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Theorem brfldext 33976
Description: The field extension relation explicited. (Contributed by Thierry Arnoux, 29-Jul-2023.)
Assertion
Ref Expression
brfldext ((𝐸 ∈ Field ∧ 𝐹 ∈ Field) → (𝐸/FldExt𝐹 ↔ (𝐹 = (𝐸s (Base‘𝐹)) ∧ (Base‘𝐹) ∈ (SubRing‘𝐸))))

Proof of Theorem brfldext
Dummy variables 𝑒 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 487 . . . . . 6 ((𝑒 = 𝐸𝑓 = 𝐹) → 𝑒 = 𝐸)
21eleq1d 2854 . . . . 5 ((𝑒 = 𝐸𝑓 = 𝐹) → (𝑒 ∈ Field ↔ 𝐸 ∈ Field))
3 simpr 489 . . . . . 6 ((𝑒 = 𝐸𝑓 = 𝐹) → 𝑓 = 𝐹)
43eleq1d 2854 . . . . 5 ((𝑒 = 𝐸𝑓 = 𝐹) → (𝑓 ∈ Field ↔ 𝐹 ∈ Field))
52, 4anbi12d 643 . . . 4 ((𝑒 = 𝐸𝑓 = 𝐹) → ((𝑒 ∈ Field ∧ 𝑓 ∈ Field) ↔ (𝐸 ∈ Field ∧ 𝐹 ∈ Field)))
63fveq2d 6883 . . . . . . 7 ((𝑒 = 𝐸𝑓 = 𝐹) → (Base‘𝑓) = (Base‘𝐹))
71, 6oveq12d 7426 . . . . . 6 ((𝑒 = 𝐸𝑓 = 𝐹) → (𝑒s (Base‘𝑓)) = (𝐸s (Base‘𝐹)))
83, 7eqeq12d 2785 . . . . 5 ((𝑒 = 𝐸𝑓 = 𝐹) → (𝑓 = (𝑒s (Base‘𝑓)) ↔ 𝐹 = (𝐸s (Base‘𝐹))))
91fveq2d 6883 . . . . . 6 ((𝑒 = 𝐸𝑓 = 𝐹) → (SubRing‘𝑒) = (SubRing‘𝐸))
106, 9eleq12d 2863 . . . . 5 ((𝑒 = 𝐸𝑓 = 𝐹) → ((Base‘𝑓) ∈ (SubRing‘𝑒) ↔ (Base‘𝐹) ∈ (SubRing‘𝐸)))
118, 10anbi12d 643 . . . 4 ((𝑒 = 𝐸𝑓 = 𝐹) → ((𝑓 = (𝑒s (Base‘𝑓)) ∧ (Base‘𝑓) ∈ (SubRing‘𝑒)) ↔ (𝐹 = (𝐸s (Base‘𝐹)) ∧ (Base‘𝐹) ∈ (SubRing‘𝐸))))
125, 11anbi12d 643 . . 3 ((𝑒 = 𝐸𝑓 = 𝐹) → (((𝑒 ∈ Field ∧ 𝑓 ∈ Field) ∧ (𝑓 = (𝑒s (Base‘𝑓)) ∧ (Base‘𝑓) ∈ (SubRing‘𝑒))) ↔ ((𝐸 ∈ Field ∧ 𝐹 ∈ Field) ∧ (𝐹 = (𝐸s (Base‘𝐹)) ∧ (Base‘𝐹) ∈ (SubRing‘𝐸)))))
13 df-fldext 33972 . . 3 /FldExt = {⟨𝑒, 𝑓⟩ ∣ ((𝑒 ∈ Field ∧ 𝑓 ∈ Field) ∧ (𝑓 = (𝑒s (Base‘𝑓)) ∧ (Base‘𝑓) ∈ (SubRing‘𝑒)))}
1412, 13brabga 5516 . 2 ((𝐸 ∈ Field ∧ 𝐹 ∈ Field) → (𝐸/FldExt𝐹 ↔ ((𝐸 ∈ Field ∧ 𝐹 ∈ Field) ∧ (𝐹 = (𝐸s (Base‘𝐹)) ∧ (Base‘𝐹) ∈ (SubRing‘𝐸)))))
1514bianabs 550 1 ((𝐸 ∈ Field ∧ 𝐹 ∈ Field) → (𝐸/FldExt𝐹 ↔ (𝐹 = (𝐸s (Base‘𝐹)) ∧ (Base‘𝐹) ∈ (SubRing‘𝐸))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 400   = wceq 1567  wcel 2149   class class class wbr 5110  cfv 6533  (class class class)co 7408  Basecbs 17265  s cress 17286  SubRingcsubrg 20650  Fieldcfield 20810  /FldExtcfldext 33969
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5258  ax-pr 5402
This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5111  df-opab 5175  df-iota 6489  df-fv 6541  df-ov 7411  df-fldext 33972
This theorem is referenced by:  ccfldextrr  33977  fldextsubrg  33980  sdrgfldext  33981  fldextress  33982  fldexttr  33989  fldextid  33990  fldsdrgfldext  33992  extdgmul  33994  fldgenfldext  33999  fldextrspunlem1  34006  algextdeglem4  34051
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