Theorem irngnzply1 32657

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 2732	. . . . . . . 8 ⊢ (𝐸 ↾s 𝐹) = (𝐸 ↾s 𝐹)
3		eqid 2732	. . . . . . . 8 ⊢ (Base‘𝐸) = (Base‘𝐸)
4		irngnzply1.1	. . . . . . . 8 ⊢ 0 = (0g‘𝐸)
5		irngnzply1.e	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ Field)
6	5	fldcrngd 20279	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ CRing)
7		irngnzply1.f	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))
8		issdrg 20355	. . . . . . . . . 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 1143	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ (SubRing‘𝐸))
11	1, 2, 3, 4, 6, 10	elirng 32652	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (𝐸 IntgRing 𝐹) ↔ (𝑥 ∈ (Base‘𝐸) ∧ ∃𝑝 ∈ (Monic1p‘(𝐸 ↾s 𝐹))((𝑂‘𝑝)‘𝑥) = 0 )))
12	11	biimpa 477	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐸 IntgRing 𝐹)) → (𝑥 ∈ (Base‘𝐸) ∧ ∃𝑝 ∈ (Monic1p‘(𝐸 ↾s 𝐹))((𝑂‘𝑝)‘𝑥) = 0 ))
13	12	simprd 496	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐸 IntgRing 𝐹)) → ∃𝑝 ∈ (Monic1p‘(𝐸 ↾s 𝐹))((𝑂‘𝑝)‘𝑥) = 0 )
14		eqid 2732	. . . . . . . . . 10 ⊢ (Poly1‘(𝐸 ↾s 𝐹)) = (Poly1‘(𝐸 ↾s 𝐹))
15		eqid 2732	. . . . . . . . . 10 ⊢ (Base‘(Poly1‘(𝐸 ↾s 𝐹))) = (Base‘(Poly1‘(𝐸 ↾s 𝐹)))
16		eqid 2732	. . . . . . . . . 10 ⊢ (Monic1p‘(𝐸 ↾s 𝐹)) = (Monic1p‘(𝐸 ↾s 𝐹))
17	14, 15, 16	mon1pcl 25593	. . . . . . . . 9 ⊢ (𝑝 ∈ (Monic1p‘(𝐸 ↾s 𝐹)) → 𝑝 ∈ (Base‘(Poly1‘(𝐸 ↾s 𝐹))))
18	17	adantl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ (Monic1p‘(𝐸 ↾s 𝐹))) → 𝑝 ∈ (Base‘(Poly1‘(𝐸 ↾s 𝐹))))
19		eqid 2732	. . . . . . . . . . . . 13 ⊢ (𝐸 ↑s (Base‘𝐸)) = (𝐸 ↑s (Base‘𝐸))
20	1, 3, 19, 2, 14	evls1rhm 21772	. . . . . . . . . . . 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 2732	. . . . . . . . . . . 12 ⊢ (Base‘(𝐸 ↑s (Base‘𝐸))) = (Base‘(𝐸 ↑s (Base‘𝐸)))
23	15, 22	rhmf 20215	. . . . . . . . . . 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 6716	. . . . . . . . 9 ⊢ (𝜑 → dom 𝑂 = (Base‘(Poly1‘(𝐸 ↾s 𝐹))))
26	25	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ (Monic1p‘(𝐸 ↾s 𝐹))) → dom 𝑂 = (Base‘(Poly1‘(𝐸 ↾s 𝐹))))
27	18, 26	eleqtrrd 2836	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ (Monic1p‘(𝐸 ↾s 𝐹))) → 𝑝 ∈ dom 𝑂)
28		eqid 2732	. . . . . . . . . 10 ⊢ (0g‘(Poly1‘(𝐸 ↾s 𝐹))) = (0g‘(Poly1‘(𝐸 ↾s 𝐹)))
29	14, 28, 16	mon1pn0 25595	. . . . . . . . 9 ⊢ (𝑝 ∈ (Monic1p‘(𝐸 ↾s 𝐹)) → 𝑝 ≠ (0g‘(Poly1‘(𝐸 ↾s 𝐹))))
30	29	adantl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ (Monic1p‘(𝐸 ↾s 𝐹))) → 𝑝 ≠ (0g‘(Poly1‘(𝐸 ↾s 𝐹))))
31		eqid 2732	. . . . . . . . . 10 ⊢ (Poly1‘𝐸) = (Poly1‘𝐸)
32		irngnzply1.z	. . . . . . . . . 10 ⊢ 𝑍 = (0g‘(Poly1‘𝐸))
33	31, 2, 14, 15, 10, 32	ressply10g 32560	. . . . . . . . 9 ⊢ (𝜑 → 𝑍 = (0g‘(Poly1‘(𝐸 ↾s 𝐹))))
34	33	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ (Monic1p‘(𝐸 ↾s 𝐹))) → 𝑍 = (0g‘(Poly1‘(𝐸 ↾s 𝐹))))
