Theorem irngnzply1 33045

Description: In the case of a field 𝐸, the roots of nonzero polynomials 𝑝 with coefficients in a subfield 𝐹 are exactly the integral elements over 𝐹. Roots of nonzero polynomials are called algebraic numbers, so this shows that in the case of a field, elements integral over 𝐹 are exactly the algebraic numbers. In this formula, dom 𝑂 represents the polynomials, and 𝑍 the zero polynomial. (Contributed by Thierry Arnoux, 5-Feb-2025.)

Hypotheses

Ref	Expression
irngnzply1.o	⊢ 𝑂 = (𝐸 evalSub1 𝐹)
irngnzply1.z	⊢ 𝑍 = (0g‘(Poly1‘𝐸))
irngnzply1.1	⊢ 0 = (0g‘𝐸)
irngnzply1.e	⊢ (𝜑 → 𝐸 ∈ Field)
irngnzply1.f	⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))

Assertion

Ref	Expression
irngnzply1	⊢ (𝜑 → (𝐸 IntgRing 𝐹) = ∪ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})(◡(𝑂‘𝑝) “ { 0 }))

Distinct variable groups:   𝐸,𝑝   𝐹,𝑝   𝑂,𝑝   𝜑,𝑝

Allowed substitution hints:   0 (𝑝)   𝑍(𝑝)

