irngnzply1lem - Mathbox for Thierry Arnoux

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Theorem irngnzply1lem 33300

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 20665	. . . . . . 7 ⊢ (𝐹 ∈ (SubDRing‘𝐸) ↔ (𝐸 ∈ DivRing ∧ 𝐹 ∈ (SubRing‘𝐸) ∧ (𝐸 ↾_s 𝐹) ∈ DivRing))
4	3	simp3bi 1145	. . . . . 6 ⊢ (𝐹 ∈ (SubDRing‘𝐸) → (𝐸 ↾_s 𝐹) ∈ DivRing)
5	2, 4	syl 17	. . . . 5 ⊢ (𝜑 → (𝐸 ↾_s 𝐹) ∈ DivRing)
6	5	drngringd 20621	. . . 4 ⊢ (𝜑 → (𝐸 ↾_s 𝐹) ∈ Ring)
7		irngnzply1lem.1	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ dom 𝑂)
8		irngnzply1.e	. . . . . . . . . 10 ⊢ (𝜑 → 𝐸 ∈ Field)
9	8	fldcrngd 20626	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ CRing)
10	2, 3	sylib 217	. . . . . . . . . 10 ⊢ (𝜑 → (𝐸 ∈ DivRing ∧ 𝐹 ∈ (SubRing‘𝐸) ∧ (𝐸 ↾_s 𝐹) ∈ DivRing))
11	10	simp2d 1141	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ (SubRing‘𝐸))
12		irngnzply1.o	. . . . . . . . . 10 ⊢ 𝑂 = (𝐸 evalSub₁ 𝐹)
13		irngnzply1lem.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝐸)
14		eqid 2727	. . . . . . . . . 10 ⊢ (𝐸 ↑_s 𝐵) = (𝐸 ↑_s 𝐵)
15		eqid 2727	. . . . . . . . . 10 ⊢ (𝐸 ↾_s 𝐹) = (𝐸 ↾_s 𝐹)
16		eqid 2727	. . . . . . . . . 10 ⊢ (Poly₁‘(𝐸 ↾_s 𝐹)) = (Poly₁‘(𝐸 ↾_s 𝐹))
17	12, 13, 14, 15, 16	evls1rhm 22228	. . . . . . . . 9 ⊢ ((𝐸 ∈ CRing ∧ 𝐹 ∈ (SubRing‘𝐸)) → 𝑂 ∈ ((Poly₁‘(𝐸 ↾_s 𝐹)) RingHom (𝐸 ↑_s 𝐵)))
18	9, 11, 17	syl2anc 583	. . . . . . . 8 ⊢ (𝜑 → 𝑂 ∈ ((Poly₁‘(𝐸 ↾_s 𝐹)) RingHom (𝐸 ↑_s 𝐵)))
19		eqid 2727	. . . . . . . . 9 ⊢ (Base‘(Poly₁‘(𝐸 ↾_s 𝐹))) = (Base‘(Poly₁‘(𝐸 ↾_s 𝐹)))
20		eqid 2727	. . . . . . . . 9 ⊢ (Base‘(𝐸 ↑_s 𝐵)) = (Base‘(𝐸 ↑_s 𝐵))
21	19, 20	rhmf 20413	. . . . . . . 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 6727	. . . . . 6 ⊢ (𝜑 → dom 𝑂 = (Base‘(Poly₁‘(𝐸 ↾_s 𝐹))))
24	7, 23	eleqtrd 2830	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ (Base‘(Poly₁‘(𝐸 ↾_s 𝐹))))
25		irngnzply1lem.2	. . . . . 6 ⊢ (𝜑 → 𝑃 ≠ 𝑍)
26		eqid 2727	. . . . . . 7 ⊢ (Poly₁‘𝐸) = (Poly₁‘𝐸)
27		irngnzply1.z	. . . . . . 7 ⊢ 𝑍 = (0_g‘(Poly₁‘𝐸))
28	26, 15, 16, 19, 11, 27	ressply10g 33178	. . . . . 6 ⊢ (𝜑 → 𝑍 = (0_g‘(Poly₁‘(𝐸 ↾_s 𝐹))))
29	25, 28	neeqtrd 3005	. . . . 5 ⊢ (𝜑 → 𝑃 ≠ (0_g‘(Poly₁‘(𝐸 ↾_s 𝐹))))
30		eqid 2727	. . . . . 6 ⊢ (0_g‘(Poly₁‘(𝐸 ↾_s 𝐹))) = (0_g‘(Poly₁‘(𝐸 ↾_s 𝐹)))