35	30, 34	neeqtrrd 3015	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ (Monic1p‘(𝐸 ↾s 𝐹))) → 𝑝 ≠ 𝑍)
36		eldifsn 4784	. . . . . . 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 745	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 IntgRing 𝐹)) ∧ (𝑝 ∈ (Monic1p‘(𝐸 ↾s 𝐹)) ∧ ((𝑂‘𝑝)‘𝑥) = 0 )) → 𝑝 ∈ (dom 𝑂 ∖ {𝑍}))
39	5	ad2antrr 724	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 IntgRing 𝐹)) ∧ (𝑝 ∈ (Monic1p‘(𝐸 ↾s 𝐹)) ∧ ((𝑂‘𝑝)‘𝑥) = 0 )) → 𝐸 ∈ Field)
40		fvexd 6894	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 IntgRing 𝐹)) ∧ (𝑝 ∈ (Monic1p‘(𝐸 ↾s 𝐹)) ∧ ((𝑂‘𝑝)‘𝑥) = 0 )) → (Base‘𝐸) ∈ V)
41	24	ad2antrr 724	. . . . . . . . 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 726	. . . . . . . . 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 7073	. . . . . . . 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 17419	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 IntgRing 𝐹)) ∧ (𝑝 ∈ (Monic1p‘(𝐸 ↾s 𝐹)) ∧ ((𝑂‘𝑝)‘𝑥) = 0 )) → (𝑂‘𝑝):(Base‘𝐸)⟶(Base‘𝐸))
45	44	ffnd 6706	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 IntgRing 𝐹)) ∧ (𝑝 ∈ (Monic1p‘(𝐸 ↾s 𝐹)) ∧ ((𝑂‘𝑝)‘𝑥) = 0 )) → (𝑂‘𝑝) Fn (Base‘𝐸))
46	12	simpld 495	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐸 IntgRing 𝐹)) → 𝑥 ∈ (Base‘𝐸))
47	46	adantr 481	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 IntgRing 𝐹)) ∧ (𝑝 ∈ (Monic1p‘(𝐸 ↾s 𝐹)) ∧ ((𝑂‘𝑝)‘𝑥) = 0 )) → 𝑥 ∈ (Base‘𝐸))
48		simprr 771	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 IntgRing 𝐹)) ∧ (𝑝 ∈ (Monic1p‘(𝐸 ↾s 𝐹)) ∧ ((𝑂‘𝑝)‘𝑥) = 0 )) → ((𝑂‘𝑝)‘𝑥) = 0 )
49		fniniseg 7047	. . . . . . 7 ⊢ ((𝑂‘𝑝) Fn (Base‘𝐸) → (𝑥 ∈ (◡(𝑂‘𝑝) “ { 0 }) ↔ (𝑥 ∈ (Base‘𝐸) ∧ ((𝑂‘𝑝)‘𝑥) = 0 )))
50	49	biimpar 478	. . . . . 6 ⊢ (((𝑂‘𝑝) Fn (Base‘𝐸) ∧ (𝑥 ∈ (Base‘𝐸) ∧ ((𝑂‘𝑝)‘𝑥) = 0 )) → 𝑥 ∈ (◡(𝑂‘𝑝) “ { 0 }))
51	45, 47, 48, 50	syl12anc 835	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 IntgRing 𝐹)) ∧ (𝑝 ∈ (Monic1p‘(𝐸 ↾s 𝐹)) ∧ ((𝑂‘𝑝)‘𝑥) = 0 )) → 𝑥 ∈ (◡(𝑂‘𝑝) “ { 0 }))
52	13, 38, 51	reximssdv 3172	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐸 IntgRing 𝐹)) → ∃𝑝 ∈ (dom 𝑂 ∖ {𝑍})𝑥 ∈ (◡(𝑂‘𝑝) “ { 0 }))
53		eliun 4995	. . . 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 1917	. . . . 5 ⊢ Ⅎ𝑝𝜑
56		nfiu1 5025	. . . . . 6 ⊢ Ⅎ𝑝∪ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})(◡(𝑂‘𝑝) “ { 0 })
57	56	nfcri 2890	. . . . 5 ⊢ Ⅎ𝑝 𝑥 ∈ ∪ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})(◡(𝑂‘𝑝) “ { 0 })
58	55, 57	nfan 1902	. . . 4 ⊢ Ⅎ𝑝(𝜑 ∧ 𝑥 ∈ ∪ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})(◡(𝑂‘𝑝) “ { 0 }))
59	5	ad2antrr 724	. . . . . 6 ⊢ (((𝜑 ∧ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})) ∧ 𝑥 ∈ (◡(𝑂‘𝑝) “ { 0 })) → 𝐸 ∈ Field)
60	7	ad2antrr 724	. . . . . 6 ⊢ (((𝜑 ∧ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})) ∧ 𝑥 ∈ (◡(𝑂‘𝑝) “ { 0 })) → 𝐹 ∈ (SubDRing‘𝐸))
61		eldifi 4123	. . . . . . . 8 ⊢ (𝑝 ∈ (dom 𝑂 ∖ {𝑍}) → 𝑝 ∈ dom 𝑂)
