Theorem brfinext 34351

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 34346	. . 3 ⊢ (𝐸/FldExt𝐹 → 𝐸 ∈ Field)
2		fldextfld2 34347	. . 3 ⊢ (𝐸/FldExt𝐹 → 𝐹 ∈ Field)
3		breq12 5111	. . . . 5 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (𝑒/FldExt𝑓 ↔ 𝐸/FldExt𝐹))
4		oveq12 7449	. . . . . 6 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (𝑒[:]𝑓) = (𝐸[:]𝐹))
5	4	eleq1d 2849	. . . . 5 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → ((𝑒[:]𝑓) ∈ ℕ0 ↔ (𝐸[:]𝐹) ∈ ℕ0))
6	3, 5	anbi12d 644	. . . 4 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → ((𝑒/FldExt𝑓 ∧ (𝑒[:]𝑓) ∈ ℕ0) ↔ (𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) ∈ ℕ0)))
7		df-finext 34342	. . . 4 ⊢ /FinExt = {⟨𝑒, 𝑓⟩ ∣ (𝑒/FldExt𝑓 ∧ (𝑒[:]𝑓) ∈ ℕ0)}
8	6, 7	brabga 5512	. . 3 ⊢ ((𝐸 ∈ Field ∧ 𝐹 ∈ Field) → (𝐸/FinExt𝐹 ↔ (𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) ∈ ℕ0)))
9	1, 2, 8	syl2anc 596	. 2 ⊢ (𝐸/FldExt𝐹 → (𝐸/FinExt𝐹 ↔ (𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) ∈ ℕ0)))
10	9	bianabs 551	1 ⊢ (𝐸/FldExt𝐹 → (𝐸/FinExt𝐹 ↔ (𝐸[:]𝐹) ∈ ℕ0))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147   class class class wbr 5106  (class class class)co 7440  ℕ₀cn0 12630  Fieldcfield 21008  /_FldExtcfldext 34337  /_FinExtcfinext 34338  [:]cextdg 34339

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5252  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-xp 5661  df-iota 6509  df-fv 6563  df-ov 7443  df-fldext 34340  df-finext 34342

This theorem is used by:  finexttrb  34364  finextalg  34397