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Theorem cnheiborlem 25275

Description: Lemma for cnheibor 25276. (Contributed by Mario Carneiro, 14-Sep-2014.)

**Hypotheses**
Ref	Expression
cnheibor.2	⊢ 𝐽 = (TopOpen‘ℂ_fld)
cnheibor.3	⊢ 𝑇 = (𝐽 ↾_t 𝑋)
cnheibor.4	⊢ 𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + (i · 𝑦)))
cnheibor.5	⊢ 𝑌 = (𝐹 “ (( -𝑅[,]𝑅) × ( -𝑅[,]𝑅)))

**Assertion**
Ref	Expression
cnheiborlem	⊢ ((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) → 𝑇 ∈ Comp)

Distinct variable groups: 𝑧,𝐹 𝑧,𝑅 𝑥,𝑦,𝑧,𝑇 𝑥,𝐽,𝑦,𝑧 𝑥,𝑋,𝑦,𝑧 Allowed substitution hints: 𝑅(𝑥, 𝑦) 𝐹(𝑥, 𝑦) 𝑌(𝑥, 𝑦, 𝑧)

Proof of Theorem cnheiborlem Dummy variable 𝑢 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnheibor.2	. . . . 5 ⊢ 𝐽 = (TopOpen‘ℂ_fld)
2	1	cnfldtop 25102	. . . 4 ⊢ 𝐽 ∈ Top
3		cnheibor.4	. . . . . . . . . 10 ⊢ 𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + (i · 𝑦)))
4	3	cnref1o 13113	. . . . . . . . 9 ⊢ 𝐹:(ℝ × ℝ)–1-1-onto→ℂ
5		f1ofn 6825	. . . . . . . . 9 ⊢ (𝐹:(ℝ × ℝ)–1-1-onto→ℂ → 𝐹 Fn (ℝ × ℝ))
6		elpreima 7057	. . . . . . . . 9 ⊢ (𝐹 Fn (ℝ × ℝ) → (𝑢 ∈ (^◡𝐹 “ 𝑋) ↔ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)))
7	4, 5, 6	mp2b 10	. . . . . . . 8 ⊢ (𝑢 ∈ (^◡𝐹 “ 𝑋) ↔ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋))
8		1st2nd2 8040	. . . . . . . . . . 11 ⊢ (𝑢 ∈ (ℝ × ℝ) → 𝑢 = ⟨(1^st ‘𝑢), (2^nd ‘𝑢)⟩)
9	8	ad2antrl 741	. . . . . . . . . 10 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → 𝑢 = ⟨(1^st ‘𝑢), (2^nd ‘𝑢)⟩)
10		xp1st 8033	. . . . . . . . . . . . 13 ⊢ (𝑢 ∈ (ℝ × ℝ) → (1^st ‘𝑢) ∈ ℝ)
11	10	ad2antrl 741	. . . . . . . . . . . 12 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → (1^st ‘𝑢) ∈ ℝ)
12	11	recnd 11337	. . . . . . . . . . . . . . . 16 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → (1^st ‘𝑢) ∈ ℂ)
13	12	abscld 15606	. . . . . . . . . . . . . . 15 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → (abs‘(1^st ‘𝑢)) ∈ ℝ)
14	1	cnfldtopon 25101	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝐽 ∈ (TopOn‘ℂ)
15	14	toponunii 23234	. . . . . . . . . . . . . . . . . . . 20 ⊢ ℂ = ∪ 𝐽
16	15	cldss 23347	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑋 ∈ (Clsd‘𝐽) → 𝑋 ⊆ ℂ)
17	16	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) → 𝑋 ⊆ ℂ)
18	17	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → 𝑋 ⊆ ℂ)