Proof of Theorem irngnzply1

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		irngnzply1.o	. . . . . . . 8 ⊢ 𝑂 = (𝐸 evalSub1 𝐹)
2		eqid 2731	. . . . . . . 8 ⊢ (𝐸 ↾s 𝐹) = (𝐸 ↾s 𝐹)
3		eqid 2731	. . . . . . . 8 ⊢ (Base‘𝐸) = (Base‘𝐸)
4		irngnzply1.1	. . . . . . . 8 ⊢ 0 = (0g‘𝐸)
5		irngnzply1.e	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ Field)
6	5	fldcrngd 20514	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ CRing)
7		irngnzply1.f	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))
8		issdrg 20548	. . . . . . . . . 10 ⊢ (𝐹 ∈ (SubDRing‘𝐸) ↔ (𝐸 ∈ DivRing ∧ 𝐹 ∈ (SubRing‘𝐸) ∧ (𝐸 ↾s 𝐹) ∈ DivRing))
9	7, 8	sylib 217	. . . . . . . . 9 ⊢ (𝜑 → (𝐸 ∈ DivRing ∧ 𝐹 ∈ (SubRing‘𝐸) ∧ (𝐸 ↾s 𝐹) ∈ DivRing))
10	9	simp2d 1142	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ (SubRing‘𝐸))
11	1, 2, 3, 4, 6, 10	elirng 33040	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (𝐸 IntgRing 𝐹) ↔ (𝑥 ∈ (Base‘𝐸) ∧ ∃𝑝 ∈ (Monic1p‘(𝐸 ↾s 𝐹))((𝑂‘𝑝)‘𝑥) = 0 )))
12	11	biimpa 476	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐸 IntgRing 𝐹)) → (𝑥 ∈ (Base‘𝐸) ∧ ∃𝑝 ∈ (Monic1p‘(𝐸 ↾s 𝐹))((𝑂‘𝑝)‘𝑥) = 0 ))
13	12	simprd 495	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐸 IntgRing 𝐹)) → ∃𝑝 ∈ (Monic1p‘(𝐸 ↾s 𝐹))((𝑂‘𝑝)‘𝑥) = 0 )
14		eqid 2731	. . . . . . . . . 10 ⊢ (Poly1‘(𝐸 ↾s 𝐹)) = (Poly1‘(𝐸 ↾s 𝐹))
15		eqid 2731	. . . . . . . . . 10 ⊢ (Base‘(Poly1‘(𝐸 ↾s 𝐹))) = (Base‘(Poly1‘(𝐸 ↾s 𝐹)))
16		eqid 2731	. . . . . . . . . 10 ⊢ (Monic1p‘(𝐸 ↾s 𝐹)) = (Monic1p‘(𝐸 ↾s 𝐹))
17	14, 15, 16	mon1pcl 25898	. . . . . . . . 9 ⊢ (𝑝 ∈ (Monic1p‘(𝐸 ↾s 𝐹)) → 𝑝 ∈ (Base‘(Poly1‘(𝐸 ↾s 𝐹))))
18	17	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ (Monic1p‘(𝐸 ↾s 𝐹))) → 𝑝 ∈ (Base‘(Poly1‘(𝐸 ↾s 𝐹))))
19		eqid 2731	. . . . . . . . . . . . 13 ⊢ (𝐸 ↑s (Base‘𝐸)) = (𝐸 ↑s (Base‘𝐸))
20	1, 3, 19, 2, 14	evls1rhm 22062	. . . . . . . . . . . 12 ⊢ ((𝐸 ∈ CRing ∧ 𝐹 ∈ (SubRing‘𝐸)) → 𝑂 ∈ ((Poly1‘(𝐸 ↾s 𝐹)) RingHom (𝐸 ↑s (Base‘𝐸))))
21	6, 10, 20	syl2anc 583	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑂 ∈ ((Poly1‘(𝐸 ↾s 𝐹)) RingHom (𝐸 ↑s (Base‘𝐸))))
22		eqid 2731	. . . . . . . . . . . 12 ⊢ (Base‘(𝐸 ↑s (Base‘𝐸))) = (Base‘(𝐸 ↑s (Base‘𝐸)))
23	15, 22	rhmf 20377	. . . . . . . . . . 11 ⊢ (𝑂 ∈ ((Poly1‘(𝐸 ↾s 𝐹)) RingHom (𝐸 ↑s (Base‘𝐸))) → 𝑂:(Base‘(Poly1‘(𝐸 ↾s 𝐹)))⟶(Base‘(𝐸 ↑s (Base‘𝐸))))
24	21, 23	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑂:(Base‘(Poly1‘(𝐸 ↾s 𝐹)))⟶(Base‘(𝐸 ↑s (Base‘𝐸))))
25	24	fdmd 6728	. . . . . . . . 9 ⊢ (𝜑 → dom 𝑂 = (Base‘(Poly1‘(𝐸 ↾s 𝐹))))
26	25	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ (Monic1p‘(𝐸 ↾s 𝐹))) → dom 𝑂 = (Base‘(Poly1‘(𝐸 ↾s 𝐹))))
