Theorem cnheiborlem 24999

Description: Lemma for cnheibor 25000. (Contributed by Mario Carneiro, 14-Sep-2014.)

Hypotheses

Ref	Expression
cnheibor.2	⊢ 𝐽 = (TopOpen‘ℂfld)
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‘ℂfld)
2	1	cnfldtop 24819	. . . 4 ⊢ 𝐽 ∈ Top
3		cnheibor.4	. . . . . . . . . 10 ⊢ 𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + (i · 𝑦)))
4	3	cnref1o 13024	. . . . . . . . 9 ⊢ 𝐹:(ℝ × ℝ)–1-1-onto→ℂ
5		f1ofn 6849	. . . . . . . . 9 ⊢ (𝐹:(ℝ × ℝ)–1-1-onto→ℂ → 𝐹 Fn (ℝ × ℝ))
6		elpreima 7077	. . . . . . . . 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 8051	. . . . . . . . . . 11 ⊢ (𝑢 ∈ (ℝ × ℝ) → 𝑢 = ⟨(1st ‘𝑢), (2nd ‘𝑢)⟩)
9	8	ad2antrl 728	. . . . . . . . . 10 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → 𝑢 = ⟨(1st ‘𝑢), (2nd ‘𝑢)⟩)
10		xp1st 8044	. . . . . . . . . . . . 13 ⊢ (𝑢 ∈ (ℝ × ℝ) → (1st ‘𝑢) ∈ ℝ)
11	10	ad2antrl 728	. . . . . . . . . . . 12 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → (1st ‘𝑢) ∈ ℝ)
12	11	recnd 11286	. . . . . . . . . . . . . . . 16 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → (1st ‘𝑢) ∈ ℂ)
13	12	abscld 15471	. . . . . . . . . . . . . . 15 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → (abs‘(1st ‘𝑢)) ∈ ℝ)
14	1	cnfldtopon 24818	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝐽 ∈ (TopOn‘ℂ)
15	14	toponunii 22937	. . . . . . . . . . . . . . . . . . . 20 ⊢ ℂ = ∪ 𝐽
16	15	cldss 23052	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑋 ∈ (Clsd‘𝐽) → 𝑋 ⊆ ℂ)
17	16	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) → 𝑋 ⊆ ℂ)
18	17	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → 𝑋 ⊆ ℂ)
19		simprr 773	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → (𝐹‘𝑢) ∈ 𝑋)
20	18, 19	sseldd 3995	. . . . . . . . . . . . . . . 16 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → (𝐹‘𝑢) ∈ ℂ)
21	20	abscld 15471	. . . . . . . . . . . . . . 15 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → (abs‘(𝐹‘𝑢)) ∈ ℝ)
22		simplrl 777	. . . . . . . . . . . . . . 15 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → 𝑅 ∈ ℝ)
23		simprl 771	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → 𝑢 ∈ (ℝ × ℝ))
24		f1ocnvfv1 7295	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐹:(ℝ × ℝ)–1-1-onto→ℂ ∧ 𝑢 ∈ (ℝ × ℝ)) → (◡𝐹‘(𝐹‘𝑢)) = 𝑢)
25	4, 23, 24	sylancr 587	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → (◡𝐹‘(𝐹‘𝑢)) = 𝑢)
26		fveq2 6906	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑧 = (𝐹‘𝑢) → (ℜ‘𝑧) = (ℜ‘(𝐹‘𝑢)))
27		fveq2 6906	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑧 = (𝐹‘𝑢) → (ℑ‘𝑧) = (ℑ‘(𝐹‘𝑢)))
28	26, 27	opeq12d 4885	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑧 = (𝐹‘𝑢) → ⟨(ℜ‘𝑧), (ℑ‘𝑧)⟩ = ⟨(ℜ‘(𝐹‘𝑢)), (ℑ‘(𝐹‘𝑢))⟩)