19		simprr 785	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → (𝐹‘𝑢) ∈ 𝑋)
20	18, 19	sseldd 3909	. . . . . . . . . . . . . . . 16 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → (𝐹‘𝑢) ∈ ℂ)
21	20	abscld 15606	. . . . . . . . . . . . . . 15 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → (abs‘(𝐹‘𝑢)) ∈ ℝ)
22		simplrl 789	. . . . . . . . . . . . . . 15 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → 𝑅 ∈ ℝ)
23		simprl 783	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → 𝑢 ∈ (ℝ × ℝ))
24		f1ocnvfv1 7284	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐹:(ℝ × ℝ)–1-1-onto→ℂ ∧ 𝑢 ∈ (ℝ × ℝ)) → (^◡𝐹‘(𝐹‘𝑢)) = 𝑢)
25	4, 23, 24	sylancr 599	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → (^◡𝐹‘(𝐹‘𝑢)) = 𝑢)
26		fveq2 6885	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑧 = (𝐹‘𝑢) → (ℜ‘𝑧) = (ℜ‘(𝐹‘𝑢)))
27		fveq2 6885	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑧 = (𝐹‘𝑢) → (ℑ‘𝑧) = (ℑ‘(𝐹‘𝑢)))
28	26, 27	opeq12d 4840	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑧 = (𝐹‘𝑢) → ⟨(ℜ‘𝑧), (ℑ‘𝑧)⟩ = ⟨(ℜ‘(𝐹‘𝑢)), (ℑ‘(𝐹‘𝑢))⟩)
29	3	cnrecnv 15332	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ^◡𝐹 = (𝑧 ∈ ℂ ↦ ⟨(ℜ‘𝑧), (ℑ‘𝑧)⟩)
30		opex 5431	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ⟨(ℜ‘(𝐹‘𝑢)), (ℑ‘(𝐹‘𝑢))⟩ ∈ V
31	28, 29, 30	fvmpt 6993	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐹‘𝑢) ∈ ℂ → (^◡𝐹‘(𝐹‘𝑢)) = ⟨(ℜ‘(𝐹‘𝑢)), (ℑ‘(𝐹‘𝑢))⟩)
32	20, 31	syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → (^◡𝐹‘(𝐹‘𝑢)) = ⟨(ℜ‘(𝐹‘𝑢)), (ℑ‘(𝐹‘𝑢))⟩)
33	25, 32	eqtr3d 2797	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → 𝑢 = ⟨(ℜ‘(𝐹‘𝑢)), (ℑ‘(𝐹‘𝑢))⟩)
34	33	fveq2d 6889	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → (1^st ‘𝑢) = (1^st ‘⟨(ℜ‘(𝐹‘𝑢)), (ℑ‘(𝐹‘𝑢))⟩))
35		fvex 6898	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℜ‘(𝐹‘𝑢)) ∈ V
36		fvex 6898	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℑ‘(𝐹‘𝑢)) ∈ V
37	35, 36	op1st 8009	. . . . . . . . . . . . . . . . . 18 ⊢ (1^st ‘⟨(ℜ‘(𝐹‘𝑢)), (ℑ‘(𝐹‘𝑢))⟩) = (ℜ‘(𝐹‘𝑢))
38	34, 37	eqtrdi 2811	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → (1^st ‘𝑢) = (ℜ‘(𝐹‘𝑢)))
39	38	fveq2d 6889	. . . . . . . . . . . . . . . 16 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → (abs‘(1^st ‘𝑢)) = (abs‘(ℜ‘(𝐹‘𝑢))))
40		absrele 15475	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹‘𝑢) ∈ ℂ → (abs‘(ℜ‘(𝐹‘𝑢))) ≤ (abs‘(𝐹‘𝑢)))
