Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > stoweidlem9 | Structured version Visualization version GIF version |
Description: Lemma for stoweid 38956: 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 4665 | . . . . 5 ⊢ (𝑇 = ∅ → (𝑡 ∈ 𝑇 ↦ 1) = (𝑡 ∈ ∅ ↦ 1)) | |
3 | mpt0 5934 | . . . . 5 ⊢ (𝑡 ∈ ∅ ↦ 1) = ∅ | |
4 | 2, 3 | syl6eq 2660 | . . . 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 2689 | . 2 ⊢ (𝜑 → ∅ ∈ 𝐴) |
8 | rzal 4025 | . . 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 6102 | . . . . . . 7 ⊢ (𝑔 = ∅ → (𝑔‘𝑡) = (∅‘𝑡)) | |
11 | 10 | oveq1d 6564 | . . . . . 6 ⊢ (𝑔 = ∅ → ((𝑔‘𝑡) − (𝐹‘𝑡)) = ((∅‘𝑡) − (𝐹‘𝑡))) |
12 | 11 | fveq2d 6107 | . . . . 5 ⊢ (𝑔 = ∅ → (abs‘((𝑔‘𝑡) − (𝐹‘𝑡))) = (abs‘((∅‘𝑡) − (𝐹‘𝑡)))) |
13 | 12 | breq1d 4593 | . . . 4 ⊢ (𝑔 = ∅ → ((abs‘((𝑔‘𝑡) − (𝐹‘𝑡))) < 𝐸 ↔ (abs‘((∅‘𝑡) − (𝐹‘𝑡))) < 𝐸)) |
14 | 13 | ralbidv 2969 | . . 3 ⊢ (𝑔 = ∅ → (∀𝑡 ∈ 𝑇 (abs‘((𝑔‘𝑡) − (𝐹‘𝑡))) < 𝐸 ↔ ∀𝑡 ∈ 𝑇 (abs‘((∅‘𝑡) − (𝐹‘𝑡))) < 𝐸)) |
15 | 14 | rspcev 3282 | . 2 ⊢ ((∅ ∈ 𝐴 ∧ ∀𝑡 ∈ 𝑇 (abs‘((∅‘𝑡) − (𝐹‘𝑡))) < 𝐸) → ∃𝑔 ∈ 𝐴 ∀𝑡 ∈ 𝑇 (abs‘((𝑔‘𝑡) − (𝐹‘𝑡))) < 𝐸) |
16 | 7, 9, 15 | syl2anc 691 | 1 ⊢ (𝜑 → ∃𝑔 ∈ 𝐴 ∀𝑡 ∈ 𝑇 (abs‘((𝑔‘𝑡) − (𝐹‘𝑡))) < 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∃wrex 2897 ∅c0 3874 class class class wbr 4583 ↦ cmpt 4643 ‘cfv 5804 (class class class)co 6549 1c1 9816 < clt 9953 − cmin 10145 abscabs 13822 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fn 5807 df-fv 5812 df-ov 6552 |
This theorem is referenced by: stoweid 38956 |
Copyright terms: Public domain | W3C validator |