31		eqid 2727	. . . . . 6 ⊢ (Unic_1p‘(𝐸 ↾_s 𝐹)) = (Unic_1p‘(𝐸 ↾_s 𝐹))
32	16, 19, 30, 31	drnguc1p 26095	. . . . 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 1369	. . . 4 ⊢ (𝜑 → 𝑃 ∈ (Unic_1p‘(𝐸 ↾_s 𝐹)))
34		eqid 2727	. . . . 5 ⊢ (Monic_1p‘(𝐸 ↾_s 𝐹)) = (Monic_1p‘(𝐸 ↾_s 𝐹))
35		eqid 2727	. . . . 5 ⊢ (._r‘(Poly₁‘(𝐸 ↾_s 𝐹))) = (._r‘(Poly₁‘(𝐸 ↾_s 𝐹)))
36		eqid 2727	. . . . 5 ⊢ (algSc‘(Poly₁‘(𝐸 ↾_s 𝐹))) = (algSc‘(Poly₁‘(𝐸 ↾_s 𝐹)))
37		eqid 2727	. . . . 5 ⊢ ( deg₁ ‘(𝐸 ↾_s 𝐹)) = ( deg₁ ‘(𝐸 ↾_s 𝐹))
38		eqid 2727	. . . . 5 ⊢ (inv_r‘(𝐸 ↾_s 𝐹)) = (inv_r‘(𝐸 ↾_s 𝐹))
39	31, 34, 16, 35, 36, 37, 38	uc1pmon1p 26074	. . . 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 583	. . 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 484	. . . . . 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 6895	. . . . 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 6893	. . . 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 2729	. . 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 2727	. . . . 5 ⊢ (._r‘𝐸) = (._r‘𝐸)
46		eqid 2727	. . . . . . 7 ⊢ (Scalar‘(Poly₁‘(𝐸 ↾_s 𝐹))) = (Scalar‘(Poly₁‘(𝐸 ↾_s 𝐹)))
47		fldsdrgfld 20675	. . . . . . . . . . 11 ⊢ ((𝐸 ∈ Field ∧ 𝐹 ∈ (SubDRing‘𝐸)) → (𝐸 ↾_s 𝐹) ∈ Field)
48	8, 2, 47	syl2anc 583	. . . . . . . . . 10 ⊢ (𝜑 → (𝐸 ↾_s 𝐹) ∈ Field)
49	48	fldcrngd 20626	. . . . . . . . 9 ⊢ (𝜑 → (𝐸 ↾_s 𝐹) ∈ CRing)
50	16	ply1assa 22105	. . . . . . . . 9 ⊢ ((𝐸 ↾_s 𝐹) ∈ CRing → (Poly₁‘(𝐸 ↾_s 𝐹)) ∈ AssAlg)
51	49, 50	syl 17	. . . . . . . 8 ⊢ (𝜑 → (Poly₁‘(𝐸 ↾_s 𝐹)) ∈ AssAlg)
52		assaring 21782	. . . . . . . 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 22157	. . . . . . . 8 ⊢ ((𝐸 ↾_s 𝐹) ∈ Ring → (Poly₁‘(𝐸 ↾_s 𝐹)) ∈ LMod)
55	6, 54	syl 17	. . . . . . 7 ⊢ (𝜑 → (Poly₁‘(𝐸 ↾_s 𝐹)) ∈ LMod)
56		eqid 2727	. . . . . . 7 ⊢ (Base‘(Scalar‘(Poly₁‘(𝐸 ↾_s 𝐹)))) = (Base‘(Scalar‘(Poly₁‘(𝐸 ↾_s 𝐹))))
57	36, 46, 53, 55, 56, 19	asclf 21802	. . . . . 6 ⊢ (𝜑 → (algSc‘(Poly₁‘(𝐸 ↾_s 𝐹))):(Base‘(Scalar‘(Poly₁‘(𝐸 ↾_s 𝐹))))⟶(Base‘(Poly₁‘(𝐸 ↾_s 𝐹))))
58		eqid 2727	. . . . . . . 8 ⊢ (Base‘(𝐸 ↾_s 𝐹)) = (Base‘(𝐸 ↾_s 𝐹))
59		eqid 2727	. . . . . . . 8 ⊢ (0_g‘(𝐸 ↾_s 𝐹)) = (0_g‘(𝐸 ↾_s 𝐹))
60	37, 16, 30, 19	deg1nn0cl 26011	. . . . . . . . . 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 1369	. . . . . . . . 9 ⊢ (𝜑 → (( deg₁ ‘(𝐸 ↾_s 𝐹))‘𝑃) ∈ ℕ₀)