29	3	cnrecnv 15200	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ◡𝐹 = (𝑧 ∈ ℂ ↦ ⟨(ℜ‘𝑧), (ℑ‘𝑧)⟩)
30		opex 5474	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ⟨(ℜ‘(𝐹‘𝑢)), (ℑ‘(𝐹‘𝑢))⟩ ∈ V
31	28, 29, 30	fvmpt 7015	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐹‘𝑢) ∈ ℂ → (◡𝐹‘(𝐹‘𝑢)) = ⟨(ℜ‘(𝐹‘𝑢)), (ℑ‘(𝐹‘𝑢))⟩)
32	20, 31	syl 17	. . . . . . . . . . . . . . . . . . . 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 2776	. . . . . . . . . . . . . . . . . . 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 6910	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → (1st ‘𝑢) = (1st ‘⟨(ℜ‘(𝐹‘𝑢)), (ℑ‘(𝐹‘𝑢))⟩))
35		fvex 6919	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℜ‘(𝐹‘𝑢)) ∈ V
36		fvex 6919	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℑ‘(𝐹‘𝑢)) ∈ V
37	35, 36	op1st 8020	. . . . . . . . . . . . . . . . . 18 ⊢ (1st ‘⟨(ℜ‘(𝐹‘𝑢)), (ℑ‘(𝐹‘𝑢))⟩) = (ℜ‘(𝐹‘𝑢))
38	34, 37	eqtrdi 2790	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → (1st ‘𝑢) = (ℜ‘(𝐹‘𝑢)))
39	38	fveq2d 6910	. . . . . . . . . . . . . . . 16 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → (abs‘(1st ‘𝑢)) = (abs‘(ℜ‘(𝐹‘𝑢))))
40		absrele 15343	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹‘𝑢) ∈ ℂ → (abs‘(ℜ‘(𝐹‘𝑢))) ≤ (abs‘(𝐹‘𝑢)))
41	20, 40	syl 17	. . . . . . . . . . . . . . . 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 5169	. . . . . . . . . . . . . . 15 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → (abs‘(1st ‘𝑢)) ≤ (abs‘(𝐹‘𝑢)))
43		fveq2 6906	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = (𝐹‘𝑢) → (abs‘𝑧) = (abs‘(𝐹‘𝑢)))
44	43	breq1d 5157	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = (𝐹‘𝑢) → ((abs‘𝑧) ≤ 𝑅 ↔ (abs‘(𝐹‘𝑢)) ≤ 𝑅))
45		simplrr 778	. . . . . . . . . . . . . . . 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 3622	. . . . . . . . . . . . . . 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 11415	. . . . . . . . . . . . . 14 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → (abs‘(1st ‘𝑢)) ≤ 𝑅)
48	11, 22	absled 15465	. . . . . . . . . . . . . 14 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → ((abs‘(1st ‘𝑢)) ≤ 𝑅 ↔ (-𝑅 ≤ (1st ‘𝑢) ∧ (1st ‘𝑢) ≤ 𝑅)))
49	47, 48	mpbid 232	. . . . . . . . . . . . 13 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → (-𝑅 ≤ (1st ‘𝑢) ∧ (1st ‘𝑢) ≤ 𝑅))
50	49	simpld 494	. . . . . . . . . . . 12 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → -𝑅 ≤ (1st ‘𝑢))
51	49	simprd 495	. . . . . . . . . . . 12 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → (1st ‘𝑢) ≤ 𝑅)
52		renegcl 11569	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ ℝ → -𝑅 ∈ ℝ)
