irngnzply1lem - Mathbox for Thierry Arnoux

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Theorem irngnzply1lem 32742

Description: In the case of a field 𝐸, a root 𝑋 of some nonzero polynomial 𝑃 with coefficients in a subfield 𝐹 is integral over 𝐹. (Contributed by Thierry Arnoux, 5-Feb-2025.)

**Hypotheses**
Ref	Expression
irngnzply1.o	⊢ 𝑂 = (𝐸 evalSub₁ 𝐹)
irngnzply1.z	⊢ 𝑍 = (0_g‘(Poly₁‘𝐸))
irngnzply1.1	⊢ 0 = (0_g‘𝐸)
irngnzply1.e	⊢ (𝜑 → 𝐸 ∈ Field)
irngnzply1.f	⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))
irngnzply1lem.b	⊢ 𝐵 = (Base‘𝐸)
irngnzply1lem.1	⊢ (𝜑 → 𝑃 ∈ dom 𝑂)
irngnzply1lem.2	⊢ (𝜑 → 𝑃 ≠ 𝑍)
irngnzply1lem.3	⊢ (𝜑 → ((𝑂‘𝑃)‘𝑋) = 0 )
irngnzply1lem.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)

**Assertion**
Ref	Expression
irngnzply1lem	⊢ (𝜑 → 𝑋 ∈ (𝐸 IntgRing 𝐹))

Proof of Theorem irngnzply1lem Dummy variable 𝑝 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		irngnzply1lem.x	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
2		irngnzply1.f	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))
3		issdrg 20396	. . . . . . 7 ⊢ (𝐹 ∈ (SubDRing‘𝐸) ↔ (𝐸 ∈ DivRing ∧ 𝐹 ∈ (SubRing‘𝐸) ∧ (𝐸 ↾_s 𝐹) ∈ DivRing))
4	3	simp3bi 1147	. . . . . 6 ⊢ (𝐹 ∈ (SubDRing‘𝐸) → (𝐸 ↾_s 𝐹) ∈ DivRing)
5	2, 4	syl 17	. . . . 5 ⊢ (𝜑 → (𝐸 ↾_s 𝐹) ∈ DivRing)
6	5	drngringd 20315	. . . 4 ⊢ (𝜑 → (𝐸 ↾_s 𝐹) ∈ Ring)
7		irngnzply1lem.1	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ dom 𝑂)
8		irngnzply1.e	. . . . . . . . . 10 ⊢ (𝜑 → 𝐸 ∈ Field)
9	8	fldcrngd 20320	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ CRing)
10	2, 3	sylib 217	. . . . . . . . . 10 ⊢ (𝜑 → (𝐸 ∈ DivRing ∧ 𝐹 ∈ (SubRing‘𝐸) ∧ (𝐸 ↾_s 𝐹) ∈ DivRing))
11	10	simp2d 1143	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ (SubRing‘𝐸))
12		irngnzply1.o	. . . . . . . . . 10 ⊢ 𝑂 = (𝐸 evalSub₁ 𝐹)
