Theorem irngss 32653

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 32655). (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 483	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝜑)
2		elirng.s	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝑅))
3		irngval.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑅)
4	3	subrgss 20315	. . . . . 6 ⊢ (𝑆 ∈ (SubRing‘𝑅) → 𝑆 ⊆ 𝐵)
5	2, 4	syl 17	. . . . 5 ⊢ (𝜑 → 𝑆 ⊆ 𝐵)
6	5	sselda 3979	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝐵)
7		eqid 2732	. . . . . . . . . 10 ⊢ (Poly1‘𝑅) = (Poly1‘𝑅)
8		irngval.u	. . . . . . . . . 10 ⊢ 𝑈 = (𝑅 ↾s 𝑆)
9		eqid 2732	. . . . . . . . . 10 ⊢ (Poly1‘𝑈) = (Poly1‘𝑈)
10		eqid 2732	. . . . . . . . . 10 ⊢ (Base‘(Poly1‘𝑈)) = (Base‘(Poly1‘𝑈))
11	2	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑆 ∈ (SubRing‘𝑅))
12		eqid 2732	. . . . . . . . . 10 ⊢ ((Poly1‘𝑅) ↾s (Base‘(Poly1‘𝑈))) = ((Poly1‘𝑅) ↾s (Base‘(Poly1‘𝑈)))
13		eqid 2732	. . . . . . . . . . 11 ⊢ (var1‘𝑅) = (var1‘𝑅)
14	13, 11, 8, 9, 10	subrgvr1cl 21717	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (var1‘𝑅) ∈ (Base‘(Poly1‘𝑈)))
15		eqid 2732	. . . . . . . . . . 11 ⊢ (algSc‘(Poly1‘𝑅)) = (algSc‘(Poly1‘𝑅))
16		simpr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝑆)
17	15, 8, 7, 9, 10, 11, 16	asclply1subcl 32564	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((algSc‘(Poly1‘𝑅))‘𝑥) ∈ (Base‘(Poly1‘𝑈)))
18	7, 8, 9, 10, 11, 12, 14, 17	ressply1sub 32563	. . . . . . . . 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 21688	. . . . . . . . . . . 12 ⊢ (𝑆 ∈ (SubRing‘𝑅) → (Base‘(Poly1‘𝑈)) ∈ (SubRing‘(Poly1‘𝑅)))
20		subrgsubg 20320	. . . . . . . . . . . 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 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (Base‘(Poly1‘𝑈)) ∈ (SubGrp‘(Poly1‘𝑅)))
23		eqid 2732	. . . . . . . . . . 11 ⊢ (-g‘(Poly1‘𝑅)) = (-g‘(Poly1‘𝑅))
24		eqid 2732	. . . . . . . . . . 11 ⊢ (-g‘((Poly1‘𝑅) ↾s (Base‘(Poly1‘𝑈)))) = (-g‘((Poly1‘𝑅) ↾s (Base‘(Poly1‘𝑈))))
25	23, 12, 24	subgsub 18992	. . . . . . . . . 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 1371	. . . . . . . . 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 2775	. . . . . . . 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 20318	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑅)) → 𝑈 ∈ CRing)
30	28, 2, 29	syl2anc 584	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑈 ∈ CRing)
31	9	ply1crng 21653	. . . . . . . . . . . 12 ⊢ (𝑈 ∈ CRing → (Poly1‘𝑈) ∈ CRing)
32	30, 31	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (Poly1‘𝑈) ∈ CRing)
33	32	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (Poly1‘𝑈) ∈ CRing)
34	33	crnggrpd 20030	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (Poly1‘𝑈) ∈ Grp)
35		eqid 2732	. . . . . . . . . 10 ⊢ (-g‘(Poly1‘𝑈)) = (-g‘(Poly1‘𝑈))
