finextfldext - Mathbox for Thierry Arnoux

	Mathbox for Thierry Arnoux		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > finextfldext		Structured version Visualization version GIF version

Theorem finextfldext 33650

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 33629	. . . . . . 7 ⊢ /_FinExt = {⟨𝑒, 𝑓⟩ ∣ (𝑒/_FldExt𝑓 ∧ (𝑒[:]𝑓) ∈ ℕ₀)}
3	2	relopabiv 5767	. . . . . 6 ⊢ Rel /_FinExt
4	3	brrelex1i 5679	. . . . 5 ⊢ (𝐸/_FinExt𝐹 → 𝐸 ∈ V)
5	1, 4	syl 17	. . . 4 ⊢ (𝜑 → 𝐸 ∈ V)
6	3	brrelex2i 5680	. . . . 5 ⊢ (𝐸/_FinExt𝐹 → 𝐹 ∈ V)
7	1, 6	syl 17	. . . 4 ⊢ (𝜑 → 𝐹 ∈ V)
8		breq12 5100	. . . . . 6 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (𝑒/_FldExt𝑓 ↔ 𝐸/_FldExt𝐹))
9		oveq12 7362	. . . . . . 7 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (𝑒[:]𝑓) = (𝐸[:]𝐹))
10	9	eleq1d 2813	. . . . . 6 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → ((𝑒[:]𝑓) ∈ ℕ₀ ↔ (𝐸[:]𝐹) ∈ ℕ₀))
11	8, 10	anbi12d 632	. . . . 5 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → ((𝑒/_FldExt𝑓 ∧ (𝑒[:]𝑓) ∈ ℕ₀) ↔ (𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) ∈ ℕ₀)))
12	11, 2	brabga 5481	. . . 4 ⊢ ((𝐸 ∈ V ∧ 𝐹 ∈ V) → (𝐸/_FinExt𝐹 ↔ (𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) ∈ ℕ₀)))
13	5, 7, 12	syl2anc 584	. . 3 ⊢ (𝜑 → (𝐸/_FinExt𝐹 ↔ (𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) ∈ ℕ₀)))
14	1, 13	mpbid 232	. 2 ⊢ (𝜑 → (𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) ∈ ℕ₀))
15	14	simpld 494	1 ⊢ (𝜑 → 𝐸/_FldExt𝐹)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2109 Vcvv 3438 class class class wbr 5095 (class class class)co 7353 ℕ₀cn0 12403 /_FldExtcfldext 33624 /_FinExtcfinext 33625 [:]cextdg 33626

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-ext 2701 ax-sep 5238 ax-nul 5248 ax-pr 5374

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2066 df-clab 2708 df-cleq 2721 df-clel 2803 df-ral 3045 df-rex 3054 df-rab 3397 df-v 3440 df-dif 3908 df-un 3910 df-ss 3922 df-nul 4287 df-if 4479 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4862 df-br 5096 df-opab 5158 df-xp 5629 df-rel 5630 df-iota 6442 df-fv 6494 df-ov 7356 df-finext 33629

This theorem is referenced by: finextalg 33684

W3C validator