53	22, 52	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → -𝑅 ∈ ℝ)
54		elicc2 13448	. . . . . . . . . . . . 13 ⊢ ((-𝑅 ∈ ℝ ∧ 𝑅 ∈ ℝ) → ((1st ‘𝑢) ∈ (-𝑅[,]𝑅) ↔ ((1st ‘𝑢) ∈ ℝ ∧ -𝑅 ≤ (1st ‘𝑢) ∧ (1st ‘𝑢) ≤ 𝑅)))
55	53, 22, 54	syl2anc 584	. . . . . . . . . . . 12 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → ((1st ‘𝑢) ∈ (-𝑅[,]𝑅) ↔ ((1st ‘𝑢) ∈ ℝ ∧ -𝑅 ≤ (1st ‘𝑢) ∧ (1st ‘𝑢) ≤ 𝑅)))
56	11, 50, 51, 55	mpbir3and 1341	. . . . . . . . . . 11 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → (1st ‘𝑢) ∈ (-𝑅[,]𝑅))
57		xp2nd 8045	. . . . . . . . . . . . 13 ⊢ (𝑢 ∈ (ℝ × ℝ) → (2nd ‘𝑢) ∈ ℝ)
58	57	ad2antrl 728	. . . . . . . . . . . 12 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → (2nd ‘𝑢) ∈ ℝ)
59	58	recnd 11286	. . . . . . . . . . . . . . . 16 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → (2nd ‘𝑢) ∈ ℂ)
60	59	abscld 15471	. . . . . . . . . . . . . . 15 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → (abs‘(2nd ‘𝑢)) ∈ ℝ)
61	33	fveq2d 6910	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → (2nd ‘𝑢) = (2nd ‘⟨(ℜ‘(𝐹‘𝑢)), (ℑ‘(𝐹‘𝑢))⟩))
62	35, 36	op2nd 8021	. . . . . . . . . . . . . . . . . 18 ⊢ (2nd ‘⟨(ℜ‘(𝐹‘𝑢)), (ℑ‘(𝐹‘𝑢))⟩) = (ℑ‘(𝐹‘𝑢))
63	61, 62	eqtrdi 2790	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → (2nd ‘𝑢) = (ℑ‘(𝐹‘𝑢)))
64	63	fveq2d 6910	. . . . . . . . . . . . . . . 16 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → (abs‘(2nd ‘𝑢)) = (abs‘(ℑ‘(𝐹‘𝑢))))
65		absimle 15344	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹‘𝑢) ∈ ℂ → (abs‘(ℑ‘(𝐹‘𝑢))) ≤ (abs‘(𝐹‘𝑢)))
66	20, 65	syl 17	. . . . . . . . . . . . . . . 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 5169	. . . . . . . . . . . . . . 15 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → (abs‘(2nd ‘𝑢)) ≤ (abs‘(𝐹‘𝑢)))
68	60, 21, 22, 67, 46	letrd 11415	. . . . . . . . . . . . . 14 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → (abs‘(2nd ‘𝑢)) ≤ 𝑅)
69	58, 22	absled 15465	. . . . . . . . . . . . . 14 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → ((abs‘(2nd ‘𝑢)) ≤ 𝑅 ↔ (-𝑅 ≤ (2nd ‘𝑢) ∧ (2nd ‘𝑢) ≤ 𝑅)))
70	68, 69	mpbid 232	. . . . . . . . . . . . 13 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → (-𝑅 ≤ (2nd ‘𝑢) ∧ (2nd ‘𝑢) ≤ 𝑅))
71	70	simpld 494	. . . . . . . . . . . 12 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → -𝑅 ≤ (2nd ‘𝑢))
72	70	simprd 495	. . . . . . . . . . . 12 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → (2nd ‘𝑢) ≤ 𝑅)
73		elicc2 13448	. . . . . . . . . . . . 13 ⊢ ((-𝑅 ∈ ℝ ∧ 𝑅 ∈ ℝ) → ((2nd ‘𝑢) ∈ (-𝑅[,]𝑅) ↔ ((2nd ‘𝑢) ∈ ℝ ∧ -𝑅 ≤ (2nd ‘𝑢) ∧ (2nd ‘𝑢) ≤ 𝑅)))
74	53, 22, 73	syl2anc 584	. . . . . . . . . . . 12 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → ((2nd ‘𝑢) ∈ (-𝑅[,]𝑅) ↔ ((2nd ‘𝑢) ∈ ℝ ∧ -𝑅 ≤ (2nd ‘𝑢) ∧ (2nd ‘𝑢) ≤ 𝑅)))