27	18, 26	eleqtrrd 2835	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ (Monic1p‘(𝐸 ↾s 𝐹))) → 𝑝 ∈ dom 𝑂)
28		eqid 2731	. . . . . . . . . 10 ⊢ (0g‘(Poly1‘(𝐸 ↾s 𝐹))) = (0g‘(Poly1‘(𝐸 ↾s 𝐹)))
29	14, 28, 16	mon1pn0 25900	. . . . . . . . 9 ⊢ (𝑝 ∈ (Monic1p‘(𝐸 ↾s 𝐹)) → 𝑝 ≠ (0g‘(Poly1‘(𝐸 ↾s 𝐹))))
30	29	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ (Monic1p‘(𝐸 ↾s 𝐹))) → 𝑝 ≠ (0g‘(Poly1‘(𝐸 ↾s 𝐹))))
31		eqid 2731	. . . . . . . . . 10 ⊢ (Poly1‘𝐸) = (Poly1‘𝐸)
32		irngnzply1.z	. . . . . . . . . 10 ⊢ 𝑍 = (0g‘(Poly1‘𝐸))
33	31, 2, 14, 15, 10, 32	ressply10g 32931	. . . . . . . . 9 ⊢ (𝜑 → 𝑍 = (0g‘(Poly1‘(𝐸 ↾s 𝐹))))
34	33	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ (Monic1p‘(𝐸 ↾s 𝐹))) → 𝑍 = (0g‘(Poly1‘(𝐸 ↾s 𝐹))))
35	30, 34	neeqtrrd 3014	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ (Monic1p‘(𝐸 ↾s 𝐹))) → 𝑝 ≠ 𝑍)
36		eldifsn 4790	. . . . . . 7 ⊢ (𝑝 ∈ (dom 𝑂 ∖ {𝑍}) ↔ (𝑝 ∈ dom 𝑂 ∧ 𝑝 ≠ 𝑍))
37	27, 35, 36	sylanbrc 582	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ (Monic1p‘(𝐸 ↾s 𝐹))) → 𝑝 ∈ (dom 𝑂 ∖ {𝑍}))
38	37	ad2ant2r 744	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 IntgRing 𝐹)) ∧ (𝑝 ∈ (Monic1p‘(𝐸 ↾s 𝐹)) ∧ ((𝑂‘𝑝)‘𝑥) = 0 )) → 𝑝 ∈ (dom 𝑂 ∖ {𝑍}))
39	5	ad2antrr 723	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 IntgRing 𝐹)) ∧ (𝑝 ∈ (Monic1p‘(𝐸 ↾s 𝐹)) ∧ ((𝑂‘𝑝)‘𝑥) = 0 )) → 𝐸 ∈ Field)
40		fvexd 6906	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 IntgRing 𝐹)) ∧ (𝑝 ∈ (Monic1p‘(𝐸 ↾s 𝐹)) ∧ ((𝑂‘𝑝)‘𝑥) = 0 )) → (Base‘𝐸) ∈ V)
41	24	ad2antrr 723	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 IntgRing 𝐹)) ∧ (𝑝 ∈ (Monic1p‘(𝐸 ↾s 𝐹)) ∧ ((𝑂‘𝑝)‘𝑥) = 0 )) → 𝑂:(Base‘(Poly1‘(𝐸 ↾s 𝐹)))⟶(Base‘(𝐸 ↑s (Base‘𝐸))))
42	17	ad2antrl 725	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 IntgRing 𝐹)) ∧ (𝑝 ∈ (Monic1p‘(𝐸 ↾s 𝐹)) ∧ ((𝑂‘𝑝)‘𝑥) = 0 )) → 𝑝 ∈ (Base‘(Poly1‘(𝐸 ↾s 𝐹))))
43	41, 42	ffvelcdmd 7087	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 IntgRing 𝐹)) ∧ (𝑝 ∈ (Monic1p‘(𝐸 ↾s 𝐹)) ∧ ((𝑂‘𝑝)‘𝑥) = 0 )) → (𝑂‘𝑝) ∈ (Base‘(𝐸 ↑s (Base‘𝐸))))
44	19, 3, 22, 39, 40, 43	pwselbas 17440	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 IntgRing 𝐹)) ∧ (𝑝 ∈ (Monic1p‘(𝐸 ↾s 𝐹)) ∧ ((𝑂‘𝑝)‘𝑥) = 0 )) → (𝑂‘𝑝):(Base‘𝐸)⟶(Base‘𝐸))
45	44	ffnd 6718	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 IntgRing 𝐹)) ∧ (𝑝 ∈ (Monic1p‘(𝐸 ↾s 𝐹)) ∧ ((𝑂‘𝑝)‘𝑥) = 0 )) → (𝑂‘𝑝) Fn (Base‘𝐸))
46	12	simpld 494	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐸 IntgRing 𝐹)) → 𝑥 ∈ (Base‘𝐸))
47	46	adantr 480	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 IntgRing 𝐹)) ∧ (𝑝 ∈ (Monic1p‘(𝐸 ↾s 𝐹)) ∧ ((𝑂‘𝑝)‘𝑥) = 0 )) → 𝑥 ∈ (Base‘𝐸))
