Theorem irngss 33844

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 33846). (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 20505	. . . . . 6 ⊢ (𝑆 ∈ (SubRing‘𝑅) → 𝑆 ⊆ 𝐵)
5	2, 4	syl 17	. . . . 5 ⊢ (𝜑 → 𝑆 ⊆ 𝐵)
6	5	sselda 3933	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝐵)
7		eqid 2736	. . . . . . . . . 10 ⊢ (Poly1‘𝑅) = (Poly1‘𝑅)
8		irngval.u	. . . . . . . . . 10 ⊢ 𝑈 = (𝑅 ↾s 𝑆)
9		eqid 2736	. . . . . . . . . 10 ⊢ (Poly1‘𝑈) = (Poly1‘𝑈)
10		eqid 2736	. . . . . . . . . 10 ⊢ (Base‘(Poly1‘𝑈)) = (Base‘(Poly1‘𝑈))
11	2	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑆 ∈ (SubRing‘𝑅))
12		eqid 2736	. . . . . . . . . 10 ⊢ ((Poly1‘𝑅) ↾s (Base‘(Poly1‘𝑈))) = ((Poly1‘𝑅) ↾s (Base‘(Poly1‘𝑈)))
13		eqid 2736	. . . . . . . . . . 11 ⊢ (var1‘𝑅) = (var1‘𝑅)
14	13, 11, 8, 9, 10	subrgvr1cl 22204	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (var1‘𝑅) ∈ (Base‘(Poly1‘𝑈)))
15		eqid 2736	. . . . . . . . . . 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 22318	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((algSc‘(Poly1‘𝑅))‘𝑥) ∈ (Base‘(Poly1‘𝑈)))
18	7, 8, 9, 10, 11, 12, 14, 17	ressply1sub 33651	. . . . . . . . 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 22173	. . . . . . . . . . . 12 ⊢ (𝑆 ∈ (SubRing‘𝑅) → (Base‘(Poly1‘𝑈)) ∈ (SubRing‘(Poly1‘𝑅)))
20		subrgsubg 20510	. . . . . . . . . . . 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 2736	. . . . . . . . . . 11 ⊢ (-g‘(Poly1‘𝑅)) = (-g‘(Poly1‘𝑅))
24		eqid 2736	. . . . . . . . . . 11 ⊢ (-g‘((Poly1‘𝑅) ↾s (Base‘(Poly1‘𝑈)))) = (-g‘((Poly1‘𝑅) ↾s (Base‘(Poly1‘𝑈))))
25	23, 12, 24	subgsub 19068	. . . . . . . . . 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 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 20508	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑅)) → 𝑈 ∈ CRing)
30	28, 2, 29	syl2anc 584	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑈 ∈ CRing)
31	9	ply1crng 22139	. . . . . . . . . . . 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 20182	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (Poly1‘𝑈) ∈ Grp)
35		eqid 2736	. . . . . . . . . 10 ⊢ (-g‘(Poly1‘𝑈)) = (-g‘(Poly1‘𝑈))
36	10, 35	grpsubcl 18950	. . . . . . . . 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 2837	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((var1‘𝑅)(-g‘(Poly1‘𝑅))((algSc‘(Poly1‘𝑅))‘𝑥)) ∈ (Base‘(Poly1‘𝑈)))
39		eqid 2736	. . . . . . . . 9 ⊢ (Base‘(Poly1‘𝑅)) = (Base‘(Poly1‘𝑅))
40		eqid 2736	. . . . . . . . 9 ⊢ ((var1‘𝑅)(-g‘(Poly1‘𝑅))((algSc‘(Poly1‘𝑅))‘𝑥)) = ((var1‘𝑅)(-g‘(Poly1‘𝑅))((algSc‘(Poly1‘𝑅))‘𝑥))
41		eqid 2736	. . . . . . . . 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 2736	. . . . . . . . 9 ⊢ (Monic1p‘𝑅) = (Monic1p‘𝑅)
46		eqid 2736	. . . . . . . . 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 26126	. . . . . . . 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 4152	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((var1‘𝑅)(-g‘(Poly1‘𝑅))((algSc‘(Poly1‘𝑅))‘𝑥)) ∈ ((Base‘(Poly1‘𝑈)) ∩ (Monic1p‘𝑅)))
51		eqid 2736	. . . . . . . 8 ⊢ (Monic1p‘𝑈) = (Monic1p‘𝑈)
52	7, 8, 9, 10, 2, 45, 51	ressply1mon1p 33649	. . . . . . 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 2839	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((var1‘𝑅)(-g‘(Poly1‘𝑅))((algSc‘(Poly1‘𝑅))‘𝑥)) ∈ (Monic1p‘𝑈))
55		fveq2 6834	. . . . . . . 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 6836	. . . . . . 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 2738	. . . . . 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 22314	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑂 = ((eval1‘𝑅) ↾ (Base‘(Poly1‘𝑈))))
61	60	fveq1d 6836	. . . . . . . 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 6854	. . . . . . . 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 6836	. . . . . 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 2736	. . . . . . . . 9 ⊢ (𝑅 ↑s 𝐵) = (𝑅 ↑s 𝐵)
66		eqid 2736	. . . . . . . . 9 ⊢ (Base‘(𝑅 ↑s 𝐵)) = (Base‘(𝑅 ↑s 𝐵))
67	3	fvexi 6848	. . . . . . . . . 10 ⊢ 𝐵 ∈ V
68	67	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐵 ∈ V)
69	41, 7, 65, 3	evl1rhm 22276	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ CRing → (eval1‘𝑅) ∈ ((Poly1‘𝑅) RingHom (𝑅 ↑s 𝐵)))
70	39, 66	rhmf 20420	. . . . . . . . . . . 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 2736	. . . . . . . . . . . . . 14 ⊢ (PwSer1‘𝑈) = (PwSer1‘𝑈)
74		eqid 2736	. . . . . . . . . . . . . 14 ⊢ (Base‘(PwSer1‘𝑈)) = (Base‘(PwSer1‘𝑈))
75	7, 8, 9, 10, 2, 73, 74, 39	ressply1bas2 22168	. . . . . . . . . . . . 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 2838	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((var1‘𝑅)(-g‘(Poly1‘𝑅))((algSc‘(Poly1‘𝑅))‘𝑥)) ∈ ((Base‘(PwSer1‘𝑈)) ∩ (Base‘(Poly1‘𝑅))))
78	77	elin2d 4157	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((var1‘𝑅)(-g‘(Poly1‘𝑅))((algSc‘(Poly1‘𝑅))‘𝑥)) ∈ (Base‘(Poly1‘𝑅)))
79	72, 78	ffvelcdmd 7030	. . . . . . . . 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 17409	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((eval1‘𝑅)‘((var1‘𝑅)(-g‘(Poly1‘𝑅))((algSc‘(Poly1‘𝑅))‘𝑥))):𝐵⟶𝐵)
81	80	ffnd 6663	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((eval1‘𝑅)‘((var1‘𝑅)(-g‘(Poly1‘𝑅))((algSc‘(Poly1‘𝑅))‘𝑥))) Fn 𝐵)
82		vsnid 4620	. . . . . . . 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 2843	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ (◡((eval1‘𝑅)‘((var1‘𝑅)(-g‘(Poly1‘𝑅))((algSc‘(Poly1‘𝑅))‘𝑥))) “ { 0 }))
85		fniniseg 7005	. . . . . . . 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 2771	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((𝑂‘((var1‘𝑅)(-g‘(Poly1‘𝑅))((algSc‘(Poly1‘𝑅))‘𝑥)))‘𝑥) = 0 )
89	54, 58, 88	rspcedvd 3578	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ∃𝑓 ∈ (Monic1p‘𝑈)((𝑂‘𝑓)‘𝑥) = 0 )
90	59, 8, 3, 47, 28, 2	elirng 33843	. . . . 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 3939	1 ⊢ (𝜑 → 𝑆 ⊆ (𝑅 IntgRing 𝑆))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∃wrex 3060  Vcvv 3440   ∩ cin 3900   ⊆ wss 3901  {csn 4580  ^◡ccnv 5623   ↾ cres 5626   “ cima 5627   Fn wfn 6487  ⟶wf 6488  ‘cfv 6492  (class class class)co 7358  1c1 11027  Basecbs 17136   ↾_s cress 17157  0_gc0g 17359   ↑_s cpws 17366  Grpcgrp 18863  -_gcsg 18865  SubGrpcsubg 19050  CRingccrg 20169   RingHom crh 20405  NzRingcnzr 20445  SubRingcsubrg 20502  algSccascl 21807  PwSer₁cps1 22115  var₁cv1 22116  Poly₁cpl1 22117   evalSub₁ ces1 22257  eval₁ce1 22258  deg₁cdg1 26015  Monic_1pcmn1 26087   IntgRing cirng 33840

