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Theorem cnheiborlem 25302

Description: Lemma for cnheibor 25303. (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 25129	. . . 4 ⊢ 𝐽 ∈ Top
3		cnheibor.4	. . . . . . . . . 10 ⊢ 𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + (i · 𝑦)))
4	3	cnref1o 13137	. . . . . . . . 9 ⊢ 𝐹:(ℝ × ℝ)⟶1-1ontoℂ
5		f1ofn 6844	. . . . . . . . 9 ⊢ (𝐹:(ℝ × ℝ)⟶1-1ontoℂ → 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 8061	. . . . . . . . . . 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 8054	. . . . . . . . . . . . 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 11361	. . . . . . . . . . . . . . . 16 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → (1^st ‘𝑢) ∈ ℂ)
13	12	abscld 15632	. . . . . . . . . . . . . . 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 25128	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝐽 ∈ (TopOn‘ℂ)
15	14	toponunii 23261	. . . . . . . . . . . . . . . . . . . 20 ⊢ ℂ = ∪ 𝐽
16	15	cldss 23374	. . . . . . . . . . . . . . . . . . 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 3914	. . . . . . . . . . . . . . . 16 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → (𝐹‘𝑢) ∈ ℂ)
21	20	abscld 15632	. . . . . . . . . . . . . . 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 7304	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐹:(ℝ × ℝ)⟶1-1ontoℂ ∧ 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 6905	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑧 = (𝐹‘𝑢) → (ℜ‘𝑧) = (ℜ‘(𝐹‘𝑢)))
27		fveq2 6905	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑧 = (𝐹‘𝑢) → (ℑ‘𝑧) = (ℑ‘(𝐹‘𝑢)))
28	26, 27	opeq12d 4851	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑧 = (𝐹‘𝑢) → ⟨(ℜ‘𝑧), (ℑ‘𝑧)⟩ = ⟨(ℜ‘(𝐹‘𝑢)), (ℑ‘(𝐹‘𝑢))⟩)
29	3	cnrecnv 15358	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ^◡𝐹 = (𝑧 ∈ ℂ ↦ ⟨(ℜ‘𝑧), (ℑ‘𝑧)⟩)
30		opex 5436	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ⟨(ℜ‘(𝐹‘𝑢)), (ℑ‘(𝐹‘𝑢))⟩ ∈ V
31	28, 29, 30	fvmpt 7013	. . . . . . . . . . . . . . . . . . . . 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 2800	. . . . . . . . . . . . . . . . . . 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 6909	. . . . . . . . . . . . . . . . . 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 6918	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℜ‘(𝐹‘𝑢)) ∈ V
36		fvex 6918	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℑ‘(𝐹‘𝑢)) ∈ V
37	35, 36	op1st 8030	. . . . . . . . . . . . . . . . . 18 ⊢ (1^st ‘⟨(ℜ‘(𝐹‘𝑢)), (ℑ‘(𝐹‘𝑢))⟩) = (ℜ‘(𝐹‘𝑢))
38	34, 37	eqtrdi 2814	. . . . . . . . . . . . . . . . 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 6909	. . . . . . . . . . . . . . . 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 15501	. . . . . . . . . . . . . . . . 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 5130	. . . . . . . . . . . . . . 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 6905	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = (𝐹‘𝑢) → (abs‘𝑧) = (abs‘(𝐹‘𝑢)))
