Theorem cnheiborlem 24943

Description: Lemma for cnheibor 24944. (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 24770	. . . 4 ⊢ 𝐽 ∈ Top
3		cnheibor.4	. . . . . . . . . 10 ⊢ 𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + (i · 𝑦)))
4	3	cnref1o 12930	. . . . . . . . 9 ⊢ 𝐹:(ℝ × ℝ)–1-1-onto→ℂ
5		f1ofn 6772	. . . . . . . . 9 ⊢ (𝐹:(ℝ × ℝ)–1-1-onto→ℂ → 𝐹 Fn (ℝ × ℝ))
6		elpreima 7003	. . . . . . . . 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 7974	. . . . . . . . . . 11 ⊢ (𝑢 ∈ (ℝ × ℝ) → 𝑢 = ⟨(1st ‘𝑢), (2nd ‘𝑢)⟩)
9	8	ad2antrl 735	. . . . . . . . . 10 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → 𝑢 = ⟨(1st ‘𝑢), (2nd ‘𝑢)⟩)
10		xp1st 7967	. . . . . . . . . . . . 13 ⊢ (𝑢 ∈ (ℝ × ℝ) → (1st ‘𝑢) ∈ ℝ)
11	10	ad2antrl 735	. . . . . . . . . . . 12 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → (1st ‘𝑢) ∈ ℝ)
12	11	recnd 11168	. . . . . . . . . . . . . . . 16 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → (1st ‘𝑢) ∈ ℂ)
13	12	abscld 15396	. . . . . . . . . . . . . . 15 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → (abs‘(1st ‘𝑢)) ∈ ℝ)
14	1	cnfldtopon 24769	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝐽 ∈ (TopOn‘ℂ)
15	14	toponunii 22903	. . . . . . . . . . . . . . . . . . . 20 ⊢ ℂ = ∪ 𝐽
16	15	cldss 23016	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑋 ∈ (Clsd‘𝐽) → 𝑋 ⊆ ℂ)
17	16	adantr 482	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) → 𝑋 ⊆ ℂ)
18	17	adantr 482	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → 𝑋 ⊆ ℂ)
19		simprr 779	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → (𝐹‘𝑢) ∈ 𝑋)
20	18, 19	sseldd 3918	. . . . . . . . . . . . . . . 16 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → (𝐹‘𝑢) ∈ ℂ)
21	20	abscld 15396	. . . . . . . . . . . . . . 15 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → (abs‘(𝐹‘𝑢)) ∈ ℝ)
22		simplrl 783	. . . . . . . . . . . . . . 15 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → 𝑅 ∈ ℝ)
23		simprl 777	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → 𝑢 ∈ (ℝ × ℝ))
24		f1ocnvfv1 7224	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐹:(ℝ × ℝ)–1-1-onto→ℂ ∧ 𝑢 ∈ (ℝ × ℝ)) → (◡𝐹‘(𝐹‘𝑢)) = 𝑢)
25	4, 23, 24	sylancr 594	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → (◡𝐹‘(𝐹‘𝑢)) = 𝑢)
26		fveq2 6831	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑧 = (𝐹‘𝑢) → (ℜ‘𝑧) = (ℜ‘(𝐹‘𝑢)))
27		fveq2 6831	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑧 = (𝐹‘𝑢) → (ℑ‘𝑧) = (ℑ‘(𝐹‘𝑢)))
28	26, 27	opeq12d 4815	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑧 = (𝐹‘𝑢) → ⟨(ℜ‘𝑧), (ℑ‘𝑧)⟩ = ⟨(ℜ‘(𝐹‘𝑢)), (ℑ‘(𝐹‘𝑢))⟩)
29	3	cnrecnv 15122	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ◡𝐹 = (𝑧 ∈ ℂ ↦ ⟨(ℜ‘𝑧), (ℑ‘𝑧)⟩)
30		opex 5406	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ⟨(ℜ‘(𝐹‘𝑢)), (ℑ‘(𝐹‘𝑢))⟩ ∈ V
31	28, 29, 30	fvmpt 6939	. . . . . . . . . . . . . . . . . . . . 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 2778	. . . . . . . . . . . . . . . . . . 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 6835	. . . . . . . . . . . . . . . . . 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 6844	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℜ‘(𝐹‘𝑢)) ∈ V
