finextfldext - Mathbox for Thierry Arnoux

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Theorem finextfldext 33848

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 33827	. . . . . . 7 ⊢ /_FinExt = {⟨𝑒, 𝑓⟩ ∣ (𝑒/_FldExt𝑓 ∧ (𝑒[:]𝑓) ∈ ℕ₀)}
3	2	relopabiv 5763	. . . . . 6 ⊢ Rel /_FinExt
4	3	brrelex1i 5674	. . . . 5 ⊢ (𝐸/_FinExt𝐹 → 𝐸 ∈ V)
5	1, 4	syl 17	. . . 4 ⊢ (𝜑 → 𝐸 ∈ V)
6	3	brrelex2i 5675	. . . . 5 ⊢ (𝐸/_FinExt𝐹 → 𝐹 ∈ V)
7	1, 6	syl 17	. . . 4 ⊢ (𝜑 → 𝐹 ∈ V)
8		breq12 5077	. . . . . 6 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (𝑒/_FldExt𝑓 ↔ 𝐸/_FldExt𝐹))
9		oveq12 7365	. . . . . . 7 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (𝑒[:]𝑓) = (𝐸[:]𝐹))
10	9	eleq1d 2824	. . . . . 6 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → ((𝑒[:]𝑓) ∈ ℕ₀ ↔ (𝐸[:]𝐹) ∈ ℕ₀))
11	8, 10	anbi12d 638	. . . . 5 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → ((𝑒/_FldExt𝑓 ∧ (𝑒[:]𝑓) ∈ ℕ₀) ↔ (𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) ∈ ℕ₀)))
12	11, 2	brabga 5476	. . . 4 ⊢ ((𝐸 ∈ V ∧ 𝐹 ∈ V) → (𝐸/_FinExt𝐹 ↔ (𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) ∈ ℕ₀)))
13	5, 7, 12	syl2anc 590	. . 3 ⊢ (𝜑 → (𝐸/_FinExt𝐹 ↔ (𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) ∈ ℕ₀)))
14	1, 13	mpbid 233	. 2 ⊢ (𝜑 → (𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) ∈ ℕ₀))
15	14	simpld 495	1 ⊢ (𝜑 → 𝐸/_FldExt𝐹)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1547 ∈ wcel 2119 Vcvv 3431 class class class wbr 5072 (class class class)co 7356 ℕ₀cn0 12428 /_FldExtcfldext 33822 /_FinExtcfinext 33823 [:]cextdg 33824

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-ext 2711 ax-sep 5218 ax-pr 5362

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-sb 2074 df-clab 2718 df-cleq 2731 df-clel 2814 df-ral 3054 df-rex 3064 df-rab 3392 df-v 3433 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4262 df-if 4455 df-sn 4556 df-pr 4558 df-op 4562 df-uni 4839 df-br 5073 df-opab 5135 df-xp 5624 df-rel 5625 df-iota 6441 df-fv 6493 df-ov 7359 df-finext 33827

This theorem is referenced by: finextalg 33882

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