Theorem cnheiborlem 24880

Description: Lemma for cnheibor 24881. (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 24698	. . . 4 ⊢ 𝐽 ∈ Top
3		cnheibor.4	. . . . . . . . . 10 ⊢ 𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + (i · 𝑦)))
4	3	cnref1o 12883	. . . . . . . . 9 ⊢ 𝐹:(ℝ × ℝ)–1-1-onto→ℂ
5		f1ofn 6764	. . . . . . . . 9 ⊢ (𝐹:(ℝ × ℝ)–1-1-onto→ℂ → 𝐹 Fn (ℝ × ℝ))
6		elpreima 6991	. . . . . . . . 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 7960	. . . . . . . . . . 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 7953	. . . . . . . . . . . . 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 11140	. . . . . . . . . . . . . . . 16 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → (1st ‘𝑢) ∈ ℂ)
13	12	abscld 15346	. . . . . . . . . . . . . . 15 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → (abs‘(1st ‘𝑢)) ∈ ℝ)
14	1	cnfldtopon 24697	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝐽 ∈ (TopOn‘ℂ)
15	14	toponunii 22831	. . . . . . . . . . . . . . . . . . . 20 ⊢ ℂ = ∪ 𝐽
16	15	cldss 22944	. . . . . . . . . . . . . . . . . . 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 772	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → (𝐹‘𝑢) ∈ 𝑋)
20	18, 19	sseldd 3930	. . . . . . . . . . . . . . . 16 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → (𝐹‘𝑢) ∈ ℂ)
21	20	abscld 15346	. . . . . . . . . . . . . . 15 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → (abs‘(𝐹‘𝑢)) ∈ ℝ)
22		simplrl 776	. . . . . . . . . . . . . . 15 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → 𝑅 ∈ ℝ)
23		simprl 770	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → 𝑢 ∈ (ℝ × ℝ))
24		f1ocnvfv1 7210	. . . . . . . . . . . . . . . . . . . . 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 6822	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑧 = (𝐹‘𝑢) → (ℜ‘𝑧) = (ℜ‘(𝐹‘𝑢)))
27		fveq2 6822	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑧 = (𝐹‘𝑢) → (ℑ‘𝑧) = (ℑ‘(𝐹‘𝑢)))
28	26, 27	opeq12d 4830	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑧 = (𝐹‘𝑢) → ⟨(ℜ‘𝑧), (ℑ‘𝑧)⟩ = ⟨(ℜ‘(𝐹‘𝑢)), (ℑ‘(𝐹‘𝑢))⟩)
29	3	cnrecnv 15072	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ◡𝐹 = (𝑧 ∈ ℂ ↦ ⟨(ℜ‘𝑧), (ℑ‘𝑧)⟩)
30		opex 5402	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ⟨(ℜ‘(𝐹‘𝑢)), (ℑ‘(𝐹‘𝑢))⟩ ∈ V
31	28, 29, 30	fvmpt 6929	. . . . . . . . . . . . . . . . . . . . 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 2768	. . . . . . . . . . . . . . . . . . 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 6826	. . . . . . . . . . . . . . . . . 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 6835	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℜ‘(𝐹‘𝑢)) ∈ V
36		fvex 6835	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℑ‘(𝐹‘𝑢)) ∈ V
37	35, 36	op1st 7929	. . . . . . . . . . . . . . . . . 18 ⊢ (1st ‘⟨(ℜ‘(𝐹‘𝑢)), (ℑ‘(𝐹‘𝑢))⟩) = (ℜ‘(𝐹‘𝑢))