41	20, 40	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → (abs‘(ℜ‘(𝐹‘𝑢))) ≤ (abs‘(𝐹‘𝑢)))
42	39, 41	eqbrtrd 5126	. . . . . . . . . . . . . . 15 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → (abs‘(1^st ‘𝑢)) ≤ (abs‘(𝐹‘𝑢)))
43		fveq2 6885	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = (𝐹‘𝑢) → (abs‘𝑧) = (abs‘(𝐹‘𝑢)))
44	43	breq1d 5112	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = (𝐹‘𝑢) → ((abs‘𝑧) ≤ 𝑅 ↔ (abs‘(𝐹‘𝑢)) ≤ 𝑅))
45		simplrr 790	. . . . . . . . . . . . . . . 16 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)
46	44, 45, 19	rspcdva 3577	. . . . . . . . . . . . . . 15 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → (abs‘(𝐹‘𝑢)) ≤ 𝑅)
47	13, 21, 22, 42, 46	letrd 11467	. . . . . . . . . . . . . 14 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → (abs‘(1^st ‘𝑢)) ≤ 𝑅)
48	11, 22	absled 15600	. . . . . . . . . . . . . 14 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → ((abs‘(1^st ‘𝑢)) ≤ 𝑅 ↔ ( -𝑅 ≤ (1^st ‘𝑢) ∧ (1^st ‘𝑢) ≤ 𝑅)))
49	47, 48	mpbid 235	. . . . . . . . . . . . 13 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → ( -𝑅 ≤ (1^st ‘𝑢) ∧ (1^st ‘𝑢) ≤ 𝑅))
50	49	simpld 500	. . . . . . . . . . . 12 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → -𝑅 ≤ (1^st ‘𝑢))
51	49	simprd 501	. . . . . . . . . . . 12 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → (1^st ‘𝑢) ≤ 𝑅)
52		renegcl 11621	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ ℝ → -𝑅 ∈ ℝ)
53	22, 52	syl 18	. . . . . . . . . . . . 13 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → -𝑅 ∈ ℝ)
54		elicc2 13542	. . . . . . . . . . . . 13 ⊢ (( -𝑅 ∈ ℝ ∧ 𝑅 ∈ ℝ) → ((1^st ‘𝑢) ∈ ( -𝑅[,]𝑅) ↔ ((1^st ‘𝑢) ∈ ℝ ∧ -𝑅 ≤ (1^st ‘𝑢) ∧ (1^st ‘𝑢) ≤ 𝑅)))
55	53, 22, 54	syl2anc 596	. . . . . . . . . . . 12 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → ((1^st ‘𝑢) ∈ ( -𝑅[,]𝑅) ↔ ((1^st ‘𝑢) ∈ ℝ ∧ -𝑅 ≤ (1^st ‘𝑢) ∧ (1^st ‘𝑢) ≤ 𝑅)))
56	11, 50, 51, 55	mpbir3and 1361	. . . . . . . . . . 11 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → (1^st ‘𝑢) ∈ ( -𝑅[,]𝑅))
57		xp2nd 8034	. . . . . . . . . . . . 13 ⊢ (𝑢 ∈ (ℝ × ℝ) → (2^nd ‘𝑢) ∈ ℝ)
58	57	ad2antrl 741	. . . . . . . . . . . 12 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → (2^nd ‘𝑢) ∈ ℝ)
59	58	recnd 11337	. . . . . . . . . . . . . . . 16 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → (2^nd ‘𝑢) ∈ ℂ)