62	61	adantl 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})) → 𝑝 ∈ dom 𝑂)
63	62	adantr 481	. . . . . 6 ⊢ (((𝜑 ∧ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})) ∧ 𝑥 ∈ (◡(𝑂‘𝑝) “ { 0 })) → 𝑝 ∈ dom 𝑂)
64		eldifsni 4787	. . . . . . . 8 ⊢ (𝑝 ∈ (dom 𝑂 ∖ {𝑍}) → 𝑝 ≠ 𝑍)
65	64	adantl 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})) → 𝑝 ≠ 𝑍)
66	65	adantr 481	. . . . . 6 ⊢ (((𝜑 ∧ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})) ∧ 𝑥 ∈ (◡(𝑂‘𝑝) “ { 0 })) → 𝑝 ≠ 𝑍)
67	5	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})) → 𝐸 ∈ Field)
68		fvexd 6894	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})) → (Base‘𝐸) ∈ V)
69	24	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})) → 𝑂:(Base‘(Poly1‘(𝐸 ↾s 𝐹)))⟶(Base‘(𝐸 ↑s (Base‘𝐸))))
70	25	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})) → dom 𝑂 = (Base‘(Poly1‘(𝐸 ↾s 𝐹))))
71	62, 70	eleqtrd 2835	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})) → 𝑝 ∈ (Base‘(Poly1‘(𝐸 ↾s 𝐹))))
72	69, 71	ffvelcdmd 7073	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})) → (𝑂‘𝑝) ∈ (Base‘(𝐸 ↑s (Base‘𝐸))))
73	19, 3, 22, 67, 68, 72	pwselbas 17419	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})) → (𝑂‘𝑝):(Base‘𝐸)⟶(Base‘𝐸))
74	73	ffnd 6706	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})) → (𝑂‘𝑝) Fn (Base‘𝐸))
75	49	biimpa 477	. . . . . . . 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 496	. . . . . 6 ⊢ (((𝜑 ∧ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})) ∧ 𝑥 ∈ (◡(𝑂‘𝑝) “ { 0 })) → ((𝑂‘𝑝)‘𝑥) = 0 )
78	76	simpld 495	. . . . . 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 32656	. . . . 5 ⊢ (((𝜑 ∧ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})) ∧ 𝑥 ∈ (◡(𝑂‘𝑝) “ { 0 })) → 𝑥 ∈ (𝐸 IntgRing 𝐹))
80	79	adantllr 717	. . . 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 482	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})(◡(𝑂‘𝑝) “ { 0 })) → ∃𝑝 ∈ (dom 𝑂 ∖ {𝑍})𝑥 ∈ (◡(𝑂‘𝑝) “ { 0 }))
83	58, 80, 82	r19.29af 3265	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})(◡(𝑂‘𝑝) “ { 0 })) → 𝑥 ∈ (𝐸 IntgRing 𝐹))
84	54, 83	impbida 799	. 2 ⊢ (𝜑 → (𝑥 ∈ (𝐸 IntgRing 𝐹) ↔ 𝑥 ∈ ∪ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})(◡(𝑂‘𝑝) “ { 0 })))
85	84	eqrdv 2730	1 ⊢ (𝜑 → (𝐸 IntgRing 𝐹) = ∪ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})(◡(𝑂‘𝑝) “ { 0 }))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   ≠ wne 2940  ∃wrex 3070  Vcvv 3474   ∖ cdif 3942  {csn 4623  ∪ ciun 4991  ^◡ccnv 5669  dom cdm 5670   “ cima 5673   Fn wfn 6528  ⟶wf 6529  ‘cfv 6533  (class class class)co 7394  Basecbs 17128   ↾_s cress 17157  0_gc0g 17369   ↑_s cpws 17376  CRingccrg 20017   RingHom crh 20200  DivRingcdr 20267  Fieldcfield 20268  SubRingcsubrg 20310  SubDRingcsdrg 20353  Poly₁cpl1 21632   evalSub₁ ces1 21763  Monic_1pcmn1 25574   IntgRing cirng 32649