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 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-pre-mulgt0 11103  ax-pre-sup 11104  ax-addf 11105

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-tp 4585  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-iin 4949  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-of 7622  df-ofr 7623  df-om 7809  df-1st 7933  df-2nd 7934  df-supp 8103  df-tpos 8168  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-2o 8398  df-er 8635  df-map 8765  df-pm 8766  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9265  df-sup 9345  df-oi 9415  df-card 9851  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-sub 11366  df-neg 11367  df-nn 12146  df-2 12208  df-3 12209  df-4 12210  df-5 12211  df-6 12212  df-7 12213  df-8 12214  df-9 12215  df-n0 12402  df-z 12489  df-dec 12608  df-uz 12752  df-fz 13424  df-fzo 13571  df-seq 13925  df-hash 14254  df-struct 17074  df-sets 17091  df-slot 17109  df-ndx 17121  df-base 17137  df-ress 17158  df-plusg 17190  df-mulr 17191  df-starv 17192  df-sca 17193  df-vsca 17194  df-ip 17195  df-tset 17196  df-ple 17197  df-ds 17199  df-unif 17200  df-hom 17201  df-cco 17202  df-0g 17361  df-gsum 17362  df-prds 17367  df-pws 17369  df-mre 17505  df-mrc 17506  df-acs 17508  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-mhm 18708  df-submnd 18709  df-grp 18866  df-minusg 18867  df-sbg 18868  df-mulg 18998  df-subg 19053  df-ghm 19142  df-cntz 19246  df-cmn 19711  df-abl 19712  df-mgp 20076  df-rng 20088  df-ur 20117  df-srg 20122  df-ring 20170  df-cring 20171  df-oppr 20273  df-dvdsr 20293  df-unit 20294  df-invr 20324  df-rhm 20408  df-nzr 20446  df-subrng 20479  df-subrg 20503  df-rlreg 20627  df-lmod 20813  df-lss 20883  df-lsp 20923  df-cnfld 21310  df-assa 21808  df-asp 21809  df-ascl 21810  df-psr 21865  df-mvr 21866  df-mpl 21867  df-opsr 21869  df-evls 22029  df-evl 22030  df-psr1 22120  df-vr1 22121  df-ply1 22122  df-coe1 22123  df-evls1 22259  df-evl1 22260  df-mdeg 26016  df-deg1 26017  df-mon1 26092  df-irng 33841

This theorem is referenced by: (None)