60	59	abscld 15606	. . . . . . . . . . . . . . 15 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → (abs‘(2^nd ‘𝑢)) ∈ ℝ)
61	33	fveq2d 6889	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → (2^nd ‘𝑢) = (2^nd ‘⟨(ℜ‘(𝐹‘𝑢)), (ℑ‘(𝐹‘𝑢))⟩))
62	35, 36	op2nd 8010	. . . . . . . . . . . . . . . . . 18 ⊢ (2^nd ‘⟨(ℜ‘(𝐹‘𝑢)), (ℑ‘(𝐹‘𝑢))⟩) = (ℑ‘(𝐹‘𝑢))
63	61, 62	eqtrdi 2811	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → (2^nd ‘𝑢) = (ℑ‘(𝐹‘𝑢)))
64	63	fveq2d 6889	. . . . . . . . . . . . . . . 16 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → (abs‘(2^nd ‘𝑢)) = (abs‘(ℑ‘(𝐹‘𝑢))))
65		absimle 15476	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹‘𝑢) ∈ ℂ → (abs‘(ℑ‘(𝐹‘𝑢))) ≤ (abs‘(𝐹‘𝑢)))
66	20, 65	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → (abs‘(ℑ‘(𝐹‘𝑢))) ≤ (abs‘(𝐹‘𝑢)))
67	64, 66	eqbrtrd 5126	. . . . . . . . . . . . . . 15 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → (abs‘(2^nd ‘𝑢)) ≤ (abs‘(𝐹‘𝑢)))
68	60, 21, 22, 67, 46	letrd 11467	. . . . . . . . . . . . . 14 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → (abs‘(2^nd ‘𝑢)) ≤ 𝑅)
69	58, 22	absled 15600	. . . . . . . . . . . . . 14 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → ((abs‘(2^nd ‘𝑢)) ≤ 𝑅 ↔ ( -𝑅 ≤ (2^nd ‘𝑢) ∧ (2^nd ‘𝑢) ≤ 𝑅)))
70	68, 69	mpbid 235	. . . . . . . . . . . . 13 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → ( -𝑅 ≤ (2^nd ‘𝑢) ∧ (2^nd ‘𝑢) ≤ 𝑅))
71	70	simpld 500	. . . . . . . . . . . 12 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → -𝑅 ≤ (2^nd ‘𝑢))
72	70	simprd 501	. . . . . . . . . . . 12 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → (2^nd ‘𝑢) ≤ 𝑅)
73		elicc2 13542	. . . . . . . . . . . . 13 ⊢ (( -𝑅 ∈ ℝ ∧ 𝑅 ∈ ℝ) → ((2^nd ‘𝑢) ∈ ( -𝑅[,]𝑅) ↔ ((2^nd ‘𝑢) ∈ ℝ ∧ -𝑅 ≤ (2^nd ‘𝑢) ∧ (2^nd ‘𝑢) ≤ 𝑅)))
74	53, 22, 73	syl2anc 596	. . . . . . . . . . . 12 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → ((2^nd ‘𝑢) ∈ ( -𝑅[,]𝑅) ↔ ((2^nd ‘𝑢) ∈ ℝ ∧ -𝑅 ≤ (2^nd ‘𝑢) ∧ (2^nd ‘𝑢) ≤ 𝑅)))
75	58, 71, 72, 74	mpbir3and 1361	. . . . . . . . . . 11 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → (2^nd ‘𝑢) ∈ ( -𝑅[,]𝑅))
76	56, 75	opelxpd 5689	. . . . . . . . . 10 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → ⟨(1^st ‘𝑢), (2^nd ‘𝑢)⟩ ∈ (( -𝑅[,]𝑅) × ( -𝑅[,]𝑅)))
77	9, 76	eqeltrd 2860	. . . . . . . . 9 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → 𝑢 ∈ (( -𝑅[,]𝑅) × ( -𝑅[,]𝑅)))