36		fvex 6844	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℑ‘(𝐹‘𝑢)) ∈ V
37	35, 36	op1st 7943	. . . . . . . . . . . . . . . . . 18 ⊢ (1st ‘⟨(ℜ‘(𝐹‘𝑢)), (ℑ‘(𝐹‘𝑢))⟩) = (ℜ‘(𝐹‘𝑢))
38	34, 37	eqtrdi 2792	. . . . . . . . . . . . . . . . 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 6835	. . . . . . . . . . . . . . . 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 15265	. . . . . . . . . . . . . . . . 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 5097	. . . . . . . . . . . . . . 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 6831	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = (𝐹‘𝑢) → (abs‘𝑧) = (abs‘(𝐹‘𝑢)))
44	43	breq1d 5085	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = (𝐹‘𝑢) → ((abs‘𝑧) ≤ 𝑅 ↔ (abs‘(𝐹‘𝑢)) ≤ 𝑅))
45		simplrr 784	. . . . . . . . . . . . . . . 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 3563	. . . . . . . . . . . . . . 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 11298	. . . . . . . . . . . . . 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 15390	. . . . . . . . . . . . . 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 234	. . . . . . . . . . . . 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 496	. . . . . . . . . . . 12 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → -𝑅 ≤ (1st ‘𝑢))
51	49	simprd 497	. . . . . . . . . . . 12 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → (1st ‘𝑢) ≤ 𝑅)
52		renegcl 11452	. . . . . . . . . . . . . 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 13359	. . . . . . . . . . . . 13 ⊢ ((-𝑅 ∈ ℝ ∧ 𝑅 ∈ ℝ) → ((1st ‘𝑢) ∈ (-𝑅[,]𝑅) ↔ ((1st ‘𝑢) ∈ ℝ ∧ -𝑅 ≤ (1st ‘𝑢) ∧ (1st ‘𝑢) ≤ 𝑅)))
55	53, 22, 54	syl2anc 591	. . . . . . . . . . . 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 1350	. . . . . . . . . . 11 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → (1st ‘𝑢) ∈ (-𝑅[,]𝑅))
57		xp2nd 7968	. . . . . . . . . . . . 13 ⊢ (𝑢 ∈ (ℝ × ℝ) → (2nd ‘𝑢) ∈ ℝ)
58	57	ad2antrl 735	. . . . . . . . . . . 12 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → (2nd ‘𝑢) ∈ ℝ)
59	58	recnd 11168	. . . . . . . . . . . . . . . 16 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → (2nd ‘𝑢) ∈ ℂ)
60	59	abscld 15396	. . . . . . . . . . . . . . 15 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → (abs‘(2nd ‘𝑢)) ∈ ℝ)
61	33	fveq2d 6835	. . . . . . . . . . . . . . . . . 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 7944	. . . . . . . . . . . . . . . . . 18 ⊢ (2nd ‘⟨(ℜ‘(𝐹‘𝑢)), (ℑ‘(𝐹‘𝑢))⟩) = (ℑ‘(𝐹‘𝑢))
63	61, 62	eqtrdi 2792	. . . . . . . . . . . . . . . . 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 6835	. . . . . . . . . . . . . . . 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 15266	. . . . . . . . . . . . . . . . 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 5097	. . . . . . . . . . . . . . 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 11298	. . . . . . . . . . . . . 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 15390	. . . . . . . . . . . . . 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 234	. . . . . . . . . . . . 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 496	. . . . . . . . . . . 12 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → -𝑅 ≤ (2nd ‘𝑢))
