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Theorem cnheiborlem 23537
Description: Lemma for cnheibor 23538. (Contributed by Mario Carneiro, 14-Sep-2014.)
Hypotheses
Ref Expression
cnheibor.2 𝐽 = (TopOpen‘ℂfld)
cnheibor.3 𝑇 = (𝐽t 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 ∧ ∀𝑧𝑋 (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.
StepHypRef Expression
1 cnheibor.2 . . . . 5 𝐽 = (TopOpen‘ℂfld)
21cnfldtop 23367 . . . 4 𝐽 ∈ Top
3 cnheibor.4 . . . . . . . . . 10 𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + (i · 𝑦)))
43cnref1o 12362 . . . . . . . . 9 𝐹:(ℝ × ℝ)–1-1-onto→ℂ
5 f1ofn 6589 . . . . . . . . 9 (𝐹:(ℝ × ℝ)–1-1-onto→ℂ → 𝐹 Fn (ℝ × ℝ))
6 elpreima 6801 . . . . . . . . 9 (𝐹 Fn (ℝ × ℝ) → (𝑢 ∈ (𝐹𝑋) ↔ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹𝑢) ∈ 𝑋)))
74, 5, 6mp2b 10 . . . . . . . 8 (𝑢 ∈ (𝐹𝑋) ↔ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹𝑢) ∈ 𝑋))
8 1st2nd2 7703 . . . . . . . . . . 11 (𝑢 ∈ (ℝ × ℝ) → 𝑢 = ⟨(1st𝑢), (2nd𝑢)⟩)
98ad2antrl 727 . . . . . . . . . 10 (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹𝑢) ∈ 𝑋)) → 𝑢 = ⟨(1st𝑢), (2nd𝑢)⟩)
10 xp1st 7696 . . . . . . . . . . . . 13 (𝑢 ∈ (ℝ × ℝ) → (1st𝑢) ∈ ℝ)
1110ad2antrl 727 . . . . . . . . . . . 12 (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹𝑢) ∈ 𝑋)) → (1st𝑢) ∈ ℝ)
1211recnd 10646 . . . . . . . . . . . . . . . 16 (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹𝑢) ∈ 𝑋)) → (1st𝑢) ∈ ℂ)
1312abscld 14775 . . . . . . . . . . . . . . 15 (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹𝑢) ∈ 𝑋)) → (abs‘(1st𝑢)) ∈ ℝ)
141cnfldtopon 23366 . . . . . . . . . . . . . . . . . . . . 21 𝐽 ∈ (TopOn‘ℂ)
1514toponunii 21499 . . . . . . . . . . . . . . . . . . . 20 ℂ = 𝐽
1615cldss 21612 . . . . . . . . . . . . . . . . . . 19 (𝑋 ∈ (Clsd‘𝐽) → 𝑋 ⊆ ℂ)
1716adantr 484 . . . . . . . . . . . . . . . . . 18 ((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧𝑋 (abs‘𝑧) ≤ 𝑅)) → 𝑋 ⊆ ℂ)
1817adantr 484 . . . . . . . . . . . . . . . . 17 (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹𝑢) ∈ 𝑋)) → 𝑋 ⊆ ℂ)
19 simprr 772 . . . . . . . . . . . . . . . . 17 (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹𝑢) ∈ 𝑋)) → (𝐹𝑢) ∈ 𝑋)
2018, 19sseldd 3944 . . . . . . . . . . . . . . . 16 (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹𝑢) ∈ 𝑋)) → (𝐹𝑢) ∈ ℂ)
2120abscld 14775 . . . . . . . . . . . . . . 15 (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹𝑢) ∈ 𝑋)) → (abs‘(𝐹𝑢)) ∈ ℝ)