38	34, 37	eqtrdi 2782	. . . . . . . . . . . . . . . . 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 6826	. . . . . . . . . . . . . . . 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 15215	. . . . . . . . . . . . . . . . 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 5111	. . . . . . . . . . . . . . 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 6822	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = (𝐹‘𝑢) → (abs‘𝑧) = (abs‘(𝐹‘𝑢)))
44	43	breq1d 5099	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = (𝐹‘𝑢) → ((abs‘𝑧) ≤ 𝑅 ↔ (abs‘(𝐹‘𝑢)) ≤ 𝑅))
45		simplrr 777	. . . . . . . . . . . . . . . 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 3573	. . . . . . . . . . . . . . 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 11270	. . . . . . . . . . . . . 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 15340	. . . . . . . . . . . . . 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 11424	. . . . . . . . . . . . . 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 13311	. . . . . . . . . . . . 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 1343	. . . . . . . . . . 11 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → (1st ‘𝑢) ∈ (-𝑅[,]𝑅))
57		xp2nd 7954	. . . . . . . . . . . . 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 11140	. . . . . . . . . . . . . . . 16 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → (2nd ‘𝑢) ∈ ℂ)
60	59	abscld 15346	. . . . . . . . . . . . . . 15 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → (abs‘(2nd ‘𝑢)) ∈ ℝ)
61	33	fveq2d 6826	. . . . . . . . . . . . . . . . . 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 7930	. . . . . . . . . . . . . . . . . 18 ⊢ (2nd ‘⟨(ℜ‘(𝐹‘𝑢)), (ℑ‘(𝐹‘𝑢))⟩) = (ℑ‘(𝐹‘𝑢))
63	61, 62	eqtrdi 2782	. . . . . . . . . . . . . . . . 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 6826	. . . . . . . . . . . . . . . 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 15216	. . . . . . . . . . . . . . . . 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 5111	. . . . . . . . . . . . . . 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 11270	. . . . . . . . . . . . . 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 15340	. . . . . . . . . . . . . 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 13311	. . . . . . . . . . . . 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 1343	. . . . . . . . . . 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 5653	. . . . . . . . . 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 2831	. . . . . . . . 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 3935	. . . . . 6 ⊢ ((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) → (◡𝐹 “ 𝑋) ⊆ ((-𝑅[,]𝑅) × (-𝑅[,]𝑅)))
81		f1ofun 6765	. . . . . . . 8 ⊢ (𝐹:(ℝ × ℝ)–1-1-onto→ℂ → Fun 𝐹)
82	4, 81	ax-mp 5	. . . . . . 7 ⊢ Fun 𝐹
83		f1ofo 6770	. . . . . . . . 9 ⊢ (𝐹:(ℝ × ℝ)–1-1-onto→ℂ → 𝐹:(ℝ × ℝ)–onto→ℂ)
84		forn 6738	. . . . . . . . 9 ⊢ (𝐹:(ℝ × ℝ)–onto→ℂ → ran 𝐹 = ℂ)
85	4, 83, 84	mp2b 10	. . . . . . . 8 ⊢ ran 𝐹 = ℂ
86	17, 85	sseqtrrdi 3971	. . . . . . 7 ⊢ ((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) → 𝑋 ⊆ ran 𝐹)
87		funimass1 6563	. . . . . . 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 3971	. . . 4 ⊢ ((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) → 𝑋 ⊆ 𝑌)