72	70	simprd 497	. . . . . . . . . . . 12 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → (2nd ‘𝑢) ≤ 𝑅)
73		elicc2 13359	. . . . . . . . . . . . 13 ⊢ ((-𝑅 ∈ ℝ ∧ 𝑅 ∈ ℝ) → ((2nd ‘𝑢) ∈ (-𝑅[,]𝑅) ↔ ((2nd ‘𝑢) ∈ ℝ ∧ -𝑅 ≤ (2nd ‘𝑢) ∧ (2nd ‘𝑢) ≤ 𝑅)))
74	53, 22, 73	syl2anc 591	. . . . . . . . . . . 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 1350	. . . . . . . . . . 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 5660	. . . . . . . . . 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 2841	. . . . . . . . 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 414	. . . . . . . 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 244	. . . . . . 7 ⊢ ((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) → (𝑢 ∈ (◡𝐹 “ 𝑋) → 𝑢 ∈ ((-𝑅[,]𝑅) × (-𝑅[,]𝑅))))
80	79	ssrdv 3923	. . . . . 6 ⊢ ((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) → (◡𝐹 “ 𝑋) ⊆ ((-𝑅[,]𝑅) × (-𝑅[,]𝑅)))
81		f1ofun 6773	. . . . . . . 8 ⊢ (𝐹:(ℝ × ℝ)–1-1-onto→ℂ → Fun 𝐹)
82	4, 81	ax-mp 5	. . . . . . 7 ⊢ Fun 𝐹
83		f1ofo 6778	. . . . . . . . 9 ⊢ (𝐹:(ℝ × ℝ)–1-1-onto→ℂ → 𝐹:(ℝ × ℝ)–onto→ℂ)
84		forn 6746	. . . . . . . . 9 ⊢ (𝐹:(ℝ × ℝ)–onto→ℂ → ran 𝐹 = ℂ)
85	4, 83, 84	mp2b 10	. . . . . . . 8 ⊢ ran 𝐹 = ℂ
86	17, 85	sseqtrrdi 3958	. . . . . . 7 ⊢ ((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) → 𝑋 ⊆ ran 𝐹)
87		funimass1 6571	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝑋 ⊆ ran 𝐹) → ((◡𝐹 “ 𝑋) ⊆ ((-𝑅[,]𝑅) × (-𝑅[,]𝑅)) → 𝑋 ⊆ (𝐹 “ ((-𝑅[,]𝑅) × (-𝑅[,]𝑅)))))
88	82, 86, 87	sylancr 594	. . . . . 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 3958	. . . 4 ⊢ ((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) → 𝑋 ⊆ 𝑌)
92		eqid 2741	. . . . . . . 8 ⊢ (topGen‘ran (,)) = (topGen‘ran (,))
93	3, 92, 1	cnrehmeo 24942	. . . . . . 7 ⊢ 𝐹 ∈ (((topGen‘ran (,)) ×t (topGen‘ran (,)))Homeo𝐽)
94		imaexg 7857	. . . . . . 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 2837	. . . . 5 ⊢ 𝑌 ∈ V
97	96	a1i 11	. . . 4 ⊢ ((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) → 𝑌 ∈ V)
98		restabs 23152	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ∈ V) → ((𝐽 ↾t 𝑌) ↾t 𝑋) = (𝐽 ↾t 𝑋))
99	2, 91, 97, 98	mp3an2i 1475	. . 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 2794	. 2 ⊢ ((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) → ((𝐽 ↾t 𝑌) ↾t 𝑋) = 𝑇)
102	90	oveq2i 7371	. . . . 5 ⊢ (𝐽 ↾t 𝑌) = (𝐽 ↾t (𝐹 “ ((-𝑅[,]𝑅) × (-𝑅[,]𝑅))))
103		ishmeo 23746	. . . . . . . 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 232	. . . . . . 7 ⊢ (𝐹 ∈ (((topGen‘ran (,)) ×t (topGen‘ran (,))) Cn 𝐽) ∧ ◡𝐹 ∈ (𝐽 Cn ((topGen‘ran (,)) ×t (topGen‘ran (,)))))
105	104	simpli 485	. . . . . 6 ⊢ 𝐹 ∈ (((topGen‘ran (,)) ×t (topGen‘ran (,))) Cn 𝐽)