44	43	breq1d 5116	. . . . . . . . . . . . . . . 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 3581	. . . . . . . . . . . . . . 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 11491	. . . . . . . . . . . . . 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 15626	. . . . . . . . . . . . . 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 11645	. . . . . . . . . . . . . 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 13566	. . . . . . . . . . . . 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 8055	. . . . . . . . . . . . 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 11361	. . . . . . . . . . . . . . . 16 ⊢ (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹‘𝑢) ∈ 𝑋)) → (2^nd ‘𝑢) ∈ ℂ)
60	59	abscld 15632	. . . . . . . . . . . . . . 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 6909	. . . . . . . . . . . . . . . . . 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 8031	. . . . . . . . . . . . . . . . . 18 ⊢ (2^nd ‘⟨(ℜ‘(𝐹‘𝑢)), (ℑ‘(𝐹‘𝑢))⟩) = (ℑ‘(𝐹‘𝑢))
63	61, 62	eqtrdi 2814	. . . . . . . . . . . . . . . . 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 6909	. . . . . . . . . . . . . . . 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 15502	. . . . . . . . . . . . . . . . 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 5130	. . . . . . . . . . . . . . 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 11491	. . . . . . . . . . . . . 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 15626	. . . . . . . . . . . . . 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 13566	. . . . . . . . . . . . 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 5694	. . . . . . . . . 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 2864	. . . . . . . . 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 3919	. . . . . 6 ⊢ ((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) → (^◡𝐹 “ 𝑋) ⊆ (( -𝑅[,]𝑅) × ( -𝑅[,]𝑅)))
81		f1ofun 6845	. . . . . . . 8 ⊢ (𝐹:(ℝ × ℝ)⟶1-1ontoℂ → Fun 𝐹)
82	4, 81	ax-mp 5	. . . . . . 7 ⊢ Fun 𝐹
83		f1ofo 6851	. . . . . . . . 9 ⊢ (𝐹:(ℝ × ℝ)⟶1-1ontoℂ → 𝐹:(ℝ × ℝ)⟶ontoℂ)
84		forn 6818	. . . . . . . . 9 ⊢ (𝐹:(ℝ × ℝ)⟶ontoℂ → ran 𝐹 = ℂ)
85	4, 83, 84	mp2b 10	. . . . . . . 8 ⊢ ran 𝐹 = ℂ
86	17, 85	sseqtrrdi 3954	. . . . . . 7 ⊢ ((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) → 𝑋 ⊆ ran 𝐹)
87		funimass1 6641	. . . . . . 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 3954	. . . 4 ⊢ ((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) → 𝑋 ⊆ 𝑌)
92		eqid 2763	. . . . . . . 8 ⊢ (topGen‘ran (,)) = (topGen‘ran (,))
93	3, 92, 1	cnrehmeo 25301	. . . . . . 7 ⊢ 𝐹 ∈ (((topGen‘ran (,)) ×_t (topGen‘ran (,)))Homeo𝐽)
