Theorem irngss 33728

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 33730). (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 = (0g‘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 20532	. . . . . 6 ⊢ (𝑆 ∈ (SubRing‘𝑅) → 𝑆 ⊆ 𝐵)
5	2, 4	syl 17	. . . . 5 ⊢ (𝜑 → 𝑆 ⊆ 𝐵)
6	5	sselda 3958	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝐵)
7		eqid 2735	. . . . . . . . . 10 ⊢ (Poly1‘𝑅) = (Poly1‘𝑅)
8		irngval.u	. . . . . . . . . 10 ⊢ 𝑈 = (𝑅 ↾s 𝑆)
9		eqid 2735	. . . . . . . . . 10 ⊢ (Poly1‘𝑈) = (Poly1‘𝑈)
10		eqid 2735	. . . . . . . . . 10 ⊢ (Base‘(Poly1‘𝑈)) = (Base‘(Poly1‘𝑈))
11	2	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑆 ∈ (SubRing‘𝑅))
12		eqid 2735	. . . . . . . . . 10 ⊢ ((Poly1‘𝑅) ↾s (Base‘(Poly1‘𝑈))) = ((Poly1‘𝑅) ↾s (Base‘(Poly1‘𝑈)))
13		eqid 2735	. . . . . . . . . . 11 ⊢ (var1‘𝑅) = (var1‘𝑅)
14	13, 11, 8, 9, 10	subrgvr1cl 22199	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (var1‘𝑅) ∈ (Base‘(Poly1‘𝑈)))
15		eqid 2735	. . . . . . . . . . 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 22312	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((algSc‘(Poly1‘𝑅))‘𝑥) ∈ (Base‘(Poly1‘𝑈)))
18	7, 8, 9, 10, 11, 12, 14, 17	ressply1sub 33583	. . . . . . . . 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 22168	. . . . . . . . . . . 12 ⊢ (𝑆 ∈ (SubRing‘𝑅) → (Base‘(Poly1‘𝑈)) ∈ (SubRing‘(Poly1‘𝑅)))
20		subrgsubg 20537	. . . . . . . . . . . 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 2735	. . . . . . . . . . 11 ⊢ (-g‘(Poly1‘𝑅)) = (-g‘(Poly1‘𝑅))
24		eqid 2735	. . . . . . . . . . 11 ⊢ (-g‘((Poly1‘𝑅) ↾s (Base‘(Poly1‘𝑈)))) = (-g‘((Poly1‘𝑅) ↾s (Base‘(Poly1‘𝑈))))
25	23, 12, 24	subgsub 19121	. . . . . . . . . 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 1373	. . . . . . . . 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 2773	. . . . . . . 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 20535	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑅)) → 𝑈 ∈ CRing)
30	28, 2, 29	syl2anc 584	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑈 ∈ CRing)
31	9	ply1crng 22134	. . . . . . . . . . . 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 20207	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (Poly1‘𝑈) ∈ Grp)
35		eqid 2735	. . . . . . . . . 10 ⊢ (-g‘(Poly1‘𝑈)) = (-g‘(Poly1‘𝑈))
36	10, 35	grpsubcl 19003	. . . . . . . . 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 1373	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((var1‘𝑅)(-g‘(Poly1‘𝑈))((algSc‘(Poly1‘𝑅))‘𝑥)) ∈ (Base‘(Poly1‘𝑈)))
38	27, 37	eqeltrrd 2835	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((var1‘𝑅)(-g‘(Poly1‘𝑅))((algSc‘(Poly1‘𝑅))‘𝑥)) ∈ (Base‘(Poly1‘𝑈)))
39		eqid 2735	. . . . . . . . 9 ⊢ (Base‘(Poly1‘𝑅)) = (Base‘(Poly1‘𝑅))
40		eqid 2735	. . . . . . . . 9 ⊢ ((var1‘𝑅)(-g‘(Poly1‘𝑅))((algSc‘(Poly1‘𝑅))‘𝑥)) = ((var1‘𝑅)(-g‘(Poly1‘𝑅))((algSc‘(Poly1‘𝑅))‘𝑥))
41		eqid 2735	. . . . . . . . 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 2735	. . . . . . . . 9 ⊢ (Monic1p‘𝑅) = (Monic1p‘𝑅)
46		eqid 2735	. . . . . . . . 9 ⊢ (deg1‘𝑅) = (deg1‘𝑅)
47		irngval.0	. . . . . . . . 9 ⊢ 0 = (0g‘𝑅)
48	7, 39, 3, 13, 23, 15, 40, 41, 43, 44, 6, 45, 46, 47	ply1remlem 26122	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (((var1‘𝑅)(-g‘(Poly1‘𝑅))((algSc‘(Poly1‘𝑅))‘𝑥)) ∈ (Monic1p‘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 1142	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((var1‘𝑅)(-g‘(Poly1‘𝑅))((algSc‘(Poly1‘𝑅))‘𝑥)) ∈ (Monic1p‘𝑅))
