Theorem fldextfld2 34047

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

Assertion

Ref	Expression
fldextfld2	⊢ (𝐸/FldExt𝐹 → 𝐹 ∈ Field)

Proof of Theorem fldextfld2

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

Step	Hyp	Ref	Expression
1		opabssxp 5752	. . 3 ⊢ {⟨𝑒, 𝑓⟩ ∣ ((𝑒 ∈ Field ∧ 𝑓 ∈ Field) ∧ (𝑓 = (𝑒 ↾s (Base‘𝑓)) ∧ (Base‘𝑓) ∈ (SubRing‘𝑒)))} ⊆ (Field × Field)
2		df-br 5109	. . . . 5 ⊢ (𝐸/FldExt𝐹 ↔ ⟨𝐸, 𝐹⟩ ∈ /FldExt)
3	2	biimpi 219	. . . 4 ⊢ (𝐸/FldExt𝐹 → ⟨𝐸, 𝐹⟩ ∈ /FldExt)
4		df-fldext 34040	. . . 4 ⊢ /FldExt = {⟨𝑒, 𝑓⟩ ∣ ((𝑒 ∈ Field ∧ 𝑓 ∈ Field) ∧ (𝑓 = (𝑒 ↾s (Base‘𝑓)) ∧ (Base‘𝑓) ∈ (SubRing‘𝑒)))}
5	3, 4	eleqtrdi 2872	. . 3 ⊢ (𝐸/FldExt𝐹 → ⟨𝐸, 𝐹⟩ ∈ {⟨𝑒, 𝑓⟩ ∣ ((𝑒 ∈ Field ∧ 𝑓 ∈ Field) ∧ (𝑓 = (𝑒 ↾s (Base‘𝑓)) ∧ (Base‘𝑓) ∈ (SubRing‘𝑒)))})
6	1, 5	sselid 3934	. 2 ⊢ (𝐸/FldExt𝐹 → ⟨𝐸, 𝐹⟩ ∈ (Field × Field))
7		opelxp2 5703	. 2 ⊢ (⟨𝐸, 𝐹⟩ ∈ (Field × Field) → 𝐹 ∈ Field)
8	6, 7	syl 18	1 ⊢ (𝐸/FldExt𝐹 → 𝐹 ∈ Field)

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   = wceq 1569   ∈ wcel 2142  ⟨cop 4594   class class class wbr 5108  {copab 5172   × cxp 5658  ‘cfv 6536  (class class class)co 7412  Basecbs 17275   ↾_s cress 17296  SubRingcsubrg 20679  Fieldcfield 20839  /_FldExtcfldext 34037

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5256  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-br 5109  df-opab 5173  df-xp 5666  df-fldext 34040

This theorem is used by:  fldextsubrg  34048  fldextress  34050  brfinext  34051  fldextsdrg  34053  fldextsralvec  34054  extdgcl  34055  extdggt0  34056  fldexttr  34057  extdgmul  34062  extdg1id  34065  extdg1b  34066  finextalg  34097