tpr2rico - Mathbox for Thierry Arnoux

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Theorem tpr2rico 29286

Description: For any point of an open set of the usual topology on (ℝ × ℝ) there is an open square which contains that point and is entirely in the open set. This is square is actually a ball by the (𝑙↑+∞) norm 𝑋. (Contributed by Thierry Arnoux, 21-Sep-2017.)

**Hypotheses**
Ref	Expression
tpr2rico.0	⊢ 𝐽 = (topGen‘ran (,))
tpr2rico.1	⊢ 𝐺 = (𝑢 ∈ ℝ, 𝑣 ∈ ℝ ↦ (𝑢 + (i · 𝑣)))
tpr2rico.2	⊢ 𝐵 = ran (𝑥 ∈ ran (,), 𝑦 ∈ ran (,) ↦ (𝑥 × 𝑦))

**Assertion**
Ref	Expression
tpr2rico	⊢ ((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 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 12050	. . . . . . . . . 10 ⊢ (,) = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)})
2	1	ixxf 12056	. . . . . . . . 9 ⊢ (,):(ℝ^* × ℝ^)⟶𝒫 ℝ^
3		ffn 5958	. . . . . . . . 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 4403	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ (𝐽 ×_t 𝐽) → 𝐴 ⊆ ∪ (𝐽 ×_t 𝐽))
6		tpr2rico.0	. . . . . . . . . . . . . . . 16 ⊢ 𝐽 = (topGen‘ran (,))
7		retop 22375	. . . . . . . . . . . . . . . 16 ⊢ (topGen‘ran (,)) ∈ Top
8	6, 7	eqeltri 2684	. . . . . . . . . . . . . . 15 ⊢ 𝐽 ∈ Top
9		uniretop 22376	. . . . . . . . . . . . . . . 16 ⊢ ℝ = ∪ (topGen‘ran (,))
10	6	unieqi 4381	. . . . . . . . . . . . . . . 16 ⊢ ∪ 𝐽 = ∪ (topGen‘ran (,))
11	9, 10	eqtr4i 2635	. . . . . . . . . . . . . . 15 ⊢ ℝ = ∪ 𝐽
12	8, 8, 11, 11	txunii 21206	. . . . . . . . . . . . . 14 ⊢ (ℝ × ℝ) = ∪ (𝐽 ×_t 𝐽)
13	5, 12	syl6sseqr 3615	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ (𝐽 ×_t 𝐽) → 𝐴 ⊆ (ℝ × ℝ))
14	13	ad2antrr 758	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → 𝐴 ⊆ (ℝ × ℝ))
15		simplr 788	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → 𝑋 ∈ 𝐴)
16	14, 15	sseldd 3569	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → 𝑋 ∈ (ℝ × ℝ))
17		xp1st 7089	. . . . . . . . . . 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 476	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → 𝑑 ∈ ℝ⁺)
20	19	rpred 11748	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → 𝑑 ∈ ℝ)
21	20	rehalfcld 11156	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → (𝑑 / 2) ∈ ℝ)
22	18, 21	resubcld 10337	. . . . . . . . 9 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → ((1^st ‘𝑋) − (𝑑 / 2)) ∈ ℝ)
23	22	rexrd 9968	. . . . . . . 8 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → ((1^st ‘𝑋) − (𝑑 / 2)) ∈ ℝ^*)
24	18, 21	readdcld 9948	. . . . . . . . 9 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → ((1^st ‘𝑋) + (𝑑 / 2)) ∈ ℝ)
25	24	rexrd 9968	. . . . . . . 8 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → ((1^st ‘𝑋) + (𝑑 / 2)) ∈ ℝ^*)
26		fnovrn 6707	. . . . . . . 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 1318	. . . . . . 7 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → (((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) ∈ ran (,))
28		xp2nd 7090	. . . . . . . . . . 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 10337	. . . . . . . . 9 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → ((2^nd ‘𝑋) − (𝑑 / 2)) ∈ ℝ)
31	30	rexrd 9968	. . . . . . . 8 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → ((2^nd ‘𝑋) − (𝑑 / 2)) ∈ ℝ^*)
32	29, 21	readdcld 9948	. . . . . . . . 9 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → ((2^nd ‘𝑋) + (𝑑 / 2)) ∈ ℝ)
33	32	rexrd 9968	. . . . . . . 8 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → ((2^nd ‘𝑋) + (𝑑 / 2)) ∈ ℝ^*)
34		fnovrn 6707	. . . . . . . 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 1318	. . . . . . 7 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2))) ∈ ran (,))