22 simplrl 776 . . . . . . . . . . . . . . 15 (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹𝑢) ∈ 𝑋)) → 𝑅 ∈ ℝ)
23 simprl 770 . . . . . . . . . . . . . . . . . . . . 21 (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹𝑢) ∈ 𝑋)) → 𝑢 ∈ (ℝ × ℝ))
24 f1ocnvfv1 7007 . . . . . . . . . . . . . . . . . . . . 21 ((𝐹:(ℝ × ℝ)–1-1-onto→ℂ ∧ 𝑢 ∈ (ℝ × ℝ)) → (𝐹‘(𝐹𝑢)) = 𝑢)
254, 23, 24sylancr 590 . . . . . . . . . . . . . . . . . . . 20 (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹𝑢) ∈ 𝑋)) → (𝐹‘(𝐹𝑢)) = 𝑢)
26 fveq2 6643 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑧 = (𝐹𝑢) → (ℜ‘𝑧) = (ℜ‘(𝐹𝑢)))
27 fveq2 6643 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑧 = (𝐹𝑢) → (ℑ‘𝑧) = (ℑ‘(𝐹𝑢)))
2826, 27opeq12d 4784 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧 = (𝐹𝑢) → ⟨(ℜ‘𝑧), (ℑ‘𝑧)⟩ = ⟨(ℜ‘(𝐹𝑢)), (ℑ‘(𝐹𝑢))⟩)
293cnrecnv 14503 . . . . . . . . . . . . . . . . . . . . . 22 𝐹 = (𝑧 ∈ ℂ ↦ ⟨(ℜ‘𝑧), (ℑ‘𝑧)⟩)
30 opex 5329 . . . . . . . . . . . . . . . . . . . . . 22 ⟨(ℜ‘(𝐹𝑢)), (ℑ‘(𝐹𝑢))⟩ ∈ V
3128, 29, 30fvmpt 6741 . . . . . . . . . . . . . . . . . . . . 21 ((𝐹𝑢) ∈ ℂ → (𝐹‘(𝐹𝑢)) = ⟨(ℜ‘(𝐹𝑢)), (ℑ‘(𝐹𝑢))⟩)
3220, 31syl 17 . . . . . . . . . . . . . . . . . . . 20 (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹𝑢) ∈ 𝑋)) → (𝐹‘(𝐹𝑢)) = ⟨(ℜ‘(𝐹𝑢)), (ℑ‘(𝐹𝑢))⟩)
3325, 32eqtr3d 2858 . . . . . . . . . . . . . . . . . . 19 (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹𝑢) ∈ 𝑋)) → 𝑢 = ⟨(ℜ‘(𝐹𝑢)), (ℑ‘(𝐹𝑢))⟩)
3433fveq2d 6647 . . . . . . . . . . . . . . . . . 18 (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹𝑢) ∈ 𝑋)) → (1st𝑢) = (1st ‘⟨(ℜ‘(𝐹𝑢)), (ℑ‘(𝐹𝑢))⟩))
35 fvex 6656 . . . . . . . . . . . . . . . . . . 19 (ℜ‘(𝐹𝑢)) ∈ V
36 fvex 6656 . . . . . . . . . . . . . . . . . . 19 (ℑ‘(𝐹𝑢)) ∈ V
3735, 36op1st 7672 . . . . . . . . . . . . . . . . . 18 (1st ‘⟨(ℜ‘(𝐹𝑢)), (ℑ‘(𝐹𝑢))⟩) = (ℜ‘(𝐹𝑢))
3834, 37syl6eq 2872 . . . . . . . . . . . . . . . . 17 (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹𝑢) ∈ 𝑋)) → (1st𝑢) = (ℜ‘(𝐹𝑢)))
3938fveq2d 6647 . . . . . . . . . . . . . . . 16 (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹𝑢) ∈ 𝑋)) → (abs‘(1st𝑢)) = (abs‘(ℜ‘(𝐹𝑢))))
40 absrele 14647 . . . . . . . . . . . . . . . . 17 ((𝐹𝑢) ∈ ℂ → (abs‘(ℜ‘(𝐹𝑢))) ≤ (abs‘(𝐹𝑢)))
4120, 40syl 17 . . . . . . . . . . . . . . . 16 (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹𝑢) ∈ 𝑋)) → (abs‘(ℜ‘(𝐹𝑢))) ≤ (abs‘(𝐹𝑢)))