78	77	ex 418	. . . . . . . 8 ⊢ ((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) → ((𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋) → 𝑢 ∈ (( -𝑅[,]𝑅) × ( -𝑅[,]𝑅))))
79	7, 78	biimtrid 245	. . . . . . 7 ⊢ ((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) → (𝑢 ∈ (^◡𝐹 “ 𝑋) → 𝑢 ∈ (( -𝑅[,]𝑅) × ( -𝑅[,]𝑅))))
80	79	ssrdv 3914	. . . . . 6 ⊢ ((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) → (^◡𝐹 “ 𝑋) ⊆ (( -𝑅[,]𝑅) × ( -𝑅[,]𝑅)))
81		f1ofun 6826	. . . . . . . 8 ⊢ (𝐹:(ℝ × ℝ)–1-1-onto→ℂ → Fun 𝐹)
82	4, 81	ax-mp 5	. . . . . . 7 ⊢ Fun 𝐹
83		f1ofo 6832	. . . . . . . . 9 ⊢ (𝐹:(ℝ × ℝ)–1-1-onto→ℂ → 𝐹:(ℝ × ℝ)–onto→ℂ)
84		forn 6799	. . . . . . . . 9 ⊢ (𝐹:(ℝ × ℝ)–onto→ℂ → ran 𝐹 = ℂ)
85	4, 83, 84	mp2b 10	. . . . . . . 8 ⊢ ran 𝐹 = ℂ
86	17, 85	sseqtrrdi 3949	. . . . . . 7 ⊢ ((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) → 𝑋 ⊆ ran 𝐹)
87		funimass1 6622	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝑋 ⊆ ran 𝐹) → ((^◡𝐹 “ 𝑋) ⊆ (( -𝑅[,]𝑅) × ( -𝑅[,]𝑅)) → 𝑋 ⊆ (𝐹 “ (( -𝑅[,]𝑅) × ( -𝑅[,]𝑅)))))
88	82, 86, 87	sylancr 599	. . . . . 6 ⊢ ((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) → ((^◡𝐹 “ 𝑋) ⊆ (( -𝑅[,]𝑅) × ( -𝑅[,]𝑅)) → 𝑋 ⊆ (𝐹 “ (( -𝑅[,]𝑅) × ( -𝑅[,]𝑅)))))
89	80, 88	mpd 16	. . . . 5 ⊢ ((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) → 𝑋 ⊆ (𝐹 “ (( -𝑅[,]𝑅) × ( -𝑅[,]𝑅))))
90		cnheibor.5	. . . . 5 ⊢ 𝑌 = (𝐹 “ (( -𝑅[,]𝑅) × ( -𝑅[,]𝑅)))
91	89, 90	sseqtrrdi 3949	. . . 4 ⊢ ((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) → 𝑋 ⊆ 𝑌)
92		eqid 2760	. . . . . . . 8 ⊢ (topGen‘ran (,)) = (topGen‘ran (,))
93	3, 92, 1	cnrehmeo 25274	. . . . . . 7 ⊢ 𝐹 ∈ (((topGen‘ran (,)) ×_t (topGen‘ran (,)))Homeo𝐽)
94		imaexg 7925	. . . . . . 7 ⊢ (𝐹 ∈ (((topGen‘ran (,)) ×_t (topGen‘ran (,)))Homeo𝐽) → (𝐹 “ (( -𝑅[,]𝑅) × ( -𝑅[,]𝑅))) ∈ V)
95	93, 94	ax-mp 5	. . . . . 6 ⊢ (𝐹 “ (( -𝑅[,]𝑅) × ( -𝑅[,]𝑅))) ∈ V
96	90, 95	eqeltri 2856	. . . . 5 ⊢ 𝑌 ∈ V
97	96	a1i 11	. . . 4 ⊢ ((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) → 𝑌 ∈ V)
98		restabs 23483	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ∈ V) → ((𝐽 ↾_t 𝑌) ↾_t 𝑋) = (𝐽 ↾_t 𝑋))
99	2, 91, 97, 98	mp3an2i 1495	. . 3 ⊢ ((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) → ((𝐽 ↾_t 𝑌) ↾_t 𝑋) = (𝐽 ↾_t 𝑋))
100		cnheibor.3	. . 3 ⊢ 𝑇 = (𝐽 ↾_t 𝑋)
101	99, 100	eqtr4di 2813	. 2 ⊢ ((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) → ((𝐽 ↾_t 𝑌) ↾_t 𝑋) = 𝑇)