13		irngnzply1lem.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝐸)
14		eqid 2732	. . . . . . . . . 10 ⊢ (𝐸 ↑_s 𝐵) = (𝐸 ↑_s 𝐵)
15		eqid 2732	. . . . . . . . . 10 ⊢ (𝐸 ↾_s 𝐹) = (𝐸 ↾_s 𝐹)
16		eqid 2732	. . . . . . . . . 10 ⊢ (Poly₁‘(𝐸 ↾_s 𝐹)) = (Poly₁‘(𝐸 ↾_s 𝐹))
17	12, 13, 14, 15, 16	evls1rhm 21832	. . . . . . . . 9 ⊢ ((𝐸 ∈ CRing ∧ 𝐹 ∈ (SubRing‘𝐸)) → 𝑂 ∈ ((Poly₁‘(𝐸 ↾_s 𝐹)) RingHom (𝐸 ↑_s 𝐵)))
18	9, 11, 17	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → 𝑂 ∈ ((Poly₁‘(𝐸 ↾_s 𝐹)) RingHom (𝐸 ↑_s 𝐵)))
19		eqid 2732	. . . . . . . . 9 ⊢ (Base‘(Poly₁‘(𝐸 ↾_s 𝐹))) = (Base‘(Poly₁‘(𝐸 ↾_s 𝐹)))
20		eqid 2732	. . . . . . . . 9 ⊢ (Base‘(𝐸 ↑_s 𝐵)) = (Base‘(𝐸 ↑_s 𝐵))
21	19, 20	rhmf 20255	. . . . . . . 8 ⊢ (𝑂 ∈ ((Poly₁‘(𝐸 ↾_s 𝐹)) RingHom (𝐸 ↑_s 𝐵)) → 𝑂:(Base‘(Poly₁‘(𝐸 ↾_s 𝐹)))⟶(Base‘(𝐸 ↑_s 𝐵)))
22	18, 21	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑂:(Base‘(Poly₁‘(𝐸 ↾_s 𝐹)))⟶(Base‘(𝐸 ↑_s 𝐵)))
23	22	fdmd 6725	. . . . . 6 ⊢ (𝜑 → dom 𝑂 = (Base‘(Poly₁‘(𝐸 ↾_s 𝐹))))
24	7, 23	eleqtrd 2835	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ (Base‘(Poly₁‘(𝐸 ↾_s 𝐹))))
25		irngnzply1lem.2	. . . . . 6 ⊢ (𝜑 → 𝑃 ≠ 𝑍)
26		eqid 2732	. . . . . . 7 ⊢ (Poly₁‘𝐸) = (Poly₁‘𝐸)
27		irngnzply1.z	. . . . . . 7 ⊢ 𝑍 = (0_g‘(Poly₁‘𝐸))
28	26, 15, 16, 19, 11, 27	ressply10g 32644	. . . . . 6 ⊢ (𝜑 → 𝑍 = (0_g‘(Poly₁‘(𝐸 ↾_s 𝐹))))
29	25, 28	neeqtrd 3010	. . . . 5 ⊢ (𝜑 → 𝑃 ≠ (0_g‘(Poly₁‘(𝐸 ↾_s 𝐹))))
30		eqid 2732	. . . . . 6 ⊢ (0_g‘(Poly₁‘(𝐸 ↾_s 𝐹))) = (0_g‘(Poly₁‘(𝐸 ↾_s 𝐹)))
31		eqid 2732	. . . . . 6 ⊢ (Unic_1p‘(𝐸 ↾_s 𝐹)) = (Unic_1p‘(𝐸 ↾_s 𝐹))
32	16, 19, 30, 31	drnguc1p 25679	. . . . 5 ⊢ (((𝐸 ↾_s 𝐹) ∈ DivRing ∧ 𝑃 ∈ (Base‘(Poly₁‘(𝐸 ↾_s 𝐹))) ∧ 𝑃 ≠ (0_g‘(Poly₁‘(𝐸 ↾_s 𝐹)))) → 𝑃 ∈ (Unic_1p‘(𝐸 ↾_s 𝐹)))
33	5, 24, 29, 32	syl3anc 1371	. . . 4 ⊢ (𝜑 → 𝑃 ∈ (Unic_1p‘(𝐸 ↾_s 𝐹)))
34		eqid 2732	. . . . 5 ⊢ (Monic_1p‘(𝐸 ↾_s 𝐹)) = (Monic_1p‘(𝐸 ↾_s 𝐹))
