irngss - Mathbox for Thierry Arnoux

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Theorem irngss 33207

Description: All elements of a subring 𝑆 are integral over 𝑆. This is only true in the case of a nonzero ring, since there are no integral elements in a zero ring (see 0ringirng 33209). (Contributed by Thierry Arnoux, 28-Jan-2025.)

**Hypotheses**
Ref	Expression
irngval.o	⊢ 𝑂 = (𝑅 evalSub₁ 𝑆)
irngval.u	⊢ 𝑈 = (𝑅 ↾_s 𝑆)
irngval.b	⊢ 𝐵 = (Base‘𝑅)
irngval.0	⊢ 0 = (0_g‘𝑅)
elirng.r	⊢ (𝜑 → 𝑅 ∈ CRing)
elirng.s	⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝑅))
irngss.1	⊢ (𝜑 → 𝑅 ∈ NzRing)

**Assertion**
Ref	Expression
irngss	⊢ (𝜑 → 𝑆 ⊆ (𝑅 IntgRing 𝑆))

Proof of Theorem irngss Dummy variables 𝑓 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 482	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝜑)
2		elirng.s	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝑅))
3		irngval.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑅)
4	3	subrgss 20470	. . . . . 6 ⊢ (𝑆 ∈ (SubRing‘𝑅) → 𝑆 ⊆ 𝐵)
5	2, 4	syl 17	. . . . 5 ⊢ (𝜑 → 𝑆 ⊆ 𝐵)
6	5	sselda 3982	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝐵)
7		eqid 2731	. . . . . . . . . 10 ⊢ (Poly₁‘𝑅) = (Poly₁‘𝑅)
8		irngval.u	. . . . . . . . . 10 ⊢ 𝑈 = (𝑅 ↾_s 𝑆)
9		eqid 2731	. . . . . . . . . 10 ⊢ (Poly₁‘𝑈) = (Poly₁‘𝑈)
10		eqid 2731	. . . . . . . . . 10 ⊢ (Base‘(Poly₁‘𝑈)) = (Base‘(Poly₁‘𝑈))
11	2	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑆 ∈ (SubRing‘𝑅))
12		eqid 2731	. . . . . . . . . 10 ⊢ ((Poly₁‘𝑅) ↾_s (Base‘(Poly₁‘𝑈))) = ((Poly₁‘𝑅) ↾_s (Base‘(Poly₁‘𝑈)))
13		eqid 2731	. . . . . . . . . . 11 ⊢ (var₁‘𝑅) = (var₁‘𝑅)
14	13, 11, 8, 9, 10	subrgvr1cl 22104	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (var₁‘𝑅) ∈ (Base‘(Poly₁‘𝑈)))
15		eqid 2731	. . . . . . . . . . 11 ⊢ (algSc‘(Poly₁‘𝑅)) = (algSc‘(Poly₁‘𝑅))
16		simpr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝑆)
17	15, 8, 7, 9, 10, 11, 16	asclply1subcl 33101	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((algSc‘(Poly₁‘𝑅))‘𝑥) ∈ (Base‘(Poly₁‘𝑈)))
18	7, 8, 9, 10, 11, 12, 14, 17	ressply1sub 33100	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((var₁‘𝑅)(-_g‘(Poly₁‘𝑈))((algSc‘(Poly₁‘𝑅))‘𝑥)) = ((var₁‘𝑅)(-_g‘((Poly₁‘𝑅) ↾_s (Base‘(Poly₁‘𝑈))))((algSc‘(Poly₁‘𝑅))‘𝑥)))
19	7, 8, 9, 10	subrgply1 22075	. . . . . . . . . . . 12 ⊢ (𝑆 ∈ (SubRing‘𝑅) → (Base‘(Poly₁‘𝑈)) ∈ (SubRing‘(Poly₁‘𝑅)))