36		eqidd 2611	. . . . . . 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 5052	. . . . . . . . 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 2620	. . . . . . . 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 5053	. . . . . . . . 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 2620	. . . . . . . 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 3295	. . . . . . 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 1318	. . . . . 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 2610	. . . . . . 7 ⊢ (𝑥 ∈ ran (,), 𝑦 ∈ ran (,) ↦ (𝑥 × 𝑦)) = (𝑥 ∈ ran (,), 𝑦 ∈ ran (,) ↦ (𝑥 × 𝑦))
44		vex 3176	. . . . . . . 8 ⊢ 𝑥 ∈ V
45		vex 3176	. . . . . . . 8 ⊢ 𝑦 ∈ V
46	44, 45	xpex 6860	. . . . . . 7 ⊢ (𝑥 × 𝑦) ∈ V
47	43, 46	elrnmpt2 6671	. . . . . 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 223	. . . . 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	syl6eleqr 2699	. . . 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 2949	. . 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 5149	. . . . . . 7 ⊢ (ℝ × ℝ) ⊆ (V × V)
53	52, 16	sseldi 3566	. . . . . 6 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → 𝑋 ∈ (V × V))
54	18	rexrd 9968	. . . . . . . 8 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → (1^st ‘𝑋) ∈ ℝ^*)
55	19	rphalfcld 11760	. . . . . . . . 9 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → (𝑑 / 2) ∈ ℝ⁺)
56	18, 55	ltsubrpd 11780	. . . . . . . 8 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → ((1^st ‘𝑋) − (𝑑 / 2)) < (1^st ‘𝑋))
57	18, 55	ltaddrpd 11781	. . . . . . . 8 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → (1^st ‘𝑋) < ((1^st ‘𝑋) + (𝑑 / 2)))
58		elioo1 12086	. . . . . . . . 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 691	. . . . . . . 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 1238	. . . . . . 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 9968	. . . . . . . 8 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → (2^nd ‘𝑋) ∈ ℝ^*)
62	29, 55	ltsubrpd 11780	. . . . . . . 8 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → ((2^nd ‘𝑋) − (𝑑 / 2)) < (2^nd ‘𝑋))
63	29, 55	ltaddrpd 11781	. . . . . . . 8 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → (2^nd ‘𝑋) < ((2^nd ‘𝑋) + (𝑑 / 2)))
64		elioo1 12086	. . . . . . . . 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 691	. . . . . . . 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 1238	. . . . . . 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 553	. . . . . 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 7092	. . . . . 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 695	. . . . 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 2949	. . . 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 11845	. . . . . . . . . . . . . . . . . 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 11840	. . . . . . . . . . . . . . . . . 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 9975	. . . . . . . . . . . . . . . . . 18 ⊢ -∞ ∈ ℝ^*
76		pnfxr 9971	. . . . . . . . . . . . . . . . . 18 ⊢ +∞ ∈ ℝ^*
77		ioossioo 12136	. . . . . . . . . . . . . . . . . 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 714	. . . . . . . . . . . . . . . . 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 691	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → (((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) ⊆ (-∞(,)+∞))
80		ioomax 12119	. . . . . . . . . . . . . . . 16 ⊢ (-∞(,)+∞) = ℝ
81	79, 80	syl6sseq 3614	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → (((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) ⊆ ℝ)
82		mnfle 11845	. . . . . . . . . . . . . . . . . 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 11840	. . . . . . . . . . . . . . . . . 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 12136	. . . . . . . . . . . . . . . . . 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 714	. . . . . . . . . . . . . . . . 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 691	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2))) ⊆ (-∞(,)+∞))
