Theorem fldextfld1 31724

Description: A field extension is only defined if the extension is a field. (Contributed by Thierry Arnoux, 29-Jul-2023.)

Assertion

Ref	Expression
fldextfld1	⊢ (𝐸/FldExt𝐹 → 𝐸 ∈ Field)

Proof of Theorem fldextfld1

Dummy variables 𝑒 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		opabssxp 5679	. . 3 ⊢ {⟨𝑒, 𝑓⟩ ∣ ((𝑒 ∈ Field ∧ 𝑓 ∈ Field) ∧ (𝑓 = (𝑒 ↾s (Base‘𝑓)) ∧ (Base‘𝑓) ∈ (SubRing‘𝑒)))} ⊆ (Field × Field)
2		df-br 5075	. . . . 5 ⊢ (𝐸/FldExt𝐹 ↔ ⟨𝐸, 𝐹⟩ ∈ /FldExt)
3	2	biimpi 215	. . . 4 ⊢ (𝐸/FldExt𝐹 → ⟨𝐸, 𝐹⟩ ∈ /FldExt)
4		df-fldext 31717	. . . 4 ⊢ /FldExt = {⟨𝑒, 𝑓⟩ ∣ ((𝑒 ∈ Field ∧ 𝑓 ∈ Field) ∧ (𝑓 = (𝑒 ↾s (Base‘𝑓)) ∧ (Base‘𝑓) ∈ (SubRing‘𝑒)))}
5	3, 4	eleqtrdi 2849	. . 3 ⊢ (𝐸/FldExt𝐹 → ⟨𝐸, 𝐹⟩ ∈ {⟨𝑒, 𝑓⟩ ∣ ((𝑒 ∈ Field ∧ 𝑓 ∈ Field) ∧ (𝑓 = (𝑒 ↾s (Base‘𝑓)) ∧ (Base‘𝑓) ∈ (SubRing‘𝑒)))})
6	1, 5	sselid 3919	. 2 ⊢ (𝐸/FldExt𝐹 → ⟨𝐸, 𝐹⟩ ∈ (Field × Field))
7		opelxp1 5630	. 2 ⊢ (⟨𝐸, 𝐹⟩ ∈ (Field × Field) → 𝐸 ∈ Field)
8	6, 7	syl 17	1 ⊢ (𝐸/FldExt𝐹 → 𝐸 ∈ Field)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ⟨cop 4567   class class class wbr 5074  {copab 5136   × cxp 5587  ‘cfv 6433  (class class class)co 7275  Basecbs 16912   ↾_s cress 16941  Fieldcfield 19992  SubRingcsubrg 20020  /_FldExtcfldext 31713

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-xp 5595  df-fldext 31717

This theorem is referenced by:  fldextsubrg  31726  fldextress  31727  brfinext  31728  fldextsralvec  31730  extdgcl  31731  extdggt0  31732  fldexttr  31733  extdgmul  31736  extdg1id  31738