92		eqid 2731	. . . . . . . 8 ⊢ (topGen‘ran (,)) = (topGen‘ran (,))
93	3, 92, 1	cnrehmeo 24878	. . . . . . 7 ⊢ 𝐹 ∈ (((topGen‘ran (,)) ×t (topGen‘ran (,)))Homeo𝐽)
94		imaexg 7843	. . . . . . 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 2827	. . . . 5 ⊢ 𝑌 ∈ V
97	96	a1i 11	. . . 4 ⊢ ((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) → 𝑌 ∈ V)
98		restabs 23080	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ∈ V) → ((𝐽 ↾t 𝑌) ↾t 𝑋) = (𝐽 ↾t 𝑋))
99	2, 91, 97, 98	mp3an2i 1468	. . 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 2784	. 2 ⊢ ((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) → ((𝐽 ↾t 𝑌) ↾t 𝑋) = 𝑇)
102	90	oveq2i 7357	. . . . 5 ⊢ (𝐽 ↾t 𝑌) = (𝐽 ↾t (𝐹 “ ((-𝑅[,]𝑅) × (-𝑅[,]𝑅))))
103		ishmeo 23674	. . . . . . . 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 13329	. . . . . . . . . . 11 ⊢ ((-𝑅 ∈ ℝ ∧ 𝑅 ∈ ℝ) → (-𝑅[,]𝑅) ⊆ ℝ)
107	52, 106	mpancom 688	. . . . . . . . . 10 ⊢ (𝑅 ∈ ℝ → (-𝑅[,]𝑅) ⊆ ℝ)
108	1, 92	rerest 24719	. . . . . . . . . 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 7364	. . . . . . . 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 24676	. . . . . . . . 9 ⊢ (topGen‘ran (,)) ∈ Top
112		ovex 7379	. . . . . . . . 9 ⊢ (-𝑅[,]𝑅) ∈ V
113		txrest 23546	. . . . . . . . 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 2784	. . . . . . 7 ⊢ (𝑅 ∈ ℝ → ((𝐽 ↾t (-𝑅[,]𝑅)) ×t (𝐽 ↾t (-𝑅[,]𝑅))) = (((topGen‘ran (,)) ×t (topGen‘ran (,))) ↾t ((-𝑅[,]𝑅) × (-𝑅[,]𝑅))))
116		eqid 2731	. . . . . . . . . . 11 ⊢ ((topGen‘ran (,)) ↾t (-𝑅[,]𝑅)) = ((topGen‘ran (,)) ↾t (-𝑅[,]𝑅))
117	92, 116	icccmp 24741	. . . . . . . . . 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 2831	. . . . . . . 8 ⊢ (𝑅 ∈ ℝ → (𝐽 ↾t (-𝑅[,]𝑅)) ∈ Comp)
120		txcmp 23558	. . . . . . . 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 2832	. . . . . 6 ⊢ (𝑅 ∈ ℝ → (((topGen‘ran (,)) ×t (topGen‘ran (,))) ↾t ((-𝑅[,]𝑅) × (-𝑅[,]𝑅))) ∈ Comp)
123		imacmp 23312	. . . . . 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 2835	. . . 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 6019	. . . . . 6 ⊢ (𝐹 “ ((-𝑅[,]𝑅) × (-𝑅[,]𝑅))) ⊆ ran 𝐹
128	90, 127	eqsstri 3976	. . . . 5 ⊢ 𝑌 ⊆ ran 𝐹
129		f1of 6763	. . . . . 6 ⊢ (𝐹:(ℝ × ℝ)–1-1-onto→ℂ → 𝐹:(ℝ × ℝ)⟶ℂ)
130		frn 6658	. . . . . 6 ⊢ (𝐹:(ℝ × ℝ)⟶ℂ → ran 𝐹 ⊆ ℂ)
131	4, 129, 130	mp2b 10	. . . . 5 ⊢ ran 𝐹 ⊆ ℂ
132	128, 131	sstri 3939	. . . 4 ⊢ 𝑌 ⊆ ℂ
133		simpl 482	. . . 4 ⊢ ((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) → 𝑋 ∈ (Clsd‘𝐽))
134	15	restcldi 23088	. . . 4 ⊢ ((𝑌 ⊆ ℂ ∧ 𝑋 ∈ (Clsd‘𝐽) ∧ 𝑋 ⊆ 𝑌) → 𝑋 ∈ (Clsd‘(𝐽 ↾t 𝑌)))
135	132, 133, 91, 134	mp3an2i 1468	. . 3 ⊢ ((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) → 𝑋 ∈ (Clsd‘(𝐽 ↾t 𝑌)))
