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Theorem brfinext 31466
Description: The finite field extension relation explicited. (Contributed by Thierry Arnoux, 29-Jul-2023.)
Assertion
Ref Expression
brfinext (𝐸/FldExt𝐹 → (𝐸/FinExt𝐹 ↔ (𝐸[:]𝐹) ∈ ℕ0))

Proof of Theorem brfinext
Dummy variables 𝑒 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fldextfld1 31462 . . 3 (𝐸/FldExt𝐹𝐸 ∈ Field)
2 fldextfld2 31463 . . 3 (𝐸/FldExt𝐹𝐹 ∈ Field)
3 breq12 5072 . . . . 5 ((𝑒 = 𝐸𝑓 = 𝐹) → (𝑒/FldExt𝑓𝐸/FldExt𝐹))
4 oveq12 7240 . . . . . 6 ((𝑒 = 𝐸𝑓 = 𝐹) → (𝑒[:]𝑓) = (𝐸[:]𝐹))
54eleq1d 2823 . . . . 5 ((𝑒 = 𝐸𝑓 = 𝐹) → ((𝑒[:]𝑓) ∈ ℕ0 ↔ (𝐸[:]𝐹) ∈ ℕ0))
63, 5anbi12d 634 . . . 4 ((𝑒 = 𝐸𝑓 = 𝐹) → ((𝑒/FldExt𝑓 ∧ (𝑒[:]𝑓) ∈ ℕ0) ↔ (𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) ∈ ℕ0)))
7 df-finext 31457 . . . 4 /FinExt = {⟨𝑒, 𝑓⟩ ∣ (𝑒/FldExt𝑓 ∧ (𝑒[:]𝑓) ∈ ℕ0)}
86, 7brabga 5429 . . 3 ((𝐸 ∈ Field ∧ 𝐹 ∈ Field) → (𝐸/FinExt𝐹 ↔ (𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) ∈ ℕ0)))
91, 2, 8syl2anc 587 . 2 (𝐸/FldExt𝐹 → (𝐸/FinExt𝐹 ↔ (𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) ∈ ℕ0)))
109bianabs 545 1 (𝐸/FldExt𝐹 → (𝐸/FinExt𝐹 ↔ (𝐸[:]𝐹) ∈ ℕ0))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1543  wcel 2111   class class class wbr 5067  (class class class)co 7231  0cn0 12114  Fieldcfield 19792  /FldExtcfldext 31451  /FinExtcfinext 31452  [:]cextdg 31454
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-ext 2709  ax-sep 5206  ax-nul 5213  ax-pr 5336
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2072  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3067  df-rex 3068  df-rab 3071  df-v 3422  df-dif 3883  df-un 3885  df-in 3887  df-ss 3897  df-nul 4252  df-if 4454  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4834  df-br 5068  df-opab 5130  df-xp 5571  df-iota 6355  df-fv 6405  df-ov 7234  df-fldext 31455  df-finext 31457
This theorem is referenced by:  finexttrb  31475
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