89	88, 80	syl6sseq 3614	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2))) ⊆ ℝ)
90		xpss12 5148	. . . . . . . . . . . . . . 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 691	. . . . . . . . . . . . . 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 3568	. . . . . . . . . . . . 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 450	. . . . . . . . . . . 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 574	. . . . . . . . . . 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 723	. . . . . . . . . 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 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ (𝑋 ∈ 𝐴 ∧ 𝑑 ∈ ℝ⁺ ∧ 𝑥 ∈ (ℝ × ℝ))) → 𝐴 ⊆ (ℝ × ℝ))
97		simpr1 1060	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ (𝑋 ∈ 𝐴 ∧ 𝑑 ∈ ℝ⁺ ∧ 𝑥 ∈ (ℝ × ℝ))) → 𝑋 ∈ 𝐴)
98	96, 97	sseldd 3569	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ (𝑋 ∈ 𝐴 ∧ 𝑑 ∈ ℝ⁺ ∧ 𝑥 ∈ (ℝ × ℝ))) → 𝑋 ∈ (ℝ × ℝ))
99	98	3anassrs 1282	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑥 ∈ (ℝ × ℝ)) → 𝑋 ∈ (ℝ × ℝ))
100		simpr 476	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑥 ∈ (ℝ × ℝ)) → 𝑥 ∈ (ℝ × ℝ))
101		simplr 788	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑥 ∈ (ℝ × ℝ)) → 𝑑 ∈ ℝ⁺)
102	101	rphalfcld 11760	. . . . . . . . . . . . . 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 29285	. . . . . . . . . . . . . 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 1318	. . . . . . . . . . . . 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 2610	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
107	103, 6, 106	cnrehmeo 22560	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝐺 ∈ ((𝐽 ×_t 𝐽)Homeo(TopOpen‘ℂ_fld))
108	106	cnfldtopon 22396	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
109	108	toponunii 20547	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ℂ = ∪ (TopOpen‘ℂ_fld)
110	12, 109	hmeof1o 21377	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐺 ∈ ((𝐽 ×_t 𝐽)Homeo(TopOpen‘ℂ_fld)) → 𝐺:(ℝ × ℝ)–1-1-onto→ℂ)
111		f1of 6050	. . . . . . . . . . . . . . . . . . . 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 6268	. . . . . . . . . . . . . . . . 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 6267	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑥 ∈ (ℝ × ℝ)) → (𝐺‘𝑥) ∈ ℂ)
117		sqsscirc2 29283	. . . . . . . . . . . . . . . . 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 1317	. . . . . . . . . . . . . . . 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 444	. . . . . . . . . . . . . . 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 11749	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑥 ∈ (ℝ × ℝ)) → 𝑑 ∈ ℝ^*)
121	120	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑥 ∈ (ℝ × ℝ)) ∧ (abs‘((𝐺‘𝑥) − (𝐺‘𝑋))) < 𝑑) → 𝑑 ∈ ℝ^*)
122		cnxmet 22386	. . . . . . . . . . . . . . . . 17 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
123	121, 122	jctil 558	. . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑥 ∈ (ℝ × ℝ)) ∧ (abs‘((𝐺‘𝑥) − (𝐺‘𝑋))) < 𝑑) → (𝐺‘𝑋) ∈ ℂ)
125	116	adantr 480	. . . . . . . . . . . . . . . . 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 553	. . . . . . . . . . . . . . . 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 2610	. . . . . . . . . . . . . . . . . . 19 ⊢ (abs ∘ − ) = (abs ∘ − )
128	127	cnmetdval 22384	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐺‘𝑥) ∈ ℂ ∧ (𝐺‘𝑋) ∈ ℂ) → ((𝐺‘𝑥)(abs ∘ − )(𝐺‘𝑋)) = (abs‘((𝐺‘𝑥) − (𝐺‘𝑋))))