36	10, 35	grpsubcl 18879	. . . . . . . . 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 1371	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((var1‘𝑅)(-g‘(Poly1‘𝑈))((algSc‘(Poly1‘𝑅))‘𝑥)) ∈ (Base‘(Poly1‘𝑈)))
38	27, 37	eqeltrrd 2834	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((var1‘𝑅)(-g‘(Poly1‘𝑅))((algSc‘(Poly1‘𝑅))‘𝑥)) ∈ (Base‘(Poly1‘𝑈)))
39		eqid 2732	. . . . . . . . 9 ⊢ (Base‘(Poly1‘𝑅)) = (Base‘(Poly1‘𝑅))
40		eqid 2732	. . . . . . . . 9 ⊢ ((var1‘𝑅)(-g‘(Poly1‘𝑅))((algSc‘(Poly1‘𝑅))‘𝑥)) = ((var1‘𝑅)(-g‘(Poly1‘𝑅))((algSc‘(Poly1‘𝑅))‘𝑥))
41		eqid 2732	. . . . . . . . 9 ⊢ (eval1‘𝑅) = (eval1‘𝑅)
42		irngss.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ NzRing)
43	42	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑅 ∈ NzRing)
44	28	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑅 ∈ CRing)
45		eqid 2732	. . . . . . . . 9 ⊢ (Monic1p‘𝑅) = (Monic1p‘𝑅)
46		eqid 2732	. . . . . . . . 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 25611	. . . . . . . 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 4191	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((var1‘𝑅)(-g‘(Poly1‘𝑅))((algSc‘(Poly1‘𝑅))‘𝑥)) ∈ ((Base‘(Poly1‘𝑈)) ∩ (Monic1p‘𝑅)))
51		eqid 2732	. . . . . . . 8 ⊢ (Monic1p‘𝑈) = (Monic1p‘𝑈)
52	7, 8, 9, 10, 2, 45, 51	ressply1mon1p 32561	. . . . . . 7 ⊢ (𝜑 → (Monic1p‘𝑈) = ((Base‘(Poly1‘𝑈)) ∩ (Monic1p‘𝑅)))
53	52	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (Monic1p‘𝑈) = ((Base‘(Poly1‘𝑈)) ∩ (Monic1p‘𝑅)))
54	50, 53	eleqtrrd 2836	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((var1‘𝑅)(-g‘(Poly1‘𝑅))((algSc‘(Poly1‘𝑅))‘𝑥)) ∈ (Monic1p‘𝑈))
55		fveq2 6879	. . . . . . . 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 6881	. . . . . . 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 2734	. . . . . 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 482	. . . . 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 32554	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑂 = ((eval1‘𝑅) ↾ (Base‘(Poly1‘𝑈))))
61	60	fveq1d 6881	. . . . . . . 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 6899	. . . . . . . 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 2772	. . . . . . 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 6881	. . . . . 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 2732	. . . . . . . . 9 ⊢ (𝑅 ↑s 𝐵) = (𝑅 ↑s 𝐵)
66		eqid 2732	. . . . . . . . 9 ⊢ (Base‘(𝑅 ↑s 𝐵)) = (Base‘(𝑅 ↑s 𝐵))
67	3	fvexi 6893	. . . . . . . . . 10 ⊢ 𝐵 ∈ V
68	67	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐵 ∈ V)
69	41, 7, 65, 3	evl1rhm 21782	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ CRing → (eval1‘𝑅) ∈ ((Poly1‘𝑅) RingHom (𝑅 ↑s 𝐵)))
70	39, 66	rhmf 20215	. . . . . . . . . . . 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 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (eval1‘𝑅):(Base‘(Poly1‘𝑅))⟶(Base‘(𝑅 ↑s 𝐵)))