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5357  ax-pr 5421  ax-un 7709  ax-cnex 11150  ax-resscn 11151  ax-1cn 11152  ax-icn 11153  ax-addcl 11154  ax-addrcl 11155  ax-mulcl 11156  ax-mulrcl 11157  ax-mulcom 11158  ax-addass 11159  ax-mulass 11160  ax-distr 11161  ax-i2m1 11162  ax-1ne0 11163  ax-1rid 11164  ax-rnegex 11165  ax-rrecex 11166  ax-cnre 11167  ax-pre-lttri 11168  ax-pre-lttrn 11169  ax-pre-ltadd 11170  ax-pre-mulgt0 11171  ax-addf 11173  ax-mulf 11174

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3775  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3964  df-nul 4320  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-tp 4628  df-op 4630  df-uni 4903  df-int 4945  df-iun 4993  df-iin 4994  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5568  df-eprel 5574  df-po 5582  df-so 5583  df-fr 5625  df-se 5626  df-we 5627  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-res 5682  df-ima 5683  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-isom 6542  df-riota 7350  df-ov 7397  df-oprab 7398  df-mpo 7399  df-of 7654  df-ofr 7655  df-om 7840  df-1st 7959  df-2nd 7960  df-supp 8131  df-tpos 8195  df-frecs 8250  df-wrecs 8281  df-recs 8355  df-rdg 8394  df-1o 8450  df-er 8688  df-map 8807  df-pm 8808  df-ixp 8877  df-en 8925  df-dom 8926  df-sdom 8927  df-fin 8928  df-fsupp 9347  df-sup 9421  df-oi 9489  df-card 9918  df-pnf 11234  df-mnf 11235  df-xr 11236  df-ltxr 11237  df-le 11238  df-sub 11430  df-neg 11431  df-nn 12197  df-2 12259  df-3 12260  df-4 12261  df-5 12262  df-6 12263  df-7 12264  df-8 12265  df-9 12266  df-n0 12457  df-z 12543  df-dec 12662  df-uz 12807  df-fz 13469  df-fzo 13612  df-seq 13951  df-hash 14275  df-struct 17064  df-sets 17081  df-slot 17099  df-ndx 17111  df-base 17129  df-ress 17158  df-plusg 17194  df-mulr 17195  df-starv 17196  df-sca 17197  df-vsca 17198  df-ip 17199  df-tset 17200  df-ple 17201  df-ds 17203  df-unif 17204  df-hom 17205  df-cco 17206  df-0g 17371  df-gsum 17372  df-prds 17377  df-pws 17379  df-mre 17514  df-mrc 17515  df-acs 17517  df-mgm 18545  df-sgrp 18594  df-mnd 18605  df-mhm 18649  df-submnd 18650  df-grp 18799  df-minusg 18800  df-sbg 18801  df-mulg 18925  df-subg 18977  df-ghm 19058  df-cntz 19149  df-cmn 19616  df-abl 19617  df-mgp 19949  df-ur 19966  df-srg 19970  df-ring 20018  df-cring 20019  df-oppr 20104  df-dvdsr 20125  df-unit 20126  df-invr 20156  df-rnghom 20203  df-drng 20269  df-field 20270  df-subrg 20312  df-sdrg 20354  df-lmod 20424  df-lss 20494  df-lsp 20534  df-rlreg 20837  df-cnfld 20881  df-assa 21343  df-asp 21344  df-ascl 21345  df-psr 21395  df-mvr 21396  df-mpl 21397  df-opsr 21399  df-evls 21566  df-evl 21567  df-psr1 21635  df-vr1 21636  df-ply1 21637  df-coe1 21638  df-evls1 21765  df-evl1 21766  df-mdeg 25501  df-deg1 25502  df-mon1 25579  df-uc1p 25580  df-irng 32650

This theorem is referenced by: (None)