106		iccssre 13377	. . . . . . . . . . 11 ⊢ ((-𝑅 ∈ ℝ ∧ 𝑅 ∈ ℝ) → (-𝑅[,]𝑅) ⊆ ℝ)
107	52, 106	mpancom 695	. . . . . . . . . 10 ⊢ (𝑅 ∈ ℝ → (-𝑅[,]𝑅) ⊆ ℝ)
108	1, 92	rerest 24791	. . . . . . . . . 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 7378	. . . . . . . 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 24748	. . . . . . . . 9 ⊢ (topGen‘ran (,)) ∈ Top
112		ovex 7393	. . . . . . . . 9 ⊢ (-𝑅[,]𝑅) ∈ V
113		txrest 23618	. . . . . . . . 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 700	. . . . . . . 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 2794	. . . . . . 7 ⊢ (𝑅 ∈ ℝ → ((𝐽 ↾t (-𝑅[,]𝑅)) ×t (𝐽 ↾t (-𝑅[,]𝑅))) = (((topGen‘ran (,)) ×t (topGen‘ran (,))) ↾t ((-𝑅[,]𝑅) × (-𝑅[,]𝑅))))
116		eqid 2741	. . . . . . . . . . 11 ⊢ ((topGen‘ran (,)) ↾t (-𝑅[,]𝑅)) = ((topGen‘ran (,)) ↾t (-𝑅[,]𝑅))
117	92, 116	icccmp 24813	. . . . . . . . . 10 ⊢ ((-𝑅 ∈ ℝ ∧ 𝑅 ∈ ℝ) → ((topGen‘ran (,)) ↾t (-𝑅[,]𝑅)) ∈ Comp)
118	52, 117	mpancom 695	. . . . . . . . 9 ⊢ (𝑅 ∈ ℝ → ((topGen‘ran (,)) ↾t (-𝑅[,]𝑅)) ∈ Comp)
119	109, 118	eqeltrd 2841	. . . . . . . 8 ⊢ (𝑅 ∈ ℝ → (𝐽 ↾t (-𝑅[,]𝑅)) ∈ Comp)
120		txcmp 23630	. . . . . . . 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 591	. . . . . . 7 ⊢ (𝑅 ∈ ℝ → ((𝐽 ↾t (-𝑅[,]𝑅)) ×t (𝐽 ↾t (-𝑅[,]𝑅))) ∈ Comp)
122	115, 121	eqeltrrd 2842	. . . . . 6 ⊢ (𝑅 ∈ ℝ → (((topGen‘ran (,)) ×t (topGen‘ran (,))) ↾t ((-𝑅[,]𝑅) × (-𝑅[,]𝑅))) ∈ Comp)
123		imacmp 23384	. . . . . 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 594	. . . . 5 ⊢ (𝑅 ∈ ℝ → (𝐽 ↾t (𝐹 “ ((-𝑅[,]𝑅) × (-𝑅[,]𝑅)))) ∈ Comp)
125	102, 124	eqeltrid 2845	. . . 4 ⊢ (𝑅 ∈ ℝ → (𝐽 ↾t 𝑌) ∈ Comp)
126	125	ad2antrl 735	. . 3 ⊢ ((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) → (𝐽 ↾t 𝑌) ∈ Comp)
127		imassrn 6030	. . . . . 6 ⊢ (𝐹 “ ((-𝑅[,]𝑅) × (-𝑅[,]𝑅))) ⊆ ran 𝐹
128	90, 127	eqsstri 3963	. . . . 5 ⊢ 𝑌 ⊆ ran 𝐹
129		f1of 6771	. . . . . 6 ⊢ (𝐹:(ℝ × ℝ)–1-1-onto→ℂ → 𝐹:(ℝ × ℝ)⟶ℂ)
130		frn 6666	. . . . . 6 ⊢ (𝐹:(ℝ × ℝ)⟶ℂ → ran 𝐹 ⊆ ℂ)
131	4, 129, 130	mp2b 10	. . . . 5 ⊢ ran 𝐹 ⊆ ℂ
132	128, 131	sstri 3926	. . . 4 ⊢ 𝑌 ⊆ ℂ
133		simpl 484	. . . 4 ⊢ ((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) → 𝑋 ∈ (Clsd‘𝐽))
134	15	restcldi 23160	. . . 4 ⊢ ((𝑌 ⊆ ℂ ∧ 𝑋 ∈ (Clsd‘𝐽) ∧ 𝑋 ⊆ 𝑌) → 𝑋 ∈ (Clsd‘(𝐽 ↾t 𝑌)))
135	132, 133, 91, 134	mp3an2i 1475	. . 3 ⊢ ((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) → 𝑋 ∈ (Clsd‘(𝐽 ↾t 𝑌)))
136		cmpcld 23389	. . 3 ⊢ (((𝐽 ↾t 𝑌) ∈ Comp ∧ 𝑋 ∈ (Clsd‘(𝐽 ↾t 𝑌))) → ((𝐽 ↾t 𝑌) ↾t 𝑋) ∈ Comp)
137	126, 135, 136	syl2anc 591	. 2 ⊢ ((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) → ((𝐽 ↾t 𝑌) ↾t 𝑋) ∈ Comp)
138	101, 137	eqeltrrd 2842	1 ⊢ ((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) → 𝑇 ∈ Comp)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 397   ∧ w3a 1093   = wceq 1548   ∈ wcel 2121  ∀wral 3055  Vcvv 3433   ⊆ wss 3885  ⟨cop 4564   class class class wbr 5075   × cxp 5619  ^◡ccnv 5620  ran crn 5622   “ cima 5624  Fun wfun 6483   Fn wfn 6484  ⟶wf 6485  –onto→wfo 6487  –1-1-onto→wf1o 6488  ‘cfv 6489  (class class class)co 7360   ∈ cmpo 7362  1^st c1st 7933  2^nd c2nd 7934  ℂcc 11031  ℝcr 11032  ici 11035   + caddc 11036   · cmul 11038   ≤ cle 11175  -cneg 11373  (,)cioo 13293  [,]cicc 13296  ℜcre 15054  ℑcim 15055  abscabs 15191   ↾_t crest 17378  TopOpenctopn 17379  topGenctg 17395  ℂ_fldccnfld 21351  Topctop 22880  Clsdccld 23003   Cn ccn 23211  Compccmp 23373   ×_t ctx 23547  Homeochmeo 23740

