Theorem irngnzply1 33680

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 2729	. . . . . . . 8 ⊢ (𝐸 ↾s 𝐹) = (𝐸 ↾s 𝐹)
3		eqid 2729	. . . . . . . 8 ⊢ (Base‘𝐸) = (Base‘𝐸)
4		irngnzply1.1	. . . . . . . 8 ⊢ 0 = (0g‘𝐸)
5		irngnzply1.e	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ Field)
6	5	fldcrngd 20663	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ CRing)
7		irngnzply1.f	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))
8		issdrg 20709	. . . . . . . . . 10 ⊢ (𝐹 ∈ (SubDRing‘𝐸) ↔ (𝐸 ∈ DivRing ∧ 𝐹 ∈ (SubRing‘𝐸) ∧ (𝐸 ↾s 𝐹) ∈ DivRing))
9	7, 8	sylib 218	. . . . . . . . 9 ⊢ (𝜑 → (𝐸 ∈ DivRing ∧ 𝐹 ∈ (SubRing‘𝐸) ∧ (𝐸 ↾s 𝐹) ∈ DivRing))
10	9	simp2d 1143	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ (SubRing‘𝐸))
11	1, 2, 3, 4, 6, 10	elirng 33675	. . . . . . 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 2729	. . . . . . . . . 10 ⊢ (Poly1‘(𝐸 ↾s 𝐹)) = (Poly1‘(𝐸 ↾s 𝐹))
15		eqid 2729	. . . . . . . . . 10 ⊢ (Base‘(Poly1‘(𝐸 ↾s 𝐹))) = (Base‘(Poly1‘(𝐸 ↾s 𝐹)))
16		eqid 2729	. . . . . . . . . 10 ⊢ (Monic1p‘(𝐸 ↾s 𝐹)) = (Monic1p‘(𝐸 ↾s 𝐹))
17	14, 15, 16	mon1pcl 26084	. . . . . . . . 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 2729	. . . . . . . . . . . . 13 ⊢ (𝐸 ↑s (Base‘𝐸)) = (𝐸 ↑s (Base‘𝐸))
20	1, 3, 19, 2, 14	evls1rhm 22243	. . . . . . . . . . . 12 ⊢ ((𝐸 ∈ CRing ∧ 𝐹 ∈ (SubRing‘𝐸)) → 𝑂 ∈ ((Poly1‘(𝐸 ↾s 𝐹)) RingHom (𝐸 ↑s (Base‘𝐸))))
21	6, 10, 20	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑂 ∈ ((Poly1‘(𝐸 ↾s 𝐹)) RingHom (𝐸 ↑s (Base‘𝐸))))
22		eqid 2729	. . . . . . . . . . . 12 ⊢ (Base‘(𝐸 ↑s (Base‘𝐸))) = (Base‘(𝐸 ↑s (Base‘𝐸)))
23	15, 22	rhmf 20406	. . . . . . . . . . 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 6680	. . . . . . . . 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 2831	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ (Monic1p‘(𝐸 ↾s 𝐹))) → 𝑝 ∈ dom 𝑂)
28		eqid 2729	. . . . . . . . . 10 ⊢ (0g‘(Poly1‘(𝐸 ↾s 𝐹))) = (0g‘(Poly1‘(𝐸 ↾s 𝐹)))
29	14, 28, 16	mon1pn0 26086	. . . . . . . . 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 2729	. . . . . . . . . 10 ⊢ (Poly1‘𝐸) = (Poly1‘𝐸)
32		irngnzply1.z	. . . . . . . . . 10 ⊢ 𝑍 = (0g‘(Poly1‘𝐸))
33	31, 2, 14, 15, 10, 32	ressply10g 33530	. . . . . . . . 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 2999	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ (Monic1p‘(𝐸 ↾s 𝐹))) → 𝑝 ≠ 𝑍)
36		eldifsn 4746	. . . . . . 7 ⊢ (𝑝 ∈ (dom 𝑂 ∖ {𝑍}) ↔ (𝑝 ∈ dom 𝑂 ∧ 𝑝 ≠ 𝑍))
37	27, 35, 36	sylanbrc 583	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ (Monic1p‘(𝐸 ↾s 𝐹))) → 𝑝 ∈ (dom 𝑂 ∖ {𝑍}))
38	37	ad2ant2r 747	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 IntgRing 𝐹)) ∧ (𝑝 ∈ (Monic1p‘(𝐸 ↾s 𝐹)) ∧ ((𝑂‘𝑝)‘𝑥) = 0 )) → 𝑝 ∈ (dom 𝑂 ∖ {𝑍}))
39	5	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 IntgRing 𝐹)) ∧ (𝑝 ∈ (Monic1p‘(𝐸 ↾s 𝐹)) ∧ ((𝑂‘𝑝)‘𝑥) = 0 )) → 𝐸 ∈ Field)
40		fvexd 6855	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 IntgRing 𝐹)) ∧ (𝑝 ∈ (Monic1p‘(𝐸 ↾s 𝐹)) ∧ ((𝑂‘𝑝)‘𝑥) = 0 )) → (Base‘𝐸) ∈ V)
