Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > stoweidlem61 | Structured version Visualization version GIF version |
Description: This lemma proves that there exists a function 𝑔 as in the proof in [BrosowskiDeutsh] p. 92: 𝑔 is in the subalgebra, and for all 𝑡 in 𝑇, abs( f(t) - g(t) ) < 2*ε. Here 𝐹 is used to represent f in the paper, and 𝐸 is used to represent ε. For this lemma there's the further assumption that the function 𝐹 to be approximated is nonnegative (this assumption is removed in a later theorem). (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
Ref | Expression |
---|---|
stoweidlem61.1 | ⊢ Ⅎ𝑡𝐹 |
stoweidlem61.2 | ⊢ Ⅎ𝑡𝜑 |
stoweidlem61.3 | ⊢ 𝐾 = (topGen‘ran (,)) |
stoweidlem61.4 | ⊢ (𝜑 → 𝐽 ∈ Comp) |
stoweidlem61.5 | ⊢ 𝑇 = ∪ 𝐽 |
stoweidlem61.6 | ⊢ (𝜑 → 𝑇 ≠ ∅) |
stoweidlem61.7 | ⊢ 𝐶 = (𝐽 Cn 𝐾) |
stoweidlem61.8 | ⊢ (𝜑 → 𝐴 ⊆ 𝐶) |
stoweidlem61.9 | ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴) |
stoweidlem61.10 | ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴) |
stoweidlem61.11 | ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴) |
stoweidlem61.12 | ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑡 ∈ 𝑇 ∧ 𝑟 ≠ 𝑡)) → ∃𝑞 ∈ 𝐴 (𝑞‘𝑟) ≠ (𝑞‘𝑡)) |
stoweidlem61.13 | ⊢ (𝜑 → 𝐹 ∈ 𝐶) |
stoweidlem61.14 | ⊢ (𝜑 → ∀𝑡 ∈ 𝑇 0 ≤ (𝐹‘𝑡)) |
stoweidlem61.15 | ⊢ (𝜑 → 𝐸 ∈ ℝ+) |
stoweidlem61.16 | ⊢ (𝜑 → 𝐸 < (1 / 3)) |
Ref | Expression |
---|---|
stoweidlem61 | ⊢ (𝜑 → ∃𝑔 ∈ 𝐴 ∀𝑡 ∈ 𝑇 (abs‘((𝑔‘𝑡) − (𝐹‘𝑡))) < (2 · 𝐸)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | stoweidlem61.1 | . . 3 ⊢ Ⅎ𝑡𝐹 | |
2 | stoweidlem61.2 | . . 3 ⊢ Ⅎ𝑡𝜑 | |
3 | stoweidlem61.3 | . . 3 ⊢ 𝐾 = (topGen‘ran (,)) | |
4 | stoweidlem61.5 | . . 3 ⊢ 𝑇 = ∪ 𝐽 | |
5 | stoweidlem61.7 | . . 3 ⊢ 𝐶 = (𝐽 Cn 𝐾) | |
6 | eqid 2821 | . . 3 ⊢ (𝑗 ∈ (0...𝑛) ↦ {𝑡 ∈ 𝑇 ∣ (𝐹‘𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)}) = (𝑗 ∈ (0...𝑛) ↦ {𝑡 ∈ 𝑇 ∣ (𝐹‘𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)}) | |
7 | eqid 2821 | . . 3 ⊢ (𝑗 ∈ (0...𝑛) ↦ {𝑡 ∈ 𝑇 ∣ ((𝑗 + (1 / 3)) · 𝐸) ≤ (𝐹‘𝑡)}) = (𝑗 ∈ (0...𝑛) ↦ {𝑡 ∈ 𝑇 ∣ ((𝑗 + (1 / 3)) · 𝐸) ≤ (𝐹‘𝑡)}) | |
8 | stoweidlem61.4 | . . 3 ⊢ (𝜑 → 𝐽 ∈ Comp) | |
9 | stoweidlem61.6 | . . 3 ⊢ (𝜑 → 𝑇 ≠ ∅) | |
10 | stoweidlem61.8 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝐶) | |
11 | stoweidlem61.9 | . . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴) | |
12 | stoweidlem61.10 | . . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴) | |
13 | stoweidlem61.11 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴) | |
14 | stoweidlem61.12 | . . 3 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑡 ∈ 𝑇 ∧ 𝑟 ≠ 𝑡)) → ∃𝑞 ∈ 𝐴 (𝑞‘𝑟) ≠ (𝑞‘𝑡)) | |
15 | stoweidlem61.13 | . . 3 ⊢ (𝜑 → 𝐹 ∈ 𝐶) | |
16 | stoweidlem61.14 | . . 3 ⊢ (𝜑 → ∀𝑡 ∈ 𝑇 0 ≤ (𝐹‘𝑡)) | |
17 | stoweidlem61.15 | . . 3 ⊢ (𝜑 → 𝐸 ∈ ℝ+) | |
18 | stoweidlem61.16 | . . 3 ⊢ (𝜑 → 𝐸 < (1 / 3)) | |
19 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18 | stoweidlem60 42344 | . 2 ⊢ (𝜑 → ∃𝑔 ∈ 𝐴 ∀𝑡 ∈ 𝑇 ∃𝑗 ∈ ℝ ((((𝑗 − (4 / 3)) · 𝐸) < (𝐹‘𝑡) ∧ (𝐹‘𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) ∧ ((𝑔‘𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < (𝑔‘𝑡)))) |
20 | nfv 1911 | . . . . 5 ⊢ Ⅎ𝑡 𝑔 ∈ 𝐴 | |
21 | 2, 20 | nfan 1896 | . . . 4 ⊢ Ⅎ𝑡(𝜑 ∧ 𝑔 ∈ 𝐴) |
22 | 17 | ad2antrr 724 | . . . . 5 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → 𝐸 ∈ ℝ+) |
23 | 3, 4, 5, 15 | fcnre 41280 | . . . . . . 7 ⊢ (𝜑 → 𝐹:𝑇⟶ℝ) |
24 | 23 | ffvelrnda 6850 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐹‘𝑡) ∈ ℝ) |
25 | 24 | adantlr 713 | . . . . 5 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → (𝐹‘𝑡) ∈ ℝ) |
26 | 10 | sselda 3966 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐴) → 𝑔 ∈ 𝐶) |
27 | 3, 4, 5, 26 | fcnre 41280 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐴) → 𝑔:𝑇⟶ℝ) |
28 | 27 | ffvelrnda 6850 | . . . . 5 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → (𝑔‘𝑡) ∈ ℝ) |
29 | simpll1 1208 | . . . . . . 7 ⊢ ((((𝐸 ∈ ℝ+ ∧ (𝐹‘𝑡) ∈ ℝ ∧ (𝑔‘𝑡) ∈ ℝ) ∧ 𝑗 ∈ ℝ) ∧ ((((𝑗 − (4 / 3)) · 𝐸) < (𝐹‘𝑡) ∧ (𝐹‘𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) ∧ ((𝑔‘𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < (𝑔‘𝑡)))) → 𝐸 ∈ ℝ+) | |
30 | simpll2 1209 | . . . . . . 7 ⊢ ((((𝐸 ∈ ℝ+ ∧ (𝐹‘𝑡) ∈ ℝ ∧ (𝑔‘𝑡) ∈ ℝ) ∧ 𝑗 ∈ ℝ) ∧ ((((𝑗 − (4 / 3)) · 𝐸) < (𝐹‘𝑡) ∧ (𝐹‘𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) ∧ ((𝑔‘𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < (𝑔‘𝑡)))) → (𝐹‘𝑡) ∈ ℝ) | |
31 | simpll3 1210 | . . . . . . 7 ⊢ ((((𝐸 ∈ ℝ+ ∧ (𝐹‘𝑡) ∈ ℝ ∧ (𝑔‘𝑡) ∈ ℝ) ∧ 𝑗 ∈ ℝ) ∧ ((((𝑗 − (4 / 3)) · 𝐸) < (𝐹‘𝑡) ∧ (𝐹‘𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) ∧ ((𝑔‘𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < (𝑔‘𝑡)))) → (𝑔‘𝑡) ∈ ℝ) | |
32 | simplr 767 | . . . . . . 7 ⊢ ((((𝐸 ∈ ℝ+ ∧ (𝐹‘𝑡) ∈ ℝ ∧ (𝑔‘𝑡) ∈ ℝ) ∧ 𝑗 ∈ ℝ) ∧ ((((𝑗 − (4 / 3)) · 𝐸) < (𝐹‘𝑡) ∧ (𝐹‘𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) ∧ ((𝑔‘𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < (𝑔‘𝑡)))) → 𝑗 ∈ ℝ) | |
33 | simprll 777 | . . . . . . 7 ⊢ ((((𝐸 ∈ ℝ+ ∧ (𝐹‘𝑡) ∈ ℝ ∧ (𝑔‘𝑡) ∈ ℝ) ∧ 𝑗 ∈ ℝ) ∧ ((((𝑗 − (4 / 3)) · 𝐸) < (𝐹‘𝑡) ∧ (𝐹‘𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) ∧ ((𝑔‘𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < (𝑔‘𝑡)))) → ((𝑗 − (4 / 3)) · 𝐸) < (𝐹‘𝑡)) | |
