Theorem irngval 33206

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 3494	. 2 ⊢ (𝜑 → 𝑅 ∈ V)
3		irngval.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑅)
4	3	fvexi 6905	. . . 4 ⊢ 𝐵 ∈ V
5	4	a1i 11	. . 3 ⊢ (𝜑 → 𝐵 ∈ V)
6		irngval.s	. . 3 ⊢ (𝜑 → 𝑆 ⊆ 𝐵)
7	5, 6	ssexd 5324	. 2 ⊢ (𝜑 → 𝑆 ∈ V)
8		fvexd 6906	. . 3 ⊢ (𝜑 → (Monic1p‘𝑈) ∈ V)
9		fvex 6904	. . . . . 6 ⊢ (𝑂‘𝑓) ∈ V
10	9	cnvex 7920	. . . . 5 ⊢ ◡(𝑂‘𝑓) ∈ V
11	10	imaex 7911	. . . 4 ⊢ (◡(𝑂‘𝑓) “ { 0 }) ∈ V
12	11	rgenw 3064	. . 3 ⊢ ∀𝑓 ∈ (Monic1p‘𝑈)(◡(𝑂‘𝑓) “ { 0 }) ∈ V
13		iunexg 7954	. . 3 ⊢ (((Monic1p‘𝑈) ∈ V ∧ ∀𝑓 ∈ (Monic1p‘𝑈)(◡(𝑂‘𝑓) “ { 0 }) ∈ V) → ∪ 𝑓 ∈ (Monic1p‘𝑈)(◡(𝑂‘𝑓) “ { 0 }) ∈ V)
14	8, 12, 13	sylancl 585	. 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 2789	. . . . 5 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (𝑟 ↾s 𝑠) = 𝑈)
18	17	fveq2d 6895	. . . 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 2789	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (𝑟 evalSub1 𝑠) = 𝑂)
22	21	fveq1d 6893	. . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → ((𝑟 evalSub1 𝑠)‘𝑓) = (𝑂‘𝑓))
23	22	cnveqd 5875	. . . . 5 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → ◡((𝑟 evalSub1 𝑠)‘𝑓) = ◡(𝑂‘𝑓))
24		simpl 482	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → 𝑟 = 𝑅)
25	24	fveq2d 6895	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (0g‘𝑟) = (0g‘𝑅))
26		irngval.0	. . . . . . 7 ⊢ 0 = (0g‘𝑅)
27	25, 26	eqtr4di 2789	. . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (0g‘𝑟) = 0 )
28	27	sneqd 4640	. . . . 5 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → {(0g‘𝑟)} = { 0 })
29	23, 28	imaeq12d 6060	. . . 4 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (◡((𝑟 evalSub1 𝑠)‘𝑓) “ {(0g‘𝑟)}) = (◡(𝑂‘𝑓) “ { 0 }))
30	18, 29	iuneq12d 5025	. . 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 33205	. . 3 ⊢ IntgRing = (𝑟 ∈ V, 𝑠 ∈ V ↦ ∪ 𝑓 ∈ (Monic1p‘(𝑟 ↾s 𝑠))(◡((𝑟 evalSub1 𝑠)‘𝑓) “ {(0g‘𝑟)}))
32	30, 31	ovmpoga 7565	. 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 1370	1 ⊢ (𝜑 → (𝑅 IntgRing 𝑆) = ∪ 𝑓 ∈ (Monic1p‘𝑈)(◡(𝑂‘𝑓) “ { 0 }))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2105  ∀wral 3060  Vcvv 3473   ⊆ wss 3948  {csn 4628  ∪ ciun 4997  ^◡ccnv 5675   “ cima 5679  ‘cfv 6543  (class class class)co 7412  Basecbs 17151   ↾_s cress 17180  0_gc0g 17392  Ringcrg 20134   evalSub₁ ces1 22153  Monic_1pcmn1 25982   IntgRing cirng 33204

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-irng 33205

This theorem is referenced by:  elirng  33207