20		subrgsubg 20475	. . . . . . . . . . . 12 ⊢ ((Base‘(Poly₁‘𝑈)) ∈ (SubRing‘(Poly₁‘𝑅)) → (Base‘(Poly₁‘𝑈)) ∈ (SubGrp‘(Poly₁‘𝑅)))
21	2, 19, 20	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → (Base‘(Poly₁‘𝑈)) ∈ (SubGrp‘(Poly₁‘𝑅)))
22	21	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (Base‘(Poly₁‘𝑈)) ∈ (SubGrp‘(Poly₁‘𝑅)))
23		eqid 2731	. . . . . . . . . . 11 ⊢ (-_g‘(Poly₁‘𝑅)) = (-_g‘(Poly₁‘𝑅))
24		eqid 2731	. . . . . . . . . . 11 ⊢ (-_g‘((Poly₁‘𝑅) ↾_s (Base‘(Poly₁‘𝑈)))) = (-_g‘((Poly₁‘𝑅) ↾_s (Base‘(Poly₁‘𝑈))))
25	23, 12, 24	subgsub 19061	. . . . . . . . . 10 ⊢ (((Base‘(Poly₁‘𝑈)) ∈ (SubGrp‘(Poly₁‘𝑅)) ∧ (var₁‘𝑅) ∈ (Base‘(Poly₁‘𝑈)) ∧ ((algSc‘(Poly₁‘𝑅))‘𝑥) ∈ (Base‘(Poly₁‘𝑈))) → ((var₁‘𝑅)(-_g‘(Poly₁‘𝑅))((algSc‘(Poly₁‘𝑅))‘𝑥)) = ((var₁‘𝑅)(-_g‘((Poly₁‘𝑅) ↾_s (Base‘(Poly₁‘𝑈))))((algSc‘(Poly₁‘𝑅))‘𝑥)))
26	22, 14, 17, 25	syl3anc 1370	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((var₁‘𝑅)(-_g‘(Poly₁‘𝑅))((algSc‘(Poly₁‘𝑅))‘𝑥)) = ((var₁‘𝑅)(-_g‘((Poly₁‘𝑅) ↾_s (Base‘(Poly₁‘𝑈))))((algSc‘(Poly₁‘𝑅))‘𝑥)))
27	18, 26	eqtr4d 2774	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((var₁‘𝑅)(-_g‘(Poly₁‘𝑈))((algSc‘(Poly₁‘𝑅))‘𝑥)) = ((var₁‘𝑅)(-_g‘(Poly₁‘𝑅))((algSc‘(Poly₁‘𝑅))‘𝑥)))
28		elirng.r	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑅 ∈ CRing)
29	8	subrgcrng 20473	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑅)) → 𝑈 ∈ CRing)
30	28, 2, 29	syl2anc 583	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑈 ∈ CRing)
31	9	ply1crng 22041	. . . . . . . . . . . 12 ⊢ (𝑈 ∈ CRing → (Poly₁‘𝑈) ∈ CRing)
32	30, 31	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (Poly₁‘𝑈) ∈ CRing)
33	32	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (Poly₁‘𝑈) ∈ CRing)
34	33	crnggrpd 20148	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (Poly₁‘𝑈) ∈ Grp)
35		eqid 2731	. . . . . . . . . 10 ⊢ (-_g‘(Poly₁‘𝑈)) = (-_g‘(Poly₁‘𝑈))
36	10, 35	grpsubcl 18946	. . . . . . . . 9 ⊢ (((Poly₁‘𝑈) ∈ Grp ∧ (var₁‘𝑅) ∈ (Base‘(Poly₁‘𝑈)) ∧ ((algSc‘(Poly₁‘𝑅))‘𝑥) ∈ (Base‘(Poly₁‘𝑈))) → ((var₁‘𝑅)(-_g‘(Poly₁‘𝑈))((algSc‘(Poly₁‘𝑅))‘𝑥)) ∈ (Base‘(Poly₁‘𝑈)))
37	34, 14, 17, 36	syl3anc 1370	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((var₁‘𝑅)(-_g‘(Poly₁‘𝑈))((algSc‘(Poly₁‘𝑅))‘𝑥)) ∈ (Base‘(Poly₁‘𝑈)))
38	27, 37	eqeltrrd 2833	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((var₁‘𝑅)(-_g‘(Poly₁‘𝑅))((algSc‘(Poly₁‘𝑅))‘𝑥)) ∈ (Base‘(Poly₁‘𝑈)))
