![]() |
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 41075: 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 4961 | . . . . 5 ⊢ (𝑇 = ∅ → (𝑡 ∈ 𝑇 ↦ 1) = (𝑡 ∈ ∅ ↦ 1)) | |
3 | mpt0 6255 | . . . . 5 ⊢ (𝑡 ∈ ∅ ↦ 1) = ∅ | |
4 | 2, 3 | syl6eq 2878 | . . . 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 2908 | . 2 ⊢ (𝜑 → ∅ ∈ 𝐴) |
8 | rzal 4296 | . . 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 6433 | . . . . . . 7 ⊢ (𝑔 = ∅ → (𝑔‘𝑡) = (∅‘𝑡)) | |
11 | 10 | oveq1d 6921 | . . . . . 6 ⊢ (𝑔 = ∅ → ((𝑔‘𝑡) − (𝐹‘𝑡)) = ((∅‘𝑡) − (𝐹‘𝑡))) |
12 | 11 | fveq2d 6438 | . . . . 5 ⊢ (𝑔 = ∅ → (abs‘((𝑔‘𝑡) − (𝐹‘𝑡))) = (abs‘((∅‘𝑡) − (𝐹‘𝑡)))) |
13 | 12 | breq1d 4884 | . . . 4 ⊢ (𝑔 = ∅ → ((abs‘((𝑔‘𝑡) − (𝐹‘𝑡))) < 𝐸 ↔ (abs‘((∅‘𝑡) − (𝐹‘𝑡))) < 𝐸)) |
14 | 13 | ralbidv 3196 | . . 3 ⊢ (𝑔 = ∅ → (∀𝑡 ∈ 𝑇 (abs‘((𝑔‘𝑡) − (𝐹‘𝑡))) < 𝐸 ↔ ∀𝑡 ∈ 𝑇 (abs‘((∅‘𝑡) − (𝐹‘𝑡))) < 𝐸)) |
15 | 14 | rspcev 3527 | . 2 ⊢ ((∅ ∈ 𝐴 ∧ ∀𝑡 ∈ 𝑇 (abs‘((∅‘𝑡) − (𝐹‘𝑡))) < 𝐸) → ∃𝑔 ∈ 𝐴 ∀𝑡 ∈ 𝑇 (abs‘((𝑔‘𝑡) − (𝐹‘𝑡))) < 𝐸) |
16 | 7, 9, 15 | syl2anc 581 | 1 ⊢ (𝜑 → ∃𝑔 ∈ 𝐴 ∀𝑡 ∈ 𝑇 (abs‘((𝑔‘𝑡) − (𝐹‘𝑡))) < 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1658 ∈ wcel 2166 ∀wral 3118 ∃wrex 3119 ∅c0 4145 class class class wbr 4874 ↦ cmpt 4953 ‘cfv 6124 (class class class)co 6906 1c1 10254 < clt 10392 − cmin 10586 abscabs 14352 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-sep 5006 ax-nul 5014 ax-pr 5128 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-ral 3123 df-rex 3124 df-rab 3127 df-v 3417 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-nul 4146 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4660 df-br 4875 df-opab 4937 df-mpt 4954 df-id 5251 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-iota 6087 df-fun 6126 df-fn 6127 df-fv 6132 df-ov 6909 |
This theorem is referenced by: stoweid 41075 |
Copyright terms: Public domain | W3C validator |