4239, 41eqbrtrd 5061 . . . . . . . . . . . . . . 15 (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹𝑢) ∈ 𝑋)) → (abs‘(1st𝑢)) ≤ (abs‘(𝐹𝑢)))
43 fveq2 6643 . . . . . . . . . . . . . . . . 17 (𝑧 = (𝐹𝑢) → (abs‘𝑧) = (abs‘(𝐹𝑢)))
4443breq1d 5049 . . . . . . . . . . . . . . . 16 (𝑧 = (𝐹𝑢) → ((abs‘𝑧) ≤ 𝑅 ↔ (abs‘(𝐹𝑢)) ≤ 𝑅))
45 simplrr 777 . . . . . . . . . . . . . . . 16 (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹𝑢) ∈ 𝑋)) → ∀𝑧𝑋 (abs‘𝑧) ≤ 𝑅)
4644, 45, 19rspcdva 3602 . . . . . . . . . . . . . . 15 (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹𝑢) ∈ 𝑋)) → (abs‘(𝐹𝑢)) ≤ 𝑅)
4713, 21, 22, 42, 46letrd 10774 . . . . . . . . . . . . . 14 (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹𝑢) ∈ 𝑋)) → (abs‘(1st𝑢)) ≤ 𝑅)
4811, 22absled 14769 . . . . . . . . . . . . . 14 (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹𝑢) ∈ 𝑋)) → ((abs‘(1st𝑢)) ≤ 𝑅 ↔ (-𝑅 ≤ (1st𝑢) ∧ (1st𝑢) ≤ 𝑅)))
4947, 48mpbid 235 . . . . . . . . . . . . 13 (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹𝑢) ∈ 𝑋)) → (-𝑅 ≤ (1st𝑢) ∧ (1st𝑢) ≤ 𝑅))
5049simpld 498 . . . . . . . . . . . 12 (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹𝑢) ∈ 𝑋)) → -𝑅 ≤ (1st𝑢))
5149simprd 499 . . . . . . . . . . . 12 (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹𝑢) ∈ 𝑋)) → (1st𝑢) ≤ 𝑅)
52 renegcl 10926 . . . . . . . . . . . . . 14 (𝑅 ∈ ℝ → -𝑅 ∈ ℝ)
5322, 52syl 17 . . . . . . . . . . . . 13 (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹𝑢) ∈ 𝑋)) → -𝑅 ∈ ℝ)
54 elicc2 12780 . . . . . . . . . . . . 13 ((-𝑅 ∈ ℝ ∧ 𝑅 ∈ ℝ) → ((1st𝑢) ∈ (-𝑅[,]𝑅) ↔ ((1st𝑢) ∈ ℝ ∧ -𝑅 ≤ (1st𝑢) ∧ (1st𝑢) ≤ 𝑅)))
5553, 22, 54syl2anc 587 . . . . . . . . . . . 12 (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹𝑢) ∈ 𝑋)) → ((1st𝑢) ∈ (-𝑅[,]𝑅) ↔ ((1st𝑢) ∈ ℝ ∧ -𝑅 ≤ (1st𝑢) ∧ (1st𝑢) ≤ 𝑅)))
5611, 50, 51, 55mpbir3and 1339 . . . . . . . . . . 11 (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹𝑢) ∈ 𝑋)) → (1st𝑢) ∈ (-𝑅[,]𝑅))
57 xp2nd 7697 . . . . . . . . . . . . 13 (𝑢 ∈ (ℝ × ℝ) → (2nd𝑢) ∈ ℝ)
5857ad2antrl 727 . . . . . . . . . . . 12 (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹𝑢) ∈ 𝑋)) → (2nd𝑢) ∈ ℝ)
5958recnd 10646 . . . . . . . . . . . . . . . 16 (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹𝑢) ∈ 𝑋)) → (2nd𝑢) ∈ ℂ)
6059abscld 14775 . . . . . . . . . . . . . . 15 (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹𝑢) ∈ 𝑋)) → (abs‘(2nd𝑢)) ∈ ℝ)