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5202  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7682  ax-cnex 11089  ax-resscn 11090  ax-1cn 11091  ax-icn 11092  ax-addcl 11093  ax-addrcl 11094  ax-mulcl 11095  ax-mulrcl 11096  ax-mulcom 11097  ax-addass 11098  ax-mulass 11099  ax-distr 11100  ax-i2m1 11101  ax-1ne0 11102  ax-1rid 11103  ax-rnegex 11104  ax-rrecex 11105  ax-cnre 11106  ax-pre-lttri 11107  ax-pre-lttrn 11108  ax-pre-ltadd 11109  ax-pre-mulgt0 11110  ax-pre-sup 11111  ax-addf 11112

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-nel 3041  df-ral 3056  df-rex 3066  df-rmo 3346  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4842  df-int 4881  df-iun 4926  df-iin 4927  df-br 5076  df-opab 5138  df-mpt 5157  df-tr 5183  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-se 5575  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-isom 6498  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-of 7624  df-om 7811  df-1st 7935  df-2nd 7936  df-supp 8105  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-2o 8400  df-er 8637  df-map 8769  df-ixp 8840  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891  df-fsupp 9269  df-fi 9318  df-sup 9349  df-inf 9350  df-oi 9419  df-card 9858  df-pnf 11176  df-mnf 11177  df-xr 11178  df-ltxr 11179  df-le 11180  df-sub 11374  df-neg 11375  df-div 11803  df-nn 12170  df-2 12239  df-3 12240  df-4 12241  df-5 12242  df-6 12243  df-7 12244  df-8 12245  df-9 12246  df-n0 12433  df-z 12520  df-dec 12640  df-uz 12784  df-q 12894  df-rp 12938  df-xneg 13058  df-xadd 13059  df-xmul 13060  df-ioo 13297  df-icc 13300  df-fz 13457  df-fzo 13604  df-seq 13959  df-exp 14019  df-hash 14288  df-cj 15056  df-re 15057  df-im 15058  df-sqrt 15192  df-abs 15193  df-struct 17112  df-sets 17129  df-slot 17147  df-ndx 17159  df-base 17175  df-ress 17196  df-plusg 17228  df-mulr 17229  df-starv 17230  df-sca 17231  df-vsca 17232  df-ip 17233  df-tset 17234  df-ple 17235  df-ds 17237  df-unif 17238  df-hom 17239  df-cco 17240  df-rest 17380  df-topn 17381  df-0g 17399  df-gsum 17400  df-topgen 17401  df-pt 17402  df-prds 17405  df-xrs 17461  df-qtop 17466  df-imas 17467  df-xps 17469  df-mre 17543  df-mrc 17544  df-acs 17546  df-mgm 18603  df-sgrp 18682  df-mnd 18698  df-submnd 18747  df-mulg 19039  df-cntz 19287  df-cmn 19752  df-psmet 21343  df-xmet 21344  df-met 21345  df-bl 21346  df-mopn 21347  df-cnfld 21352  df-top 22881  df-topon 22898  df-topsp 22920  df-bases 22933  df-cld 23006  df-cn 23214  df-cnp 23215  df-cmp 23374  df-tx 23549  df-hmeo 23742  df-xms 24307  df-ms 24308  df-tms 24309  df-cncf 24867

This theorem is referenced by:  cnheibor  24944