df-psmet 20212 df-xmet 20213 df-met 20214 df-bl 20215 df-mopn 20216 df-cnfld 20221 df-top 21648 df-topon 21665 df-topsp 21687 df-bases 21700 df-cn 21981 df-cnp 21982 df-tx 22316 df-hmeo 22509 df-xms 23076 df-ms 23077 df-tms 23078 df-cncf 23633

This theorem is referenced by: dya2iocnei 31822

W3C validator