Theorem irngss 34077

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 34079). (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 487	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝜑)
2		elirng.s	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝑅))
3		irngval.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑅)
4	3	subrgss 20671	. . . . . 6 ⊢ (𝑆 ∈ (SubRing‘𝑅) → 𝑆 ⊆ 𝐵)
5	2, 4	syl 18	. . . . 5 ⊢ (𝜑 → 𝑆 ⊆ 𝐵)
6	5	sselda 3937	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝐵)
7		eqid 2763	. . . . . . . . . 10 ⊢ (Poly1‘𝑅) = (Poly1‘𝑅)
8		irngval.u	. . . . . . . . . 10 ⊢ 𝑈 = (𝑅 ↾s 𝑆)
9		eqid 2763	. . . . . . . . . 10 ⊢ (Poly1‘𝑈) = (Poly1‘𝑈)
10		eqid 2763	. . . . . . . . . 10 ⊢ (Base‘(Poly1‘𝑈)) = (Base‘(Poly1‘𝑈))
11	2	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑆 ∈ (SubRing‘𝑅))
12		eqid 2763	. . . . . . . . . 10 ⊢ ((Poly1‘𝑅) ↾s (Base‘(Poly1‘𝑈))) = ((Poly1‘𝑅) ↾s (Base‘(Poly1‘𝑈)))
13		eqid 2763	. . . . . . . . . . 11 ⊢ (var1‘𝑅) = (var1‘𝑅)
14	13, 11, 8, 9, 10	subrgvr1cl 22423	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (var1‘𝑅) ∈ (Base‘(Poly1‘𝑈)))
15		eqid 2763	. . . . . . . . . . 11 ⊢ (algSc‘(Poly1‘𝑅)) = (algSc‘(Poly1‘𝑅))
16		simpr 489	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝑆)
17	15, 8, 7, 9, 10, 11, 16	asclply1subcl 22534	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((algSc‘(Poly1‘𝑅))‘𝑥) ∈ (Base‘(Poly1‘𝑈)))
18	7, 8, 9, 10, 11, 12, 14, 17	ressply1sub 33860	. . . . . . . . 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 22392	. . . . . . . . . . . 12 ⊢ (𝑆 ∈ (SubRing‘𝑅) → (Base‘(Poly1‘𝑈)) ∈ (SubRing‘(Poly1‘𝑅)))
20		subrgsubg 20676	. . . . . . . . . . . 12 ⊢ ((Base‘(Poly1‘𝑈)) ∈ (SubRing‘(Poly1‘𝑅)) → (Base‘(Poly1‘𝑈)) ∈ (SubGrp‘(Poly1‘𝑅)))
21	2, 19, 20	3syl 19	. . . . . . . . . . 11 ⊢ (𝜑 → (Base‘(Poly1‘𝑈)) ∈ (SubGrp‘(Poly1‘𝑅)))
22	21	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (Base‘(Poly1‘𝑈)) ∈ (SubGrp‘(Poly1‘𝑅)))
23		eqid 2763	. . . . . . . . . . 11 ⊢ (-g‘(Poly1‘𝑅)) = (-g‘(Poly1‘𝑅))
24		eqid 2763	. . . . . . . . . . 11 ⊢ (-g‘((Poly1‘𝑅) ↾s (Base‘(Poly1‘𝑈)))) = (-g‘((Poly1‘𝑅) ↾s (Base‘(Poly1‘𝑈))))
25	23, 12, 24	subgsub 19200	. . . . . . . . . 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 1398	. . . . . . . . 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 2801	. . . . . . . 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 20674	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑅)) → 𝑈 ∈ CRing)
30	28, 2, 29	syl2anc 595	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑈 ∈ CRing)
31	9	ply1crng 22358	. . . . . . . . . . . 12 ⊢ (𝑈 ∈ CRing → (Poly1‘𝑈) ∈ CRing)
32	30, 31	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → (Poly1‘𝑈) ∈ CRing)
33	32	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (Poly1‘𝑈) ∈ CRing)
34	33	crnggrpd 20324	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (Poly1‘𝑈) ∈ Grp)