75	58, 71, 72, 74	mpbir3and 1341	. . . . . . . . . . 11 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → (2nd ‘𝑢) ∈ (-𝑅[,]𝑅))
76	56, 75	opelxpd 5727	. . . . . . . . . 10 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → ⟨(1st ‘𝑢), (2nd ‘𝑢)⟩ ∈ ((-𝑅[,]𝑅) × (-𝑅[,]𝑅)))
77	9, 76	eqeltrd 2838	. . . . . . . . 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 412	. . . . . . . 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 242	. . . . . . 7 ⊢ ((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) → (𝑢 ∈ (◡𝐹 “ 𝑋) → 𝑢 ∈ ((-𝑅[,]𝑅) × (-𝑅[,]𝑅))))
80	79	ssrdv 4000	. . . . . 6 ⊢ ((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) → (◡𝐹 “ 𝑋) ⊆ ((-𝑅[,]𝑅) × (-𝑅[,]𝑅)))
81		f1ofun 6850	. . . . . . . 8 ⊢ (𝐹:(ℝ × ℝ)–1-1-onto→ℂ → Fun 𝐹)
82	4, 81	ax-mp 5	. . . . . . 7 ⊢ Fun 𝐹
83		f1ofo 6855	. . . . . . . . 9 ⊢ (𝐹:(ℝ × ℝ)–1-1-onto→ℂ → 𝐹:(ℝ × ℝ)–onto→ℂ)
84		forn 6823	. . . . . . . . 9 ⊢ (𝐹:(ℝ × ℝ)–onto→ℂ → ran 𝐹 = ℂ)
85	4, 83, 84	mp2b 10	. . . . . . . 8 ⊢ ran 𝐹 = ℂ
86	17, 85	sseqtrrdi 4046	. . . . . . 7 ⊢ ((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) → 𝑋 ⊆ ran 𝐹)
87		funimass1 6649	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝑋 ⊆ ran 𝐹) → ((◡𝐹 “ 𝑋) ⊆ ((-𝑅[,]𝑅) × (-𝑅[,]𝑅)) → 𝑋 ⊆ (𝐹 “ ((-𝑅[,]𝑅) × (-𝑅[,]𝑅)))))
88	82, 86, 87	sylancr 587	. . . . . 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 15	. . . . 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 4046	. . . 4 ⊢ ((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) → 𝑋 ⊆ 𝑌)
92		eqid 2734	. . . . . . . 8 ⊢ (topGen‘ran (,)) = (topGen‘ran (,))
93	3, 92, 1	cnrehmeo 24997	. . . . . . 7 ⊢ 𝐹 ∈ (((topGen‘ran (,)) ×t (topGen‘ran (,)))Homeo𝐽)
94		imaexg 7935	. . . . . . 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 2834	. . . . 5 ⊢ 𝑌 ∈ V
97	96	a1i 11	. . . 4 ⊢ ((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) → 𝑌 ∈ V)
98		restabs 23188	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ∈ V) → ((𝐽 ↾t 𝑌) ↾t 𝑋) = (𝐽 ↾t 𝑋))
99	2, 91, 97, 98	mp3an2i 1465	. . 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 2792	. 2 ⊢ ((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) → ((𝐽 ↾t 𝑌) ↾t 𝑋) = 𝑇)
102	90	oveq2i 7441	. . . . 5 ⊢ (𝐽 ↾t 𝑌) = (𝐽 ↾t (𝐹 “ ((-𝑅[,]𝑅) × (-𝑅[,]𝑅))))
103		ishmeo 23782	. . . . . . . 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 230	. . . . . . 7 ⊢ (𝐹 ∈ (((topGen‘ran (,)) ×t (topGen‘ran (,))) Cn 𝐽) ∧ ◡𝐹 ∈ (𝐽 Cn ((topGen‘ran (,)) ×t (topGen‘ran (,)))))
105	104	simpli 483	. . . . . 6 ⊢ 𝐹 ∈ (((topGen‘ran (,)) ×t (topGen‘ran (,))) Cn 𝐽)
106		iccssre 13465	. . . . . . . . . . 11 ⊢ ((-𝑅 ∈ ℝ ∧ 𝑅 ∈ ℝ) → (-𝑅[,]𝑅) ⊆ ℝ)
