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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 2737 | . . 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 2737 | . . 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 46075 | . 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 1914 | . . . . 5 ⊢ Ⅎ𝑡 𝑔 ∈ 𝐴 | |
| 21 | 2, 20 | nfan 1899 | . . . 4 ⊢ Ⅎ𝑡(𝜑 ∧ 𝑔 ∈ 𝐴) | 
| 22 | 17 | ad2antrr 726 | . . . . 5 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → 𝐸 ∈ ℝ+) | 
| 23 | 3, 4, 5, 15 | fcnre 45030 | . . . . . . 7 ⊢ (𝜑 → 𝐹:𝑇⟶ℝ) | 
| 24 | 23 | ffvelcdmda 7104 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐹‘𝑡) ∈ ℝ) | 
| 25 | 24 | adantlr 715 | . . . . 5 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → (𝐹‘𝑡) ∈ ℝ) | 
| 26 | 10 | sselda 3983 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐴) → 𝑔 ∈ 𝐶) | 
| 27 | 3, 4, 5, 26 | fcnre 45030 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐴) → 𝑔:𝑇⟶ℝ) | 
| 28 | 27 | ffvelcdmda 7104 | . . . . 5 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → (𝑔‘𝑡) ∈ ℝ) | 
| 29 | simpll1 1213 | . . . . . . 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 1214 | . . . . . . 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 1215 | . . . . . . 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 769 | . . . . . . 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 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)))) → ((𝑗 − (4 / 3)) · 𝐸) < (𝐹‘𝑡)) | |
| 34 | simprlr 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)))) → (𝐹‘𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) | |
| 35 | simprrr 782 | . . . . . . 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 781 | . . . . . . 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 46028 | . . . . . 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 3157 | . . . . 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 1373 | . . . 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 3260 | . . 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 3168 | . 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 395 ∧ w3a 1087 = wceq 1540 Ⅎwnf 1783 ∈ wcel 2108 Ⅎwnfc 2890 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 {crab 3436 ⊆ wss 3951 ∅c0 4333 ∪ cuni 4907 class class class wbr 5143 ↦ cmpt 5225 ran crn 5686 ‘cfv 6561 (class class class)co 7431 ℝcr 11154 0cc0 11155 1c1 11156 + caddc 11158 · cmul 11160 < clt 11295 ≤ cle 11296 − cmin 11492 / cdiv 11920 2c2 12321 3c3 12322 4c4 12323 ℝ+crp 13034 (,)cioo 13387 ...cfz 13547 abscabs 15273 topGenctg 17482 Cn ccn 23232 Compccmp 23394 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-pm 8869 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-fi 9451 df-sup 9482 df-inf 9483 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ioo 13391 df-ioc 13392 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-fl 13832 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-clim 15524 df-rlim 15525 df-sum 15723 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17467 df-topn 17468 df-0g 17486 df-gsum 17487 df-topgen 17488 df-pt 17489 df-prds 17492 df-xrs 17547 df-qtop 17552 df-imas 17553 df-xps 17555 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-mulg 19086 df-cntz 19335 df-cmn 19800 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-cnfld 21365 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-cld 23027 df-cn 23235 df-cnp 23236 df-cmp 23395 df-tx 23570 df-hmeo 23763 df-xms 24330 df-ms 24331 df-tms 24332 | 
| This theorem is referenced by: stoweidlem62 46077 | 
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