48		simprr 770	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 IntgRing 𝐹)) ∧ (𝑝 ∈ (Monic1p‘(𝐸 ↾s 𝐹)) ∧ ((𝑂‘𝑝)‘𝑥) = 0 )) → ((𝑂‘𝑝)‘𝑥) = 0 )
49		fniniseg 7061	. . . . . . 7 ⊢ ((𝑂‘𝑝) Fn (Base‘𝐸) → (𝑥 ∈ (◡(𝑂‘𝑝) “ { 0 }) ↔ (𝑥 ∈ (Base‘𝐸) ∧ ((𝑂‘𝑝)‘𝑥) = 0 )))
50	49	biimpar 477	. . . . . 6 ⊢ (((𝑂‘𝑝) Fn (Base‘𝐸) ∧ (𝑥 ∈ (Base‘𝐸) ∧ ((𝑂‘𝑝)‘𝑥) = 0 )) → 𝑥 ∈ (◡(𝑂‘𝑝) “ { 0 }))
51	45, 47, 48, 50	syl12anc 834	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 IntgRing 𝐹)) ∧ (𝑝 ∈ (Monic1p‘(𝐸 ↾s 𝐹)) ∧ ((𝑂‘𝑝)‘𝑥) = 0 )) → 𝑥 ∈ (◡(𝑂‘𝑝) “ { 0 }))
52	13, 38, 51	reximssdv 3171	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐸 IntgRing 𝐹)) → ∃𝑝 ∈ (dom 𝑂 ∖ {𝑍})𝑥 ∈ (◡(𝑂‘𝑝) “ { 0 }))
53		eliun 5001	. . . 4 ⊢ (𝑥 ∈ ∪ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})(◡(𝑂‘𝑝) “ { 0 }) ↔ ∃𝑝 ∈ (dom 𝑂 ∖ {𝑍})𝑥 ∈ (◡(𝑂‘𝑝) “ { 0 }))
54	52, 53	sylibr 233	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐸 IntgRing 𝐹)) → 𝑥 ∈ ∪ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})(◡(𝑂‘𝑝) “ { 0 }))
55		nfv 1916	. . . . 5 ⊢ Ⅎ𝑝𝜑
56		nfiu1 5031	. . . . . 6 ⊢ Ⅎ𝑝∪ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})(◡(𝑂‘𝑝) “ { 0 })
57	56	nfcri 2889	. . . . 5 ⊢ Ⅎ𝑝 𝑥 ∈ ∪ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})(◡(𝑂‘𝑝) “ { 0 })
58	55, 57	nfan 1901	. . . 4 ⊢ Ⅎ𝑝(𝜑 ∧ 𝑥 ∈ ∪ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})(◡(𝑂‘𝑝) “ { 0 }))
59	5	ad2antrr 723	. . . . . 6 ⊢ (((𝜑 ∧ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})) ∧ 𝑥 ∈ (◡(𝑂‘𝑝) “ { 0 })) → 𝐸 ∈ Field)
60	7	ad2antrr 723	. . . . . 6 ⊢ (((𝜑 ∧ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})) ∧ 𝑥 ∈ (◡(𝑂‘𝑝) “ { 0 })) → 𝐹 ∈ (SubDRing‘𝐸))
61		eldifi 4126	. . . . . . . 8 ⊢ (𝑝 ∈ (dom 𝑂 ∖ {𝑍}) → 𝑝 ∈ dom 𝑂)
62	61	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})) → 𝑝 ∈ dom 𝑂)
63	62	adantr 480	. . . . . 6 ⊢ (((𝜑 ∧ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})) ∧ 𝑥 ∈ (◡(𝑂‘𝑝) “ { 0 })) → 𝑝 ∈ dom 𝑂)
64		eldifsni 4793	. . . . . . . 8 ⊢ (𝑝 ∈ (dom 𝑂 ∖ {𝑍}) → 𝑝 ≠ 𝑍)
65	64	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})) → 𝑝 ≠ 𝑍)
66	65	adantr 480	. . . . . 6 ⊢ (((𝜑 ∧ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})) ∧ 𝑥 ∈ (◡(𝑂‘𝑝) “ { 0 })) → 𝑝 ≠ 𝑍)
67	5	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})) → 𝐸 ∈ Field)
68		fvexd 6906	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})) → (Base‘𝐸) ∈ V)
69	24	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})) → 𝑂:(Base‘(Poly1‘(𝐸 ↾s 𝐹)))⟶(Base‘(𝐸 ↑s (Base‘𝐸))))
70	25	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})) → dom 𝑂 = (Base‘(Poly1‘(𝐸 ↾s 𝐹))))