136		cmpcld 23317	. . 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 2832	1 ⊢ ((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) → 𝑇 ∈ Comp)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111  ∀wral 3047  Vcvv 3436   ⊆ wss 3897  ⟨cop 4579   class class class wbr 5089   × cxp 5612  ^◡ccnv 5613  ran crn 5615   “ cima 5617  Fun wfun 6475   Fn wfn 6476  ⟶wf 6477  –onto→wfo 6479  –1-1-onto→wf1o 6480  ‘cfv 6481  (class class class)co 7346   ∈ cmpo 7348  1^st c1st 7919  2^nd c2nd 7920  ℂcc 11004  ℝcr 11005  ici 11008   + caddc 11009   · cmul 11011   ≤ cle 11147  -cneg 11345  (,)cioo 13245  [,]cicc 13248  ℜcre 15004  ℑcim 15005  abscabs 15141   ↾_t crest 17324  TopOpenctopn 17325  topGenctg 17341  ℂ_fldccnfld 21291  Topctop 22808  Clsdccld 22931   Cn ccn 23139  Compccmp 23301   ×_t ctx 23475  Homeochmeo 23668

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668  ax-cnex 11062  ax-resscn 11063  ax-1cn 11064  ax-icn 11065  ax-addcl 11066  ax-addrcl 11067  ax-mulcl 11068  ax-mulrcl 11069  ax-mulcom 11070  ax-addass 11071  ax-mulass 11072  ax-distr 11073  ax-i2m1 11074  ax-1ne0 11075  ax-1rid 11076  ax-rnegex 11077  ax-rrecex 11078  ax-cnre 11079  ax-pre-lttri 11080  ax-pre-lttrn 11081  ax-pre-ltadd 11082  ax-pre-mulgt0 11083  ax-pre-sup 11084  ax-addf 11085

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-tp 4578  df-op 4580  df-uni 4857  df-int 4896  df-iun 4941  df-iin 4942  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-se 5568  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-isom 6490  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-of 7610  df-om 7797  df-1st 7921  df-2nd 7922  df-supp 8091  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-2o 8386  df-er 8622  df-map 8752  df-ixp 8822  df-en 8870  df-dom 8871  df-sdom 8872  df-fin 8873  df-fsupp 9246  df-fi 9295  df-sup 9326  df-inf 9327  df-oi 9396  df-card 9832  df-pnf 11148  df-mnf 11149  df-xr 11150  df-ltxr 11151  df-le 11152  df-sub 11346  df-neg 11347  df-div 11775  df-nn 12126  df-2 12188  df-3 12189  df-4 12190  df-5 12191  df-6 12192  df-7 12193  df-8 12194  df-9 12195  df-n0 12382  df-z 12469  df-dec 12589  df-uz 12733  df-q 12847  df-rp 12891  df-xneg 13011  df-xadd 13012  df-xmul 13013  df-ioo 13249  df-icc 13252  df-fz 13408  df-fzo 13555  df-seq 13909  df-exp 13969  df-hash 14238  df-cj 15006  df-re 15007  df-im 15008  df-sqrt 15142  df-abs 15143  df-struct 17058  df-sets 17075  df-slot 17093  df-ndx 17105  df-base 17121  df-ress 17142  df-plusg 17174  df-mulr 17175  df-starv 17176  df-sca 17177  df-vsca 17178  df-ip 17179  df-tset 17180  df-ple 17181  df-ds 17183  df-unif 17184  df-hom 17185  df-cco 17186  df-rest 17326  df-topn 17327  df-0g 17345  df-gsum 17346  df-topgen 17347  df-pt 17348  df-prds 17351  df-xrs 17406  df-qtop 17411  df-imas 17412  df-xps 17414  df-mre 17488  df-mrc 17489  df-acs 17491  df-mgm 18548  df-sgrp 18627  df-mnd 18643  df-submnd 18692  df-mulg 18981  df-cntz 19229  df-cmn 19694  df-psmet 21283  df-xmet 21284  df-met 21285  df-bl 21286  df-mopn 21287  df-cnfld 21292  df-top 22809  df-topon 22826  df-topsp 22848  df-bases 22861  df-cld 22934  df-cn 23142  df-cnp 23143  df-cmp 23302  df-tx 23477  df-hmeo 23670  df-xms 24235  df-ms 24236  df-tms 24237  df-cncf 24798

This theorem is referenced by:  cnheibor  24881