35		eqid 2732	. . . . 5 ⊢ (._r‘(Poly₁‘(𝐸 ↾_s 𝐹))) = (._r‘(Poly₁‘(𝐸 ↾_s 𝐹)))
36		eqid 2732	. . . . 5 ⊢ (algSc‘(Poly₁‘(𝐸 ↾_s 𝐹))) = (algSc‘(Poly₁‘(𝐸 ↾_s 𝐹)))
37		eqid 2732	. . . . 5 ⊢ ( deg₁ ‘(𝐸 ↾_s 𝐹)) = ( deg₁ ‘(𝐸 ↾_s 𝐹))
38		eqid 2732	. . . . 5 ⊢ (inv_r‘(𝐸 ↾_s 𝐹)) = (inv_r‘(𝐸 ↾_s 𝐹))
39	31, 34, 16, 35, 36, 37, 38	uc1pmon1p 25660	. . . 4 ⊢ (((𝐸 ↾_s 𝐹) ∈ Ring ∧ 𝑃 ∈ (Unic_1p‘(𝐸 ↾_s 𝐹))) → (((algSc‘(Poly₁‘(𝐸 ↾_s 𝐹)))‘((inv_r‘(𝐸 ↾_s 𝐹))‘((coe₁‘𝑃)‘(( deg₁ ‘(𝐸 ↾_s 𝐹))‘𝑃))))(._r‘(Poly₁‘(𝐸 ↾_s 𝐹)))𝑃) ∈ (Monic_1p‘(𝐸 ↾_s 𝐹)))
40	6, 33, 39	syl2anc 584	. . 3 ⊢ (𝜑 → (((algSc‘(Poly₁‘(𝐸 ↾_s 𝐹)))‘((inv_r‘(𝐸 ↾_s 𝐹))‘((coe₁‘𝑃)‘(( deg₁ ‘(𝐸 ↾_s 𝐹))‘𝑃))))(._r‘(Poly₁‘(𝐸 ↾_s 𝐹)))𝑃) ∈ (Monic_1p‘(𝐸 ↾_s 𝐹)))
41		simpr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 = (((algSc‘(Poly₁‘(𝐸 ↾_s 𝐹)))‘((inv_r‘(𝐸 ↾_s 𝐹))‘((coe₁‘𝑃)‘(( deg₁ ‘(𝐸 ↾_s 𝐹))‘𝑃))))(._r‘(Poly₁‘(𝐸 ↾_s 𝐹)))𝑃)) → 𝑝 = (((algSc‘(Poly₁‘(𝐸 ↾_s 𝐹)))‘((inv_r‘(𝐸 ↾_s 𝐹))‘((coe₁‘𝑃)‘(( deg₁ ‘(𝐸 ↾_s 𝐹))‘𝑃))))(._r‘(Poly₁‘(𝐸 ↾_s 𝐹)))𝑃))
42	41	fveq2d 6892	. . . . 5 ⊢ ((𝜑 ∧ 𝑝 = (((algSc‘(Poly₁‘(𝐸 ↾_s 𝐹)))‘((inv_r‘(𝐸 ↾_s 𝐹))‘((coe₁‘𝑃)‘(( deg₁ ‘(𝐸 ↾_s 𝐹))‘𝑃))))(._r‘(Poly₁‘(𝐸 ↾_s 𝐹)))𝑃)) → (𝑂‘𝑝) = (𝑂‘(((algSc‘(Poly₁‘(𝐸 ↾_s 𝐹)))‘((inv_r‘(𝐸 ↾_s 𝐹))‘((coe₁‘𝑃)‘(( deg₁ ‘(𝐸 ↾_s 𝐹))‘𝑃))))(._r‘(Poly₁‘(𝐸 ↾_s 𝐹)))𝑃)))
43	42	fveq1d 6890	. . . 4 ⊢ ((𝜑 ∧ 𝑝 = (((algSc‘(Poly₁‘(𝐸 ↾_s 𝐹)))‘((inv_r‘(𝐸 ↾_s 𝐹))‘((coe₁‘𝑃)‘(( deg₁ ‘(𝐸 ↾_s 𝐹))‘𝑃))))(._r‘(Poly₁‘(𝐸 ↾_s 𝐹)))𝑃)) → ((𝑂‘𝑝)‘𝑋) = ((𝑂‘(((algSc‘(Poly₁‘(𝐸 ↾_s 𝐹)))‘((inv_r‘(𝐸 ↾_s 𝐹))‘((coe₁‘𝑃)‘(( deg₁ ‘(𝐸 ↾_s 𝐹))‘𝑃))))(._r‘(Poly₁‘(𝐸 ↾_s 𝐹)))𝑃))‘𝑋))
44	43	eqeq1d 2734	. . 3 ⊢ ((𝜑 ∧ 𝑝 = (((algSc‘(Poly₁‘(𝐸 ↾_s 𝐹)))‘((inv_r‘(𝐸 ↾_s 𝐹))‘((coe₁‘𝑃)‘(( deg₁ ‘(𝐸 ↾_s 𝐹))‘𝑃))))(._r‘(Poly₁‘(𝐸 ↾_s 𝐹)))𝑃)) → (((𝑂‘𝑝)‘𝑋) = 0 ↔ ((𝑂‘(((algSc‘(Poly₁‘(𝐸 ↾_s 𝐹)))‘((inv_r‘(𝐸 ↾_s 𝐹))‘((coe₁‘𝑃)‘(( deg₁ ‘(𝐸 ↾_s 𝐹))‘𝑃))))(._r‘(Poly₁‘(𝐸 ↾_s 𝐹)))𝑃))‘𝑋) = 0 ))