71	62, 70	eleqtrd 2834	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})) → 𝑝 ∈ (Base‘(Poly1‘(𝐸 ↾s 𝐹))))
72	69, 71	ffvelcdmd 7087	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})) → (𝑂‘𝑝) ∈ (Base‘(𝐸 ↑s (Base‘𝐸))))
73	19, 3, 22, 67, 68, 72	pwselbas 17440	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})) → (𝑂‘𝑝):(Base‘𝐸)⟶(Base‘𝐸))
74	73	ffnd 6718	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})) → (𝑂‘𝑝) Fn (Base‘𝐸))
75	49	biimpa 476	. . . . . . . 8 ⊢ (((𝑂‘𝑝) Fn (Base‘𝐸) ∧ 𝑥 ∈ (◡(𝑂‘𝑝) “ { 0 })) → (𝑥 ∈ (Base‘𝐸) ∧ ((𝑂‘𝑝)‘𝑥) = 0 ))
76	74, 75	sylan 579	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})) ∧ 𝑥 ∈ (◡(𝑂‘𝑝) “ { 0 })) → (𝑥 ∈ (Base‘𝐸) ∧ ((𝑂‘𝑝)‘𝑥) = 0 ))
77	76	simprd 495	. . . . . 6 ⊢ (((𝜑 ∧ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})) ∧ 𝑥 ∈ (◡(𝑂‘𝑝) “ { 0 })) → ((𝑂‘𝑝)‘𝑥) = 0 )
78	76	simpld 494	. . . . . 6 ⊢ (((𝜑 ∧ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})) ∧ 𝑥 ∈ (◡(𝑂‘𝑝) “ { 0 })) → 𝑥 ∈ (Base‘𝐸))
79	1, 32, 4, 59, 60, 3, 63, 66, 77, 78	irngnzply1lem 33044	. . . . 5 ⊢ (((𝜑 ∧ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})) ∧ 𝑥 ∈ (◡(𝑂‘𝑝) “ { 0 })) → 𝑥 ∈ (𝐸 IntgRing 𝐹))
80	79	adantllr 716	. . . 4 ⊢ ((((𝜑 ∧ 𝑥 ∈ ∪ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})(◡(𝑂‘𝑝) “ { 0 })) ∧ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})) ∧ 𝑥 ∈ (◡(𝑂‘𝑝) “ { 0 })) → 𝑥 ∈ (𝐸 IntgRing 𝐹))
81	53	biimpi 215	. . . . 5 ⊢ (𝑥 ∈ ∪ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})(◡(𝑂‘𝑝) “ { 0 }) → ∃𝑝 ∈ (dom 𝑂 ∖ {𝑍})𝑥 ∈ (◡(𝑂‘𝑝) “ { 0 }))
82	81	adantl 481	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})(◡(𝑂‘𝑝) “ { 0 })) → ∃𝑝 ∈ (dom 𝑂 ∖ {𝑍})𝑥 ∈ (◡(𝑂‘𝑝) “ { 0 }))
83	58, 80, 82	r19.29af 3264	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})(◡(𝑂‘𝑝) “ { 0 })) → 𝑥 ∈ (𝐸 IntgRing 𝐹))
84	54, 83	impbida 798	. 2 ⊢ (𝜑 → (𝑥 ∈ (𝐸 IntgRing 𝐹) ↔ 𝑥 ∈ ∪ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})(◡(𝑂‘𝑝) “ { 0 })))
85	84	eqrdv 2729	1 ⊢ (𝜑 → (𝐸 IntgRing 𝐹) = ∪ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})(◡(𝑂‘𝑝) “ { 0 }))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105   ≠ wne 2939  ∃wrex 3069  Vcvv 3473   ∖ cdif 3945  {csn 4628  ∪ ciun 4997  ^◡ccnv 5675  dom cdm 5676   “ cima 5679   Fn wfn 6538  ⟶wf 6539  ‘cfv 6543  (class class class)co 7412  Basecbs 17149   ↾_s cress 17178  0_gc0g 17390   ↑_s cpws 17397  CRingccrg 20129   RingHom crh 20361  SubRingcsubrg 20458  DivRingcdr 20501  Fieldcfield 20502  SubDRingcsdrg 20546  Poly₁cpl1 21921   evalSub₁ ces1 22053  Monic_1pcmn1 25879   IntgRing cirng 33037