35		eqid 2763	. . . . . . . . . 10 ⊢ (-g‘(Poly1‘𝑈)) = (-g‘(Poly1‘𝑈))
36	10, 35	grpsubcl 19081	. . . . . . . . 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 1398	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((var1‘𝑅)(-g‘(Poly1‘𝑈))((algSc‘(Poly1‘𝑅))‘𝑥)) ∈ (Base‘(Poly1‘𝑈)))
38	27, 37	eqeltrrd 2864	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((var1‘𝑅)(-g‘(Poly1‘𝑅))((algSc‘(Poly1‘𝑅))‘𝑥)) ∈ (Base‘(Poly1‘𝑈)))
39		eqid 2763	. . . . . . . . 9 ⊢ (Base‘(Poly1‘𝑅)) = (Base‘(Poly1‘𝑅))
40		eqid 2763	. . . . . . . . 9 ⊢ ((var1‘𝑅)(-g‘(Poly1‘𝑅))((algSc‘(Poly1‘𝑅))‘𝑥)) = ((var1‘𝑅)(-g‘(Poly1‘𝑅))((algSc‘(Poly1‘𝑅))‘𝑥))
41		eqid 2763	. . . . . . . . 9 ⊢ (eval1‘𝑅) = (eval1‘𝑅)
42		irngss.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ NzRing)
43	42	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑅 ∈ NzRing)
44	28	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑅 ∈ CRing)
45		eqid 2763	. . . . . . . . 9 ⊢ (Monic1p‘𝑅) = (Monic1p‘𝑅)
46		eqid 2763	. . . . . . . . 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 26322	. . . . . . . 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 1160	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((var1‘𝑅)(-g‘(Poly1‘𝑅))((algSc‘(Poly1‘𝑅))‘𝑥)) ∈ (Monic1p‘𝑅))
50	38, 49	elind 4153	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((var1‘𝑅)(-g‘(Poly1‘𝑅))((algSc‘(Poly1‘𝑅))‘𝑥)) ∈ ((Base‘(Poly1‘𝑈)) ∩ (Monic1p‘𝑅)))
51		eqid 2763	. . . . . . . 8 ⊢ (Monic1p‘𝑈) = (Monic1p‘𝑈)
52	7, 8, 9, 10, 2, 45, 51	ressply1mon1p 33858	. . . . . . 7 ⊢ (𝜑 → (Monic1p‘𝑈) = ((Base‘(Poly1‘𝑈)) ∩ (Monic1p‘𝑅)))
53	52	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (Monic1p‘𝑈) = ((Base‘(Poly1‘𝑈)) ∩ (Monic1p‘𝑅)))
54	50, 53	eleqtrrd 2866	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((var1‘𝑅)(-g‘(Poly1‘𝑅))((algSc‘(Poly1‘𝑅))‘𝑥)) ∈ (Monic1p‘𝑈))
55		fveq2 6881	. . . . . . . 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 6883	. . . . . . 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 2765	. . . . . 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 486	. . . . 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 22530	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑂 = ((eval1‘𝑅) ↾ (Base‘(Poly1‘𝑈))))
61	60	fveq1d 6883	. . . . . . . 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 6901	. . . . . . . 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 2798	. . . . . . 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 6883	. . . . . 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 2763	. . . . . . . . 9 ⊢ (𝑅 ↑s 𝐵) = (𝑅 ↑s 𝐵)
66		eqid 2763	. . . . . . . . 9 ⊢ (Base‘(𝑅 ↑s 𝐵)) = (Base‘(𝑅 ↑s 𝐵))
67	3	fvexi 6895	. . . . . . . . . 10 ⊢ 𝐵 ∈ V
68	67	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐵 ∈ V)
69	41, 7, 65, 3	evl1rhm 22492	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ CRing → (eval1‘𝑅) ∈ ((Poly1‘𝑅) RingHom (𝑅 ↑s 𝐵)))
