Theorem brfinext 33809

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 33804	. . 3 ⊢ (𝐸/FldExt𝐹 → 𝐸 ∈ Field)
2		fldextfld2 33805	. . 3 ⊢ (𝐸/FldExt𝐹 → 𝐹 ∈ Field)
3		breq12 5103	. . . . 5 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (𝑒/FldExt𝑓 ↔ 𝐸/FldExt𝐹))
4		oveq12 7367	. . . . . 6 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (𝑒[:]𝑓) = (𝐸[:]𝐹))
5	4	eleq1d 2821	. . . . 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 33800	. . . 4 ⊢ /FinExt = {⟨𝑒, 𝑓⟩ ∣ (𝑒/FldExt𝑓 ∧ (𝑒[:]𝑓) ∈ ℕ0)}
8	6, 7	brabga 5482	. . 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 1541   ∈ wcel 2113   class class class wbr 5098  (class class class)co 7358  ℕ₀cn0 12401  Fieldcfield 20663  /_FldExtcfldext 33795  /_FinExtcfinext 33796  [:]cextdg 33797

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-xp 5630  df-iota 6448  df-fv 6500  df-ov 7361  df-fldext 33798  df-finext 33800

This theorem is referenced by:  finexttrb  33822  finextalg  33855