Theorem irngnzply1 32577

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 20274	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ CRing)
7		irngnzply1.f	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))
8		issdrg 20348	. . . . . . . . . 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 32572	. . . . . . 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 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 25586	. . . . . . . . 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 2731	. . . . . . . . . . . . 13 ⊢ (𝐸 ↑s (Base‘𝐸)) = (𝐸 ↑s (Base‘𝐸))
20	1, 3, 19, 2, 14	evls1rhm 21765	. . . . . . . . . . . 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 2731	. . . . . . . . . . . 12 ⊢ (Base‘(𝐸 ↑s (Base‘𝐸))) = (Base‘(𝐸 ↑s (Base‘𝐸)))
23	15, 22	rhmf 20210	. . . . . . . . . . 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 6712	. . . . . . . . 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 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 25588	. . . . . . . . 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 2731	. . . . . . . . . 10 ⊢ (Poly1‘𝐸) = (Poly1‘𝐸)
32		irngnzply1.z	. . . . . . . . . 10 ⊢ 𝑍 = (0g‘(Poly1‘𝐸))
33	31, 2, 14, 15, 10, 32	ressply10g 32480	. . . . . . . . 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 3014	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ (Monic1p‘(𝐸 ↾s 𝐹))) → 𝑝 ≠ 𝑍)
36		eldifsn 4780	. . . . . . 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 6890	. . . . . . . 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 7069	. . . . . . . 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 17414	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 IntgRing 𝐹)) ∧ (𝑝 ∈ (Monic1p‘(𝐸 ↾s 𝐹)) ∧ ((𝑂‘𝑝)‘𝑥) = 0 )) → (𝑂‘𝑝):(Base‘𝐸)⟶(Base‘𝐸))
45	44	ffnd 6702	. . . . . 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 7043	. . . . . . 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 3171	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐸 IntgRing 𝐹)) → ∃𝑝 ∈ (dom 𝑂 ∖ {𝑍})𝑥 ∈ (◡(𝑂‘𝑝) “ { 0 }))
53		eliun 4991	. . . 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 5021	. . . . . 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 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 4119	. . . . . . . 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 4783	. . . . . . . 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 6890	. . . . . . . . . 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 2834	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})) → 𝑝 ∈ (Base‘(Poly1‘(𝐸 ↾s 𝐹))))
72	69, 71	ffvelcdmd 7069	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})) → (𝑂‘𝑝) ∈ (Base‘(𝐸 ↑s (Base‘𝐸))))
73	19, 3, 22, 67, 68, 72	pwselbas 17414	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})) → (𝑂‘𝑝):(Base‘𝐸)⟶(Base‘𝐸))
74	73	ffnd 6702	. . . . . . . 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 32576	. . . . 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 3264	. . 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 2729	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 2939  ∃wrex 3069  Vcvv 3470   ∖ cdif 3938  {csn 4619  ∪ ciun 4987  ^◡ccnv 5665  dom cdm 5666   “ cima 5669   Fn wfn 6524  ⟶wf 6525  ‘cfv 6529  (class class class)co 7390  Basecbs 17123   ↾_s cress 17152  0_gc0g 17364   ↑_s cpws 17371  CRingccrg 20012   RingHom crh 20195  DivRingcdr 20262  Fieldcfield 20263  SubRingcsubrg 20303  SubDRingcsdrg 20346  Poly₁cpl1 21625   evalSub₁ ces1 21756  Monic_1pcmn1 25567   IntgRing cirng 32569

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 2702  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7705  ax-cnex 11145  ax-resscn 11146  ax-1cn 11147  ax-icn 11148  ax-addcl 11149  ax-addrcl 11150  ax-mulcl 11151  ax-mulrcl 11152  ax-mulcom 11153  ax-addass 11154  ax-mulass 11155  ax-distr 11156  ax-i2m1 11157  ax-1ne0 11158  ax-1rid 11159  ax-rnegex 11160  ax-rrecex 11161  ax-cnre 11162  ax-pre-lttri 11163  ax-pre-lttrn 11164  ax-pre-ltadd 11165  ax-pre-mulgt0 11166  ax-addf 11168  ax-mulf 11169

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 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 3430  df-v 3472  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3960  df-nul 4316  df-if 4520  df-pw 4595  df-sn 4620  df-pr 4622  df-tp 4624  df-op 4626  df-uni 4899  df-int 4941  df-iun 4989  df-iin 4990  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-se 5622  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6286  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6531  df-fn 6532  df-f 6533  df-f1 6534  df-fo 6535  df-f1o 6536  df-fv 6537  df-isom 6538  df-riota 7346  df-ov 7393  df-oprab 7394  df-mpo 7395  df-of 7650  df-ofr 7651  df-om 7836  df-1st 7954  df-2nd 7955  df-supp 8126  df-tpos 8190  df-frecs 8245  df-wrecs 8276  df-recs 8350  df-rdg 8389  df-1o 8445  df-er 8683  df-map 8802  df-pm 8803  df-ixp 8872  df-en 8920  df-dom 8921  df-sdom 8922  df-fin 8923  df-fsupp 9342  df-sup 9416  df-oi 9484  df-card 9913  df-pnf 11229  df-mnf 11230  df-xr 11231  df-ltxr 11232  df-le 11233  df-sub 11425  df-neg 11426  df-nn 12192  df-2 12254  df-3 12255  df-4 12256  df-5 12257  df-6 12258  df-7 12259  df-8 12260  df-9 12261  df-n0 12452  df-z 12538  df-dec 12657  df-uz 12802  df-fz 13464  df-fzo 13607  df-seq 13946  df-hash 14270  df-struct 17059  df-sets 17076  df-slot 17094  df-ndx 17106  df-base 17124  df-ress 17153  df-plusg 17189  df-mulr 17190  df-starv 17191  df-sca 17192  df-vsca 17193  df-ip 17194  df-tset 17195  df-ple 17196  df-ds 17198  df-unif 17199  df-hom 17200  df-cco 17201  df-0g 17366  df-gsum 17367  df-prds 17372  df-pws 17374  df-mre 17509  df-mrc 17510  df-acs 17512  df-mgm 18540  df-sgrp 18589  df-mnd 18600  df-mhm 18644  df-submnd 18645  df-grp 18794  df-minusg 18795  df-sbg 18796  df-mulg 18920  df-subg 18972  df-ghm 19053  df-cntz 19144  df-cmn 19611  df-abl 19612  df-mgp 19944  df-ur 19961  df-srg 19965  df-ring 20013  df-cring 20014  df-oppr 20099  df-dvdsr 20120  df-unit 20121  df-invr 20151  df-rnghom 20198  df-drng 20264  df-field 20265  df-subrg 20305  df-sdrg 20347  df-lmod 20417  df-lss 20487  df-lsp 20527  df-rlreg 20830  df-cnfld 20874  df-assa 21336  df-asp 21337  df-ascl 21338  df-psr 21388  df-mvr 21389  df-mpl 21390  df-opsr 21392  df-evls 21559  df-evl 21560  df-psr1 21628  df-vr1 21629  df-ply1 21630  df-coe1 21631  df-evls1 21758  df-evl1 21759  df-mdeg 25494  df-deg1 25495  df-mon1 25572  df-uc1p 25573  df-irng 32570

This theorem is referenced by: (None)