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Mirrors > Home > MPE Home > Th. List > Mathboxes > stoweidlem9 | Structured version Visualization version GIF version |
Description: Lemma for stoweid 46051: here the Stone Weierstrass theorem is proven for the trivial case, T is the empty set. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
Ref | Expression |
---|---|
stoweidlem9.1 | ⊢ (𝜑 → 𝑇 = ∅) |
stoweidlem9.2 | ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ 1) ∈ 𝐴) |
Ref | Expression |
---|---|
stoweidlem9 | ⊢ (𝜑 → ∃𝑔 ∈ 𝐴 ∀𝑡 ∈ 𝑇 (abs‘((𝑔‘𝑡) − (𝐹‘𝑡))) < 𝐸) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | stoweidlem9.1 | . . . 4 ⊢ (𝜑 → 𝑇 = ∅) | |
2 | mpteq1 5233 | . . . . 5 ⊢ (𝑇 = ∅ → (𝑡 ∈ 𝑇 ↦ 1) = (𝑡 ∈ ∅ ↦ 1)) | |
3 | mpt0 6708 | . . . . 5 ⊢ (𝑡 ∈ ∅ ↦ 1) = ∅ | |
4 | 2, 3 | eqtrdi 2792 | . . . 4 ⊢ (𝑇 = ∅ → (𝑡 ∈ 𝑇 ↦ 1) = ∅) |
5 | 1, 4 | syl 17 | . . 3 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ 1) = ∅) |
6 | stoweidlem9.2 | . . 3 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ 1) ∈ 𝐴) | |
7 | 5, 6 | eqeltrrd 2841 | . 2 ⊢ (𝜑 → ∅ ∈ 𝐴) |
8 | rzal 4508 | . . 3 ⊢ (𝑇 = ∅ → ∀𝑡 ∈ 𝑇 (abs‘((∅‘𝑡) − (𝐹‘𝑡))) < 𝐸) | |
9 | 1, 8 | syl 17 | . 2 ⊢ (𝜑 → ∀𝑡 ∈ 𝑇 (abs‘((∅‘𝑡) − (𝐹‘𝑡))) < 𝐸) |
10 | fveq1 6903 | . . . . . 6 ⊢ (𝑔 = ∅ → (𝑔‘𝑡) = (∅‘𝑡)) | |
11 | 10 | fvoveq1d 7451 | . . . . 5 ⊢ (𝑔 = ∅ → (abs‘((𝑔‘𝑡) − (𝐹‘𝑡))) = (abs‘((∅‘𝑡) − (𝐹‘𝑡)))) |
12 | 11 | breq1d 5151 | . . . 4 ⊢ (𝑔 = ∅ → ((abs‘((𝑔‘𝑡) − (𝐹‘𝑡))) < 𝐸 ↔ (abs‘((∅‘𝑡) − (𝐹‘𝑡))) < 𝐸)) |
13 | 12 | ralbidv 3177 | . . 3 ⊢ (𝑔 = ∅ → (∀𝑡 ∈ 𝑇 (abs‘((𝑔‘𝑡) − (𝐹‘𝑡))) < 𝐸 ↔ ∀𝑡 ∈ 𝑇 (abs‘((∅‘𝑡) − (𝐹‘𝑡))) < 𝐸)) |
14 | 13 | rspcev 3621 | . 2 ⊢ ((∅ ∈ 𝐴 ∧ ∀𝑡 ∈ 𝑇 (abs‘((∅‘𝑡) − (𝐹‘𝑡))) < 𝐸) → ∃𝑔 ∈ 𝐴 ∀𝑡 ∈ 𝑇 (abs‘((𝑔‘𝑡) − (𝐹‘𝑡))) < 𝐸) |
15 | 7, 9, 14 | syl2anc 584 | 1 ⊢ (𝜑 → ∃𝑔 ∈ 𝐴 ∀𝑡 ∈ 𝑇 (abs‘((𝑔‘𝑡) − (𝐹‘𝑡))) < 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ∀wral 3060 ∃wrex 3069 ∅c0 4332 class class class wbr 5141 ↦ cmpt 5223 ‘cfv 6559 (class class class)co 7429 1c1 11152 < clt 11291 − cmin 11488 abscabs 15269 |
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 2707 ax-sep 5294 ax-nul 5304 ax-pr 5430 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4906 df-br 5142 df-opab 5204 df-mpt 5224 df-id 5576 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-iota 6512 df-fun 6561 df-fn 6562 df-fv 6567 df-ov 7432 |
This theorem is referenced by: stoweid 46051 |
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