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Theorem brfinext 31073
 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 31069 . . 3 (𝐸/FldExt𝐹𝐸 ∈ Field)
2 fldextfld2 31070 . . 3 (𝐸/FldExt𝐹𝐹 ∈ Field)
3 breq12 5058 . . . . 5 ((𝑒 = 𝐸𝑓 = 𝐹) → (𝑒/FldExt𝑓𝐸/FldExt𝐹))
4 oveq12 7155 . . . . . 6 ((𝑒 = 𝐸𝑓 = 𝐹) → (𝑒[:]𝑓) = (𝐸[:]𝐹))
54eleq1d 2900 . . . . 5 ((𝑒 = 𝐸𝑓 = 𝐹) → ((𝑒[:]𝑓) ∈ ℕ0 ↔ (𝐸[:]𝐹) ∈ ℕ0))
63, 5anbi12d 633 . . . 4 ((𝑒 = 𝐸𝑓 = 𝐹) → ((𝑒/FldExt𝑓 ∧ (𝑒[:]𝑓) ∈ ℕ0) ↔ (𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) ∈ ℕ0)))
7 df-finext 31064 . . . 4 /FinExt = {⟨𝑒, 𝑓⟩ ∣ (𝑒/FldExt𝑓 ∧ (𝑒[:]𝑓) ∈ ℕ0)}
86, 7brabga 5409 . . 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 1538   ∈ wcel 2115   class class class wbr 5053  (class class class)co 7146  ℕ0cn0 11892  Fieldcfield 19498  /FldExtcfldext 31058  /FinExtcfinext 31059  [:]cextdg 31061 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-ral 3138  df-rex 3139  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-xp 5549  df-iota 6303  df-fv 6352  df-ov 7149  df-fldext 31062  df-finext 31064 This theorem is referenced by:  finexttrb  31082
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