6133fveq2d 6647 . . . . . . . . . . . . . . . . . 18 (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹𝑢) ∈ 𝑋)) → (2nd𝑢) = (2nd ‘⟨(ℜ‘(𝐹𝑢)), (ℑ‘(𝐹𝑢))⟩))
6235, 36op2nd 7673 . . . . . . . . . . . . . . . . . 18 (2nd ‘⟨(ℜ‘(𝐹𝑢)), (ℑ‘(𝐹𝑢))⟩) = (ℑ‘(𝐹𝑢))
6361, 62syl6eq 2872 . . . . . . . . . . . . . . . . 17 (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹𝑢) ∈ 𝑋)) → (2nd𝑢) = (ℑ‘(𝐹𝑢)))
6463fveq2d 6647 . . . . . . . . . . . . . . . 16 (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹𝑢) ∈ 𝑋)) → (abs‘(2nd𝑢)) = (abs‘(ℑ‘(𝐹𝑢))))
65 absimle 14648 . . . . . . . . . . . . . . . . 17 ((𝐹𝑢) ∈ ℂ → (abs‘(ℑ‘(𝐹𝑢))) ≤ (abs‘(𝐹𝑢)))
6620, 65syl 17 . . . . . . . . . . . . . . . 16 (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹𝑢) ∈ 𝑋)) → (abs‘(ℑ‘(𝐹𝑢))) ≤ (abs‘(𝐹𝑢)))
6764, 66eqbrtrd 5061 . . . . . . . . . . . . . . 15 (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹𝑢) ∈ 𝑋)) → (abs‘(2nd𝑢)) ≤ (abs‘(𝐹𝑢)))
6860, 21, 22, 67, 46letrd 10774 . . . . . . . . . . . . . 14 (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹𝑢) ∈ 𝑋)) → (abs‘(2nd𝑢)) ≤ 𝑅)
6958, 22absled 14769 . . . . . . . . . . . . . 14 (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹𝑢) ∈ 𝑋)) → ((abs‘(2nd𝑢)) ≤ 𝑅 ↔ (-𝑅 ≤ (2nd𝑢) ∧ (2nd𝑢) ≤ 𝑅)))
7068, 69mpbid 235 . . . . . . . . . . . . 13 (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹𝑢) ∈ 𝑋)) → (-𝑅 ≤ (2nd𝑢) ∧ (2nd𝑢) ≤ 𝑅))
7170simpld 498 . . . . . . . . . . . 12 (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹𝑢) ∈ 𝑋)) → -𝑅 ≤ (2nd𝑢))
7270simprd 499 . . . . . . . . . . . 12 (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹𝑢) ∈ 𝑋)) → (2nd𝑢) ≤ 𝑅)
73 elicc2 12780 . . . . . . . . . . . . 13 ((-𝑅 ∈ ℝ ∧ 𝑅 ∈ ℝ) → ((2nd𝑢) ∈ (-𝑅[,]𝑅) ↔ ((2nd𝑢) ∈ ℝ ∧ -𝑅 ≤ (2nd𝑢) ∧ (2nd𝑢) ≤ 𝑅)))
7453, 22, 73syl2anc 587 . . . . . . . . . . . 12 (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹𝑢) ∈ 𝑋)) → ((2nd𝑢) ∈ (-𝑅[,]𝑅) ↔ ((2nd𝑢) ∈ ℝ ∧ -𝑅 ≤ (2nd𝑢) ∧ (2nd𝑢) ≤ 𝑅)))
7558, 71, 72, 74mpbir3and 1339 . . . . . . . . . . 11 (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹𝑢) ∈ 𝑋)) → (2nd𝑢) ∈ (-𝑅[,]𝑅))
7656, 75opelxpd 5566 . . . . . . . . . 10 (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹𝑢) ∈ 𝑋)) → ⟨(1st𝑢), (2nd𝑢)⟩ ∈ ((-𝑅[,]𝑅) × (-𝑅[,]𝑅)))
779, 76eqeltrd 2912 . . . . . . . . 9 (((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧𝑋 (abs‘𝑧) ≤ 𝑅)) ∧ (𝑢 ∈ (ℝ × ℝ) ∧ (𝐹𝑢) ∈ 𝑋)) → 𝑢 ∈ ((-𝑅[,]𝑅) × (-𝑅[,]𝑅)))