129	125, 124, 128	syl2anc 691	. . . . . . . . . . . . . . . . 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 476	. . . . . . . . . . . . . . . . 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 4605	. . . . . . . . . . . . . . . 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 22007	. . . . . . . . . . . . . . . . 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 501	. . . . . . . . . . . . . . . 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 1317	. . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . 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 449	. . . . . . . . . . . . 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 46	. . . . . . . . . . . 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 6062	. . . . . . . . . . . . . . 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 6052	. . . . . . . . . . . . . 14 ⊢ (^◡𝐺:ℂ–1-1-onto→(ℝ × ℝ) → Fun ^◡𝐺)
141	139, 140	ax-mp 5	. . . . . . . . . . . . 13 ⊢ Fun ^◡𝐺
142		f1odm 6054	. . . . . . . . . . . . . . 15 ⊢ (^◡𝐺:ℂ–1-1-onto→(ℝ × ℝ) → dom ^◡𝐺 = ℂ)
143	139, 142	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ dom ^◡𝐺 = ℂ
144	116, 143	syl6eleqr 2699	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑥 ∈ (ℝ × ℝ)) → (𝐺‘𝑥) ∈ dom ^◡𝐺)
145		funfvima 6396	. . . . . . . . . . . . 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 694	. . . . . . . . . . . 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 6432	. . . . . . . . . . . . . . 15 ⊢ ((𝐺:(ℝ × ℝ)–1-1-onto→ℂ ∧ 𝑥 ∈ (ℝ × ℝ)) → (^◡𝐺‘(𝐺‘𝑥)) = 𝑥)
149	147, 100, 148	syl2anc 691	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑥 ∈ (ℝ × ℝ)) → (^◡𝐺‘(𝐺‘𝑥)) = 𝑥)
150	149	eleq1d 2672	. . . . . . . . . . . . 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 218	. . . . . . . . . . . 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 58	. . . . . . . . . . 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 444	. . . . . . . . . 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 449	. . . . . . . 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 3574	. . . . . . 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 2949	. . . . . 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	mpt2fun 6660	. . . . . . . . . 10 ⊢ Fun 𝐺
159	158	a1i 11	. . . . . . . . 9 ⊢ ((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) → Fun 𝐺)
160	13	sselda 3568	. . . . . . . . . 10 ⊢ ((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) → 𝑋 ∈ (ℝ × ℝ))
161		f1odm 6054	. . . . . . . . . . 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	syl6eleqr 2699	. . . . . . . . 9 ⊢ ((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) → 𝑋 ∈ dom 𝐺)
164		simpr 476	. . . . . . . . 9 ⊢ ((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) → 𝑋 ∈ 𝐴)
165		funfvima 6396	. . . . . . . . . 10 ⊢ ((Fun 𝐺 ∧ 𝑋 ∈ dom 𝐺) → (𝑋 ∈ 𝐴 → (𝐺‘𝑋) ∈ (𝐺 “ 𝐴)))
166	165	imp 444	. . . . . . . . 9 ⊢ (((Fun 𝐺 ∧ 𝑋 ∈ dom 𝐺) ∧ 𝑋 ∈ 𝐴) → (𝐺‘𝑋) ∈ (𝐺 “ 𝐴))
167	159, 163, 164, 166	syl21anc 1317	. . . . . . . 8 ⊢ ((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) → (𝐺‘𝑋) ∈ (𝐺 “ 𝐴))
168		hmeoima 21378	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ ((𝐽 ×_t 𝐽)Homeo(TopOpen‘ℂ_fld)) ∧ 𝐴 ∈ (𝐽 ×_t 𝐽)) → (𝐺 “ 𝐴) ∈ (TopOpen‘ℂ_fld))
169	107, 168	mpan 702	. . . . . . . . . 10 ⊢ (𝐴 ∈ (𝐽 ×_t 𝐽) → (𝐺 “ 𝐴) ∈ (TopOpen‘ℂ_fld))
170	106	cnfldtopn 22395	. . . . . . . . . . . . 13 ⊢ (TopOpen‘ℂ_fld) = (MetOpen‘(abs ∘ − ))
171	170	elmopn2 22060	. . . . . . . . . . . 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 479	. . . . . . . . . 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 480	. . . . . . . 8 ⊢ ((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) → ∀𝑚 ∈ (𝐺 “ 𝐴)∃𝑑 ∈ ℝ⁺ (𝑚(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺 “ 𝐴))
176		oveq1 6556	. . . . . . . . . . 11 ⊢ (𝑚 = (𝐺‘𝑋) → (𝑚(ball‘(abs ∘ − ))𝑑) = ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑))
177	176	sseq1d 3595	. . . . . . . . . 10 ⊢ (𝑚 = (𝐺‘𝑋) → ((𝑚(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺 “ 𝐴) ↔ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺 “ 𝐴)))