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  ax-cnex 11170  ax-resscn 11171  ax-1cn 11172  ax-icn 11173  ax-addcl 11174  ax-addrcl 11175  ax-mulcl 11176  ax-mulrcl 11177  ax-mulcom 11178  ax-addass 11179  ax-mulass 11180  ax-distr 11181  ax-i2m1 11182  ax-1ne0 11183  ax-1rid 11184  ax-rnegex 11185  ax-rrecex 11186  ax-cnre 11187  ax-pre-lttri 11188  ax-pre-lttrn 11189  ax-pre-ltadd 11190  ax-pre-mulgt0 11191  ax-addf 11193  ax-mulf 11194

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  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-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  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-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-of 7674  df-ofr 7675  df-om 7860  df-1st 7979  df-2nd 7980  df-supp 8151  df-tpos 8215  df-frecs 8270  df-wrecs 8301  df-recs 8375  df-rdg 8414  df-1o 8470  df-er 8707  df-map 8826  df-pm 8827  df-ixp 8896  df-en 8944  df-dom 8945  df-sdom 8946  df-fin 8947  df-fsupp 9366  df-sup 9441  df-oi 9509  df-card 9938  df-pnf 11255  df-mnf 11256  df-xr 11257  df-ltxr 11258  df-le 11259  df-sub 11451  df-neg 11452  df-nn 12218  df-2 12280  df-3 12281  df-4 12282  df-5 12283  df-6 12284  df-7 12285  df-8 12286  df-9 12287  df-n0 12478  df-z 12564  df-dec 12683  df-uz 12828  df-fz 13490  df-fzo 13633  df-seq 13972  df-hash 14296  df-struct 17085  df-sets 17102  df-slot 17120  df-ndx 17132  df-base 17150  df-ress 17179  df-plusg 17215  df-mulr 17216  df-starv 17217  df-sca 17218  df-vsca 17219  df-ip 17220  df-tset 17221  df-ple 17222  df-ds 17224  df-unif 17225  df-hom 17226  df-cco 17227  df-0g 17392  df-gsum 17393  df-prds 17398  df-pws 17400  df-mre 17535  df-mrc 17536  df-acs 17538  df-mgm 18566  df-sgrp 18645  df-mnd 18661  df-mhm 18706  df-submnd 18707  df-grp 18859  df-minusg 18860  df-sbg 18861  df-mulg 18988  df-subg 19040  df-ghm 19129  df-cntz 19223  df-cmn 19692  df-abl 19693  df-mgp 20030  df-rng 20048  df-ur 20077  df-srg 20082  df-ring 20130  df-cring 20131  df-oppr 20226  df-dvdsr 20249  df-unit 20250  df-invr 20280  df-rhm 20364  df-subrng 20435  df-subrg 20460  df-drng 20503  df-field 20504  df-sdrg 20547  df-lmod 20617  df-lss 20688  df-lsp 20728  df-rlreg 21100  df-cnfld 21146  df-assa 21628  df-asp 21629  df-ascl 21630  df-psr 21682  df-mvr 21683  df-mpl 21684  df-opsr 21686  df-evls 21855  df-evl 21856  df-psr1 21924  df-vr1 21925  df-ply1 21926  df-coe1 21927  df-evls1 22055  df-evl1 22056  df-mdeg 25806  df-deg1 25807  df-mon1 25884  df-uc1p 25885  df-irng 33038

This theorem is referenced by:  irngnminplynz  33061