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Theorem brfinext 33378
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 33374 . . 3 (𝐸/FldExt𝐹𝐸 ∈ Field)
2 fldextfld2 33375 . . 3 (𝐸/FldExt𝐹𝐹 ∈ Field)
3 breq12 5157 . . . . 5 ((𝑒 = 𝐸𝑓 = 𝐹) → (𝑒/FldExt𝑓𝐸/FldExt𝐹))
4 oveq12 7435 . . . . . 6 ((𝑒 = 𝐸𝑓 = 𝐹) → (𝑒[:]𝑓) = (𝐸[:]𝐹))
54eleq1d 2814 . . . . 5 ((𝑒 = 𝐸𝑓 = 𝐹) → ((𝑒[:]𝑓) ∈ ℕ0 ↔ (𝐸[:]𝐹) ∈ ℕ0))
63, 5anbi12d 630 . . . 4 ((𝑒 = 𝐸𝑓 = 𝐹) → ((𝑒/FldExt𝑓 ∧ (𝑒[:]𝑓) ∈ ℕ0) ↔ (𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) ∈ ℕ0)))
7 df-finext 33369 . . . 4 /FinExt = {⟨𝑒, 𝑓⟩ ∣ (𝑒/FldExt𝑓 ∧ (𝑒[:]𝑓) ∈ ℕ0)}
86, 7brabga 5540 . . 3 ((𝐸 ∈ Field ∧ 𝐹 ∈ Field) → (𝐸/FinExt𝐹 ↔ (𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) ∈ ℕ0)))
91, 2, 8syl2anc 582 . 2 (𝐸/FldExt𝐹 → (𝐸/FinExt𝐹 ↔ (𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) ∈ ℕ0)))
109bianabs 540 1 (𝐸/FldExt𝐹 → (𝐸/FinExt𝐹 ↔ (𝐸[:]𝐹) ∈ ℕ0))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wcel 2098   class class class wbr 5152  (class class class)co 7426  0cn0 12510  Fieldcfield 20632  /FldExtcfldext 33363  /FinExtcfinext 33364  [:]cextdg 33366
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-xp 5688  df-iota 6505  df-fv 6561  df-ov 7429  df-fldext 33367  df-finext 33369
This theorem is referenced by:  finexttrb  33387
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