70	39, 66	rhmf 20563	. . . . . . . . . . . 12 ⊢ ((eval1‘𝑅) ∈ ((Poly1‘𝑅) RingHom (𝑅 ↑s 𝐵)) → (eval1‘𝑅):(Base‘(Poly1‘𝑅))⟶(Base‘(𝑅 ↑s 𝐵)))
71	28, 69, 70	3syl 19	. . . . . . . . . . 11 ⊢ (𝜑 → (eval1‘𝑅):(Base‘(Poly1‘𝑅))⟶(Base‘(𝑅 ↑s 𝐵)))
72	71	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (eval1‘𝑅):(Base‘(Poly1‘𝑅))⟶(Base‘(𝑅 ↑s 𝐵)))
73		eqid 2763	. . . . . . . . . . . . . 14 ⊢ (PwSer1‘𝑈) = (PwSer1‘𝑈)
74		eqid 2763	. . . . . . . . . . . . . 14 ⊢ (Base‘(PwSer1‘𝑈)) = (Base‘(PwSer1‘𝑈))
75	7, 8, 9, 10, 2, 73, 74, 39	ressply1bas2 22387	. . . . . . . . . . . . 13 ⊢ (𝜑 → (Base‘(Poly1‘𝑈)) = ((Base‘(PwSer1‘𝑈)) ∩ (Base‘(Poly1‘𝑅))))
76	75	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (Base‘(Poly1‘𝑈)) = ((Base‘(PwSer1‘𝑈)) ∩ (Base‘(Poly1‘𝑅))))
77	38, 76	eleqtrd 2865	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((var1‘𝑅)(-g‘(Poly1‘𝑅))((algSc‘(Poly1‘𝑅))‘𝑥)) ∈ ((Base‘(PwSer1‘𝑈)) ∩ (Base‘(Poly1‘𝑅))))
78	77	elin2d 4158	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((var1‘𝑅)(-g‘(Poly1‘𝑅))((algSc‘(Poly1‘𝑅))‘𝑥)) ∈ (Base‘(Poly1‘𝑅)))
79	72, 78	ffvelcdmd 7080	. . . . . . . . 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 17537	. . . . . . . 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 4629	. . . . . . . 8 ⊢ 𝑥 ∈ {𝑥}
83	48	simp3d 1162	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (◡((eval1‘𝑅)‘((var1‘𝑅)(-g‘(Poly1‘𝑅))((algSc‘(Poly1‘𝑅))‘𝑥))) “ { 0 }) = {𝑥})
84	82, 83	eleqtrrid 2870	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ (◡((eval1‘𝑅)‘((var1‘𝑅)(-g‘(Poly1‘𝑅))((algSc‘(Poly1‘𝑅))‘𝑥))) “ { 0 }))
85		fniniseg 7055	. . . . . . . 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 504	. . . . . . 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 595	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (((eval1‘𝑅)‘((var1‘𝑅)(-g‘(Poly1‘𝑅))((algSc‘(Poly1‘𝑅))‘𝑥)))‘𝑥) = 0 )
88	64, 87	eqtrd 2798	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((𝑂‘((var1‘𝑅)(-g‘(Poly1‘𝑅))((algSc‘(Poly1‘𝑅))‘𝑥)))‘𝑥) = 0 )
89	54, 58, 88	rspcedvd 3583	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ∃𝑓 ∈ (Monic1p‘𝑈)((𝑂‘𝑓)‘𝑥) = 0 )
90	59, 8, 3, 47, 28, 2	elirng 34076	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (𝑅 IntgRing 𝑆) ↔ (𝑥 ∈ 𝐵 ∧ ∃𝑓 ∈ (Monic1p‘𝑈)((𝑂‘𝑓)‘𝑥) = 0 )))
91	90	biimpar 482	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ ∃𝑓 ∈ (Monic1p‘𝑈)((𝑂‘𝑓)‘𝑥) = 0 )) → 𝑥 ∈ (𝑅 IntgRing 𝑆))
92	1, 6, 89, 91	syl12anc 849	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ (𝑅 IntgRing 𝑆))
93	92	ex 417	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑆 → 𝑥 ∈ (𝑅 IntgRing 𝑆)))