107	52, 106	mpancom 688	. . . . . . . . . 10 ⊢ (𝑅 ∈ ℝ → (-𝑅[,]𝑅) ⊆ ℝ)
108	1, 92	rerest 24839	. . . . . . . . . 10 ⊢ ((-𝑅[,]𝑅) ⊆ ℝ → (𝐽 ↾t (-𝑅[,]𝑅)) = ((topGen‘ran (,)) ↾t (-𝑅[,]𝑅)))
109	107, 108	syl 17	. . . . . . . . 9 ⊢ (𝑅 ∈ ℝ → (𝐽 ↾t (-𝑅[,]𝑅)) = ((topGen‘ran (,)) ↾t (-𝑅[,]𝑅)))
110	109, 109	oveq12d 7448	. . . . . . . 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 24797	. . . . . . . . 9 ⊢ (topGen‘ran (,)) ∈ Top
112		ovex 7463	. . . . . . . . 9 ⊢ (-𝑅[,]𝑅) ∈ V
113		txrest 23654	. . . . . . . . 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 693	. . . . . . . 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 2792	. . . . . . 7 ⊢ (𝑅 ∈ ℝ → ((𝐽 ↾t (-𝑅[,]𝑅)) ×t (𝐽 ↾t (-𝑅[,]𝑅))) = (((topGen‘ran (,)) ×t (topGen‘ran (,))) ↾t ((-𝑅[,]𝑅) × (-𝑅[,]𝑅))))
116		eqid 2734	. . . . . . . . . . 11 ⊢ ((topGen‘ran (,)) ↾t (-𝑅[,]𝑅)) = ((topGen‘ran (,)) ↾t (-𝑅[,]𝑅))
117	92, 116	icccmp 24860	. . . . . . . . . 10 ⊢ ((-𝑅 ∈ ℝ ∧ 𝑅 ∈ ℝ) → ((topGen‘ran (,)) ↾t (-𝑅[,]𝑅)) ∈ Comp)
118	52, 117	mpancom 688	. . . . . . . . 9 ⊢ (𝑅 ∈ ℝ → ((topGen‘ran (,)) ↾t (-𝑅[,]𝑅)) ∈ Comp)
119	109, 118	eqeltrd 2838	. . . . . . . 8 ⊢ (𝑅 ∈ ℝ → (𝐽 ↾t (-𝑅[,]𝑅)) ∈ Comp)
120		txcmp 23666	. . . . . . . 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 584	. . . . . . 7 ⊢ (𝑅 ∈ ℝ → ((𝐽 ↾t (-𝑅[,]𝑅)) ×t (𝐽 ↾t (-𝑅[,]𝑅))) ∈ Comp)
122	115, 121	eqeltrrd 2839	. . . . . 6 ⊢ (𝑅 ∈ ℝ → (((topGen‘ran (,)) ×t (topGen‘ran (,))) ↾t ((-𝑅[,]𝑅) × (-𝑅[,]𝑅))) ∈ Comp)
123		imacmp 23420	. . . . . 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 587	. . . . 5 ⊢ (𝑅 ∈ ℝ → (𝐽 ↾t (𝐹 “ ((-𝑅[,]𝑅) × (-𝑅[,]𝑅)))) ∈ Comp)
125	102, 124	eqeltrid 2842	. . . 4 ⊢ (𝑅 ∈ ℝ → (𝐽 ↾t 𝑌) ∈ Comp)
126	125	ad2antrl 728	. . 3 ⊢ ((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) → (𝐽 ↾t 𝑌) ∈ Comp)
127		imassrn 6090	. . . . . 6 ⊢ (𝐹 “ ((-𝑅[,]𝑅) × (-𝑅[,]𝑅))) ⊆ ran 𝐹
128	90, 127	eqsstri 4029	. . . . 5 ⊢ 𝑌 ⊆ ran 𝐹
129		f1of 6848	. . . . . 6 ⊢ (𝐹:(ℝ × ℝ)–1-1-onto→ℂ → 𝐹:(ℝ × ℝ)⟶ℂ)
130		frn 6743	. . . . . 6 ⊢ (𝐹:(ℝ × ℝ)⟶ℂ → ran 𝐹 ⊆ ℂ)
131	4, 129, 130	mp2b 10	. . . . 5 ⊢ ran 𝐹 ⊆ ℂ
132	128, 131	sstri 4004	. . . 4 ⊢ 𝑌 ⊆ ℂ
133		simpl 482	. . . 4 ⊢ ((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) → 𝑋 ∈ (Clsd‘𝐽))
134	15	restcldi 23196	. . . 4 ⊢ ((𝑌 ⊆ ℂ ∧ 𝑋 ∈ (Clsd‘𝐽) ∧ 𝑋 ⊆ 𝑌) → 𝑋 ∈ (Clsd‘(𝐽 ↾t 𝑌)))
135	132, 133, 91, 134	mp3an2i 1465	. . 3 ⊢ ((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) → 𝑋 ∈ (Clsd‘(𝐽 ↾t 𝑌)))
136		cmpcld 23425	. . 3 ⊢ (((𝐽 ↾t 𝑌) ∈ Comp ∧ 𝑋 ∈ (Clsd‘(𝐽 ↾t 𝑌))) → ((𝐽 ↾t 𝑌) ↾t 𝑋) ∈ Comp)
137	126, 135, 136	syl2anc 584	. 2 ⊢ ((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) → ((𝐽 ↾t 𝑌) ↾t 𝑋) ∈ Comp)