102	90	oveq2i 7431	. . . . 5 ⊢ (𝐽 ↾_t 𝑌) = (𝐽 ↾_t (𝐹 “ (( -𝑅[,]𝑅) × ( -𝑅[,]𝑅))))
103		ishmeo 24078	. . . . . . . 8 ⊢ (𝐹 ∈ (((topGen‘ran (,)) ×_t (topGen‘ran (,)))Homeo𝐽) ↔ (𝐹 ∈ (((topGen‘ran (,)) ×_t (topGen‘ran (,))) Cn 𝐽) ∧ ^◡𝐹 ∈ (𝐽 Cn ((topGen‘ran (,)) ×_t (topGen‘ran (,))))))
104	93, 103	mpbi 233	. . . . . . 7 ⊢ (𝐹 ∈ (((topGen‘ran (,)) ×_t (topGen‘ran (,))) Cn 𝐽) ∧ ^◡𝐹 ∈ (𝐽 Cn ((topGen‘ran (,)) ×_t (topGen‘ran (,)))))
105	104	simpli 489	. . . . . 6 ⊢ 𝐹 ∈ (((topGen‘ran (,)) ×_t (topGen‘ran (,))) Cn 𝐽)
106		iccssre 13560	. . . . . . . . . . 11 ⊢ (( -𝑅 ∈ ℝ ∧ 𝑅 ∈ ℝ) → ( -𝑅[,]𝑅) ⊆ ℝ)
107	52, 106	mpancom 701	. . . . . . . . . 10 ⊢ (𝑅 ∈ ℝ → ( -𝑅[,]𝑅) ⊆ ℝ)
108	1, 92	rerest 25123	. . . . . . . . . 10 ⊢ (( -𝑅[,]𝑅) ⊆ ℝ → (𝐽 ↾_t ( -𝑅[,]𝑅)) = ((topGen‘ran (,)) ↾_t ( -𝑅[,]𝑅)))
109	107, 108	syl 18	. . . . . . . . 9 ⊢ (𝑅 ∈ ℝ → (𝐽 ↾_t ( -𝑅[,]𝑅)) = ((topGen‘ran (,)) ↾_t ( -𝑅[,]𝑅)))
110	109, 109	oveq12d 7438	. . . . . . . 8 ⊢ (𝑅 ∈ ℝ → ((𝐽 ↾_t ( -𝑅[,]𝑅)) ×_t (𝐽 ↾_t ( -𝑅[,]𝑅))) = (((topGen‘ran (,)) ↾_t ( -𝑅[,]𝑅)) ×_t ((topGen‘ran (,)) ↾_t ( -𝑅[,]𝑅))))
111		retop 25080	. . . . . . . . 9 ⊢ (topGen‘ran (,)) ∈ Top
112		ovex 7453	. . . . . . . . 9 ⊢ ( -𝑅[,]𝑅) ∈ V
113		txrest 23950	. . . . . . . . 9 ⊢ ((((topGen‘ran (,)) ∈ Top ∧ (topGen‘ran (,)) ∈ Top) ∧ (( -𝑅[,]𝑅) ∈ V ∧ ( -𝑅[,]𝑅) ∈ V)) → (((topGen‘ran (,)) ×_t (topGen‘ran (,))) ↾_t (( -𝑅[,]𝑅) × ( -𝑅[,]𝑅))) = (((topGen‘ran (,)) ↾_t ( -𝑅[,]𝑅)) ×_t ((topGen‘ran (,)) ↾_t ( -𝑅[,]𝑅))))
114	111, 111, 112, 112, 113	mp4an 706	. . . . . . . 8 ⊢ (((topGen‘ran (,)) ×_t (topGen‘ran (,))) ↾_t (( -𝑅[,]𝑅) × ( -𝑅[,]𝑅))) = (((topGen‘ran (,)) ↾_t ( -𝑅[,]𝑅)) ×_t ((topGen‘ran (,)) ↾_t ( -𝑅[,]𝑅)))
115	110, 114	eqtr4di 2813	. . . . . . 7 ⊢ (𝑅 ∈ ℝ → ((𝐽 ↾_t ( -𝑅[,]𝑅)) ×_t (𝐽 ↾_t ( -𝑅[,]𝑅))) = (((topGen‘ran (,)) ×_t (topGen‘ran (,))) ↾_t (( -𝑅[,]𝑅) × ( -𝑅[,]𝑅))))
116		eqid 2760	. . . . . . . . . . 11 ⊢ ((topGen‘ran (,)) ↾_t ( -𝑅[,]𝑅)) = ((topGen‘ran (,)) ↾_t ( -𝑅[,]𝑅))
117	92, 116	icccmp 25145	. . . . . . . . . 10 ⊢ (( -𝑅 ∈ ℝ ∧ 𝑅 ∈ ℝ) → ((topGen‘ran (,)) ↾_t ( -𝑅[,]𝑅)) ∈ Comp)
118	52, 117	mpancom 701	. . . . . . . . 9 ⊢ (𝑅 ∈ ℝ → ((topGen‘ran (,)) ↾_t ( -𝑅[,]𝑅)) ∈ Comp)
119	109, 118	eqeltrd 2860	. . . . . . . 8 ⊢ (𝑅 ∈ ℝ → (𝐽 ↾_t ( -𝑅[,]𝑅)) ∈ Comp)
120		txcmp 23962	. . . . . . . 8 ⊢ (((𝐽 ↾_t ( -𝑅[,]𝑅)) ∈ Comp ∧ (𝐽 ↾_t ( -𝑅[,]𝑅)) ∈ Comp) → ((𝐽 ↾_t ( -𝑅[,]𝑅)) ×_t (𝐽 ↾_t ( -𝑅[,]𝑅))) ∈ Comp)
