Theorem irngval 34053

Description: The elements of a field 𝑅 integral over a subset 𝑆. In the case of a subfield, those are the algebraic numbers over the field 𝑆 within the field 𝑅. That is, the numbers 𝑋 which are roots of monic polynomials 𝑃(𝑋) with coefficients in 𝑆. (Contributed by Thierry Arnoux, 28-Jan-2025.)

Hypotheses

Ref	Expression
irngval.o	⊢ 𝑂 = (𝑅 evalSub1 𝑆)
irngval.u	⊢ 𝑈 = (𝑅 ↾s 𝑆)
irngval.b	⊢ 𝐵 = (Base‘𝑅)
irngval.0	⊢ 0 = (0g‘𝑅)
irngval.r	⊢ (𝜑 → 𝑅 ∈ Ring)
irngval.s	⊢ (𝜑 → 𝑆 ⊆ 𝐵)

Assertion

Ref	Expression
irngval	⊢ (𝜑 → (𝑅 IntgRing 𝑆) = ∪ 𝑓 ∈ (Monic1p‘𝑈)(◡(𝑂‘𝑓) “ { 0 }))

Distinct variable groups:   𝐵,𝑓   𝑓,𝑂   𝑅,𝑓   𝑆,𝑓   𝑈,𝑓   𝜑,𝑓

Allowed substitution hint:   0 (𝑓)

Proof of Theorem irngval

Dummy variables 𝑟 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		irngval.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ Ring)
2	1	elexd 3478	. 2 ⊢ (𝜑 → 𝑅 ∈ V)
3		irngval.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑅)
4	3	fvexi 6897	. . . 4 ⊢ 𝐵 ∈ V
5	4	a1i 11	. . 3 ⊢ (𝜑 → 𝐵 ∈ V)
6		irngval.s	. . 3 ⊢ (𝜑 → 𝑆 ⊆ 𝐵)
7	5, 6	ssexd 5296	. 2 ⊢ (𝜑 → 𝑆 ∈ V)
8		fvexd 6898	. . 3 ⊢ (𝜑 → (Monic1p‘𝑈) ∈ V)
9		fvex 6896	. . . . . 6 ⊢ (𝑂‘𝑓) ∈ V
10	9	cnvex 7923	. . . . 5 ⊢ ◡(𝑂‘𝑓) ∈ V
11	10	imaex 7912	. . . 4 ⊢ (◡(𝑂‘𝑓) “ { 0 }) ∈ V
12	11	rgenw 3083	. . 3 ⊢ ∀𝑓 ∈ (Monic1p‘𝑈)(◡(𝑂‘𝑓) “ { 0 }) ∈ V
13		iunexg 7961	. . 3 ⊢ (((Monic1p‘𝑈) ∈ V ∧ ∀𝑓 ∈ (Monic1p‘𝑈)(◡(𝑂‘𝑓) “ { 0 }) ∈ V) → ∪ 𝑓 ∈ (Monic1p‘𝑈)(◡(𝑂‘𝑓) “ { 0 }) ∈ V)
14	8, 12, 13	sylancl 597	. 2 ⊢ (𝜑 → ∪ 𝑓 ∈ (Monic1p‘𝑈)(◡(𝑂‘𝑓) “ { 0 }) ∈ V)
15		oveq12 7421	. . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (𝑟 ↾s 𝑠) = (𝑅 ↾s 𝑆))
16		irngval.u	. . . . . 6 ⊢ 𝑈 = (𝑅 ↾s 𝑆)
17	15, 16	eqtr4di 2816	. . . . 5 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (𝑟 ↾s 𝑠) = 𝑈)
18	17	fveq2d 6887	. . . 4 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (Monic1p‘(𝑟 ↾s 𝑠)) = (Monic1p‘𝑈))
19		oveq12 7421	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (𝑟 evalSub1 𝑠) = (𝑅 evalSub1 𝑆))
20		irngval.o	. . . . . . . 8 ⊢ 𝑂 = (𝑅 evalSub1 𝑆)
21	19, 20	eqtr4di 2816	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (𝑟 evalSub1 𝑠) = 𝑂)
22	21	fveq1d 6885	. . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → ((𝑟 evalSub1 𝑠)‘𝑓) = (𝑂‘𝑓))
23	22	cnveqd 5863	. . . . 5 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → ◡((𝑟 evalSub1 𝑠)‘𝑓) = ◡(𝑂‘𝑓))
24		simpl 487	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → 𝑟 = 𝑅)
25	24	fveq2d 6887	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (0g‘𝑟) = (0g‘𝑅))
26		irngval.0	. . . . . . 7 ⊢ 0 = (0g‘𝑅)
27	25, 26	eqtr4di 2816	. . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (0g‘𝑟) = 0 )
28	27	sneqd 4602	. . . . 5 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → {(0g‘𝑟)} = { 0 })
29	23, 28	imaeq12d 6065	. . . 4 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (◡((𝑟 evalSub1 𝑠)‘𝑓) “ {(0g‘𝑟)}) = (◡(𝑂‘𝑓) “ { 0 }))
30	18, 29	iuneq12d 4987	. . 3 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → ∪ 𝑓 ∈ (Monic1p‘(𝑟 ↾s 𝑠))(◡((𝑟 evalSub1 𝑠)‘𝑓) “ {(0g‘𝑟)}) = ∪ 𝑓 ∈ (Monic1p‘𝑈)(◡(𝑂‘𝑓) “ { 0 }))
31		df-irng 34052	. . 3 ⊢ IntgRing = (𝑟 ∈ V, 𝑠 ∈ V ↦ ∪ 𝑓 ∈ (Monic1p‘(𝑟 ↾s 𝑠))(◡((𝑟 evalSub1 𝑠)‘𝑓) “ {(0g‘𝑟)}))
32	30, 31	ovmpoga 7566	. 2 ⊢ ((𝑅 ∈ V ∧ 𝑆 ∈ V ∧ ∪ 𝑓 ∈ (Monic1p‘𝑈)(◡(𝑂‘𝑓) “ { 0 }) ∈ V) → (𝑅 IntgRing 𝑆) = ∪ 𝑓 ∈ (Monic1p‘𝑈)(◡(𝑂‘𝑓) “ { 0 }))
33	2, 7, 14, 32	syl3anc 1398	1 ⊢ (𝜑 → (𝑅 IntgRing 𝑆) = ∪ 𝑓 ∈ (Monic1p‘𝑈)(◡(𝑂‘𝑓) “ { 0 }))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∀wral 3079  Vcvv 3455   ⊆ wss 3906  {csn 4590  ∪ ciun 4957  ^◡ccnv 5662   “ cima 5666  ‘cfv 6538  (class class class)co 7412  Basecbs 17270   ↾_s cress 17291  0_gc0g 17493  Ringcrg 20316   evalSub₁ ces1 22454  Monic_1pcmn1 26264   IntgRing cirng 34051

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3746  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fv 6546  df-ov 7415  df-oprab 7416  df-mpo 7417  df-irng 34052

This theorem is referenced by:  elirng  34054