Theorem fldextfld2 32023

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 5710	. . 3 ⊢ {⟨𝑒, 𝑓⟩ ∣ ((𝑒 ∈ Field ∧ 𝑓 ∈ Field) ∧ (𝑓 = (𝑒 ↾s (Base‘𝑓)) ∧ (Base‘𝑓) ∈ (SubRing‘𝑒)))} ⊆ (Field × Field)
2		df-br 5093	. . . . 5 ⊢ (𝐸/FldExt𝐹 ↔ ⟨𝐸, 𝐹⟩ ∈ /FldExt)
3	2	biimpi 215	. . . 4 ⊢ (𝐸/FldExt𝐹 → ⟨𝐸, 𝐹⟩ ∈ /FldExt)
4		df-fldext 32015	. . . 4 ⊢ /FldExt = {⟨𝑒, 𝑓⟩ ∣ ((𝑒 ∈ Field ∧ 𝑓 ∈ Field) ∧ (𝑓 = (𝑒 ↾s (Base‘𝑓)) ∧ (Base‘𝑓) ∈ (SubRing‘𝑒)))}
5	3, 4	eleqtrdi 2847	. . 3 ⊢ (𝐸/FldExt𝐹 → ⟨𝐸, 𝐹⟩ ∈ {⟨𝑒, 𝑓⟩ ∣ ((𝑒 ∈ Field ∧ 𝑓 ∈ Field) ∧ (𝑓 = (𝑒 ↾s (Base‘𝑓)) ∧ (Base‘𝑓) ∈ (SubRing‘𝑒)))})
6	1, 5	sselid 3930	. 2 ⊢ (𝐸/FldExt𝐹 → ⟨𝐸, 𝐹⟩ ∈ (Field × Field))
7		opelxp2 5662	. 2 ⊢ (⟨𝐸, 𝐹⟩ ∈ (Field × Field) → 𝐹 ∈ Field)
8	6, 7	syl 17	1 ⊢ (𝐸/FldExt𝐹 → 𝐹 ∈ Field)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1540   ∈ wcel 2105  ⟨cop 4579   class class class wbr 5092  {copab 5154   × cxp 5618  ‘cfv 6479  (class class class)co 7337  Basecbs 17009   ↾_s cress 17038  Fieldcfield 20094  SubRingcsubrg 20125  /_FldExtcfldext 32011

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707  ax-sep 5243  ax-nul 5250  ax-pr 5372

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-sn 4574  df-pr 4576  df-op 4580  df-br 5093  df-opab 5155  df-xp 5626  df-fldext 32015

This theorem is referenced by:  fldextsubrg  32024  fldextress  32025  brfinext  32026  fldextsralvec  32028  extdgcl  32029  extdggt0  32030  fldexttr  32031  extdgmul  32034  extdg1id  32036  extdg1b  32037