45		eqid 2732	. . . . 5 ⊢ (._r‘𝐸) = (._r‘𝐸)
46		eqid 2732	. . . . . . 7 ⊢ (Scalar‘(Poly₁‘(𝐸 ↾_s 𝐹))) = (Scalar‘(Poly₁‘(𝐸 ↾_s 𝐹)))
47		fldsdrgfld 20406	. . . . . . . . . . 11 ⊢ ((𝐸 ∈ Field ∧ 𝐹 ∈ (SubDRing‘𝐸)) → (𝐸 ↾_s 𝐹) ∈ Field)
48	8, 2, 47	syl2anc 584	. . . . . . . . . 10 ⊢ (𝜑 → (𝐸 ↾_s 𝐹) ∈ Field)
49	48	fldcrngd 20320	. . . . . . . . 9 ⊢ (𝜑 → (𝐸 ↾_s 𝐹) ∈ CRing)
50	16	ply1assa 21714	. . . . . . . . 9 ⊢ ((𝐸 ↾_s 𝐹) ∈ CRing → (Poly₁‘(𝐸 ↾_s 𝐹)) ∈ AssAlg)
51	49, 50	syl 17	. . . . . . . 8 ⊢ (𝜑 → (Poly₁‘(𝐸 ↾_s 𝐹)) ∈ AssAlg)
52		assaring 21407	. . . . . . . 8 ⊢ ((Poly₁‘(𝐸 ↾_s 𝐹)) ∈ AssAlg → (Poly₁‘(𝐸 ↾_s 𝐹)) ∈ Ring)
53	51, 52	syl 17	. . . . . . 7 ⊢ (𝜑 → (Poly₁‘(𝐸 ↾_s 𝐹)) ∈ Ring)
54	16	ply1lmod 21765	. . . . . . . 8 ⊢ ((𝐸 ↾_s 𝐹) ∈ Ring → (Poly₁‘(𝐸 ↾_s 𝐹)) ∈ LMod)
55	6, 54	syl 17	. . . . . . 7 ⊢ (𝜑 → (Poly₁‘(𝐸 ↾_s 𝐹)) ∈ LMod)
56		eqid 2732	. . . . . . 7 ⊢ (Base‘(Scalar‘(Poly₁‘(𝐸 ↾_s 𝐹)))) = (Base‘(Scalar‘(Poly₁‘(𝐸 ↾_s 𝐹))))
57	36, 46, 53, 55, 56, 19	asclf 21427	. . . . . 6 ⊢ (𝜑 → (algSc‘(Poly₁‘(𝐸 ↾_s 𝐹))):(Base‘(Scalar‘(Poly₁‘(𝐸 ↾_s 𝐹))))⟶(Base‘(Poly₁‘(𝐸 ↾_s 𝐹))))
58		eqid 2732	. . . . . . . 8 ⊢ (Base‘(𝐸 ↾_s 𝐹)) = (Base‘(𝐸 ↾_s 𝐹))
59		eqid 2732	. . . . . . . 8 ⊢ (0_g‘(𝐸 ↾_s 𝐹)) = (0_g‘(𝐸 ↾_s 𝐹))
60	37, 16, 30, 19	deg1nn0cl 25597	. . . . . . . . . 10 ⊢ (((𝐸 ↾_s 𝐹) ∈ Ring ∧ 𝑃 ∈ (Base‘(Poly₁‘(𝐸 ↾_s 𝐹))) ∧ 𝑃 ≠ (0_g‘(Poly₁‘(𝐸 ↾_s 𝐹)))) → (( deg₁ ‘(𝐸 ↾_s 𝐹))‘𝑃) ∈ ℕ₀)
61	6, 24, 29, 60	syl3anc 1371	. . . . . . . . 9 ⊢ (𝜑 → (( deg₁ ‘(𝐸 ↾_s 𝐹))‘𝑃) ∈ ℕ₀)
62		eqid 2732	. . . . . . . . . 10 ⊢ (coe₁‘𝑃) = (coe₁‘𝑃)
63	62, 19, 16, 58	coe1fvalcl 21727	. . . . . . . . 9 ⊢ ((𝑃 ∈ (Base‘(Poly₁‘(𝐸 ↾_s 𝐹))) ∧ (( deg₁ ‘(𝐸 ↾_s 𝐹))‘𝑃) ∈ ℕ₀) → ((coe₁‘𝑃)‘(( deg₁ ‘(𝐸 ↾_s 𝐹))‘𝑃)) ∈ (Base‘(𝐸 ↾_s 𝐹)))
64	24, 61, 63	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → ((coe₁‘𝑃)‘(( deg₁ ‘(𝐸 ↾_s 𝐹))‘𝑃)) ∈ (Base‘(𝐸 ↾_s 𝐹)))
