Theorem bralgext 34208

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 5108	. . . 4 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (𝑒/FldExt𝑓 ↔ 𝐸/FldExt𝐹))
4		simpl 488	. . . . . 6 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → 𝑒 = 𝐸)
5		fveq2 6879	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (Base‘𝑓) = (Base‘𝐹))
6		bralgext.c	. . . . . . . 8 ⊢ 𝐶 = (Base‘𝐹)
7	5, 6	eqtr4di 2813	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (Base‘𝑓) = 𝐶)
8	7	adantl 487	. . . . . 6 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (Base‘𝑓) = 𝐶)
9	4, 8	oveq12d 7432	. . . . 5 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (𝑒 IntgRing (Base‘𝑓)) = (𝐸 IntgRing 𝐶))
10		fveq2 6879	. . . . . . 7 ⊢ (𝑒 = 𝐸 → (Base‘𝑒) = (Base‘𝐸))
11		bralgext.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐸)
12	10, 11	eqtr4di 2813	. . . . . 6 ⊢ (𝑒 = 𝐸 → (Base‘𝑒) = 𝐵)
13	12	adantr 486	. . . . 5 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (Base‘𝑒) = 𝐵)
14	9, 13	eqeq12d 2776	. . . 4 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → ((𝑒 IntgRing (Base‘𝑓)) = (Base‘𝑒) ↔ (𝐸 IntgRing 𝐶) = 𝐵))
15	3, 14	anbi12d 644	. . 3 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → ((𝑒/FldExt𝑓 ∧ (𝑒 IntgRing (Base‘𝑓)) = (Base‘𝑒)) ↔ (𝐸/FldExt𝐹 ∧ (𝐸 IntgRing 𝐶) = 𝐵)))
16		df-algext 34207	. . 3 ⊢ /AlgExt = {⟨𝑒, 𝑓⟩ ∣ (𝑒/FldExt𝑓 ∧ (𝑒 IntgRing (Base‘𝑓)) = (Base‘𝑒))}
17	15, 16	brabga 5512	. 2 ⊢ ((𝐸 ∈ 𝑉 ∧ 𝐹 ∈ 𝑉) → (𝐸/AlgExt𝐹 ↔ (𝐸/FldExt𝐹 ∧ (𝐸 IntgRing 𝐶) = 𝐵)))
18	1, 2, 17	syl2anc 596	1 ⊢ (𝜑 → (𝐸/AlgExt𝐹 ↔ (𝐸/FldExt𝐹 ∧ (𝐸 IntgRing 𝐶) = 𝐵)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145   class class class wbr 5103  ‘cfv 6533  (class class class)co 7414  Basecbs 17302  /_FldExtcfldext 34149   IntgRing cirng 34194  /_AlgExtcalgext 34206

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5251  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-iota 6489  df-fv 6541  df-ov 7417  df-algext 34207

This theorem is used by:  finextalg  34209