94		imaexg 7946	. . . . . . 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 2860	. . . . 5 ⊢ 𝑌 ∈ V
97	96	a1i 11	. . . 4 ⊢ ((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) → 𝑌 ∈ V)
98		restabs 23510	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ∈ V) → ((𝐽 ↾_t 𝑌) ↾_t 𝑋) = (𝐽 ↾_t 𝑋))
99	2, 91, 97, 98	mp3an2i 1496	. . 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 2816	. 2 ⊢ ((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) → ((𝐽 ↾_t 𝑌) ↾_t 𝑋) = 𝑇)
102	90	oveq2i 7451	. . . . 5 ⊢ (𝐽 ↾_t 𝑌) = (𝐽 ↾_t (𝐹 “ (( -𝑅[,]𝑅) × ( -𝑅[,]𝑅))))
103		ishmeo 24105	. . . . . . . 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 13584	. . . . . . . . . . 11 ⊢ (( -𝑅 ∈ ℝ ∧ 𝑅 ∈ ℝ) → ( -𝑅[,]𝑅) ⊆ ℝ)
107	52, 106	mpancom 701	. . . . . . . . . 10 ⊢ (𝑅 ∈ ℝ → ( -𝑅[,]𝑅) ⊆ ℝ)
108	1, 92	rerest 25150	. . . . . . . . . 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 7458	. . . . . . . 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 25107	. . . . . . . . 9 ⊢ (topGen‘ran (,)) ∈ Top
112		ovex 7473	. . . . . . . . 9 ⊢ ( -𝑅[,]𝑅) ∈ V
113		txrest 23977	. . . . . . . . 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 2816	. . . . . . 7 ⊢ (𝑅 ∈ ℝ → ((𝐽 ↾_t ( -𝑅[,]𝑅)) ×_t (𝐽 ↾_t ( -𝑅[,]𝑅))) = (((topGen‘ran (,)) ×_t (topGen‘ran (,))) ↾_t (( -𝑅[,]𝑅) × ( -𝑅[,]𝑅))))
116		eqid 2763	. . . . . . . . . . 11 ⊢ ((topGen‘ran (,)) ↾_t ( -𝑅[,]𝑅)) = ((topGen‘ran (,)) ↾_t ( -𝑅[,]𝑅))
117	92, 116	icccmp 25172	. . . . . . . . . 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 2864	. . . . . . . 8 ⊢ (𝑅 ∈ ℝ → (𝐽 ↾_t ( -𝑅[,]𝑅)) ∈ Comp)
120		txcmp 23989	. . . . . . . 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 2865	. . . . . 6 ⊢ (𝑅 ∈ ℝ → (((topGen‘ran (,)) ×_t (topGen‘ran (,))) ↾_t (( -𝑅[,]𝑅) × ( -𝑅[,]𝑅))) ∈ Comp)
123		imacmp 23742	. . . . . 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 2868	. . . 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 6209	. . . . . 6 ⊢ (𝐹 “ (( -𝑅[,]𝑅) × ( -𝑅[,]𝑅))) ⊆ ran 𝐹
128	90, 127	eqsstri 3959	. . . . 5 ⊢ 𝑌 ⊆ ran 𝐹
129		f1of 6843	. . . . . 6 ⊢ (𝐹:(ℝ × ℝ)⟶1-1ontoℂ → 𝐹:(ℝ × ℝ)⟶ℂ)
130		frn 6736	. . . . . 6 ⊢ (𝐹:(ℝ × ℝ)⟶ℂ → ran 𝐹 ⊆ ℂ)
131	4, 129, 130	mp2b 10	. . . . 5 ⊢ ran 𝐹 ⊆ ℂ
132	128, 131	sstri 3922	. . . 4 ⊢ 𝑌 ⊆ ℂ
133		simpl 488	. . . 4 ⊢ ((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) → 𝑋 ∈ (Clsd‘𝐽))
134	15	restcldi 23518	. . . 4 ⊢ ((𝑌 ⊆ ℂ ∧ 𝑋 ∈ (Clsd‘𝐽) ∧ 𝑋 ⊆ 𝑌) → 𝑋 ∈ (Clsd‘(𝐽 ↾_t 𝑌)))
135	132, 133, 91, 134	mp3an2i 1496	. . 3 ⊢ ((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) → 𝑋 ∈ (Clsd‘(𝐽 ↾_t 𝑌)))
