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Theorem fldextfld2 33399
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 β†’ 𝐹 ∈ Field)

Proof of Theorem fldextfld2
Dummy variables 𝑒 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opabssxp 5764 . . 3 {βŸ¨π‘’, π‘“βŸ© ∣ ((𝑒 ∈ Field ∧ 𝑓 ∈ Field) ∧ (𝑓 = (𝑒 β†Ύs (Baseβ€˜π‘“)) ∧ (Baseβ€˜π‘“) ∈ (SubRingβ€˜π‘’)))} βŠ† (Field Γ— Field)
2 df-br 5144 . . . . 5 (𝐸/FldExt𝐹 ↔ ⟨𝐸, 𝐹⟩ ∈ /FldExt)
32biimpi 215 . . . 4 (𝐸/FldExt𝐹 β†’ ⟨𝐸, 𝐹⟩ ∈ /FldExt)
4 df-fldext 33391 . . . 4 /FldExt = {βŸ¨π‘’, π‘“βŸ© ∣ ((𝑒 ∈ Field ∧ 𝑓 ∈ Field) ∧ (𝑓 = (𝑒 β†Ύs (Baseβ€˜π‘“)) ∧ (Baseβ€˜π‘“) ∈ (SubRingβ€˜π‘’)))}
53, 4eleqtrdi 2835 . . 3 (𝐸/FldExt𝐹 β†’ ⟨𝐸, 𝐹⟩ ∈ {βŸ¨π‘’, π‘“βŸ© ∣ ((𝑒 ∈ Field ∧ 𝑓 ∈ Field) ∧ (𝑓 = (𝑒 β†Ύs (Baseβ€˜π‘“)) ∧ (Baseβ€˜π‘“) ∈ (SubRingβ€˜π‘’)))})
61, 5sselid 3970 . 2 (𝐸/FldExt𝐹 β†’ ⟨𝐸, 𝐹⟩ ∈ (Field Γ— Field))
7 opelxp2 5715 . 2 (⟨𝐸, 𝐹⟩ ∈ (Field Γ— Field) β†’ 𝐹 ∈ Field)
86, 7syl 17 1 (𝐸/FldExt𝐹 β†’ 𝐹 ∈ Field)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  βŸ¨cop 4630   class class class wbr 5143  {copab 5205   Γ— cxp 5670  β€˜cfv 6543  (class class class)co 7416  Basecbs 17179   β†Ύs cress 17208  SubRingcsubrg 20510  Fieldcfield 20629  /FldExtcfldext 33387
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5144  df-opab 5206  df-xp 5678  df-fldext 33391
This theorem is referenced by:  fldextsubrg  33400  fldextress  33401  brfinext  33402  fldextsralvec  33404  extdgcl  33405  extdggt0  33406  fldexttr  33407  extdgmul  33410  extdg1id  33412  extdg1b  33413
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