34 | simprlr 778 | . . . . . . 7 ⊢ ((((𝐸 ∈ ℝ+ ∧ (𝐹‘𝑡) ∈ ℝ ∧ (𝑔‘𝑡) ∈ ℝ) ∧ 𝑗 ∈ ℝ) ∧ ((((𝑗 − (4 / 3)) · 𝐸) < (𝐹‘𝑡) ∧ (𝐹‘𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) ∧ ((𝑔‘𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < (𝑔‘𝑡)))) → (𝐹‘𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) | |
35 | simprrr 780 | . . . . . . 7 ⊢ ((((𝐸 ∈ ℝ+ ∧ (𝐹‘𝑡) ∈ ℝ ∧ (𝑔‘𝑡) ∈ ℝ) ∧ 𝑗 ∈ ℝ) ∧ ((((𝑗 − (4 / 3)) · 𝐸) < (𝐹‘𝑡) ∧ (𝐹‘𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) ∧ ((𝑔‘𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < (𝑔‘𝑡)))) → ((𝑗 − (4 / 3)) · 𝐸) < (𝑔‘𝑡)) | |
36 | simprrl 779 | . . . . . . 7 ⊢ ((((𝐸 ∈ ℝ+ ∧ (𝐹‘𝑡) ∈ ℝ ∧ (𝑔‘𝑡) ∈ ℝ) ∧ 𝑗 ∈ ℝ) ∧ ((((𝑗 − (4 / 3)) · 𝐸) < (𝐹‘𝑡) ∧ (𝐹‘𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) ∧ ((𝑔‘𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < (𝑔‘𝑡)))) → (𝑔‘𝑡) < ((𝑗 + (1 / 3)) · 𝐸)) | |
37 | 29, 30, 31, 32, 33, 34, 35, 36 | stoweidlem13 42297 | . . . . . 6 ⊢ ((((𝐸 ∈ ℝ+ ∧ (𝐹‘𝑡) ∈ ℝ ∧ (𝑔‘𝑡) ∈ ℝ) ∧ 𝑗 ∈ ℝ) ∧ ((((𝑗 − (4 / 3)) · 𝐸) < (𝐹‘𝑡) ∧ (𝐹‘𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) ∧ ((𝑔‘𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < (𝑔‘𝑡)))) → (abs‘((𝑔‘𝑡) − (𝐹‘𝑡))) < (2 · 𝐸)) |
38 | 37 | rexlimdva2 3287 | . . . . 5 ⊢ ((𝐸 ∈ ℝ+ ∧ (𝐹‘𝑡) ∈ ℝ ∧ (𝑔‘𝑡) ∈ ℝ) → (∃𝑗 ∈ ℝ ((((𝑗 − (4 / 3)) · 𝐸) < (𝐹‘𝑡) ∧ (𝐹‘𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) ∧ ((𝑔‘𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < (𝑔‘𝑡))) → (abs‘((𝑔‘𝑡) − (𝐹‘𝑡))) < (2 · 𝐸))) |
39 | 22, 25, 28, 38 | syl3anc 1367 | . . . 4 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → (∃𝑗 ∈ ℝ ((((𝑗 − (4 / 3)) · 𝐸) < (𝐹‘𝑡) ∧ (𝐹‘𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) ∧ ((𝑔‘𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < (𝑔‘𝑡))) → (abs‘((𝑔‘𝑡) − (𝐹‘𝑡))) < (2 · 𝐸))) |
40 | 21, 39 | ralimdaa 3217 | . . 3 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐴) → (∀𝑡 ∈ 𝑇 ∃𝑗 ∈ ℝ ((((𝑗 − (4 / 3)) · 𝐸) < (𝐹‘𝑡) ∧ (𝐹‘𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) ∧ ((𝑔‘𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < (𝑔‘𝑡))) → ∀𝑡 ∈ 𝑇 (abs‘((𝑔‘𝑡) − (𝐹‘𝑡))) < (2 · 𝐸))) |
41 | 40 | reximdva 3274 | . 2 ⊢ (𝜑 → (∃𝑔 ∈ 𝐴 ∀𝑡 ∈ 𝑇 ∃𝑗 ∈ ℝ ((((𝑗 − (4 / 3)) · 𝐸) < (𝐹‘𝑡) ∧ (𝐹‘𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) ∧ ((𝑔‘𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < (𝑔‘𝑡))) → ∃𝑔 ∈ 𝐴 ∀𝑡 ∈ 𝑇 (abs‘((𝑔‘𝑡) − (𝐹‘𝑡))) < (2 · 𝐸))) |
