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Theorem brfinext 32399
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 32395 . . 3 (𝐸/FldExt𝐹𝐸 ∈ Field)
2 fldextfld2 32396 . . 3 (𝐸/FldExt𝐹𝐹 ∈ Field)
3 breq12 5111 . . . . 5 ((𝑒 = 𝐸𝑓 = 𝐹) → (𝑒/FldExt𝑓𝐸/FldExt𝐹))
4 oveq12 7367 . . . . . 6 ((𝑒 = 𝐸𝑓 = 𝐹) → (𝑒[:]𝑓) = (𝐸[:]𝐹))
54eleq1d 2819 . . . . 5 ((𝑒 = 𝐸𝑓 = 𝐹) → ((𝑒[:]𝑓) ∈ ℕ0 ↔ (𝐸[:]𝐹) ∈ ℕ0))
63, 5anbi12d 632 . . . 4 ((𝑒 = 𝐸𝑓 = 𝐹) → ((𝑒/FldExt𝑓 ∧ (𝑒[:]𝑓) ∈ ℕ0) ↔ (𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) ∈ ℕ0)))
7 df-finext 32390 . . . 4 /FinExt = {⟨𝑒, 𝑓⟩ ∣ (𝑒/FldExt𝑓 ∧ (𝑒[:]𝑓) ∈ ℕ0)}
86, 7brabga 5492 . . 3 ((𝐸 ∈ Field ∧ 𝐹 ∈ Field) → (𝐸/FinExt𝐹 ↔ (𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) ∈ ℕ0)))
91, 2, 8syl2anc 585 . 2 (𝐸/FldExt𝐹 → (𝐸/FinExt𝐹 ↔ (𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) ∈ ℕ0)))
109bianabs 543 1 (𝐸/FldExt𝐹 → (𝐸/FinExt𝐹 ↔ (𝐸[:]𝐹) ∈ ℕ0))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107   class class class wbr 5106  (class class class)co 7358  0cn0 12418  Fieldcfield 20198  /FldExtcfldext 32384  /FinExtcfinext 32385  [:]cextdg 32387
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-xp 5640  df-iota 6449  df-fv 6505  df-ov 7361  df-fldext 32388  df-finext 32390
This theorem is referenced by:  finexttrb  32408
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