39		eqid 2731	. . . . . . . . 9 ⊢ (Base‘(Poly₁‘𝑅)) = (Base‘(Poly₁‘𝑅))
40		eqid 2731	. . . . . . . . 9 ⊢ ((var₁‘𝑅)(-_g‘(Poly₁‘𝑅))((algSc‘(Poly₁‘𝑅))‘𝑥)) = ((var₁‘𝑅)(-_g‘(Poly₁‘𝑅))((algSc‘(Poly₁‘𝑅))‘𝑥))
41		eqid 2731	. . . . . . . . 9 ⊢ (eval₁‘𝑅) = (eval₁‘𝑅)
42		irngss.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ NzRing)
43	42	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑅 ∈ NzRing)
44	28	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑅 ∈ CRing)
45		eqid 2731	. . . . . . . . 9 ⊢ (Monic_1p‘𝑅) = (Monic_1p‘𝑅)
46		eqid 2731	. . . . . . . . 9 ⊢ ( deg₁ ‘𝑅) = ( deg₁ ‘𝑅)
47		irngval.0	. . . . . . . . 9 ⊢ 0 = (0_g‘𝑅)
48	7, 39, 3, 13, 23, 15, 40, 41, 43, 44, 6, 45, 46, 47	ply1remlem 26018	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (((var₁‘𝑅)(-_g‘(Poly₁‘𝑅))((algSc‘(Poly₁‘𝑅))‘𝑥)) ∈ (Monic_1p‘𝑅) ∧ (( deg₁ ‘𝑅)‘((var₁‘𝑅)(-_g‘(Poly₁‘𝑅))((algSc‘(Poly₁‘𝑅))‘𝑥))) = 1 ∧ (^◡((eval₁‘𝑅)‘((var₁‘𝑅)(-_g‘(Poly₁‘𝑅))((algSc‘(Poly₁‘𝑅))‘𝑥))) “ { 0 }) = {𝑥}))
49	48	simp1d 1141	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((var₁‘𝑅)(-_g‘(Poly₁‘𝑅))((algSc‘(Poly₁‘𝑅))‘𝑥)) ∈ (Monic_1p‘𝑅))
50	38, 49	elind 4194	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((var₁‘𝑅)(-_g‘(Poly₁‘𝑅))((algSc‘(Poly₁‘𝑅))‘𝑥)) ∈ ((Base‘(Poly₁‘𝑈)) ∩ (Monic_1p‘𝑅)))
51		eqid 2731	. . . . . . . 8 ⊢ (Monic_1p‘𝑈) = (Monic_1p‘𝑈)
52	7, 8, 9, 10, 2, 45, 51	ressply1mon1p 33098	. . . . . . 7 ⊢ (𝜑 → (Monic_1p‘𝑈) = ((Base‘(Poly₁‘𝑈)) ∩ (Monic_1p‘𝑅)))
53	52	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (Monic_1p‘𝑈) = ((Base‘(Poly₁‘𝑈)) ∩ (Monic_1p‘𝑅)))
54	50, 53	eleqtrrd 2835	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((var₁‘𝑅)(-_g‘(Poly₁‘𝑅))((algSc‘(Poly₁‘𝑅))‘𝑥)) ∈ (Monic_1p‘𝑈))
55		fveq2 6891	. . . . . . . 8 ⊢ (𝑓 = ((var₁‘𝑅)(-_g‘(Poly₁‘𝑅))((algSc‘(Poly₁‘𝑅))‘𝑥)) → (𝑂‘𝑓) = (𝑂‘((var₁‘𝑅)(-_g‘(Poly₁‘𝑅))((algSc‘(Poly₁‘𝑅))‘𝑥))))
56	55	fveq1d 6893	. . . . . . 7 ⊢ (𝑓 = ((var₁‘𝑅)(-_g‘(Poly₁‘𝑅))((algSc‘(Poly₁‘𝑅))‘𝑥)) → ((𝑂‘𝑓)‘𝑥) = ((𝑂‘((var₁‘𝑅)(-_g‘(Poly₁‘𝑅))((algSc‘(Poly₁‘𝑅))‘𝑥)))‘𝑥))
57	56	eqeq1d 2733	. . . . . 6 ⊢ (𝑓 = ((var₁‘𝑅)(-_g‘(Poly₁‘𝑅))((algSc‘(Poly₁‘𝑅))‘𝑥)) → (((𝑂‘𝑓)‘𝑥) = 0 ↔ ((𝑂‘((var₁‘𝑅)(-_g‘(Poly₁‘𝑅))((algSc‘(Poly₁‘𝑅))‘𝑥)))‘𝑥) = 0 ))
58	57	adantl 481	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑓 = ((var₁‘𝑅)(-_g‘(Poly₁‘𝑅))((algSc‘(Poly₁‘𝑅))‘𝑥))) → (((𝑂‘𝑓)‘𝑥) = 0 ↔ ((𝑂‘((var₁‘𝑅)(-_g‘(Poly₁‘𝑅))((algSc‘(Poly₁‘𝑅))‘𝑥)))‘𝑥) = 0 ))
