Theorem stoweidlem9 46458

Description: Lemma for stoweid 46512: here the Stone Weierstrass theorem is proven for the trivial case, T is the empty set. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Hypotheses

Ref	Expression
stoweidlem9.1	⊢ (𝜑 → 𝑇 = ∅)
stoweidlem9.2	⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ 1) ∈ 𝐴)

Assertion

Ref	Expression
stoweidlem9	⊢ (𝜑 → ∃𝑔 ∈ 𝐴 ∀𝑡 ∈ 𝑇 (abs‘((𝑔‘𝑡) − (𝐹‘𝑡))) < 𝐸)

Distinct variable groups:   𝐴,𝑔   𝑔,𝐸   𝑔,𝐹   𝑡,𝑔,𝑇

Allowed substitution hints:   𝜑(𝑡,𝑔)   𝐴(𝑡)   𝐸(𝑡)   𝐹(𝑡)

Proof of Theorem stoweidlem9

Step	Hyp	Ref	Expression
1		stoweidlem9.1	. . . 4 ⊢ (𝜑 → 𝑇 = ∅)
2		mpteq1 5175	. . . . 5 ⊢ (𝑇 = ∅ → (𝑡 ∈ 𝑇 ↦ 1) = (𝑡 ∈ ∅ ↦ 1))
3		mpt0 6635	. . . . 5 ⊢ (𝑡 ∈ ∅ ↦ 1) = ∅
4	2, 3	eqtrdi 2788	. . . 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 2838	. 2 ⊢ (𝜑 → ∅ ∈ 𝐴)
8		rzal 4435	. . 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 6834	. . . . . 6 ⊢ (𝑔 = ∅ → (𝑔‘𝑡) = (∅‘𝑡))
11	10	fvoveq1d 7383	. . . . 5 ⊢ (𝑔 = ∅ → (abs‘((𝑔‘𝑡) − (𝐹‘𝑡))) = (abs‘((∅‘𝑡) − (𝐹‘𝑡))))
12	11	breq1d 5096	. . . 4 ⊢ (𝑔 = ∅ → ((abs‘((𝑔‘𝑡) − (𝐹‘𝑡))) < 𝐸 ↔ (abs‘((∅‘𝑡) − (𝐹‘𝑡))) < 𝐸))
13	12	ralbidv 3161	. . 3 ⊢ (𝑔 = ∅ → (∀𝑡 ∈ 𝑇 (abs‘((𝑔‘𝑡) − (𝐹‘𝑡))) < 𝐸 ↔ ∀𝑡 ∈ 𝑇 (abs‘((∅‘𝑡) − (𝐹‘𝑡))) < 𝐸))
14	13	rspcev 3565	. 2 ⊢ ((∅ ∈ 𝐴 ∧ ∀𝑡 ∈ 𝑇 (abs‘((∅‘𝑡) − (𝐹‘𝑡))) < 𝐸) → ∃𝑔 ∈ 𝐴 ∀𝑡 ∈ 𝑇 (abs‘((𝑔‘𝑡) − (𝐹‘𝑡))) < 𝐸)
15	7, 9, 14	syl2anc 585	1 ⊢ (𝜑 → ∃𝑔 ∈ 𝐴 ∀𝑡 ∈ 𝑇 (abs‘((𝑔‘𝑡) − (𝐹‘𝑡))) < 𝐸)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062  ∅c0 4274   class class class wbr 5086   ↦ cmpt 5167  ‘cfv 6493  (class class class)co 7361  1c1 11033   < clt 11173   − cmin 11371  abscabs 15190

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-iota 6449  df-fun 6495  df-fn 6496  df-fv 6501  df-ov 7364

This theorem is referenced by:  stoweid  46512