121	119, 119, 120	syl2anc 596	. . . . . . 7 ⊢ (𝑅 ∈ ℝ → ((𝐽 ↾_t ( -𝑅[,]𝑅)) ×_t (𝐽 ↾_t ( -𝑅[,]𝑅))) ∈ Comp)
122	115, 121	eqeltrrd 2861	. . . . . 6 ⊢ (𝑅 ∈ ℝ → (((topGen‘ran (,)) ×_t (topGen‘ran (,))) ↾_t (( -𝑅[,]𝑅) × ( -𝑅[,]𝑅))) ∈ Comp)
123		imacmp 23715	. . . . . 6 ⊢ ((𝐹 ∈ (((topGen‘ran (,)) ×_t (topGen‘ran (,))) Cn 𝐽) ∧ (((topGen‘ran (,)) ×_t (topGen‘ran (,))) ↾_t (( -𝑅[,]𝑅) × ( -𝑅[,]𝑅))) ∈ Comp) → (𝐽 ↾_t (𝐹 “ (( -𝑅[,]𝑅) × ( -𝑅[,]𝑅)))) ∈ Comp)
124	105, 122, 123	sylancr 599	. . . . 5 ⊢ (𝑅 ∈ ℝ → (𝐽 ↾_t (𝐹 “ (( -𝑅[,]𝑅) × ( -𝑅[,]𝑅)))) ∈ Comp)
125	102, 124	eqeltrid 2864	. . . 4 ⊢ (𝑅 ∈ ℝ → (𝐽 ↾_t 𝑌) ∈ Comp)
126	125	ad2antrl 741	. . 3 ⊢ ((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) → (𝐽 ↾_t 𝑌) ∈ Comp)
127		imassrn 6197	. . . . . 6 ⊢ (𝐹 “ (( -𝑅[,]𝑅) × ( -𝑅[,]𝑅))) ⊆ ran 𝐹
128	90, 127	eqsstri 3954	. . . . 5 ⊢ 𝑌 ⊆ ran 𝐹
129		f1of 6824	. . . . . 6 ⊢ (𝐹:(ℝ × ℝ)–1-1-onto→ℂ → 𝐹:(ℝ × ℝ)⟶ℂ)
130		frn 6717	. . . . . 6 ⊢ (𝐹:(ℝ × ℝ)⟶ℂ → ran 𝐹 ⊆ ℂ)
131	4, 129, 130	mp2b 10	. . . . 5 ⊢ ran 𝐹 ⊆ ℂ
132	128, 131	sstri 3917	. . . 4 ⊢ 𝑌 ⊆ ℂ
133		simpl 488	. . . 4 ⊢ ((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) → 𝑋 ∈ (Clsd‘𝐽))
134	15	restcldi 23491	. . . 4 ⊢ ((𝑌 ⊆ ℂ ∧ 𝑋 ∈ (Clsd‘𝐽) ∧ 𝑋 ⊆ 𝑌) → 𝑋 ∈ (Clsd‘(𝐽 ↾_t 𝑌)))
135	132, 133, 91, 134	mp3an2i 1495	. . 3 ⊢ ((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) → 𝑋 ∈ (Clsd‘(𝐽 ↾_t 𝑌)))
136		cmpcld 23720	. . 3 ⊢ (((𝐽 ↾_t 𝑌) ∈ Comp ∧ 𝑋 ∈ (Clsd‘(𝐽 ↾_t 𝑌))) → ((𝐽 ↾_t 𝑌) ↾_t 𝑋) ∈ Comp)
137	126, 135, 136	syl2anc 596	. 2 ⊢ ((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) → ((𝐽 ↾_t 𝑌) ↾_t 𝑋) ∈ Comp)
138	101, 137	eqeltrrd 2861	1 ⊢ ((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) → 𝑇 ∈ Comp)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1569 ∈ wcel 2144 ∀wral 3076 Vcvv 3450 ⊆ wss 3895 ⟨cop 4589 class class class wbr 5102 × cxp 5648 ^◡ccnv 5649 ran crn 5651 “ cima 5653 Fun wfun 6532 Fn wfn 6533 ⟶wf 6534 –onto→wfo 6536 –1-1-onto→wf1o 6537 ‘cfv 6538 (class class class)co 7420 ∈ cmpo 7422 1^st c1st 7999 2^nd c2nd 8000 ℂcc 11198 ℝcr 11199 ici 11202 + caddc 11203 · cmul 11205 ≤ cle 11344 -cneg 11542 (,)cioo 13476 [,]cicc 13479 ℜcre 15264 ℑcim 15265 abscabs 15401 ↾_t crest 17591 TopOpenctopn 17592 topGenctg 17608 ℂ_fldccnfld 21678 Topctop 23211 Clsdccld 23334 Cn ccn 23542 Compccmp 23704 ×_t ctx 23879 Homeochmeo 24072

