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Theorem brfldext 31067
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 486 . . . . . 6 ((𝑒 = 𝐸𝑓 = 𝐹) → 𝑒 = 𝐸)
21eleq1d 2900 . . . . 5 ((𝑒 = 𝐸𝑓 = 𝐹) → (𝑒 ∈ Field ↔ 𝐸 ∈ Field))
3 simpr 488 . . . . . 6 ((𝑒 = 𝐸𝑓 = 𝐹) → 𝑓 = 𝐹)
43eleq1d 2900 . . . . 5 ((𝑒 = 𝐸𝑓 = 𝐹) → (𝑓 ∈ Field ↔ 𝐹 ∈ Field))
52, 4anbi12d 633 . . . 4 ((𝑒 = 𝐸𝑓 = 𝐹) → ((𝑒 ∈ Field ∧ 𝑓 ∈ Field) ↔ (𝐸 ∈ Field ∧ 𝐹 ∈ Field)))
63fveq2d 6663 . . . . . . 7 ((𝑒 = 𝐸𝑓 = 𝐹) → (Base‘𝑓) = (Base‘𝐹))
71, 6oveq12d 7164 . . . . . 6 ((𝑒 = 𝐸𝑓 = 𝐹) → (𝑒s (Base‘𝑓)) = (𝐸s (Base‘𝐹)))
83, 7eqeq12d 2840 . . . . 5 ((𝑒 = 𝐸𝑓 = 𝐹) → (𝑓 = (𝑒s (Base‘𝑓)) ↔ 𝐹 = (𝐸s (Base‘𝐹))))
91fveq2d 6663 . . . . . 6 ((𝑒 = 𝐸𝑓 = 𝐹) → (SubRing‘𝑒) = (SubRing‘𝐸))
106, 9eleq12d 2910 . . . . 5 ((𝑒 = 𝐸𝑓 = 𝐹) → ((Base‘𝑓) ∈ (SubRing‘𝑒) ↔ (Base‘𝐹) ∈ (SubRing‘𝐸)))
118, 10anbi12d 633 . . . 4 ((𝑒 = 𝐸𝑓 = 𝐹) → ((𝑓 = (𝑒s (Base‘𝑓)) ∧ (Base‘𝑓) ∈ (SubRing‘𝑒)) ↔ (𝐹 = (𝐸s (Base‘𝐹)) ∧ (Base‘𝐹) ∈ (SubRing‘𝐸))))
125, 11anbi12d 633 . . 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 31062 . . 3 /FldExt = {⟨𝑒, 𝑓⟩ ∣ ((𝑒 ∈ Field ∧ 𝑓 ∈ Field) ∧ (𝑓 = (𝑒s (Base‘𝑓)) ∧ (Base‘𝑓) ∈ (SubRing‘𝑒)))}
1412, 13brabga 5409 . 2 ((𝐸 ∈ Field ∧ 𝐹 ∈ Field) → (𝐸/FldExt𝐹 ↔ ((𝐸 ∈ Field ∧ 𝐹 ∈ Field) ∧ (𝐹 = (𝐸s (Base‘𝐹)) ∧ (Base‘𝐹) ∈ (SubRing‘𝐸)))))
1514bianabs 545 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 399   = wceq 1538  wcel 2115   class class class wbr 5053  cfv 6344  (class class class)co 7146  Basecbs 16481  s cress 16482  Fieldcfield 19498  SubRingcsubrg 19526  /FldExtcfldext 31058
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-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4826  df-br 5054  df-opab 5116  df-iota 6303  df-fv 6352  df-ov 7149  df-fldext 31062
This theorem is referenced by:  ccfldextrr  31068  fldextsubrg  31071  fldextress  31072  fldexttr  31078  fldextid  31079  extdgmul  31081
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