7877ex 416 . . . . . . . 8 ((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧𝑋 (abs‘𝑧) ≤ 𝑅)) → ((𝑢 ∈ (ℝ × ℝ) ∧ (𝐹𝑢) ∈ 𝑋) → 𝑢 ∈ ((-𝑅[,]𝑅) × (-𝑅[,]𝑅))))
797, 78syl5bi 245 . . . . . . 7 ((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧𝑋 (abs‘𝑧) ≤ 𝑅)) → (𝑢 ∈ (𝐹𝑋) → 𝑢 ∈ ((-𝑅[,]𝑅) × (-𝑅[,]𝑅))))
8079ssrdv 3949 . . . . . 6 ((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧𝑋 (abs‘𝑧) ≤ 𝑅)) → (𝐹𝑋) ⊆ ((-𝑅[,]𝑅) × (-𝑅[,]𝑅)))
81 f1ofun 6590 . . . . . . . 8 (𝐹:(ℝ × ℝ)–1-1-onto→ℂ → Fun 𝐹)
824, 81ax-mp 5 . . . . . . 7 Fun 𝐹
83 f1ofo 6595 . . . . . . . . 9 (𝐹:(ℝ × ℝ)–1-1-onto→ℂ → 𝐹:(ℝ × ℝ)–onto→ℂ)
84 forn 6566 . . . . . . . . 9 (𝐹:(ℝ × ℝ)–onto→ℂ → ran 𝐹 = ℂ)
854, 83, 84mp2b 10 . . . . . . . 8 ran 𝐹 = ℂ
8617, 85sseqtrrdi 3994 . . . . . . 7 ((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧𝑋 (abs‘𝑧) ≤ 𝑅)) → 𝑋 ⊆ ran 𝐹)
87 funimass1 6409 . . . . . . 7 ((Fun 𝐹𝑋 ⊆ ran 𝐹) → ((𝐹𝑋) ⊆ ((-𝑅[,]𝑅) × (-𝑅[,]𝑅)) → 𝑋 ⊆ (𝐹 “ ((-𝑅[,]𝑅) × (-𝑅[,]𝑅)))))
8882, 86, 87sylancr 590 . . . . . 6 ((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧𝑋 (abs‘𝑧) ≤ 𝑅)) → ((𝐹𝑋) ⊆ ((-𝑅[,]𝑅) × (-𝑅[,]𝑅)) → 𝑋 ⊆ (𝐹 “ ((-𝑅[,]𝑅) × (-𝑅[,]𝑅)))))
8980, 88mpd 15 . . . . 5 ((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧𝑋 (abs‘𝑧) ≤ 𝑅)) → 𝑋 ⊆ (𝐹 “ ((-𝑅[,]𝑅) × (-𝑅[,]𝑅))))
90 cnheibor.5 . . . . 5 𝑌 = (𝐹 “ ((-𝑅[,]𝑅) × (-𝑅[,]𝑅)))
9189, 90sseqtrrdi 3994 . . . 4 ((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧𝑋 (abs‘𝑧) ≤ 𝑅)) → 𝑋𝑌)
92 eqid 2821 . . . . . . . 8 (topGen‘ran (,)) = (topGen‘ran (,))
933, 92, 1cnrehmeo 23536 . . . . . . 7 𝐹 ∈ (((topGen‘ran (,)) ×t (topGen‘ran (,)))Homeo𝐽)
94 imaexg 7595 . . . . . . 7 (𝐹 ∈ (((topGen‘ran (,)) ×t (topGen‘ran (,)))Homeo𝐽) → (𝐹 “ ((-𝑅[,]𝑅) × (-𝑅[,]𝑅))) ∈ V)
9593, 94ax-mp 5 . . . . . 6 (𝐹 “ ((-𝑅[,]𝑅) × (-𝑅[,]𝑅))) ∈ V
9690, 95eqeltri 2908 . . . . 5 𝑌 ∈ V
9796a1i 11 . . . 4 ((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧𝑋 (abs‘𝑧) ≤ 𝑅)) → 𝑌 ∈ V)
98 restabs 21748 . . . 4 ((𝐽 ∈ Top ∧ 𝑋𝑌𝑌 ∈ V) → ((𝐽t 𝑌) ↾t 𝑋) = (𝐽t 𝑋))
992, 91, 97, 98mp3an2i 1463 . . 3 ((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧𝑋 (abs‘𝑧) ≤ 𝑅)) → ((𝐽t 𝑌) ↾t 𝑋) = (𝐽t 𝑋))
100 cnheibor.3 . . 3 𝑇 = (𝐽t 𝑋)
10199, 100syl6eqr 2874 . 2 ((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧𝑋 (abs‘𝑧) ≤ 𝑅)) → ((𝐽t 𝑌) ↾t 𝑋) = 𝑇)
10290oveq2i 7141 . . . . 5 (𝐽t 𝑌) = (𝐽t (𝐹 “ ((-𝑅[,]𝑅) × (-𝑅[,]𝑅))))
