Theorem bralgext 33841

Description: Express the fact that a field extension 𝐸 / 𝐹 is algebraic. (Contributed by Thierry Arnoux, 10-Jan-2026.)

Hypotheses

Ref	Expression
bralgext.b	⊢ 𝐵 = (Base‘𝐸)
bralgext.c	⊢ 𝐶 = (Base‘𝐹)
bralgext.e	⊢ (𝜑 → 𝐸 ∈ 𝑉)
bralgext.f	⊢ (𝜑 → 𝐹 ∈ 𝑉)

Assertion

Ref	Expression
bralgext	⊢ (𝜑 → (𝐸/AlgExt𝐹 ↔ (𝐸/FldExt𝐹 ∧ (𝐸 IntgRing 𝐶) = 𝐵)))

Proof of Theorem bralgext

Dummy variables 𝑒 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		bralgext.e	. 2 ⊢ (𝜑 → 𝐸 ∈ 𝑉)
2		bralgext.f	. 2 ⊢ (𝜑 → 𝐹 ∈ 𝑉)
3		breq12 5090	. . . 4 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (𝑒/FldExt𝑓 ↔ 𝐸/FldExt𝐹))
4		simpl 482	. . . . . 6 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → 𝑒 = 𝐸)
5		fveq2 6840	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (Base‘𝑓) = (Base‘𝐹))
6		bralgext.c	. . . . . . . 8 ⊢ 𝐶 = (Base‘𝐹)
7	5, 6	eqtr4di 2789	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (Base‘𝑓) = 𝐶)
8	7	adantl 481	. . . . . 6 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (Base‘𝑓) = 𝐶)
9	4, 8	oveq12d 7385	. . . . 5 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (𝑒 IntgRing (Base‘𝑓)) = (𝐸 IntgRing 𝐶))
10		fveq2 6840	. . . . . . 7 ⊢ (𝑒 = 𝐸 → (Base‘𝑒) = (Base‘𝐸))
11		bralgext.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐸)
12	10, 11	eqtr4di 2789	. . . . . 6 ⊢ (𝑒 = 𝐸 → (Base‘𝑒) = 𝐵)
13	12	adantr 480	. . . . 5 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (Base‘𝑒) = 𝐵)
14	9, 13	eqeq12d 2752	. . . 4 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → ((𝑒 IntgRing (Base‘𝑓)) = (Base‘𝑒) ↔ (𝐸 IntgRing 𝐶) = 𝐵))
15	3, 14	anbi12d 633	. . 3 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → ((𝑒/FldExt𝑓 ∧ (𝑒 IntgRing (Base‘𝑓)) = (Base‘𝑒)) ↔ (𝐸/FldExt𝐹 ∧ (𝐸 IntgRing 𝐶) = 𝐵)))
16		df-algext 33840	. . 3 ⊢ /AlgExt = {⟨𝑒, 𝑓⟩ ∣ (𝑒/FldExt𝑓 ∧ (𝑒 IntgRing (Base‘𝑓)) = (Base‘𝑒))}
17	15, 16	brabga 5489	. 2 ⊢ ((𝐸 ∈ 𝑉 ∧ 𝐹 ∈ 𝑉) → (𝐸/AlgExt𝐹 ↔ (𝐸/FldExt𝐹 ∧ (𝐸 IntgRing 𝐶) = 𝐵)))
18	1, 2, 17	syl2anc 585	1 ⊢ (𝜑 → (𝐸/AlgExt𝐹 ↔ (𝐸/FldExt𝐹 ∧ (𝐸 IntgRing 𝐶) = 𝐵)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114   class class class wbr 5085  ‘cfv 6498  (class class class)co 7367  Basecbs 17179  /_FldExtcfldext 33782   IntgRing cirng 33827  /_AlgExtcalgext 33839

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-iota 6454  df-fv 6506  df-ov 7370  df-algext 33840

This theorem is referenced by:  finextalg  33842