50	38, 49	elind 4175	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((var1‘𝑅)(-g‘(Poly1‘𝑅))((algSc‘(Poly1‘𝑅))‘𝑥)) ∈ ((Base‘(Poly1‘𝑈)) ∩ (Monic1p‘𝑅)))
51		eqid 2735	. . . . . . . 8 ⊢ (Monic1p‘𝑈) = (Monic1p‘𝑈)
52	7, 8, 9, 10, 2, 45, 51	ressply1mon1p 33581	. . . . . . 7 ⊢ (𝜑 → (Monic1p‘𝑈) = ((Base‘(Poly1‘𝑈)) ∩ (Monic1p‘𝑅)))
53	52	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (Monic1p‘𝑈) = ((Base‘(Poly1‘𝑈)) ∩ (Monic1p‘𝑅)))
54	50, 53	eleqtrrd 2837	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((var1‘𝑅)(-g‘(Poly1‘𝑅))((algSc‘(Poly1‘𝑅))‘𝑥)) ∈ (Monic1p‘𝑈))
55		fveq2 6876	. . . . . . . 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 6878	. . . . . . 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 2737	. . . . . 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 22308	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑂 = ((eval1‘𝑅) ↾ (Base‘(Poly1‘𝑈))))
61	60	fveq1d 6878	. . . . . . . 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 6896	. . . . . . . 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 2770	. . . . . . 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 6878	. . . . . 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 2735	. . . . . . . . 9 ⊢ (𝑅 ↑s 𝐵) = (𝑅 ↑s 𝐵)
66		eqid 2735	. . . . . . . . 9 ⊢ (Base‘(𝑅 ↑s 𝐵)) = (Base‘(𝑅 ↑s 𝐵))
67	3	fvexi 6890	. . . . . . . . . 10 ⊢ 𝐵 ∈ V
68	67	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐵 ∈ V)
69	41, 7, 65, 3	evl1rhm 22270	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ CRing → (eval1‘𝑅) ∈ ((Poly1‘𝑅) RingHom (𝑅 ↑s 𝐵)))
70	39, 66	rhmf 20445	. . . . . . . . . . . 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 2735	. . . . . . . . . . . . . 14 ⊢ (PwSer1‘𝑈) = (PwSer1‘𝑈)
74		eqid 2735	. . . . . . . . . . . . . 14 ⊢ (Base‘(PwSer1‘𝑈)) = (Base‘(PwSer1‘𝑈))
75	7, 8, 9, 10, 2, 73, 74, 39	ressply1bas2 22163	. . . . . . . . . . . . 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 2836	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((var1‘𝑅)(-g‘(Poly1‘𝑅))((algSc‘(Poly1‘𝑅))‘𝑥)) ∈ ((Base‘(PwSer1‘𝑈)) ∩ (Base‘(Poly1‘𝑅))))
78	77	elin2d 4180	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((var1‘𝑅)(-g‘(Poly1‘𝑅))((algSc‘(Poly1‘𝑅))‘𝑥)) ∈ (Base‘(Poly1‘𝑅)))
79	72, 78	ffvelcdmd 7075	. . . . . . . . 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 17503	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((eval1‘𝑅)‘((var1‘𝑅)(-g‘(Poly1‘𝑅))((algSc‘(Poly1‘𝑅))‘𝑥))):𝐵⟶𝐵)
81	80	ffnd 6707	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((eval1‘𝑅)‘((var1‘𝑅)(-g‘(Poly1‘𝑅))((algSc‘(Poly1‘𝑅))‘𝑥))) Fn 𝐵)
82		vsnid 4639	. . . . . . . 8 ⊢ 𝑥 ∈ {𝑥}
83	48	simp3d 1144	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (◡((eval1‘𝑅)‘((var1‘𝑅)(-g‘(Poly1‘𝑅))((algSc‘(Poly1‘𝑅))‘𝑥))) “ { 0 }) = {𝑥})
84	82, 83	eleqtrrid 2841	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ (◡((eval1‘𝑅)‘((var1‘𝑅)(-g‘(Poly1‘𝑅))((algSc‘(Poly1‘𝑅))‘𝑥))) “ { 0 }))
85		fniniseg 7050	. . . . . . . 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 584	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (((eval1‘𝑅)‘((var1‘𝑅)(-g‘(Poly1‘𝑅))((algSc‘(Poly1‘𝑅))‘𝑥)))‘𝑥) = 0 )
88	64, 87	eqtrd 2770	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((𝑂‘((var1‘𝑅)(-g‘(Poly1‘𝑅))((algSc‘(Poly1‘𝑅))‘𝑥)))‘𝑥) = 0 )
89	54, 58, 88	rspcedvd 3603	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ∃𝑓 ∈ (Monic1p‘𝑈)((𝑂‘𝑓)‘𝑥) = 0 )
90	59, 8, 3, 47, 28, 2	elirng 33727	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (𝑅 IntgRing 𝑆) ↔ (𝑥 ∈ 𝐵 ∧ ∃𝑓 ∈ (Monic1p‘𝑈)((𝑂‘𝑓)‘𝑥) = 0 )))