103 ishmeo 22342 . . . . . . . 8 (𝐹 ∈ (((topGen‘ran (,)) ×t (topGen‘ran (,)))Homeo𝐽) ↔ (𝐹 ∈ (((topGen‘ran (,)) ×t (topGen‘ran (,))) Cn 𝐽) ∧ 𝐹 ∈ (𝐽 Cn ((topGen‘ran (,)) ×t (topGen‘ran (,))))))
10493, 103mpbi 233 . . . . . . 7 (𝐹 ∈ (((topGen‘ran (,)) ×t (topGen‘ran (,))) Cn 𝐽) ∧ 𝐹 ∈ (𝐽 Cn ((topGen‘ran (,)) ×t (topGen‘ran (,)))))
105104simpli 487 . . . . . 6 𝐹 ∈ (((topGen‘ran (,)) ×t (topGen‘ran (,))) Cn 𝐽)
106 iccssre 12797 . . . . . . . . . . 11 ((-𝑅 ∈ ℝ ∧ 𝑅 ∈ ℝ) → (-𝑅[,]𝑅) ⊆ ℝ)
10752, 106mpancom 687 . . . . . . . . . 10 (𝑅 ∈ ℝ → (-𝑅[,]𝑅) ⊆ ℝ)
1081, 92rerest 23387 . . . . . . . . . 10 ((-𝑅[,]𝑅) ⊆ ℝ → (𝐽t (-𝑅[,]𝑅)) = ((topGen‘ran (,)) ↾t (-𝑅[,]𝑅)))
109107, 108syl 17 . . . . . . . . 9 (𝑅 ∈ ℝ → (𝐽t (-𝑅[,]𝑅)) = ((topGen‘ran (,)) ↾t (-𝑅[,]𝑅)))
110109, 109oveq12d 7148 . . . . . . . 8 (𝑅 ∈ ℝ → ((𝐽t (-𝑅[,]𝑅)) ×t (𝐽t (-𝑅[,]𝑅))) = (((topGen‘ran (,)) ↾t (-𝑅[,]𝑅)) ×t ((topGen‘ran (,)) ↾t (-𝑅[,]𝑅))))
111 retop 23345 . . . . . . . . 9 (topGen‘ran (,)) ∈ Top
112 ovex 7163 . . . . . . . . 9 (-𝑅[,]𝑅) ∈ V
113 txrest 22214 . . . . . . . . 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))))
114111, 111, 112, 112, 113mp4an 692 . . . . . . . 8 (((topGen‘ran (,)) ×t (topGen‘ran (,))) ↾t ((-𝑅[,]𝑅) × (-𝑅[,]𝑅))) = (((topGen‘ran (,)) ↾t (-𝑅[,]𝑅)) ×t ((topGen‘ran (,)) ↾t (-𝑅[,]𝑅)))
115110, 114syl6eqr 2874 . . . . . . 7 (𝑅 ∈ ℝ → ((𝐽t (-𝑅[,]𝑅)) ×t (𝐽t (-𝑅[,]𝑅))) = (((topGen‘ran (,)) ×t (topGen‘ran (,))) ↾t ((-𝑅[,]𝑅) × (-𝑅[,]𝑅))))
116 eqid 2821 . . . . . . . . . . 11 ((topGen‘ran (,)) ↾t (-𝑅[,]𝑅)) = ((topGen‘ran (,)) ↾t (-𝑅[,]𝑅))
11792, 116icccmp 23408 . . . . . . . . . 10 ((-𝑅 ∈ ℝ ∧ 𝑅 ∈ ℝ) → ((topGen‘ran (,)) ↾t (-𝑅[,]𝑅)) ∈ Comp)
11852, 117mpancom 687 . . . . . . . . 9 (𝑅 ∈ ℝ → ((topGen‘ran (,)) ↾t (-𝑅[,]𝑅)) ∈ Comp)
119109, 118eqeltrd 2912 . . . . . . . 8 (𝑅 ∈ ℝ → (𝐽t (-𝑅[,]𝑅)) ∈ Comp)
120 txcmp 22226 . . . . . . . 8 (((𝐽t (-𝑅[,]𝑅)) ∈ Comp ∧ (𝐽t (-𝑅[,]𝑅)) ∈ Comp) → ((𝐽t (-𝑅[,]𝑅)) ×t (𝐽t (-𝑅[,]𝑅))) ∈ Comp)
121119, 119, 120syl2anc 587 . . . . . . 7 (𝑅 ∈ ℝ → ((𝐽t (-𝑅[,]𝑅)) ×t (𝐽t (-𝑅[,]𝑅))) ∈ Comp)
122115, 121eqeltrrd 2913 . . . . . 6 (𝑅 ∈ ℝ → (((topGen‘ran (,)) ×t (topGen‘ran (,))) ↾t ((-𝑅[,]𝑅) × (-𝑅[,]𝑅))) ∈ Comp)
123 imacmp 21980 . . . . . 6 ((𝐹 ∈ (((topGen‘ran (,)) ×t (topGen‘ran (,))) Cn 𝐽) ∧ (((topGen‘ran (,)) ×t (topGen‘ran (,))) ↾t ((-𝑅[,]𝑅) × (-𝑅[,]𝑅))) ∈ Comp) → (𝐽t (𝐹 “ ((-𝑅[,]𝑅) × (-𝑅[,]𝑅)))) ∈ Comp)
124105, 122, 123sylancr 590 . . . . 5 (𝑅 ∈ ℝ → (𝐽t (𝐹 “ ((-𝑅[,]𝑅) × (-𝑅[,]𝑅)))) ∈ Comp)