94	93	ssrdv 3943	1 ⊢ (𝜑 → 𝑆 ⊆ (𝑅 IntgRing 𝑆))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∃wrex 3089  Vcvv 3455   ∩ cin 3904   ⊆ wss 3905  {csn 4589  ^◡ccnv 5660   ↾ cres 5663   “ cima 5664   Fn wfn 6531  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410  1c1 11096  Basecbs 17264   ↾_s cress 17285  0_gc0g 17487   ↑_s cpws 17494  Grpcgrp 18995  -_gcsg 18997  SubGrpcsubg 19181  CRingccrg 20311   RingHom crh 20547  NzRingcnzr 20609  SubRingcsubrg 20668  algSccascl 22002  PwSer₁cps1 22335  var₁cv1 22336  Poly₁cpl1 22337   evalSub₁ ces1 22473  eval₁ce1 22474  deg₁cdg1 26211  Monic_1pcmn1 26283   IntgRing cirng 34073

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11151  ax-resscn 11152  ax-1cn 11153  ax-icn 11154  ax-addcl 11155  ax-addrcl 11156  ax-mulcl 11157  ax-mulrcl 11158  ax-mulcom 11159  ax-addass 11160  ax-mulass 11161  ax-distr 11162  ax-i2m1 11163  ax-1ne0 11164  ax-1rid 11165  ax-rnegex 11166  ax-rrecex 11167  ax-cnre 11168  ax-pre-lttri 11169  ax-pre-lttrn 11170  ax-pre-ltadd 11171  ax-pre-mulgt0 11172  ax-pre-sup 11173  ax-addf 11174

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4873  df-int 4913  df-iun 4958  df-iin 4959  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-se 5615  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-of 7674  df-ofr 7675  df-om 7859  df-1st 7982  df-2nd 7983  df-supp 8153  df-tpos 8218  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-2o 8450  df-er 8690  df-map 8822  df-pm 8823  df-ixp 8892  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-fsupp 9318  df-sup 9398  df-oi 9468  df-card 9921  df-pnf 11240  df-mnf 11241  df-xr 11242  df-ltxr 11243  df-le 11244  df-sub 11438  df-neg 11439  df-nn 12229  df-2 12298  df-3 12299  df-4 12300  df-5 12301  df-6 12302  df-7 12303  df-8 12304  df-9 12305  df-n0 12500  df-z 12587  df-dec 12707  df-uz 12858  df-fz 13531  df-fzo 13679  df-seq 14034  df-hash 14363  df-struct 17202  df-sets 17219  df-slot 17237  df-ndx 17249  df-base 17265  df-ress 17286  df-plusg 17318  df-mulr 17319  df-starv 17320  df-sca 17321  df-vsca 17322  df-ip 17323  df-tset 17324  df-ple 17325  df-ds 17327  df-unif 17328  df-hom 17329  df-cco 17330  df-0g 17489  df-gsum 17490  df-prds 17495  df-pws 17497  df-mre 17633  df-mrc 17634  df-acs 17636  df-mgm 18693  df-sgrp 18772  df-mnd 18788  df-mhm 18836  df-submnd 18837  df-grp 18998  df-minusg 18999  df-sbg 19000  df-mulg 19129  df-subg 19184  df-ghm 19279  df-cntz 19382  df-cmn 19847  df-abl 19848  df-mgp 20212  df-rng 20226  df-ur 20259  df-srg 20264  df-ring 20312  df-cring 20313  df-oppr 20415  df-dvdsr 20435  df-unit 20436  df-invr 20466  df-rhm 20550  df-nzr 20610  df-subrng 20645  df-subrg 20669  df-rlreg 20793  df-lmod 20983  df-lss 21053  df-lsp 21093  df-cnfld 21523  df-assa 22003  df-asp 22004  df-ascl 22005  df-psr 22059  df-mvr 22060  df-mpl 22061  df-opsr 22063  df-evls 22225  df-evl 22226  df-psr1 22340  df-vr1 22341  df-ply1 22342  df-coe1 22343  df-evls1 22475  df-evl1 22476  df-mdeg 26212  df-deg1 26213  df-mon1 26288  df-irng 34074

This theorem is referenced by: (None)