finextfldext - Mathbox for Thierry Arnoux

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Theorem finextfldext 34296

Description: A finite field extension is a field extension. (Contributed by Thierry Arnoux, 10-Jan-2026.)

**Hypothesis**
Ref	Expression
finextfldext.1	⊢ (𝜑 → 𝐸/_FinExt𝐹)

**Assertion**
Ref	Expression
finextfldext	⊢ (𝜑 → 𝐸/_FldExt𝐹)

Proof of Theorem finextfldext Dummy variables 𝑒 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		finextfldext.1	. . 3 ⊢ (𝜑 → 𝐸/_FinExt𝐹)
2		df-finext 34275	. . . . . . 7 ⊢ /_FinExt = {⟨𝑒, 𝑓⟩ ∣ (𝑒/_FldExt𝑓 ∧ (𝑒[:]𝑓) ∈ ℕ₀)}
3	2	relopabiv 5797	. . . . . 6 ⊢ Rel /_FinExt
4	3	brrelex1i 5706	. . . . 5 ⊢ (𝐸/_FinExt𝐹 → 𝐸 ∈ V)
5	1, 4	syl 18	. . . 4 ⊢ (𝜑 → 𝐸 ∈ V)
6	3	brrelex2i 5707	. . . . 5 ⊢ (𝐸/_FinExt𝐹 → 𝐹 ∈ V)
7	1, 6	syl 18	. . . 4 ⊢ (𝜑 → 𝐹 ∈ V)
8		breq12 5107	. . . . . 6 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (𝑒/_FldExt𝑓 ↔ 𝐸/_FldExt𝐹))
9		oveq12 7429	. . . . . . 7 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (𝑒[:]𝑓) = (𝐸[:]𝐹))
10	9	eleq1d 2845	. . . . . 6 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → ((𝑒[:]𝑓) ∈ ℕ₀ ↔ (𝐸[:]𝐹) ∈ ℕ₀))
11	8, 10	anbi12d 644	. . . . 5 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → ((𝑒/_FldExt𝑓 ∧ (𝑒[:]𝑓) ∈ ℕ₀) ↔ (𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) ∈ ℕ₀)))
12	11, 2	brabga 5507	. . . 4 ⊢ ((𝐸 ∈ V ∧ 𝐹 ∈ V) → (𝐸/_FinExt𝐹 ↔ (𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) ∈ ℕ₀)))
13	5, 7, 12	syl2anc 596	. . 3 ⊢ (𝜑 → (𝐸/_FinExt𝐹 ↔ (𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) ∈ ℕ₀)))
14	1, 13	mpbid 235	. 2 ⊢ (𝜑 → (𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) ∈ ℕ₀))
15	14	simpld 500	1 ⊢ (𝜑 → 𝐸/_FldExt𝐹)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1569 ∈ wcel 2144 Vcvv 3450 class class class wbr 5102 (class class class)co 7420 ℕ₀cn0 12606 /_FldExtcfldext 34270 /_FinExtcfinext 34271 [:]cextdg 34272

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-ext 2732 ax-sep 5248 ax-pr 5390

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-sb 2099 df-clab 2739 df-cleq 2752 df-clel 2835 df-ral 3077 df-rex 3087 df-rab 3413 df-v 3452 df-dif 3899 df-un 3901 df-in 3903 df-ss 3914 df-nul 4279 df-if 4482 df-sn 4584 df-pr 4586 df-op 4590 df-uni 4867 df-br 5103 df-opab 5167 df-xp 5656 df-rel 5657 df-iota 6494 df-fv 6546 df-ov 7423 df-finext 34275

This theorem is used by: finextalg 34330

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