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Theorem fldextfld1 31102
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/FldExt𝐹𝐸 ∈ Field)

Proof of Theorem fldextfld1
Dummy variables 𝑒 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opabssxp 5631 . . 3 {⟨𝑒, 𝑓⟩ ∣ ((𝑒 ∈ Field ∧ 𝑓 ∈ Field) ∧ (𝑓 = (𝑒s (Base‘𝑓)) ∧ (Base‘𝑓) ∈ (SubRing‘𝑒)))} ⊆ (Field × Field)
2 df-br 5054 . . . . 5 (𝐸/FldExt𝐹 ↔ ⟨𝐸, 𝐹⟩ ∈ /FldExt)
32biimpi 219 . . . 4 (𝐸/FldExt𝐹 → ⟨𝐸, 𝐹⟩ ∈ /FldExt)
4 df-fldext 31095 . . . 4 /FldExt = {⟨𝑒, 𝑓⟩ ∣ ((𝑒 ∈ Field ∧ 𝑓 ∈ Field) ∧ (𝑓 = (𝑒s (Base‘𝑓)) ∧ (Base‘𝑓) ∈ (SubRing‘𝑒)))}
53, 4eleqtrdi 2926 . . 3 (𝐸/FldExt𝐹 → ⟨𝐸, 𝐹⟩ ∈ {⟨𝑒, 𝑓⟩ ∣ ((𝑒 ∈ Field ∧ 𝑓 ∈ Field) ∧ (𝑓 = (𝑒s (Base‘𝑓)) ∧ (Base‘𝑓) ∈ (SubRing‘𝑒)))})
61, 5sseldi 3952 . 2 (𝐸/FldExt𝐹 → ⟨𝐸, 𝐹⟩ ∈ (Field × Field))
7 opelxp1 5584 . 2 (⟨𝐸, 𝐹⟩ ∈ (Field × Field) → 𝐸 ∈ Field)
86, 7syl 17 1 (𝐸/FldExt𝐹𝐸 ∈ Field)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  cop 4557   class class class wbr 5053  {copab 5115   × cxp 5541  cfv 6344  (class class class)co 7150  Basecbs 16486  s cress 16487  Fieldcfield 19506  SubRingcsubrg 19534  /FldExtcfldext 31091
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 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pr 5318
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-v 3483  df-dif 3923  df-un 3925  df-in 3927  df-ss 3937  df-nul 4278  df-if 4452  df-sn 4552  df-pr 4554  df-op 4558  df-br 5054  df-opab 5116  df-xp 5549  df-fldext 31095
This theorem is referenced by:  fldextsubrg  31104  fldextress  31105  brfinext  31106  fldextsralvec  31108  extdgcl  31109  extdggt0  31110  fldexttr  31111  extdgmul  31114  extdg1id  31116
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