62		eqid 2727	. . . . . . . . . 10 ⊢ (coe₁‘𝑃) = (coe₁‘𝑃)
63	62, 19, 16, 58	coe1fvalcl 22118	. . . . . . . . 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 583	. . . . . . . 8 ⊢ (𝜑 → ((coe₁‘𝑃)‘(( deg₁ ‘(𝐸 ↾_s 𝐹))‘𝑃)) ∈ (Base‘(𝐸 ↾_s 𝐹)))
65	37, 16, 30, 19, 59, 62	deg1ldg 26015	. . . . . . . . 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 1369	. . . . . . . 8 ⊢ (𝜑 → ((coe₁‘𝑃)‘(( deg₁ ‘(𝐸 ↾_s 𝐹))‘𝑃)) ≠ (0_g‘(𝐸 ↾_s 𝐹)))
67	58, 59, 38, 5, 64, 66	drnginvrcld 20637	. . . . . . 7 ⊢ (𝜑 → ((inv_r‘(𝐸 ↾_s 𝐹))‘((coe₁‘𝑃)‘(( deg₁ ‘(𝐸 ↾_s 𝐹))‘𝑃))) ∈ (Base‘(𝐸 ↾_s 𝐹)))
68	16	ply1sca 22158	. . . . . . . . 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 6895	. . . . . . 7 ⊢ (𝜑 → (Base‘(𝐸 ↾_s 𝐹)) = (Base‘(Scalar‘(Poly₁‘(𝐸 ↾_s 𝐹)))))
71	67, 70	eleqtrd 2830	. . . . . 6 ⊢ (𝜑 → ((inv_r‘(𝐸 ↾_s 𝐹))‘((coe₁‘𝑃)‘(( deg₁ ‘(𝐸 ↾_s 𝐹))‘𝑃))) ∈ (Base‘(Scalar‘(Poly₁‘(𝐸 ↾_s 𝐹)))))
72	57, 71	ffvelcdmd 7089	. . . . 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 33186	. . . 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 7430	. . . 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 20177	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ Ring)
77	13	fvexi 6905	. . . . . . . 8 ⊢ 𝐵 ∈ V
78	77	a1i 11	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ V)
79	22, 72	ffvelcdmd 7089	. . . . . . 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 17462	. . . . . 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 7089	. . . . 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 20219	. . . . 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 583	. . . 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 2771	. . 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 3609	. 2 ⊢ (𝜑 → ∃𝑝 ∈ (Monic_1p‘(𝐸 ↾_s 𝐹))((𝑂‘𝑝)‘𝑋) = 0 )
87	12, 15, 13, 82, 9, 11	elirng 33296	. 2 ⊢ (𝜑 → (𝑋 ∈ (𝐸 IntgRing 𝐹) ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑝 ∈ (Monic_1p‘(𝐸 ↾_s 𝐹))((𝑂‘𝑝)‘𝑋) = 0 )))