65	37, 16, 30, 19, 59, 62	deg1ldg 25601	. . . . . . . . 9 ⊢ (((𝐸 ↾_s 𝐹) ∈ Ring ∧ 𝑃 ∈ (Base‘(Poly₁‘(𝐸 ↾_s 𝐹))) ∧ 𝑃 ≠ (0_g‘(Poly₁‘(𝐸 ↾_s 𝐹)))) → ((coe₁‘𝑃)‘(( deg₁ ‘(𝐸 ↾_s 𝐹))‘𝑃)) ≠ (0_g‘(𝐸 ↾_s 𝐹)))
66	6, 24, 29, 65	syl3anc 1371	. . . . . . . 8 ⊢ (𝜑 → ((coe₁‘𝑃)‘(( deg₁ ‘(𝐸 ↾_s 𝐹))‘𝑃)) ≠ (0_g‘(𝐸 ↾_s 𝐹)))
67	58, 59, 38, 5, 64, 66	drnginvrcld 20331	. . . . . . 7 ⊢ (𝜑 → ((inv_r‘(𝐸 ↾_s 𝐹))‘((coe₁‘𝑃)‘(( deg₁ ‘(𝐸 ↾_s 𝐹))‘𝑃))) ∈ (Base‘(𝐸 ↾_s 𝐹)))
68	16	ply1sca 21766	. . . . . . . . 9 ⊢ ((𝐸 ↾_s 𝐹) ∈ Field → (𝐸 ↾_s 𝐹) = (Scalar‘(Poly₁‘(𝐸 ↾_s 𝐹))))
69	48, 68	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝐸 ↾_s 𝐹) = (Scalar‘(Poly₁‘(𝐸 ↾_s 𝐹))))
70	69	fveq2d 6892	. . . . . . 7 ⊢ (𝜑 → (Base‘(𝐸 ↾_s 𝐹)) = (Base‘(Scalar‘(Poly₁‘(𝐸 ↾_s 𝐹)))))
71	67, 70	eleqtrd 2835	. . . . . 6 ⊢ (𝜑 → ((inv_r‘(𝐸 ↾_s 𝐹))‘((coe₁‘𝑃)‘(( deg₁ ‘(𝐸 ↾_s 𝐹))‘𝑃))) ∈ (Base‘(Scalar‘(Poly₁‘(𝐸 ↾_s 𝐹)))))
72	57, 71	ffvelcdmd 7084	. . . . 5 ⊢ (𝜑 → ((algSc‘(Poly₁‘(𝐸 ↾_s 𝐹)))‘((inv_r‘(𝐸 ↾_s 𝐹))‘((coe₁‘𝑃)‘(( deg₁ ‘(𝐸 ↾_s 𝐹))‘𝑃)))) ∈ (Base‘(Poly₁‘(𝐸 ↾_s 𝐹))))
73	12, 13, 16, 15, 19, 35, 45, 9, 11, 72, 24, 1	evls1muld 32637	. . . 4 ⊢ (𝜑 → ((𝑂‘(((algSc‘(Poly₁‘(𝐸 ↾_s 𝐹)))‘((inv_r‘(𝐸 ↾_s 𝐹))‘((coe₁‘𝑃)‘(( deg₁ ‘(𝐸 ↾_s 𝐹))‘𝑃))))(._r‘(Poly₁‘(𝐸 ↾_s 𝐹)))𝑃))‘𝑋) = (((𝑂‘((algSc‘(Poly₁‘(𝐸 ↾_s 𝐹)))‘((inv_r‘(𝐸 ↾_s 𝐹))‘((coe₁‘𝑃)‘(( deg₁ ‘(𝐸 ↾_s 𝐹))‘𝑃)))))‘𝑋)(._r‘𝐸)((𝑂‘𝑃)‘𝑋)))
74		irngnzply1lem.3	. . . . 5 ⊢ (𝜑 → ((𝑂‘𝑃)‘𝑋) = 0 )
75	74	oveq2d 7421	. . . 4 ⊢ (𝜑 → (((𝑂‘((algSc‘(Poly₁‘(𝐸 ↾_s 𝐹)))‘((inv_r‘(𝐸 ↾_s 𝐹))‘((coe₁‘𝑃)‘(( deg₁ ‘(𝐸 ↾_s 𝐹))‘𝑃)))))‘𝑋)(._r‘𝐸)((𝑂‘𝑃)‘𝑋)) = (((𝑂‘((algSc‘(Poly₁‘(𝐸 ↾_s 𝐹)))‘((inv_r‘(𝐸 ↾_s 𝐹))‘((coe₁‘𝑃)‘(( deg₁ ‘(𝐸 ↾_s 𝐹))‘𝑃)))))‘𝑋)(._r‘𝐸) 0 ))
76	9	crngringd 20062	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ Ring)
77	13	fvexi 6902	. . . . . . . 8 ⊢ 𝐵 ∈ V