138	101, 137	eqeltrrd 2839	1 ⊢ ((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) → 𝑇 ∈ Comp)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1536   ∈ wcel 2105  ∀wral 3058  Vcvv 3477   ⊆ wss 3962  ⟨cop 4636   class class class wbr 5147   × cxp 5686  ^◡ccnv 5687  ran crn 5689   “ cima 5691  Fun wfun 6556   Fn wfn 6557  ⟶wf 6558  –onto→wfo 6560  –1-1-onto→wf1o 6561  ‘cfv 6562  (class class class)co 7430   ∈ cmpo 7432  1^st c1st 8010  2^nd c2nd 8011  ℂcc 11150  ℝcr 11151  ici 11154   + caddc 11155   · cmul 11157   ≤ cle 11293  -cneg 11490  (,)cioo 13383  [,]cicc 13386  ℜcre 15132  ℑcim 15133  abscabs 15269   ↾_t crest 17466  TopOpenctopn 17467  topGenctg 17483  ℂ_fldccnfld 21381  Topctop 22914  Clsdccld 23039   Cn ccn 23247  Compccmp 23409   ×_t ctx 23583  Homeochmeo 23776

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229  ax-pre-sup 11230  ax-addf 11231

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-iin 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-of 7696  df-om 7887  df-1st 8012  df-2nd 8013  df-supp 8184  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-2o 8505  df-er 8743  df-map 8866  df-ixp 8936  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-fsupp 9399  df-fi 9448  df-sup 9479  df-inf 9480  df-oi 9547  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-div 11918  df-nn 12264  df-2 12326  df-3 12327  df-4 12328  df-5 12329  df-6 12330  df-7 12331  df-8 12332  df-9 12333  df-n0 12524  df-z 12611  df-dec 12731  df-uz 12876  df-q 12988  df-rp 13032  df-xneg 13151  df-xadd 13152  df-xmul 13153  df-ioo 13387  df-icc 13390  df-fz 13544  df-fzo 13691  df-seq 14039  df-exp 14099  df-hash 14366  df-cj 15134  df-re 15135  df-im 15136  df-sqrt 15270  df-abs 15271  df-struct 17180  df-sets 17197  df-slot 17215  df-ndx 17227  df-base 17245  df-ress 17274  df-plusg 17310  df-mulr 17311  df-starv 17312  df-sca 17313  df-vsca 17314  df-ip 17315  df-tset 17316  df-ple 17317  df-ds 17319  df-unif 17320  df-hom 17321  df-cco 17322  df-rest 17468  df-topn 17469  df-0g 17487  df-gsum 17488  df-topgen 17489  df-pt 17490  df-prds 17493  df-xrs 17548  df-qtop 17553  df-imas 17554  df-xps 17556  df-mre 17630  df-mrc 17631  df-acs 17633  df-mgm 18665  df-sgrp 18744  df-mnd 18760  df-submnd 18809  df-mulg 19098  df-cntz 19347  df-cmn 19814  df-psmet 21373  df-xmet 21374  df-met 21375  df-bl 21376  df-mopn 21377  df-cnfld 21382  df-top 22915  df-topon 22932  df-topsp 22954  df-bases 22968  df-cld 23042  df-cn 23250  df-cnp 23251  df-cmp 23410  df-tx 23585  df-hmeo 23778  df-xms 24345  df-ms 24346  df-tms 24347  df-cncf 24917

This theorem is referenced by:  cnheibor  25000