Theorem irngnzply1 33686

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 20651	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ CRing)
7		irngnzply1.f	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))
8		issdrg 20697	. . . . . . . . . 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 33681	. . . . . . 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 26050	. . . . . . . . 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 22209	. . . . . . . . . . . 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 20394	. . . . . . . . . . 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 6698	. . . . . . . . 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 26052	. . . . . . . . 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 33536	. . . . . . . . 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 4750	. . . . . . 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 6873	. . . . . . . 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 7057	. . . . . . . 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 17452	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 IntgRing 𝐹)) ∧ (𝑝 ∈ (Monic1p‘(𝐸 ↾s 𝐹)) ∧ ((𝑂‘𝑝)‘𝑥) = 0 )) → (𝑂‘𝑝):(Base‘𝐸)⟶(Base‘𝐸))
45	44	ffnd 6689	. . . . . 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 7032	. . . . . . 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 4959	. . . 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 4991	. . . . . 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 4094	. . . . . . . 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 4754	. . . . . . . 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 6873	. . . . . . . . . 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 7057	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})) → (𝑂‘𝑝) ∈ (Base‘(𝐸 ↑s (Base‘𝐸))))
73	19, 3, 22, 67, 68, 72	pwselbas 17452	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})) → (𝑂‘𝑝):(Base‘𝐸)⟶(Base‘𝐸))
74	73	ffnd 6689	. . . . . . . 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 33685	. . . . 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 3246	. . 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 3447   ∖ cdif 3911  {csn 4589  ∪ ciun 4955  ^◡ccnv 5637  dom cdm 5638   “ cima 5641   Fn wfn 6506  ⟶wf 6507  ‘cfv 6511  (class class class)co 7387  Basecbs 17179   ↾_s cress 17200  0_gc0g 17402   ↑_s cpws 17409  CRingccrg 20143   RingHom crh 20378  SubRingcsubrg 20478  DivRingcdr 20638  Fieldcfield 20639  SubDRingcsdrg 20695  Poly₁cpl1 22061   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  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145  ax-addf 11147

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 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-iin 4958  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-se 5592  df-we 5593  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-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-isom 6520  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-of 7653  df-ofr 7654  df-om 7843  df-1st 7968  df-2nd 7969  df-supp 8140  df-tpos 8205  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-2o 8435  df-er 8671  df-map 8801  df-pm 8802  df-ixp 8871  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-fsupp 9313  df-sup 9393  df-oi 9463  df-card 9892  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-nn 12187  df-2 12249  df-3 12250  df-4 12251  df-5 12252  df-6 12253  df-7 12254  df-8 12255  df-9 12256  df-n0 12443  df-z 12530  df-dec 12650  df-uz 12794  df-fz 13469  df-fzo 13616  df-seq 13967  df-hash 14296  df-struct 17117  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-ress 17201  df-plusg 17233  df-mulr 17234  df-starv 17235  df-sca 17236  df-vsca 17237  df-ip 17238  df-tset 17239  df-ple 17240  df-ds 17242  df-unif 17243  df-hom 17244  df-cco 17245  df-0g 17404  df-gsum 17405  df-prds 17410  df-pws 17412  df-mre 17547  df-mrc 17548  df-acs 17550  df-mgm 18567  df-sgrp 18646  df-mnd 18662  df-mhm 18710  df-submnd 18711  df-grp 18868  df-minusg 18869  df-sbg 18870  df-mulg 19000  df-subg 19055  df-ghm 19145  df-cntz 19249  df-cmn 19712  df-abl 19713  df-mgp 20050  df-rng 20062  df-ur 20091  df-srg 20096  df-ring 20144  df-cring 20145  df-oppr 20246  df-dvdsr 20266  df-unit 20267  df-invr 20297  df-rhm 20381  df-subrng 20455  df-subrg 20479  df-rlreg 20603  df-drng 20640  df-field 20641  df-sdrg 20696  df-lmod 20768  df-lss 20838  df-lsp 20878  df-cnfld 21265  df-assa 21762  df-asp 21763  df-ascl 21764  df-psr 21818  df-mvr 21819  df-mpl 21820  df-opsr 21822  df-evls 21981  df-evl 21982  df-psr1 22064  df-vr1 22065  df-ply1 22066  df-coe1 22067  df-evls1 22202  df-evl1 22203  df-mdeg 25960  df-deg1 25961  df-mon1 26036  df-uc1p 26037  df-irng 33679

This theorem is referenced by:  irngnminplynz  33702