Theorem brfinext 33694

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.

Step	Hyp	Ref	Expression
1		fldextfld1 33690	. . 3 ⊢ (𝐸/FldExt𝐹 → 𝐸 ∈ Field)
2		fldextfld2 33691	. . 3 ⊢ (𝐸/FldExt𝐹 → 𝐹 ∈ Field)
3		breq12 5154	. . . . 5 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (𝑒/FldExt𝑓 ↔ 𝐸/FldExt𝐹))
4		oveq12 7444	. . . . . 6 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (𝑒[:]𝑓) = (𝐸[:]𝐹))
5	4	eleq1d 2825	. . . . 5 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → ((𝑒[:]𝑓) ∈ ℕ0 ↔ (𝐸[:]𝐹) ∈ ℕ0))
6	3, 5	anbi12d 632	. . . 4 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → ((𝑒/FldExt𝑓 ∧ (𝑒[:]𝑓) ∈ ℕ0) ↔ (𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) ∈ ℕ0)))
7		df-finext 33685	. . . 4 ⊢ /FinExt = {⟨𝑒, 𝑓⟩ ∣ (𝑒/FldExt𝑓 ∧ (𝑒[:]𝑓) ∈ ℕ0)}
8	6, 7	brabga 5545	. . 3 ⊢ ((𝐸 ∈ Field ∧ 𝐹 ∈ Field) → (𝐸/FinExt𝐹 ↔ (𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) ∈ ℕ0)))
9	1, 2, 8	syl2anc 584	. 2 ⊢ (𝐸/FldExt𝐹 → (𝐸/FinExt𝐹 ↔ (𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) ∈ ℕ0)))
10	9	bianabs 541	1 ⊢ (𝐸/FldExt𝐹 → (𝐸/FinExt𝐹 ↔ (𝐸[:]𝐹) ∈ ℕ0))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1538   ∈ wcel 2107   class class class wbr 5149  (class class class)co 7435  ℕ₀cn0 12530  Fieldcfield 20753  /_FldExtcfldext 33679  /_FinExtcfinext 33680  [:]cextdg 33682

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5303  ax-nul 5313  ax-pr 5439

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1541  df-fal 1551  df-ex 1778  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3435  df-v 3481  df-dif 3967  df-un 3969  df-ss 3981  df-nul 4341  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4914  df-br 5150  df-opab 5212  df-xp 5696  df-iota 6519  df-fv 6574  df-ov 7438  df-fldext 33683  df-finext 33685

This theorem is referenced by:  finexttrb  33703