41	24	ad2antrr 726	. . . . . . . . 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 728	. . . . . . . . 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 7039	. . . . . . . 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 17429	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 IntgRing 𝐹)) ∧ (𝑝 ∈ (Monic1p‘(𝐸 ↾s 𝐹)) ∧ ((𝑂‘𝑝)‘𝑥) = 0 )) → (𝑂‘𝑝):(Base‘𝐸)⟶(Base‘𝐸))
45	44	ffnd 6671	. . . . . 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 772	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 IntgRing 𝐹)) ∧ (𝑝 ∈ (Monic1p‘(𝐸 ↾s 𝐹)) ∧ ((𝑂‘𝑝)‘𝑥) = 0 )) → ((𝑂‘𝑝)‘𝑥) = 0 )
49		fniniseg 7014	. . . . . . 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 836	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 IntgRing 𝐹)) ∧ (𝑝 ∈ (Monic1p‘(𝐸 ↾s 𝐹)) ∧ ((𝑂‘𝑝)‘𝑥) = 0 )) → 𝑥 ∈ (◡(𝑂‘𝑝) “ { 0 }))
52	13, 38, 51	reximssdv 3151	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐸 IntgRing 𝐹)) → ∃𝑝 ∈ (dom 𝑂 ∖ {𝑍})𝑥 ∈ (◡(𝑂‘𝑝) “ { 0 }))
53		eliun 4955	. . . 4 ⊢ (𝑥 ∈ ∪ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})(◡(𝑂‘𝑝) “ { 0 }) ↔ ∃𝑝 ∈ (dom 𝑂 ∖ {𝑍})𝑥 ∈ (◡(𝑂‘𝑝) “ { 0 }))
54	52, 53	sylibr 234	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐸 IntgRing 𝐹)) → 𝑥 ∈ ∪ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})(◡(𝑂‘𝑝) “ { 0 }))
55		nfv 1914	. . . . 5 ⊢ Ⅎ𝑝𝜑
56		nfiu1 4987	. . . . . 6 ⊢ Ⅎ𝑝∪ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})(◡(𝑂‘𝑝) “ { 0 })
57	56	nfcri 2883	. . . . 5 ⊢ Ⅎ𝑝 𝑥 ∈ ∪ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})(◡(𝑂‘𝑝) “ { 0 })
58	55, 57	nfan 1899	. . . 4 ⊢ Ⅎ𝑝(𝜑 ∧ 𝑥 ∈ ∪ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})(◡(𝑂‘𝑝) “ { 0 }))
59	5	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})) ∧ 𝑥 ∈ (◡(𝑂‘𝑝) “ { 0 })) → 𝐸 ∈ Field)
60	7	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})) ∧ 𝑥 ∈ (◡(𝑂‘𝑝) “ { 0 })) → 𝐹 ∈ (SubDRing‘𝐸))
61		eldifi 4090	. . . . . . . 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 4750	. . . . . . . 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 6855	. . . . . . . . . 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 2830	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})) → 𝑝 ∈ (Base‘(Poly1‘(𝐸 ↾s 𝐹))))
72	69, 71	ffvelcdmd 7039	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})) → (𝑂‘𝑝) ∈ (Base‘(𝐸 ↑s (Base‘𝐸))))
73	19, 3, 22, 67, 68, 72	pwselbas 17429	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})) → (𝑂‘𝑝):(Base‘𝐸)⟶(Base‘𝐸))
74	73	ffnd 6671	. . . . . . . 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 580	. . . . . . 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 33679	. . . . 5 ⊢ (((𝜑 ∧ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})) ∧ 𝑥 ∈ (◡(𝑂‘𝑝) “ { 0 })) → 𝑥 ∈ (𝐸 IntgRing 𝐹))
80	79	adantllr 719	. . . 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 216	. . . . 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 3244	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})(◡(𝑂‘𝑝) “ { 0 })) → 𝑥 ∈ (𝐸 IntgRing 𝐹))
84	54, 83	impbida 800	. 2 ⊢ (𝜑 → (𝑥 ∈ (𝐸 IntgRing 𝐹) ↔ 𝑥 ∈ ∪ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})(◡(𝑂‘𝑝) “ { 0 })))