59		irngval.o	. . . . . . . . . 10 ⊢ 𝑂 = (𝑅 evalSub₁ 𝑆)
60	59, 3, 9, 8, 10, 41, 44, 11	ressply1evl 33088	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑂 = ((eval₁‘𝑅) ↾ (Base‘(Poly₁‘𝑈))))
61	60	fveq1d 6893	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝑂‘((var₁‘𝑅)(-_g‘(Poly₁‘𝑅))((algSc‘(Poly₁‘𝑅))‘𝑥))) = (((eval₁‘𝑅) ↾ (Base‘(Poly₁‘𝑈)))‘((var₁‘𝑅)(-_g‘(Poly₁‘𝑅))((algSc‘(Poly₁‘𝑅))‘𝑥))))
62	38	fvresd 6911	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (((eval₁‘𝑅) ↾ (Base‘(Poly₁‘𝑈)))‘((var₁‘𝑅)(-_g‘(Poly₁‘𝑅))((algSc‘(Poly₁‘𝑅))‘𝑥))) = ((eval₁‘𝑅)‘((var₁‘𝑅)(-_g‘(Poly₁‘𝑅))((algSc‘(Poly₁‘𝑅))‘𝑥))))
63	61, 62	eqtrd 2771	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝑂‘((var₁‘𝑅)(-_g‘(Poly₁‘𝑅))((algSc‘(Poly₁‘𝑅))‘𝑥))) = ((eval₁‘𝑅)‘((var₁‘𝑅)(-_g‘(Poly₁‘𝑅))((algSc‘(Poly₁‘𝑅))‘𝑥))))
64	63	fveq1d 6893	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((𝑂‘((var₁‘𝑅)(-_g‘(Poly₁‘𝑅))((algSc‘(Poly₁‘𝑅))‘𝑥)))‘𝑥) = (((eval₁‘𝑅)‘((var₁‘𝑅)(-_g‘(Poly₁‘𝑅))((algSc‘(Poly₁‘𝑅))‘𝑥)))‘𝑥))
65		eqid 2731	. . . . . . . . 9 ⊢ (𝑅 ↑_s 𝐵) = (𝑅 ↑_s 𝐵)
66		eqid 2731	. . . . . . . . 9 ⊢ (Base‘(𝑅 ↑_s 𝐵)) = (Base‘(𝑅 ↑_s 𝐵))
67	3	fvexi 6905	. . . . . . . . . 10 ⊢ 𝐵 ∈ V
68	67	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐵 ∈ V)
69	41, 7, 65, 3	evl1rhm 22171	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ CRing → (eval₁‘𝑅) ∈ ((Poly₁‘𝑅) RingHom (𝑅 ↑_s 𝐵)))
70	39, 66	rhmf 20383	. . . . . . . . . . . 12 ⊢ ((eval₁‘𝑅) ∈ ((Poly₁‘𝑅) RingHom (𝑅 ↑_s 𝐵)) → (eval₁‘𝑅):(Base‘(Poly₁‘𝑅))⟶(Base‘(𝑅 ↑_s 𝐵)))
71	28, 69, 70	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → (eval₁‘𝑅):(Base‘(Poly₁‘𝑅))⟶(Base‘(𝑅 ↑_s 𝐵)))
72	71	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (eval₁‘𝑅):(Base‘(Poly₁‘𝑅))⟶(Base‘(𝑅 ↑_s 𝐵)))
73		eqid 2731	. . . . . . . . . . . . . 14 ⊢ (PwSer₁‘𝑈) = (PwSer₁‘𝑈)
74		eqid 2731	. . . . . . . . . . . . . 14 ⊢ (Base‘(PwSer₁‘𝑈)) = (Base‘(PwSer₁‘𝑈))
75	7, 8, 9, 10, 2, 73, 74, 39	ressply1bas2 22070	. . . . . . . . . . . . 13 ⊢ (𝜑 → (Base‘(Poly₁‘𝑈)) = ((Base‘(PwSer₁‘𝑈)) ∩ (Base‘(Poly₁‘𝑅))))
76	75	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (Base‘(Poly₁‘𝑈)) = ((Base‘(PwSer₁‘𝑈)) ∩ (Base‘(Poly₁‘𝑅))))
77	38, 76	eleqtrd 2834	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((var₁‘𝑅)(-_g‘(Poly₁‘𝑅))((algSc‘(Poly₁‘𝑅))‘𝑥)) ∈ ((Base‘(PwSer₁‘𝑈)) ∩ (Base‘(Poly₁‘𝑅))))
