Theorem irngval 32687

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 3495	. 2 ⊢ (𝜑 → 𝑅 ∈ V)
3		irngval.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑅)
4	3	fvexi 6895	. . . 4 ⊢ 𝐵 ∈ V
5	4	a1i 11	. . 3 ⊢ (𝜑 → 𝐵 ∈ V)
6		irngval.s	. . 3 ⊢ (𝜑 → 𝑆 ⊆ 𝐵)
7	5, 6	ssexd 5320	. 2 ⊢ (𝜑 → 𝑆 ∈ V)
8		fvexd 6896	. . 3 ⊢ (𝜑 → (Monic1p‘𝑈) ∈ V)
9		fvex 6894	. . . . . 6 ⊢ (𝑂‘𝑓) ∈ V
10	9	cnvex 7903	. . . . 5 ⊢ ◡(𝑂‘𝑓) ∈ V
11	10	imaex 7894	. . . 4 ⊢ (◡(𝑂‘𝑓) “ { 0 }) ∈ V
12	11	rgenw 3066	. . 3 ⊢ ∀𝑓 ∈ (Monic1p‘𝑈)(◡(𝑂‘𝑓) “ { 0 }) ∈ V
13		iunexg 7937	. . 3 ⊢ (((Monic1p‘𝑈) ∈ V ∧ ∀𝑓 ∈ (Monic1p‘𝑈)(◡(𝑂‘𝑓) “ { 0 }) ∈ V) → ∪ 𝑓 ∈ (Monic1p‘𝑈)(◡(𝑂‘𝑓) “ { 0 }) ∈ V)
14	8, 12, 13	sylancl 587	. 2 ⊢ (𝜑 → ∪ 𝑓 ∈ (Monic1p‘𝑈)(◡(𝑂‘𝑓) “ { 0 }) ∈ V)
15		oveq12 7405	. . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (𝑟 ↾s 𝑠) = (𝑅 ↾s 𝑆))
16		irngval.u	. . . . . 6 ⊢ 𝑈 = (𝑅 ↾s 𝑆)
17	15, 16	eqtr4di 2791	. . . . 5 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (𝑟 ↾s 𝑠) = 𝑈)
18	17	fveq2d 6885	. . . 4 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (Monic1p‘(𝑟 ↾s 𝑠)) = (Monic1p‘𝑈))
19		oveq12 7405	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (𝑟 evalSub1 𝑠) = (𝑅 evalSub1 𝑆))
20		irngval.o	. . . . . . . 8 ⊢ 𝑂 = (𝑅 evalSub1 𝑆)
21	19, 20	eqtr4di 2791	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (𝑟 evalSub1 𝑠) = 𝑂)
22	21	fveq1d 6883	. . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → ((𝑟 evalSub1 𝑠)‘𝑓) = (𝑂‘𝑓))
23	22	cnveqd 5870	. . . . 5 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → ◡((𝑟 evalSub1 𝑠)‘𝑓) = ◡(𝑂‘𝑓))
24		simpl 484	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → 𝑟 = 𝑅)
25	24	fveq2d 6885	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (0g‘𝑟) = (0g‘𝑅))
26		irngval.0	. . . . . . 7 ⊢ 0 = (0g‘𝑅)
27	25, 26	eqtr4di 2791	. . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (0g‘𝑟) = 0 )
28	27	sneqd 4636	. . . . 5 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → {(0g‘𝑟)} = { 0 })
29	23, 28	imaeq12d 6053	. . . 4 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (◡((𝑟 evalSub1 𝑠)‘𝑓) “ {(0g‘𝑟)}) = (◡(𝑂‘𝑓) “ { 0 }))
30	18, 29	iuneq12d 5021	. . 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 32686	. . 3 ⊢ IntgRing = (𝑟 ∈ V, 𝑠 ∈ V ↦ ∪ 𝑓 ∈ (Monic1p‘(𝑟 ↾s 𝑠))(◡((𝑟 evalSub1 𝑠)‘𝑓) “ {(0g‘𝑟)}))
32	30, 31	ovmpoga 7549	. 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 1372	1 ⊢ (𝜑 → (𝑅 IntgRing 𝑆) = ∪ 𝑓 ∈ (Monic1p‘𝑈)(◡(𝑂‘𝑓) “ { 0 }))

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  ∀wral 3062  Vcvv 3475   ⊆ wss 3946  {csn 4624  ∪ ciun 4993  ^◡ccnv 5671   “ cima 5675  ‘cfv 6535  (class class class)co 7396  Basecbs 17131   ↾_s cress 17160  0_gc0g 17372  Ringcrg 20038   evalSub₁ ces1 21801  Monic_1pcmn1 25612   IntgRing cirng 32685

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5359  ax-pr 5423  ax-un 7712

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3776  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4905  df-iun 4995  df-br 5145  df-opab 5207  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6487  df-fun 6537  df-fv 6543  df-ov 7399  df-oprab 7400  df-mpo 7401  df-irng 32686

This theorem is referenced by:  elirng  32688