Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fldextfld1 Structured version   Visualization version   GIF version

Theorem fldextfld1 32395
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 β†’ 𝐸 ∈ Field)

Proof of Theorem fldextfld1
Dummy variables 𝑒 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opabssxp 5725 . . 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 5107 . . . . 5 (𝐸/FldExt𝐹 ↔ ⟨𝐸, 𝐹⟩ ∈ /FldExt)
32biimpi 215 . . . 4 (𝐸/FldExt𝐹 β†’ ⟨𝐸, 𝐹⟩ ∈ /FldExt)
4 df-fldext 32388 . . . 4 /FldExt = {βŸ¨π‘’, π‘“βŸ© ∣ ((𝑒 ∈ Field ∧ 𝑓 ∈ Field) ∧ (𝑓 = (𝑒 β†Ύs (Baseβ€˜π‘“)) ∧ (Baseβ€˜π‘“) ∈ (SubRingβ€˜π‘’)))}
53, 4eleqtrdi 2844 . . 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 3943 . 2 (𝐸/FldExt𝐹 β†’ ⟨𝐸, 𝐹⟩ ∈ (Field Γ— Field))
7 opelxp1 5675 . 2 (⟨𝐸, 𝐹⟩ ∈ (Field Γ— Field) β†’ 𝐸 ∈ Field)
86, 7syl 17 1 (𝐸/FldExt𝐹 β†’ 𝐸 ∈ Field)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βŸ¨cop 4593   class class class wbr 5106  {copab 5168   Γ— cxp 5632  β€˜cfv 6497  (class class class)co 7358  Basecbs 17088   β†Ύs cress 17117  Fieldcfield 20198  SubRingcsubrg 20232  /FldExtcfldext 32384
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-xp 5640  df-fldext 32388
This theorem is referenced by:  fldextsubrg  32397  fldextress  32398  brfinext  32399  fldextsralvec  32401  extdgcl  32402  extdggt0  32403  fldexttr  32404  extdgmul  32407  extdg1id  32409
  Copyright terms: Public domain W3C validator