78	77	elin2d 4199	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((var₁‘𝑅)(-_g‘(Poly₁‘𝑅))((algSc‘(Poly₁‘𝑅))‘𝑥)) ∈ (Base‘(Poly₁‘𝑅)))
79	72, 78	ffvelcdmd 7087	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((eval₁‘𝑅)‘((var₁‘𝑅)(-_g‘(Poly₁‘𝑅))((algSc‘(Poly₁‘𝑅))‘𝑥))) ∈ (Base‘(𝑅 ↑_s 𝐵)))
80	65, 3, 66, 43, 68, 79	pwselbas 17442	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((eval₁‘𝑅)‘((var₁‘𝑅)(-_g‘(Poly₁‘𝑅))((algSc‘(Poly₁‘𝑅))‘𝑥))):𝐵⟶𝐵)
81	80	ffnd 6718	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((eval₁‘𝑅)‘((var₁‘𝑅)(-_g‘(Poly₁‘𝑅))((algSc‘(Poly₁‘𝑅))‘𝑥))) Fn 𝐵)
82		vsnid 4665	. . . . . . . 8 ⊢ 𝑥 ∈ {𝑥}
83	48	simp3d 1143	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (^◡((eval₁‘𝑅)‘((var₁‘𝑅)(-_g‘(Poly₁‘𝑅))((algSc‘(Poly₁‘𝑅))‘𝑥))) “ { 0 }) = {𝑥})
84	82, 83	eleqtrrid 2839	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ (^◡((eval₁‘𝑅)‘((var₁‘𝑅)(-_g‘(Poly₁‘𝑅))((algSc‘(Poly₁‘𝑅))‘𝑥))) “ { 0 }))
85		fniniseg 7061	. . . . . . . 8 ⊢ (((eval₁‘𝑅)‘((var₁‘𝑅)(-_g‘(Poly₁‘𝑅))((algSc‘(Poly₁‘𝑅))‘𝑥))) Fn 𝐵 → (𝑥 ∈ (^◡((eval₁‘𝑅)‘((var₁‘𝑅)(-_g‘(Poly₁‘𝑅))((algSc‘(Poly₁‘𝑅))‘𝑥))) “ { 0 }) ↔ (𝑥 ∈ 𝐵 ∧ (((eval₁‘𝑅)‘((var₁‘𝑅)(-_g‘(Poly₁‘𝑅))((algSc‘(Poly₁‘𝑅))‘𝑥)))‘𝑥) = 0 )))
86	85	simplbda 499	. . . . . . 7 ⊢ ((((eval₁‘𝑅)‘((var₁‘𝑅)(-_g‘(Poly₁‘𝑅))((algSc‘(Poly₁‘𝑅))‘𝑥))) Fn 𝐵 ∧ 𝑥 ∈ (^◡((eval₁‘𝑅)‘((var₁‘𝑅)(-_g‘(Poly₁‘𝑅))((algSc‘(Poly₁‘𝑅))‘𝑥))) “ { 0 })) → (((eval₁‘𝑅)‘((var₁‘𝑅)(-_g‘(Poly₁‘𝑅))((algSc‘(Poly₁‘𝑅))‘𝑥)))‘𝑥) = 0 )
87	81, 84, 86	syl2anc 583	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (((eval₁‘𝑅)‘((var₁‘𝑅)(-_g‘(Poly₁‘𝑅))((algSc‘(Poly₁‘𝑅))‘𝑥)))‘𝑥) = 0 )
88	64, 87	eqtrd 2771	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((𝑂‘((var₁‘𝑅)(-_g‘(Poly₁‘𝑅))((algSc‘(Poly₁‘𝑅))‘𝑥)))‘𝑥) = 0 )
89	54, 58, 88	rspcedvd 3614	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ∃𝑓 ∈ (Monic_1p‘𝑈)((𝑂‘𝑓)‘𝑥) = 0 )
90	59, 8, 3, 47, 28, 2	elirng 33206	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (𝑅 IntgRing 𝑆) ↔ (𝑥 ∈ 𝐵 ∧ ∃𝑓 ∈ (Monic_1p‘𝑈)((𝑂‘𝑓)‘𝑥) = 0 )))
91	90	biimpar 477	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ ∃𝑓 ∈ (Monic_1p‘𝑈)((𝑂‘𝑓)‘𝑥) = 0 )) → 𝑥 ∈ (𝑅 IntgRing 𝑆))
92	1, 6, 89, 91	syl12anc 834	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ (𝑅 IntgRing 𝑆))
93	92	ex 412	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑆 → 𝑥 ∈ (𝑅 IntgRing 𝑆)))