73		eqid 2732	. . . . . . . . . . . . . 14 ⊢ (PwSer1‘𝑈) = (PwSer1‘𝑈)
74		eqid 2732	. . . . . . . . . . . . . 14 ⊢ (Base‘(PwSer1‘𝑈)) = (Base‘(PwSer1‘𝑈))
75	7, 8, 9, 10, 2, 73, 74, 39	ressply1bas2 21683	. . . . . . . . . . . . 13 ⊢ (𝜑 → (Base‘(Poly1‘𝑈)) = ((Base‘(PwSer1‘𝑈)) ∩ (Base‘(Poly1‘𝑅))))
76	75	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (Base‘(Poly1‘𝑈)) = ((Base‘(PwSer1‘𝑈)) ∩ (Base‘(Poly1‘𝑅))))
77	38, 76	eleqtrd 2835	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((var1‘𝑅)(-g‘(Poly1‘𝑅))((algSc‘(Poly1‘𝑅))‘𝑥)) ∈ ((Base‘(PwSer1‘𝑈)) ∩ (Base‘(Poly1‘𝑅))))
78	77	elin2d 4196	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((var1‘𝑅)(-g‘(Poly1‘𝑅))((algSc‘(Poly1‘𝑅))‘𝑥)) ∈ (Base‘(Poly1‘𝑅)))
79	72, 78	ffvelcdmd 7073	. . . . . . . . 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 17419	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((eval1‘𝑅)‘((var1‘𝑅)(-g‘(Poly1‘𝑅))((algSc‘(Poly1‘𝑅))‘𝑥))):𝐵⟶𝐵)
81	80	ffnd 6706	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((eval1‘𝑅)‘((var1‘𝑅)(-g‘(Poly1‘𝑅))((algSc‘(Poly1‘𝑅))‘𝑥))) Fn 𝐵)
82		vsnid 4660	. . . . . . . 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 2840	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ (◡((eval1‘𝑅)‘((var1‘𝑅)(-g‘(Poly1‘𝑅))((algSc‘(Poly1‘𝑅))‘𝑥))) “ { 0 }))
85		fniniseg 7047	. . . . . . . 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 500	. . . . . . 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 2772	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((𝑂‘((var1‘𝑅)(-g‘(Poly1‘𝑅))((algSc‘(Poly1‘𝑅))‘𝑥)))‘𝑥) = 0 )
89	54, 58, 88	rspcedvd 3612	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ∃𝑓 ∈ (Monic1p‘𝑈)((𝑂‘𝑓)‘𝑥) = 0 )
90	59, 8, 3, 47, 28, 2	elirng 32652	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (𝑅 IntgRing 𝑆) ↔ (𝑥 ∈ 𝐵 ∧ ∃𝑓 ∈ (Monic1p‘𝑈)((𝑂‘𝑓)‘𝑥) = 0 )))
91	90	biimpar 478	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ ∃𝑓 ∈ (Monic1p‘𝑈)((𝑂‘𝑓)‘𝑥) = 0 )) → 𝑥 ∈ (𝑅 IntgRing 𝑆))
92	1, 6, 89, 91	syl12anc 835	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ (𝑅 IntgRing 𝑆))
93	92	ex 413	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑆 → 𝑥 ∈ (𝑅 IntgRing 𝑆)))
94	93	ssrdv 3985	1 ⊢ (𝜑 → 𝑆 ⊆ (𝑅 IntgRing 𝑆))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∃wrex 3070  Vcvv 3474   ∩ cin 3944   ⊆ wss 3945  {csn 4623  ^◡ccnv 5669   ↾ cres 5672   “ cima 5673   Fn wfn 6528  ⟶wf 6529  ‘cfv 6533  (class class class)co 7394  1c1 11095  Basecbs 17128   ↾_s cress 17157  0_gc0g 17369   ↑_s cpws 17376  Grpcgrp 18796  -_gcsg 18798  SubGrpcsubg 18974  CRingccrg 20017   RingHom crh 20200  NzRingcnzr 20243  SubRingcsubrg 20310  algSccascl 21342  PwSer₁cps1 21630  var₁cv1 21631  Poly₁cpl1 21632   evalSub₁ ces1 21763  eval₁ce1 21764   deg₁ cdg1 25500  Monic_1pcmn1 25574   IntgRing cirng 32649