125102, 124eqeltrid 2916 . . . 4 (𝑅 ∈ ℝ → (𝐽t 𝑌) ∈ Comp)
126125ad2antrl 727 . . 3 ((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧𝑋 (abs‘𝑧) ≤ 𝑅)) → (𝐽t 𝑌) ∈ Comp)
127 imassrn 5913 . . . . . 6 (𝐹 “ ((-𝑅[,]𝑅) × (-𝑅[,]𝑅))) ⊆ ran 𝐹
12890, 127eqsstri 3977 . . . . 5 𝑌 ⊆ ran 𝐹
129 f1of 6588 . . . . . 6 (𝐹:(ℝ × ℝ)–1-1-onto→ℂ → 𝐹:(ℝ × ℝ)⟶ℂ)
130 frn 6493 . . . . . 6 (𝐹:(ℝ × ℝ)⟶ℂ → ran 𝐹 ⊆ ℂ)
1314, 129, 130mp2b 10 . . . . 5 ran 𝐹 ⊆ ℂ
132128, 131sstri 3952 . . . 4 𝑌 ⊆ ℂ
133 simpl 486 . . . 4 ((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧𝑋 (abs‘𝑧) ≤ 𝑅)) → 𝑋 ∈ (Clsd‘𝐽))
13415restcldi 21756 . . . 4 ((𝑌 ⊆ ℂ ∧ 𝑋 ∈ (Clsd‘𝐽) ∧ 𝑋𝑌) → 𝑋 ∈ (Clsd‘(𝐽t 𝑌)))
135132, 133, 91, 134mp3an2i 1463 . . 3 ((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧𝑋 (abs‘𝑧) ≤ 𝑅)) → 𝑋 ∈ (Clsd‘(𝐽t 𝑌)))
136 cmpcld 21985 . . 3 (((𝐽t 𝑌) ∈ Comp ∧ 𝑋 ∈ (Clsd‘(𝐽t 𝑌))) → ((𝐽t 𝑌) ↾t 𝑋) ∈ Comp)
137126, 135, 136syl2anc 587 . 2 ((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧𝑋 (abs‘𝑧) ≤ 𝑅)) → ((𝐽t 𝑌) ↾t 𝑋) ∈ Comp)
138101, 137eqeltrrd 2913 1 ((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧𝑋 (abs‘𝑧) ≤ 𝑅)) → 𝑇 ∈ Comp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2115  wral 3126  Vcvv 3471  wss 3910  cop 4546   class class class wbr 5039   × cxp 5526  ccnv 5527  ran crn 5529  cima 5531  Fun wfun 6322   Fn wfn 6323  wf 6324  ontowfo 6326  1-1-ontowf1o 6327  cfv 6328  (class class class)co 7130  cmpo 7132  1st c1st 7662  2nd c2nd 7663  cc 10512  cr 10513  ici 10516   + caddc 10517   · cmul 10519  cle 10653  -cneg 10848  (,)cioo 12716  [,]cicc 12719  cre 14435  cim 14436  abscabs 14572  t crest 16672  TopOpenctopn 16673  topGenctg 16689  fldccnfld 20520  Topctop 21476  Clsdccld 21599   Cn ccn 21807  Compccmp 21969   ×t ctx 22143  Homeochmeo 22336
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-rep 5163  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436  ax-cnex 10570  ax-resscn 10571  ax-1cn 10572  ax-icn 10573  ax-addcl 10574  ax-addrcl 10575  ax-mulcl 10576  ax-mulrcl 10577  ax-mulcom 10578  ax-addass 10579  ax-mulass 10580  ax-distr 10581  ax-i2m1 10582  ax-1ne0 10583  ax-1rid 10584  ax-rnegex 10585  ax-rrecex 10586  ax-cnre 10587  ax-pre-lttri 10588  ax-pre-lttrn 10589  ax-pre-ltadd 10590  ax-pre-mulgt0 10591  ax-pre-sup 10592  ax-addf 10593  ax-mulf 10594
