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Theorem brfldext 33698
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 482 . . . . . 6 ((𝑒 = 𝐸𝑓 = 𝐹) → 𝑒 = 𝐸)
21eleq1d 2826 . . . . 5 ((𝑒 = 𝐸𝑓 = 𝐹) → (𝑒 ∈ Field ↔ 𝐸 ∈ Field))
3 simpr 484 . . . . . 6 ((𝑒 = 𝐸𝑓 = 𝐹) → 𝑓 = 𝐹)
43eleq1d 2826 . . . . 5 ((𝑒 = 𝐸𝑓 = 𝐹) → (𝑓 ∈ Field ↔ 𝐹 ∈ Field))
52, 4anbi12d 632 . . . 4 ((𝑒 = 𝐸𝑓 = 𝐹) → ((𝑒 ∈ Field ∧ 𝑓 ∈ Field) ↔ (𝐸 ∈ Field ∧ 𝐹 ∈ Field)))
63fveq2d 6910 . . . . . . 7 ((𝑒 = 𝐸𝑓 = 𝐹) → (Base‘𝑓) = (Base‘𝐹))
71, 6oveq12d 7449 . . . . . 6 ((𝑒 = 𝐸𝑓 = 𝐹) → (𝑒s (Base‘𝑓)) = (𝐸s (Base‘𝐹)))
83, 7eqeq12d 2753 . . . . 5 ((𝑒 = 𝐸𝑓 = 𝐹) → (𝑓 = (𝑒s (Base‘𝑓)) ↔ 𝐹 = (𝐸s (Base‘𝐹))))
91fveq2d 6910 . . . . . 6 ((𝑒 = 𝐸𝑓 = 𝐹) → (SubRing‘𝑒) = (SubRing‘𝐸))
106, 9eleq12d 2835 . . . . 5 ((𝑒 = 𝐸𝑓 = 𝐹) → ((Base‘𝑓) ∈ (SubRing‘𝑒) ↔ (Base‘𝐹) ∈ (SubRing‘𝐸)))
118, 10anbi12d 632 . . . 4 ((𝑒 = 𝐸𝑓 = 𝐹) → ((𝑓 = (𝑒s (Base‘𝑓)) ∧ (Base‘𝑓) ∈ (SubRing‘𝑒)) ↔ (𝐹 = (𝐸s (Base‘𝐹)) ∧ (Base‘𝐹) ∈ (SubRing‘𝐸))))
125, 11anbi12d 632 . . 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 33693 . . 3 /FldExt = {⟨𝑒, 𝑓⟩ ∣ ((𝑒 ∈ Field ∧ 𝑓 ∈ Field) ∧ (𝑓 = (𝑒s (Base‘𝑓)) ∧ (Base‘𝑓) ∈ (SubRing‘𝑒)))}
1412, 13brabga 5539 . 2 ((𝐸 ∈ Field ∧ 𝐹 ∈ Field) → (𝐸/FldExt𝐹 ↔ ((𝐸 ∈ Field ∧ 𝐹 ∈ Field) ∧ (𝐹 = (𝐸s (Base‘𝐹)) ∧ (Base‘𝐹) ∈ (SubRing‘𝐸)))))
1514bianabs 541 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 206  wa 395   = wceq 1540  wcel 2108   class class class wbr 5143  cfv 6561  (class class class)co 7431  Basecbs 17247  s cress 17274  SubRingcsubrg 20569  Fieldcfield 20730  /FldExtcfldext 33689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-iota 6514  df-fv 6569  df-ov 7434  df-fldext 33693
This theorem is referenced by:  ccfldextrr  33699  fldextsubrg  33702  fldextress  33703  fldexttr  33709  fldextid  33710  fldsdrgfldext  33712  extdgmul  33714  fldgenfldext  33718  fldextrspunlem1  33725  algextdeglem4  33761
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