136		cmpcld 23747	. . 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 2865	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 1570 ∈ wcel 2147 ∀wral 3080 Vcvv 3454 ⊆ wss 3900 ⟨cop 4838 class class class wbr 5106 × cxp 5653 ^◡ccnv 5654 ran crn 5656 “ cima 5658 Fun wfun 6549 Fn wfn 6550 ⟶wf 6551 ⟶ontowfo 6553 ⟶1-1ontowf1o 6554 ‘cfv 6555 (class class class)co 7440 ∈ cmpo 7442 1^st c1st 8020 2^nd c2nd 8021 ℂcc 11222 ℝcr 11223 ici 11226 + caddc 11227 · cmul 11229 ≤ cle 11368 -cneg 11566 (,)cioo 13500 [,]cicc 13503 ℜcre 15290 ℑcim 15291 abscabs 15427 ↾_t crest 17617 TopOpenctopn 17618 topGenctg 17634 ℂ_fldccnfld 21705 Topctop 23238 Clsdccld 23361 Cn ccn 23569 Compccmp 23731 ×_t ctx 23906 Homeochmeo 24099

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7772 ax-cnex 11280 ax-resscn 11281 ax-1cn 11282 ax-icn 11283 ax-addcl 11284 ax-addrcl 11285 ax-mulcl 11286 ax-mulrcl 11287 ax-mulcom 11288 ax-addass 11289 ax-mulass 11290 ax-distr 11291 ax-i2m1 11292 ax-1ne0 11293 ax-1rid 11294 ax-rnegex 11295 ax-rrecex 11296 ax-cnre 11297 ax-pre-lttri 11298 ax-pre-lttrn 11299 ax-pre-ltadd 11300 ax-pre-mulgt0 11301 ax-pre-sup 11302 ax-addf 11303

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4839 df-uni 4871 df-int 4911 df-iun 4956 df-iin 4957 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6319 df-ord 6380 df-on 6381 df-lim 6382 df-suc 6383 df-iota 6509 df-fun 6557 df-fn 6558 df-f 6559 df-f1 6560 df-fo 6561 df-f1o 6562 df-fv 6563 df-isom 6564 df-riota 7397 df-ov 7443 df-oprab 7444 df-mpo 7445 df-of 7714 df-om 7899 df-1st 8022 df-2nd 8023 df-supp 8202 df-frecs 8324 df-wrecs 8355 df-recs 8404 df-rdg 8443 df-1o 8501 df-2o 8502 df-er 8742 df-map 8874 df-ixp 8951 df-en 8999 df-dom 9000 df-sdom 9001 df-fin 9002 df-fsupp 9379 df-fi 9427 df-sup 9458 df-inf 9459 df-oi 9528 df-card 10044 df-pnf 11369 df-mnf 11370 df-xr 11371 df-ltxr 11372 df-le 11373 df-sub 11567 df-neg 11568 df-div 11998 df-nn 12360 df-2 12429 df-3 12430 df-4 12431 df-5 12432 df-6 12433 df-7 12434 df-8 12435 df-9 12436 df-n0 12631 df-z 12718 df-dec 12839 df-uz 12990 df-q 13100 df-rp 13145 df-xneg 13265 df-xadd 13266 df-xmul 13267 df-ioo 13504 df-icc 13507 df-fz 13664 df-fzo 13813 df-seq 14169 df-exp 14229 df-hash 14499 df-cj 15292 df-re 15293 df-im 15294 df-sqrt 15428 df-abs 15429 df-struct 17351 df-sets 17368 df-slot 17386 df-ndx 17398 df-base 17414 df-ress 17435 df-plusg 17467 df-mulr 17468 df-starv 17469 df-sca 17470 df-vsca 17471 df-ip 17472 df-tset 17473 df-ple 17474 df-ds 17476 df-unif 17477 df-hom 17478 df-cco 17479 df-rest 17619 df-topn 17620 df-0g 17638 df-gsum 17639 df-topgen 17640 df-pt 17641 df-prds 17644 df-xrs 17700 df-qtop 17705 df-imas 17706 df-xps 17708 df-mre 17782 df-mrc 17783 df-acs 17785 df-mgm 18842 df-sgrp 18934 df-mnd 18950 df-submnd 19005 df-mulg 19304 df-cntz 19558 df-cmn 20023 df-psmet 21697 df-xmet 21698 df-met 21699 df-bl 21700 df-mopn 21701 df-cnfld 21706 df-top 23239 df-topon 23256 df-topsp 23278 df-bases 23291 df-cld 23364 df-cn 23572 df-cnp 23573 df-cmp 23732 df-tx 23908 df-hmeo 24101 df-xms 24666 df-ms 24667 df-tms 24668 df-cncf 25226

This theorem is used by: cnheibor 25303

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