42 | 19, 41 | mpd 15 | 1 ⊢ (𝜑 → ∃𝑔 ∈ 𝐴 ∀𝑡 ∈ 𝑇 (abs‘((𝑔‘𝑡) − (𝐹‘𝑡))) < (2 · 𝐸)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 Ⅎwnf 1780 ∈ wcel 2110 Ⅎwnfc 2961 ≠ wne 3016 ∀wral 3138 ∃wrex 3139 {crab 3142 ⊆ wss 3935 ∅c0 4290 ∪ cuni 4837 class class class wbr 5065 ↦ cmpt 5145 ran crn 5555 ‘cfv 6354 (class class class)co 7155 ℝcr 10535 0cc0 10536 1c1 10537 + caddc 10539 · cmul 10541 < clt 10674 ≤ cle 10675 − cmin 10869 / cdiv 11296 2c2 11691 3c3 11692 4c4 11693 ℝ+crp 12388 (,)cioo 12737 ...cfz 12891 abscabs 14592 topGenctg 16710 Cn ccn 21831 Compccmp 21993 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-inf2 9103 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-pre-sup 10614 ax-mulf 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-iin 4921 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-se 5514 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-isom 6363 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-of 7408 df-om 7580 df-1st 7688 df-2nd 7689 df-supp 7830 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-2o 8102 df-oadd 8105 df-er 8288 df-map 8407 df-pm 8408 df-ixp 8461 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-fsupp 8833 df-fi 8874 df-sup 8905 df-inf 8906 df-oi 8973 df-card 9367 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-div 11297 df-nn 11638 df-2 11699 df-3 11700 df-4 11701 df-5 11702 df-6 11703 df-7 11704 df-8 11705 df-9 11706 df-n0 11897 df-z 11981 df-dec 12098 df-uz 12243 df-q 12348 df-rp 12389 df-xneg 12506 df-xadd 12507 df-xmul 12508 df-ioo 12741 df-ioc 12742 df-ico 12743 df-icc 12744 df-fz 12892 df-fzo 13033 df-fl 13161 df-seq 13369 df-exp 13429 df-hash 13690 df-cj 14457 df-re 14458 df-im 14459 df-sqrt 14593 df-abs 14594 df-clim 14844 df-rlim 14845 df-sum 15042 df-struct 16484 df-ndx 16485 df-slot 16486 df-base 16488 df-sets 16489 df-ress 16490 df-plusg 16577 df-mulr 16578 df-starv 16579 df-sca 16580 df-vsca 16581 df-ip 16582 df-tset 16583 df-ple 16584 df-ds 16586 df-unif 16587 df-hom 16588 df-cco 16589 df-rest 16695 df-topn 16696 df-0g 16714 df-gsum 16715 df-topgen 16716 df-pt 16717 df-prds 16720 df-xrs 16774 df-qtop 16779 df-imas 16780 df-xps 16782 df-mre 16856 df-mrc 16857 df-acs 16859 df-mgm 17851 df-sgrp 17900 df-mnd 17911 df-submnd 17956 df-mulg 18224 df-cntz 18446 df-cmn 18907 df-psmet 20536 df-xmet 20537 df-met 20538 df-bl 20539 df-mopn 20540 df-cnfld 20545 df-top 21501 df-topon 21518 df-topsp 21540 df-bases 21553 df-cld 21626 df-cn 21834 df-cnp 21835 df-cmp 21994 df-tx 22169 df-hmeo 22362 df-xms 22929 df-ms 22930 df-tms 22931 |
This theorem is referenced by: stoweidlem62 42346 |
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