78	77	a1i 11	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ V)
79	22, 72	ffvelcdmd 7084	. . . . . . 7 ⊢ (𝜑 → (𝑂‘((algSc‘(Poly₁‘(𝐸 ↾_s 𝐹)))‘((inv_r‘(𝐸 ↾_s 𝐹))‘((coe₁‘𝑃)‘(( deg₁ ‘(𝐸 ↾_s 𝐹))‘𝑃))))) ∈ (Base‘(𝐸 ↑_s 𝐵)))
80	14, 13, 20, 8, 78, 79	pwselbas 17431	. . . . . 6 ⊢ (𝜑 → (𝑂‘((algSc‘(Poly₁‘(𝐸 ↾_s 𝐹)))‘((inv_r‘(𝐸 ↾_s 𝐹))‘((coe₁‘𝑃)‘(( deg₁ ‘(𝐸 ↾_s 𝐹))‘𝑃))))):𝐵⟶𝐵)
81	80, 1	ffvelcdmd 7084	. . . . 5 ⊢ (𝜑 → ((𝑂‘((algSc‘(Poly₁‘(𝐸 ↾_s 𝐹)))‘((inv_r‘(𝐸 ↾_s 𝐹))‘((coe₁‘𝑃)‘(( deg₁ ‘(𝐸 ↾_s 𝐹))‘𝑃)))))‘𝑋) ∈ 𝐵)
82		irngnzply1.1	. . . . . 6 ⊢ 0 = (0_g‘𝐸)
83	13, 45, 82	ringrz 20101	. . . . 5 ⊢ ((𝐸 ∈ Ring ∧ ((𝑂‘((algSc‘(Poly₁‘(𝐸 ↾_s 𝐹)))‘((inv_r‘(𝐸 ↾_s 𝐹))‘((coe₁‘𝑃)‘(( deg₁ ‘(𝐸 ↾_s 𝐹))‘𝑃)))))‘𝑋) ∈ 𝐵) → (((𝑂‘((algSc‘(Poly₁‘(𝐸 ↾_s 𝐹)))‘((inv_r‘(𝐸 ↾_s 𝐹))‘((coe₁‘𝑃)‘(( deg₁ ‘(𝐸 ↾_s 𝐹))‘𝑃)))))‘𝑋)(._r‘𝐸) 0 ) = 0 )
84	76, 81, 83	syl2anc 584	. . . 4 ⊢ (𝜑 → (((𝑂‘((algSc‘(Poly₁‘(𝐸 ↾_s 𝐹)))‘((inv_r‘(𝐸 ↾_s 𝐹))‘((coe₁‘𝑃)‘(( deg₁ ‘(𝐸 ↾_s 𝐹))‘𝑃)))))‘𝑋)(._r‘𝐸) 0 ) = 0 )
85	73, 75, 84	3eqtrd 2776	. . 3 ⊢ (𝜑 → ((𝑂‘(((algSc‘(Poly₁‘(𝐸 ↾_s 𝐹)))‘((inv_r‘(𝐸 ↾_s 𝐹))‘((coe₁‘𝑃)‘(( deg₁ ‘(𝐸 ↾_s 𝐹))‘𝑃))))(._r‘(Poly₁‘(𝐸 ↾_s 𝐹)))𝑃))‘𝑋) = 0 )
86	40, 44, 85	rspcedvd 3614	. 2 ⊢ (𝜑 → ∃𝑝 ∈ (Monic_1p‘(𝐸 ↾_s 𝐹))((𝑂‘𝑝)‘𝑋) = 0 )
87	12, 15, 13, 82, 9, 11	elirng 32738	. 2 ⊢ (𝜑 → (𝑋 ∈ (𝐸 IntgRing 𝐹) ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑝 ∈ (Monic_1p‘(𝐸 ↾_s 𝐹))((𝑂‘𝑝)‘𝑋) = 0 )))
88	1, 86, 87	mpbir2and 711	1 ⊢ (𝜑 → 𝑋 ∈ (𝐸 IntgRing 𝐹))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ∃wrex 3070 Vcvv 3474 dom cdm 5675 ⟶wf 6536 ‘cfv 6540 (class class class)co 7405 ℕ₀cn0 12468 Basecbs 17140 ↾_s cress 17169 ._rcmulr 17194 Scalarcsca 17196 0_gc0g 17381 ↑_s cpws 17388 Ringcrg 20049 CRingccrg 20050 inv_rcinvr 20193 RingHom crh 20240 DivRingcdr 20307 Fieldcfield 20308 SubRingcsubrg 20351 SubDRingcsdrg 20394 LModclmod 20463 AssAlgcasa 21396 algSccascl 21398 Poly₁cpl1 21692 coe₁cco1 21693 evalSub₁ ces1 21823 deg₁ cdg1 25560 Monic_1pcmn1 25634 Unic_1pcuc1p 25635 IntgRing cirng 32735

