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Mirrors > Home > MPE Home > Th. List > Mathboxes > stoweidlem9 | Structured version Visualization version GIF version |
Description: Lemma for stoweid 43130: 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 5115 | . . . . 5 ⊢ (𝑇 = ∅ → (𝑡 ∈ 𝑇 ↦ 1) = (𝑡 ∈ ∅ ↦ 1)) | |
3 | mpt0 6473 | . . . . 5 ⊢ (𝑡 ∈ ∅ ↦ 1) = ∅ | |
4 | 2, 3 | eqtrdi 2789 | . . . 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 2834 | . 2 ⊢ (𝜑 → ∅ ∈ 𝐴) |
8 | rzal 4392 | . . 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 6667 | . . . . . 6 ⊢ (𝑔 = ∅ → (𝑔‘𝑡) = (∅‘𝑡)) | |
11 | 10 | fvoveq1d 7186 | . . . . 5 ⊢ (𝑔 = ∅ → (abs‘((𝑔‘𝑡) − (𝐹‘𝑡))) = (abs‘((∅‘𝑡) − (𝐹‘𝑡)))) |
12 | 11 | breq1d 5037 | . . . 4 ⊢ (𝑔 = ∅ → ((abs‘((𝑔‘𝑡) − (𝐹‘𝑡))) < 𝐸 ↔ (abs‘((∅‘𝑡) − (𝐹‘𝑡))) < 𝐸)) |
13 | 12 | ralbidv 3109 | . . 3 ⊢ (𝑔 = ∅ → (∀𝑡 ∈ 𝑇 (abs‘((𝑔‘𝑡) − (𝐹‘𝑡))) < 𝐸 ↔ ∀𝑡 ∈ 𝑇 (abs‘((∅‘𝑡) − (𝐹‘𝑡))) < 𝐸)) |
14 | 13 | rspcev 3524 | . 2 ⊢ ((∅ ∈ 𝐴 ∧ ∀𝑡 ∈ 𝑇 (abs‘((∅‘𝑡) − (𝐹‘𝑡))) < 𝐸) → ∃𝑔 ∈ 𝐴 ∀𝑡 ∈ 𝑇 (abs‘((𝑔‘𝑡) − (𝐹‘𝑡))) < 𝐸) |
15 | 7, 9, 14 | syl2anc 587 | 1 ⊢ (𝜑 → ∃𝑔 ∈ 𝐴 ∀𝑡 ∈ 𝑇 (abs‘((𝑔‘𝑡) − (𝐹‘𝑡))) < 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2113 ∀wral 3053 ∃wrex 3054 ∅c0 4209 class class class wbr 5027 ↦ cmpt 5107 ‘cfv 6333 (class class class)co 7164 1c1 10609 < clt 10746 − cmin 10941 abscabs 14676 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-sep 5164 ax-nul 5171 ax-pr 5293 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ral 3058 df-rex 3059 df-v 3399 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-nul 4210 df-if 4412 df-sn 4514 df-pr 4516 df-op 4520 df-uni 4794 df-br 5028 df-opab 5090 df-mpt 5108 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6291 df-fun 6335 df-fn 6336 df-fv 6341 df-ov 7167 |
This theorem is referenced by: stoweid 43130 |
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