94	93	ssrdv 3988	1 ⊢ (𝜑 → 𝑆 ⊆ (𝑅 IntgRing 𝑆))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ∃wrex 3069 Vcvv 3473 ∩ cin 3947 ⊆ wss 3948 {csn 4628 ^◡ccnv 5675 ↾ cres 5678 “ cima 5679 Fn wfn 6538 ⟶wf 6539 ‘cfv 6543 (class class class)co 7412 1c1 11117 Basecbs 17151 ↾_s cress 17180 0_gc0g 17392 ↑_s cpws 17399 Grpcgrp 18861 -_gcsg 18863 SubGrpcsubg 19043 CRingccrg 20135 RingHom crh 20367 NzRingcnzr 20410 SubRingcsubrg 20465 algSccascl 21717 PwSer₁cps1 22018 var₁cv1 22019 Poly₁cpl1 22020 evalSub₁ ces1 22152 eval₁ce1 22153 deg₁ cdg1 25907 Monic_1pcmn1 25981 IntgRing cirng 33203

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-pre-sup 11194 ax-addf 11195 ax-mulf 11196

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 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 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7674 df-ofr 7675 df-om 7860 df-1st 7979 df-2nd 7980 df-supp 8152 df-tpos 8217 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-er 8709 df-map 8828 df-pm 8829 df-ixp 8898 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-fsupp 9368 df-sup 9443 df-oi 9511 df-card 9940 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12480 df-z 12566 df-dec 12685 df-uz 12830 df-fz 13492 df-fzo 13635 df-seq 13974 df-hash 14298 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-starv 17219 df-sca 17220 df-vsca 17221 df-ip 17222 df-tset 17223 df-ple 17224 df-ds 17226 df-unif 17227 df-hom 17228 df-cco 17229 df-0g 17394 df-gsum 17395 df-prds 17400 df-pws 17402 df-mre 17537 df-mrc 17538 df-acs 17540 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-mhm 18711 df-submnd 18712 df-grp 18864 df-minusg 18865 df-sbg 18866 df-mulg 18994 df-subg 19046 df-ghm 19135 df-cntz 19229 df-cmn 19698 df-abl 19699 df-mgp 20036 df-rng 20054 df-ur 20083 df-srg 20088 df-ring 20136 df-cring 20137 df-oppr 20232 df-dvdsr 20255 df-unit 20256 df-invr 20286 df-rhm 20370 df-nzr 20411 df-subrng 20442 df-subrg 20467 df-lmod 20704 df-lss 20775 df-lsp 20815 df-rlreg 21188 df-cnfld 21234 df-assa 21718 df-asp 21719 df-ascl 21720 df-psr 21772 df-mvr 21773 df-mpl 21774 df-opsr 21776 df-evls 21946 df-evl 21947 df-psr1 22023 df-vr1 22024 df-ply1 22025 df-coe1 22026 df-evls1 22154 df-evl1 22155 df-mdeg 25908 df-deg1 25909 df-mon1 25986 df-irng 33204

This theorem is referenced by: (None)

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