88	1, 86, 87	mpbir2and 712	1 ⊢ (𝜑 → 𝑋 ∈ (𝐸 IntgRing 𝐹))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ≠ wne 2935 ∃wrex 3065 Vcvv 3469 dom cdm 5672 ⟶wf 6538 ‘cfv 6542 (class class class)co 7414 ℕ₀cn0 12494 Basecbs 17171 ↾_s cress 17200 ._rcmulr 17225 Scalarcsca 17227 0_gc0g 17412 ↑_s cpws 17419 Ringcrg 20164 CRingccrg 20165 inv_rcinvr 20315 RingHom crh 20397 SubRingcsubrg 20495 DivRingcdr 20613 Fieldcfield 20614 SubDRingcsdrg 20663 LModclmod 20732 AssAlgcasa 21771 algSccascl 21773 Poly₁cpl1 22083 coe₁cco1 22084 evalSub₁ ces1 22219 deg₁ cdg1 25974 Monic_1pcmn1 26048 Unic_1pcuc1p 26049 IntgRing cirng 33293

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 ax-addf 11209

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7679 df-ofr 7680 df-om 7865 df-1st 7987 df-2nd 7988 df-supp 8160 df-tpos 8225 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-map 8838 df-pm 8839 df-ixp 8908 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-fsupp 9378 df-sup 9457 df-oi 9525 df-card 9954 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-nn 12235 df-2 12297 df-3 12298 df-4 12299 df-5 12300 df-6 12301 df-7 12302 df-8 12303 df-9 12304 df-n0 12495 df-z 12581 df-dec 12700 df-uz 12845 df-fz 13509 df-fzo 13652 df-seq 13991 df-hash 14314 df-struct 17107 df-sets 17124 df-slot 17142 df-ndx 17154 df-base 17172 df-ress 17201 df-plusg 17237 df-mulr 17238 df-starv 17239 df-sca 17240 df-vsca 17241 df-ip 17242 df-tset 17243 df-ple 17244 df-ds 17246 df-unif 17247 df-hom 17248 df-cco 17249 df-0g 17414 df-gsum 17415 df-prds 17420 df-pws 17422 df-mre 17557 df-mrc 17558 df-acs 17560 df-mgm 18591 df-sgrp 18670 df-mnd 18686 df-mhm 18731 df-submnd 18732 df-grp 18884 df-minusg 18885 df-sbg 18886 df-mulg 19015 df-subg 19069 df-ghm 19159 df-cntz 19259 df-cmn 19728 df-abl 19729 df-mgp 20066 df-rng 20084 df-ur 20113 df-srg 20118 df-ring 20166 df-cring 20167 df-oppr 20262 df-dvdsr 20285 df-unit 20286 df-invr 20316 df-rhm 20400 df-subrng 20472 df-subrg 20497 df-drng 20615 df-field 20616 df-sdrg 20664 df-lmod 20734 df-lss 20805 df-lsp 20845 df-rlreg 21219 df-cnfld 21267 df-assa 21774 df-asp 21775 df-ascl 21776 df-psr 21829 df-mvr 21830 df-mpl 21831 df-opsr 21833 df-evls 22005 df-evl 22006 df-psr1 22086 df-vr1 22087 df-ply1 22088 df-coe1 22089 df-evls1 22221 df-evl1 22222 df-mdeg 25975 df-deg1 25976 df-mon1 26053 df-uc1p 26054 df-irng 33294

This theorem is referenced by: irngnzply1 33301

W3C validator