91	90	biimpar 477	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ ∃𝑓 ∈ (Monic1p‘𝑈)((𝑂‘𝑓)‘𝑥) = 0 )) → 𝑥 ∈ (𝑅 IntgRing 𝑆))
92	1, 6, 89, 91	syl12anc 836	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ (𝑅 IntgRing 𝑆))
93	92	ex 412	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑆 → 𝑥 ∈ (𝑅 IntgRing 𝑆)))
94	93	ssrdv 3964	1 ⊢ (𝜑 → 𝑆 ⊆ (𝑅 IntgRing 𝑆))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∃wrex 3060  Vcvv 3459   ∩ cin 3925   ⊆ wss 3926  {csn 4601  ^◡ccnv 5653   ↾ cres 5656   “ cima 5657   Fn wfn 6526  ⟶wf 6527  ‘cfv 6531  (class class class)co 7405  1c1 11130  Basecbs 17228   ↾_s cress 17251  0_gc0g 17453   ↑_s cpws 17460  Grpcgrp 18916  -_gcsg 18918  SubGrpcsubg 19103  CRingccrg 20194   RingHom crh 20429  NzRingcnzr 20472  SubRingcsubrg 20529  algSccascl 21812  PwSer₁cps1 22110  var₁cv1 22111  Poly₁cpl1 22112   evalSub₁ ces1 22251  eval₁ce1 22252  deg₁cdg1 26011  Monic_1pcmn1 26083   IntgRing cirng 33724

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-cnex 11185  ax-resscn 11186  ax-1cn 11187  ax-icn 11188  ax-addcl 11189  ax-addrcl 11190  ax-mulcl 11191  ax-mulrcl 11192  ax-mulcom 11193  ax-addass 11194  ax-mulass 11195  ax-distr 11196  ax-i2m1 11197  ax-1ne0 11198  ax-1rid 11199  ax-rnegex 11200  ax-rrecex 11201  ax-cnre 11202  ax-pre-lttri 11203  ax-pre-lttrn 11204  ax-pre-ltadd 11205  ax-pre-mulgt0 11206  ax-pre-sup 11207  ax-addf 11208

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-tp 4606  df-op 4608  df-uni 4884  df-int 4923  df-iun 4969  df-iin 4970  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-se 5607  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-isom 6540  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-of 7671  df-ofr 7672  df-om 7862  df-1st 7988  df-2nd 7989  df-supp 8160  df-tpos 8225  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-2o 8481  df-er 8719  df-map 8842  df-pm 8843  df-ixp 8912  df-en 8960  df-dom 8961  df-sdom 8962  df-fin 8963  df-fsupp 9374  df-sup 9454  df-oi 9524  df-card 9953  df-pnf 11271  df-mnf 11272  df-xr 11273  df-ltxr 11274  df-le 11275  df-sub 11468  df-neg 11469  df-nn 12241  df-2 12303  df-3 12304  df-4 12305  df-5 12306  df-6 12307  df-7 12308  df-8 12309  df-9 12310  df-n0 12502  df-z 12589  df-dec 12709  df-uz 12853  df-fz 13525  df-fzo 13672  df-seq 14020  df-hash 14349  df-struct 17166  df-sets 17183  df-slot 17201  df-ndx 17213  df-base 17229  df-ress 17252  df-plusg 17284  df-mulr 17285  df-starv 17286  df-sca 17287  df-vsca 17288  df-ip 17289  df-tset 17290  df-ple 17291  df-ds 17293  df-unif 17294  df-hom 17295  df-cco 17296  df-0g 17455  df-gsum 17456  df-prds 17461  df-pws 17463  df-mre 17598  df-mrc 17599  df-acs 17601  df-mgm 18618  df-sgrp 18697  df-mnd 18713  df-mhm 18761  df-submnd 18762  df-grp 18919  df-minusg 18920  df-sbg 18921  df-mulg 19051  df-subg 19106  df-ghm 19196  df-cntz 19300  df-cmn 19763  df-abl 19764  df-mgp 20101  df-rng 20113  df-ur 20142  df-srg 20147  df-ring 20195  df-cring 20196  df-oppr 20297  df-dvdsr 20317  df-unit 20318  df-invr 20348  df-rhm 20432  df-nzr 20473  df-subrng 20506  df-subrg 20530  df-rlreg 20654  df-lmod 20819  df-lss 20889  df-lsp 20929  df-cnfld 21316  df-assa 21813  df-asp 21814  df-ascl 21815  df-psr 21869  df-mvr 21870  df-mpl 21871  df-opsr 21873  df-evls 22032  df-evl 22033  df-psr1 22115  df-vr1 22116  df-ply1 22117  df-coe1 22118  df-evls1 22253  df-evl1 22254  df-mdeg 26012  df-deg1 26013  df-mon1 26088  df-irng 33725

This theorem is referenced by: (None)