Theorem irngss 33702

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 33704). (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 20589	. . . . . 6 ⊢ (𝑆 ∈ (SubRing‘𝑅) → 𝑆 ⊆ 𝐵)
5	2, 4	syl 17	. . . . 5 ⊢ (𝜑 → 𝑆 ⊆ 𝐵)
6	5	sselda 3995	. . . 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 22281	. . . . . . . . . 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 22394	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((algSc‘(Poly1‘𝑅))‘𝑥) ∈ (Base‘(Poly1‘𝑈)))
18	7, 8, 9, 10, 11, 12, 14, 17	ressply1sub 33575	. . . . . . . . 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 22250	. . . . . . . . . . . 12 ⊢ (𝑆 ∈ (SubRing‘𝑅) → (Base‘(Poly1‘𝑈)) ∈ (SubRing‘(Poly1‘𝑅)))
20		subrgsubg 20594	. . . . . . . . . . . 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 19169	. . . . . . . . . 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 2778	. . . . . . . 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 20592	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑅)) → 𝑈 ∈ CRing)
30	28, 2, 29	syl2anc 584	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑈 ∈ CRing)
31	9	ply1crng 22216	. . . . . . . . . . . 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 20265	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (Poly1‘𝑈) ∈ Grp)
35		eqid 2735	. . . . . . . . . 10 ⊢ (-g‘(Poly1‘𝑈)) = (-g‘(Poly1‘𝑈))
36	10, 35	grpsubcl 19051	. . . . . . . . 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 2840	. . . . . . 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 26219	. . . . . . . 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 1141	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((var1‘𝑅)(-g‘(Poly1‘𝑅))((algSc‘(Poly1‘𝑅))‘𝑥)) ∈ (Monic1p‘𝑅))
50	38, 49	elind 4210	. . . . . 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 33573	. . . . . . 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 2842	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((var1‘𝑅)(-g‘(Poly1‘𝑅))((algSc‘(Poly1‘𝑅))‘𝑥)) ∈ (Monic1p‘𝑈))
55		fveq2 6907	. . . . . . . 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 6909	. . . . . . 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 22390	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑂 = ((eval1‘𝑅) ↾ (Base‘(Poly1‘𝑈))))
61	60	fveq1d 6909	. . . . . . . 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 6927	. . . . . . . 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 2775	. . . . . . 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 6909	. . . . . 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 6921	. . . . . . . . . 10 ⊢ 𝐵 ∈ V
68	67	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐵 ∈ V)
69	41, 7, 65, 3	evl1rhm 22352	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ CRing → (eval1‘𝑅) ∈ ((Poly1‘𝑅) RingHom (𝑅 ↑s 𝐵)))
70	39, 66	rhmf 20502	. . . . . . . . . . . 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 22245	. . . . . . . . . . . . 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 2841	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((var1‘𝑅)(-g‘(Poly1‘𝑅))((algSc‘(Poly1‘𝑅))‘𝑥)) ∈ ((Base‘(PwSer1‘𝑈)) ∩ (Base‘(Poly1‘𝑅))))
78	77	elin2d 4215	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((var1‘𝑅)(-g‘(Poly1‘𝑅))((algSc‘(Poly1‘𝑅))‘𝑥)) ∈ (Base‘(Poly1‘𝑅)))
79	72, 78	ffvelcdmd 7105	. . . . . . . . 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 17536	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((eval1‘𝑅)‘((var1‘𝑅)(-g‘(Poly1‘𝑅))((algSc‘(Poly1‘𝑅))‘𝑥))):𝐵⟶𝐵)
81	80	ffnd 6738	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((eval1‘𝑅)‘((var1‘𝑅)(-g‘(Poly1‘𝑅))((algSc‘(Poly1‘𝑅))‘𝑥))) Fn 𝐵)
82		vsnid 4668	. . . . . . . 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 2846	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ (◡((eval1‘𝑅)‘((var1‘𝑅)(-g‘(Poly1‘𝑅))((algSc‘(Poly1‘𝑅))‘𝑥))) “ { 0 }))
85		fniniseg 7080	. . . . . . . 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 2775	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((𝑂‘((var1‘𝑅)(-g‘(Poly1‘𝑅))((algSc‘(Poly1‘𝑅))‘𝑥)))‘𝑥) = 0 )
89	54, 58, 88	rspcedvd 3624	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ∃𝑓 ∈ (Monic1p‘𝑈)((𝑂‘𝑓)‘𝑥) = 0 )
90	59, 8, 3, 47, 28, 2	elirng 33701	. . . . 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 837	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ (𝑅 IntgRing 𝑆))
93	92	ex 412	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑆 → 𝑥 ∈ (𝑅 IntgRing 𝑆)))
94	93	ssrdv 4001	1 ⊢ (𝜑 → 𝑆 ⊆ (𝑅 IntgRing 𝑆))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2106  ∃wrex 3068  Vcvv 3478   ∩ cin 3962   ⊆ wss 3963  {csn 4631  ^◡ccnv 5688   ↾ cres 5691   “ cima 5692   Fn wfn 6558  ⟶wf 6559  ‘cfv 6563  (class class class)co 7431  1c1 11154  Basecbs 17245   ↾_s cress 17274  0_gc0g 17486   ↑_s cpws 17493  Grpcgrp 18964  -_gcsg 18966  SubGrpcsubg 19151  CRingccrg 20252   RingHom crh 20486  NzRingcnzr 20529  SubRingcsubrg 20586  algSccascl 21890  PwSer₁cps1 22192  var₁cv1 22193  Poly₁cpl1 22194   evalSub₁ ces1 22333  eval₁ce1 22334  deg₁cdg1 26108  Monic_1pcmn1 26180   IntgRing cirng 33698

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231  ax-addf 11232

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-ofr 7698  df-om 7888  df-1st 8013  df-2nd 8014  df-supp 8185  df-tpos 8250  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-er 8744  df-map 8867  df-pm 8868  df-ixp 8937  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-fsupp 9400  df-sup 9480  df-oi 9548  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12612  df-dec 12732  df-uz 12877  df-fz 13545  df-fzo 13692  df-seq 14040  df-hash 14367  df-struct 17181  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-mulr 17312  df-starv 17313  df-sca 17314  df-vsca 17315  df-ip 17316  df-tset 17317  df-ple 17318  df-ds 17320  df-unif 17321  df-hom 17322  df-cco 17323  df-0g 17488  df-gsum 17489  df-prds 17494  df-pws 17496  df-mre 17631  df-mrc 17632  df-acs 17634  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-mhm 18809  df-submnd 18810  df-grp 18967  df-minusg 18968  df-sbg 18969  df-mulg 19099  df-subg 19154  df-ghm 19244  df-cntz 19348  df-cmn 19815  df-abl 19816  df-mgp 20153  df-rng 20171  df-ur 20200  df-srg 20205  df-ring 20253  df-cring 20254  df-oppr 20351  df-dvdsr 20374  df-unit 20375  df-invr 20405  df-rhm 20489  df-nzr 20530  df-subrng 20563  df-subrg 20587  df-rlreg 20711  df-lmod 20877  df-lss 20948  df-lsp 20988  df-cnfld 21383  df-assa 21891  df-asp 21892  df-ascl 21893  df-psr 21947  df-mvr 21948  df-mpl 21949  df-opsr 21951  df-evls 22116  df-evl 22117  df-psr1 22197  df-vr1 22198  df-ply1 22199  df-coe1 22200  df-evls1 22335  df-evl1 22336  df-mdeg 26109  df-deg1 26110  df-mon1 26185  df-irng 33699

This theorem is referenced by: (None)