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-nel 3112  df-ral 3131  df-rex 3132  df-reu 3133  df-rmo 3134  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-tp 4545  df-op 4547  df-uni 4812  df-int 4850  df-iun 4894  df-iin 4895  df-br 5040  df-opab 5102  df-mpt 5120  df-tr 5146  df-id 5433  df-eprel 5438  df-po 5447  df-so 5448  df-fr 5487  df-se 5488  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6121  df-ord 6167  df-on 6168  df-lim 6169  df-suc 6170  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-isom 6337  df-riota 7088  df-ov 7133  df-oprab 7134  df-mpo 7135  df-of 7384  df-om 7556  df-1st 7664  df-2nd 7665  df-supp 7806  df-wrecs 7922  df-recs 7983  df-rdg 8021  df-1o 8077  df-2o 8078  df-oadd 8081  df-er 8264  df-map 8383  df-ixp 8437  df-en 8485  df-dom 8486  df-sdom 8487  df-fin 8488  df-fsupp 8810  df-fi 8851  df-sup 8882  df-inf 8883  df-oi 8950  df-card 9344  df-pnf 10654  df-mnf 10655  df-xr 10656  df-ltxr 10657  df-le 10658  df-sub 10849  df-neg 10850  df-div 11275  df-nn 11616  df-2 11678  df-3 11679  df-4 11680  df-5 11681  df-6 11682  df-7 11683  df-8 11684  df-9 11685  df-n0 11876  df-z 11960  df-dec 12077  df-uz 12222  df-q 12327  df-rp 12368  df-xneg 12485  df-xadd 12486  df-xmul 12487  df-ioo 12720  df-icc 12723  df-fz 12876  df-fzo 13017  df-seq 13353  df-exp 13414  df-hash 13675  df-cj 14437  df-re 14438  df-im 14439  df-sqrt 14573  df-abs 14574  df-struct 16463  df-ndx 16464  df-slot 16465  df-base 16467  df-sets 16468  df-ress 16469  df-plusg 16556  df-mulr 16557  df-starv 16558  df-sca 16559  df-vsca 16560  df-ip 16561  df-tset 16562  df-ple 16563  df-ds 16565  df-unif 16566  df-hom 16567  df-cco 16568  df-rest 16674  df-topn 16675  df-0g 16693  df-gsum 16694  df-topgen 16695  df-pt 16696  df-prds 16699  df-xrs 16753  df-qtop 16758  df-imas 16759  df-xps 16761  df-mre 16835  df-mrc 16836  df-acs 16838  df-mgm 17830  df-sgrp 17879  df-mnd 17890  df-submnd 17935  df-mulg 18203  df-cntz 18425  df-cmn 18886  df-psmet 20512  df-xmet 20513  df-met 20514  df-bl 20515  df-mopn 20516  df-cnfld 20521  df-top 21477  df-topon 21494  df-topsp 21516  df-bases 21529  df-cld 21602  df-cn 21810  df-cnp 21811  df-cmp 21970  df-tx 22145  df-hmeo 22338  df-xms 22905  df-ms 22906  df-tms 22907  df-cncf 23461
This theorem is referenced by:  cnheibor  23538
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