178	177	rexbidv 3034	. . . . . . . . 9 ⊢ (𝑚 = (𝐺‘𝑋) → (∃𝑑 ∈ ℝ⁺ (𝑚(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺 “ 𝐴) ↔ ∃𝑑 ∈ ℝ⁺ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺 “ 𝐴)))
179	178	rspcva 3280	. . . . . . . 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 691	. . . . . . 7 ⊢ ((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) → ∃𝑑 ∈ ℝ⁺ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺 “ 𝐴))
181		imass2 5420	. . . . . . . . . 10 ⊢ (((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺 “ 𝐴) → (^◡𝐺 “ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑)) ⊆ (^◡𝐺 “ (𝐺 “ 𝐴)))
182		f1of1 6049	. . . . . . . . . . . . 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 6066	. . . . . . . . . . . 12 ⊢ ((𝐺:(ℝ × ℝ)–1-1→ℂ ∧ 𝐴 ⊆ (ℝ × ℝ)) → (^◡𝐺 “ (𝐺 “ 𝐴)) = 𝐴)
185	183, 13, 184	sylancr 694	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (𝐽 ×_t 𝐽) → (^◡𝐺 “ (𝐺 “ 𝐴)) = 𝐴)
186	185	sseq2d 3596	. . . . . . . . . 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 233	. . . . . . . . 9 ⊢ (𝐴 ∈ (𝐽 ×_t 𝐽) → (((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺 “ 𝐴) → (^◡𝐺 “ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑)) ⊆ 𝐴))
188	187	reximdv 2999	. . . . . . . 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 480	. . . . . . 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 3054	. . . . . 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 691	. . . . 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 3576	. . . . . 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 2994	. . . . 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 3054	. . . 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 691	. . 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 3054	. . 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 691	. 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 2677	. . . . 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 3589	. . . . 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 743	. . . 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 3282	. . 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 3011	. 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 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∃wrex 2897 Vcvv 3173 ⊆ wss 3540 𝒫 cpw 4108 ∪ cuni 4372 class class class wbr 4583 × cxp 5036 ^◡ccnv 5037 dom cdm 5038 ran crn 5039 “ cima 5041 ∘ ccom 5042 Fun wfun 5798 Fn wfn 5799 ⟶wf 5800 –1-1→wf1 5801 –1-1-onto→wf1o 5803 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 1^st c1st 7057 2^nd c2nd 7058 ℂcc 9813 ℝcr 9814 ici 9817 + caddc 9818 · cmul 9820 +∞cpnf 9950 -∞cmnf 9951 ℝ^*cxr 9952 < clt 9953 ≤ cle 9954 − cmin 10145 / cdiv 10563 2c2 10947 ℝ⁺crp 11708 (,)cioo 12046 ℜcre 13685 ℑcim 13686 abscabs 13822 TopOpenctopn 15905 topGenctg 15921 ∞Metcxmt 19552 ballcbl 19554 ℂ_fldccnfld 19567 Topctop 20517 ×_t ctx 21173 Homeochmeo 21366

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-fi 8200 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-ioo 12050 df-icc 12053 df-fz 12198 df-fzo 12335 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-hom 15793 df-cco 15794 df-rest 15906 df-topn 15907 df-0g 15925 df-gsum 15926 df-topgen 15927 df-pt 15928 df-prds 15931 df-xrs 15985 df-qtop 15990 df-imas 15991 df-xps 15993 df-mre 16069 df-mrc 16070 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-mulg 17364 df-cntz 17573 df-cmn 18018 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-cnfld 19568 df-top 20521 df-bases 20522 df-topon 20523 df-topsp 20524 df-cn 20841 df-cnp 20842 df-tx 21175 df-hmeo 21368 df-xms 21935 df-ms 21936 df-tms 21937 df-cncf 22489

This theorem is referenced by: dya2iocnei 29671

W3C validator