Theorem stoweidlem9 42301

Description: Lemma for stoweid 42355: 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 5156	. . . . 5 ⊢ (𝑇 = ∅ → (𝑡 ∈ 𝑇 ↦ 1) = (𝑡 ∈ ∅ ↦ 1))
3		mpt0 6492	. . . . 5 ⊢ (𝑡 ∈ ∅ ↦ 1) = ∅
4	2, 3	syl6eq 2874	. . . 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 2916	. 2 ⊢ (𝜑 → ∅ ∈ 𝐴)
8		rzal 4455	. . 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 6671	. . . . . 6 ⊢ (𝑔 = ∅ → (𝑔‘𝑡) = (∅‘𝑡))
11	10	fvoveq1d 7180	. . . . 5 ⊢ (𝑔 = ∅ → (abs‘((𝑔‘𝑡) − (𝐹‘𝑡))) = (abs‘((∅‘𝑡) − (𝐹‘𝑡))))
12	11	breq1d 5078	. . . 4 ⊢ (𝑔 = ∅ → ((abs‘((𝑔‘𝑡) − (𝐹‘𝑡))) < 𝐸 ↔ (abs‘((∅‘𝑡) − (𝐹‘𝑡))) < 𝐸))
13	12	ralbidv 3199	. . 3 ⊢ (𝑔 = ∅ → (∀𝑡 ∈ 𝑇 (abs‘((𝑔‘𝑡) − (𝐹‘𝑡))) < 𝐸 ↔ ∀𝑡 ∈ 𝑇 (abs‘((∅‘𝑡) − (𝐹‘𝑡))) < 𝐸))
14	13	rspcev 3625	. 2 ⊢ ((∅ ∈ 𝐴 ∧ ∀𝑡 ∈ 𝑇 (abs‘((∅‘𝑡) − (𝐹‘𝑡))) < 𝐸) → ∃𝑔 ∈ 𝐴 ∀𝑡 ∈ 𝑇 (abs‘((𝑔‘𝑡) − (𝐹‘𝑡))) < 𝐸)
15	7, 9, 14	syl2anc 586	1 ⊢ (𝜑 → ∃𝑔 ∈ 𝐴 ∀𝑡 ∈ 𝑇 (abs‘((𝑔‘𝑡) − (𝐹‘𝑡))) < 𝐸)

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2114  ∀wral 3140  ∃wrex 3141  ∅c0 4293   class class class wbr 5068   ↦ cmpt 5148  ‘cfv 6357  (class class class)co 7158  1c1 10540   < clt 10677   − cmin 10872  abscabs 14595

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-iota 6316  df-fun 6359  df-fn 6360  df-fv 6365  df-ov 7161

This theorem is referenced by:  stoweid  42355