85	84	eqrdv 2727	1 ⊢ (𝜑 → (𝐸 IntgRing 𝐹) = ∪ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})(◡(𝑂‘𝑝) “ { 0 }))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∃wrex 3053  Vcvv 3444   ∖ cdif 3908  {csn 4585  ∪ ciun 4951  ^◡ccnv 5630  dom cdm 5631   “ cima 5634   Fn wfn 6494  ⟶wf 6495  ‘cfv 6499  (class class class)co 7369  Basecbs 17156   ↾_s cress 17177  0_gc0g 17379   ↑_s cpws 17386  CRingccrg 20155   RingHom crh 20390  SubRingcsubrg 20490  DivRingcdr 20650  Fieldcfield 20651  SubDRingcsdrg 20707  Poly₁cpl1 22095   evalSub₁ ces1 22234  Monic_1pcmn1 26065   IntgRing cirng 33672

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 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-cnex 11102  ax-resscn 11103  ax-1cn 11104  ax-icn 11105  ax-addcl 11106  ax-addrcl 11107  ax-mulcl 11108  ax-mulrcl 11109  ax-mulcom 11110  ax-addass 11111  ax-mulass 11112  ax-distr 11113  ax-i2m1 11114  ax-1ne0 11115  ax-1rid 11116  ax-rnegex 11117  ax-rrecex 11118  ax-cnre 11119  ax-pre-lttri 11120  ax-pre-lttrn 11121  ax-pre-ltadd 11122  ax-pre-mulgt0 11123  ax-addf 11125

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  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-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-uni 4868  df-int 4907  df-iun 4953  df-iin 4954  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-isom 6508  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-of 7633  df-ofr 7634  df-om 7823  df-1st 7947  df-2nd 7948  df-supp 8117  df-tpos 8182  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-1o 8411  df-2o 8412  df-er 8648  df-map 8778  df-pm 8779  df-ixp 8848  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-fsupp 9289  df-sup 9369  df-oi 9439  df-card 9870  df-pnf 11188  df-mnf 11189  df-xr 11190  df-ltxr 11191  df-le 11192  df-sub 11385  df-neg 11386  df-nn 12165  df-2 12227  df-3 12228  df-4 12229  df-5 12230  df-6 12231  df-7 12232  df-8 12233  df-9 12234  df-n0 12421  df-z 12508  df-dec 12628  df-uz 12772  df-fz 13447  df-fzo 13594  df-seq 13945  df-hash 14274  df-struct 17094  df-sets 17111  df-slot 17129  df-ndx 17141  df-base 17157  df-ress 17178  df-plusg 17210  df-mulr 17211  df-starv 17212  df-sca 17213  df-vsca 17214  df-ip 17215  df-tset 17216  df-ple 17217  df-ds 17219  df-unif 17220  df-hom 17221  df-cco 17222  df-0g 17381  df-gsum 17382  df-prds 17387  df-pws 17389  df-mre 17524  df-mrc 17525  df-acs 17527  df-mgm 18550  df-sgrp 18629  df-mnd 18645  df-mhm 18693  df-submnd 18694  df-grp 18851  df-minusg 18852  df-sbg 18853  df-mulg 18983  df-subg 19038  df-ghm 19128  df-cntz 19232  df-cmn 19697  df-abl 19698  df-mgp 20062  df-rng 20074  df-ur 20103  df-srg 20108  df-ring 20156  df-cring 20157  df-oppr 20258  df-dvdsr 20278  df-unit 20279  df-invr 20309  df-rhm 20393  df-subrng 20467  df-subrg 20491  df-rlreg 20615  df-drng 20652  df-field 20653  df-sdrg 20708  df-lmod 20801  df-lss 20871  df-lsp 20911  df-cnfld 21298  df-assa 21796  df-asp 21797  df-ascl 21798  df-psr 21852  df-mvr 21853  df-mpl 21854  df-opsr 21856  df-evls 22015  df-evl 22016  df-psr1 22098  df-vr1 22099  df-ply1 22100  df-coe1 22101  df-evls1 22236  df-evl1 22237  df-mdeg 25994  df-deg1 25995  df-mon1 26070  df-uc1p 26071  df-irng 33673

This theorem is referenced by:  irngnminplynz  33696