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7751 ax-cnex 11256 ax-resscn 11257 ax-1cn 11258 ax-icn 11259 ax-addcl 11260 ax-addrcl 11261 ax-mulcl 11262 ax-mulrcl 11263 ax-mulcom 11264 ax-addass 11265 ax-mulass 11266 ax-distr 11267 ax-i2m1 11268 ax-1ne0 11269 ax-1rid 11270 ax-rnegex 11271 ax-rrecex 11272 ax-cnre 11273 ax-pre-lttri 11274 ax-pre-lttrn 11275 ax-pre-ltadd 11276 ax-pre-mulgt0 11277 ax-pre-sup 11278 ax-addf 11279

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-tp 4588 df-op 4590 df-uni 4867 df-int 4907 df-iun 4952 df-iin 4953 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-se 5604 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-isom 6547 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7693 df-om 7878 df-1st 8001 df-2nd 8002 df-supp 8178 df-frecs 8299 df-wrecs 8330 df-recs 8379 df-rdg 8418 df-1o 8476 df-2o 8477 df-er 8717 df-map 8849 df-ixp 8926 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-fsupp 9354 df-fi 9403 df-sup 9434 df-inf 9435 df-oi 9504 df-card 10020 df-pnf 11345 df-mnf 11346 df-xr 11347 df-ltxr 11348 df-le 11349 df-sub 11543 df-neg 11544 df-div 11974 df-nn 12336 df-2 12405 df-3 12406 df-4 12407 df-5 12408 df-6 12409 df-7 12410 df-8 12411 df-9 12412 df-n0 12607 df-z 12694 df-dec 12815 df-uz 12966 df-q 13076 df-rp 13121 df-xneg 13241 df-xadd 13242 df-xmul 13243 df-ioo 13480 df-icc 13483 df-fz 13640 df-fzo 13789 df-seq 14145 df-exp 14205 df-hash 14475 df-cj 15266 df-re 15267 df-im 15268 df-sqrt 15402 df-abs 15403 df-struct 17325 df-sets 17342 df-slot 17360 df-ndx 17372 df-base 17388 df-ress 17409 df-plusg 17441 df-mulr 17442 df-starv 17443 df-sca 17444 df-vsca 17445 df-ip 17446 df-tset 17447 df-ple 17448 df-ds 17450 df-unif 17451 df-hom 17452 df-cco 17453 df-rest 17593 df-topn 17594 df-0g 17612 df-gsum 17613 df-topgen 17614 df-pt 17615 df-prds 17618 df-xrs 17674 df-qtop 17679 df-imas 17680 df-xps 17682 df-mre 17756 df-mrc 17757 df-acs 17759 df-mgm 18816 df-sgrp 18908 df-mnd 18924 df-submnd 18979 df-mulg 19278 df-cntz 19531 df-cmn 19996 df-psmet 21670 df-xmet 21671 df-met 21672 df-bl 21673 df-mopn 21674 df-cnfld 21679 df-top 23212 df-topon 23229 df-topsp 23251 df-bases 23264 df-cld 23337 df-cn 23545 df-cnp 23546 df-cmp 23705 df-tx 23881 df-hmeo 24074 df-xms 24639 df-ms 24640 df-tms 24641 df-cncf 25199

This theorem is used by: cnheibor 25276

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