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 5279  ax-sep 5293  ax-nul 5300  ax-pow 5357  ax-pr 5421  ax-un 7709  ax-cnex 11150  ax-resscn 11151  ax-1cn 11152  ax-icn 11153  ax-addcl 11154  ax-addrcl 11155  ax-mulcl 11156  ax-mulrcl 11157  ax-mulcom 11158  ax-addass 11159  ax-mulass 11160  ax-distr 11161  ax-i2m1 11162  ax-1ne0 11163  ax-1rid 11164  ax-rnegex 11165  ax-rrecex 11166  ax-cnre 11167  ax-pre-lttri 11168  ax-pre-lttrn 11169  ax-pre-ltadd 11170  ax-pre-mulgt0 11171  ax-pre-sup 11172  ax-addf 11173  ax-mulf 11174

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 3775  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3964  df-nul 4320  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-tp 4628  df-op 4630  df-uni 4903  df-int 4945  df-iun 4993  df-iin 4994  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5568  df-eprel 5574  df-po 5582  df-so 5583  df-fr 5625  df-se 5626  df-we 5627  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-res 5682  df-ima 5683  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-isom 6542  df-riota 7350  df-ov 7397  df-oprab 7398  df-mpo 7399  df-of 7654  df-ofr 7655  df-om 7840  df-1st 7959  df-2nd 7960  df-supp 8131  df-tpos 8195  df-frecs 8250  df-wrecs 8281  df-recs 8355  df-rdg 8394  df-1o 8450  df-er 8688  df-map 8807  df-pm 8808  df-ixp 8877  df-en 8925  df-dom 8926  df-sdom 8927  df-fin 8928  df-fsupp 9347  df-sup 9421  df-oi 9489  df-card 9918  df-pnf 11234  df-mnf 11235  df-xr 11236  df-ltxr 11237  df-le 11238  df-sub 11430  df-neg 11431  df-nn 12197  df-2 12259  df-3 12260  df-4 12261  df-5 12262  df-6 12263  df-7 12264  df-8 12265  df-9 12266  df-n0 12457  df-z 12543  df-dec 12662  df-uz 12807  df-fz 13469  df-fzo 13612  df-seq 13951  df-hash 14275  df-struct 17064  df-sets 17081  df-slot 17099  df-ndx 17111  df-base 17129  df-ress 17158  df-plusg 17194  df-mulr 17195  df-starv 17196  df-sca 17197  df-vsca 17198  df-ip 17199  df-tset 17200  df-ple 17201  df-ds 17203  df-unif 17204  df-hom 17205  df-cco 17206  df-0g 17371  df-gsum 17372  df-prds 17377  df-pws 17379  df-mre 17514  df-mrc 17515  df-acs 17517  df-mgm 18545  df-sgrp 18594  df-mnd 18605  df-mhm 18649  df-submnd 18650  df-grp 18799  df-minusg 18800  df-sbg 18801  df-mulg 18925  df-subg 18977  df-ghm 19058  df-cntz 19149  df-cmn 19616  df-abl 19617  df-mgp 19949  df-ur 19966  df-srg 19970  df-ring 20018  df-cring 20019  df-oppr 20104  df-dvdsr 20125  df-unit 20126  df-invr 20156  df-rnghom 20203  df-nzr 20244  df-subrg 20312  df-lmod 20424  df-lss 20494  df-lsp 20534  df-rlreg 20837  df-cnfld 20881  df-assa 21343  df-asp 21344  df-ascl 21345  df-psr 21395  df-mvr 21396  df-mpl 21397  df-opsr 21399  df-evls 21566  df-evl 21567  df-psr1 21635  df-vr1 21636  df-ply1 21637  df-coe1 21638  df-evls1 21765  df-evl1 21766  df-mdeg 25501  df-deg1 25502  df-mon1 25579  df-irng 32650

This theorem is referenced by: (None)