Theorem irngval 33680

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 3471	. 2 ⊢ (𝜑 → 𝑅 ∈ V)
3		irngval.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑅)
4	3	fvexi 6872	. . . 4 ⊢ 𝐵 ∈ V
5	4	a1i 11	. . 3 ⊢ (𝜑 → 𝐵 ∈ V)
6		irngval.s	. . 3 ⊢ (𝜑 → 𝑆 ⊆ 𝐵)
7	5, 6	ssexd 5279	. 2 ⊢ (𝜑 → 𝑆 ∈ V)
8		fvexd 6873	. . 3 ⊢ (𝜑 → (Monic1p‘𝑈) ∈ V)
9		fvex 6871	. . . . . 6 ⊢ (𝑂‘𝑓) ∈ V
10	9	cnvex 7901	. . . . 5 ⊢ ◡(𝑂‘𝑓) ∈ V
11	10	imaex 7890	. . . 4 ⊢ (◡(𝑂‘𝑓) “ { 0 }) ∈ V
12	11	rgenw 3048	. . 3 ⊢ ∀𝑓 ∈ (Monic1p‘𝑈)(◡(𝑂‘𝑓) “ { 0 }) ∈ V
13		iunexg 7942	. . 3 ⊢ (((Monic1p‘𝑈) ∈ V ∧ ∀𝑓 ∈ (Monic1p‘𝑈)(◡(𝑂‘𝑓) “ { 0 }) ∈ V) → ∪ 𝑓 ∈ (Monic1p‘𝑈)(◡(𝑂‘𝑓) “ { 0 }) ∈ V)
14	8, 12, 13	sylancl 586	. 2 ⊢ (𝜑 → ∪ 𝑓 ∈ (Monic1p‘𝑈)(◡(𝑂‘𝑓) “ { 0 }) ∈ V)
15		oveq12 7396	. . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (𝑟 ↾s 𝑠) = (𝑅 ↾s 𝑆))
16		irngval.u	. . . . . 6 ⊢ 𝑈 = (𝑅 ↾s 𝑆)
17	15, 16	eqtr4di 2782	. . . . 5 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (𝑟 ↾s 𝑠) = 𝑈)
18	17	fveq2d 6862	. . . 4 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (Monic1p‘(𝑟 ↾s 𝑠)) = (Monic1p‘𝑈))
19		oveq12 7396	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (𝑟 evalSub1 𝑠) = (𝑅 evalSub1 𝑆))
20		irngval.o	. . . . . . . 8 ⊢ 𝑂 = (𝑅 evalSub1 𝑆)
21	19, 20	eqtr4di 2782	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (𝑟 evalSub1 𝑠) = 𝑂)
22	21	fveq1d 6860	. . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → ((𝑟 evalSub1 𝑠)‘𝑓) = (𝑂‘𝑓))
23	22	cnveqd 5839	. . . . 5 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → ◡((𝑟 evalSub1 𝑠)‘𝑓) = ◡(𝑂‘𝑓))
24		simpl 482	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → 𝑟 = 𝑅)
25	24	fveq2d 6862	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (0g‘𝑟) = (0g‘𝑅))
26		irngval.0	. . . . . . 7 ⊢ 0 = (0g‘𝑅)
27	25, 26	eqtr4di 2782	. . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (0g‘𝑟) = 0 )
28	27	sneqd 4601	. . . . 5 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → {(0g‘𝑟)} = { 0 })
29	23, 28	imaeq12d 6032	. . . 4 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (◡((𝑟 evalSub1 𝑠)‘𝑓) “ {(0g‘𝑟)}) = (◡(𝑂‘𝑓) “ { 0 }))
30	18, 29	iuneq12d 4985	. . 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 33679	. . 3 ⊢ IntgRing = (𝑟 ∈ V, 𝑠 ∈ V ↦ ∪ 𝑓 ∈ (Monic1p‘(𝑟 ↾s 𝑠))(◡((𝑟 evalSub1 𝑠)‘𝑓) “ {(0g‘𝑟)}))
32	30, 31	ovmpoga 7543	. 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 1373	1 ⊢ (𝜑 → (𝑅 IntgRing 𝑆) = ∪ 𝑓 ∈ (Monic1p‘𝑈)(◡(𝑂‘𝑓) “ { 0 }))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  Vcvv 3447   ⊆ wss 3914  {csn 4589  ∪ ciun 4955  ^◡ccnv 5637   “ cima 5641  ‘cfv 6511  (class class class)co 7387  Basecbs 17179   ↾_s cress 17200  0_gc0g 17402  Ringcrg 20142   evalSub₁ ces1 22200  Monic_1pcmn1 26031   IntgRing cirng 33678

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-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

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-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-irng 33679

This theorem is referenced by:  elirng  33681