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 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-addf 11185 ax-mulf 11186

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 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7666 df-ofr 7667 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8143 df-tpos 8207 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8699 df-map 8818 df-pm 8819 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-sup 9433 df-oi 9501 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-fz 13481 df-fzo 13624 df-seq 13963 df-hash 14287 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-starv 17208 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-unif 17216 df-hom 17217 df-cco 17218 df-0g 17383 df-gsum 17384 df-prds 17389 df-pws 17391 df-mre 17526 df-mrc 17527 df-acs 17529 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-mhm 18667 df-submnd 18668 df-grp 18818 df-minusg 18819 df-sbg 18820 df-mulg 18945 df-subg 18997 df-ghm 19084 df-cntz 19175 df-cmn 19644 df-abl 19645 df-mgp 19982 df-ur 19999 df-srg 20003 df-ring 20051 df-cring 20052 df-oppr 20142 df-dvdsr 20163 df-unit 20164 df-invr 20194 df-rnghom 20243 df-drng 20309 df-field 20310 df-subrg 20353 df-sdrg 20395 df-lmod 20465 df-lss 20535 df-lsp 20575 df-rlreg 20891 df-cnfld 20937 df-assa 21399 df-asp 21400 df-ascl 21401 df-psr 21453 df-mvr 21454 df-mpl 21455 df-opsr 21457 df-evls 21626 df-evl 21627 df-psr1 21695 df-vr1 21696 df-ply1 21697 df-coe1 21698 df-evls1 21825 df-evl1 21826 df-mdeg 25561 df-deg1 25562 df-mon1 25639 df-uc1p 25640 df-irng 32736

This theorem is referenced by: irngnzply1 32743

W3C validator