Theorem evls1fldgencl 34413

Description: Closure of the subring polynomial evaluation in the field extention. (Contributed by Thierry Arnoux, 2-Apr-2025.)

Hypotheses

Ref	Expression
evls1fldgencl.1	⊢ 𝐵 = (Base‘𝐸)
evls1fldgencl.2	⊢ 𝑂 = (𝐸 evalSub1 𝐹)
evls1fldgencl.3	⊢ 𝑃 = (Poly1‘(𝐸 ↾s 𝐹))
evls1fldgencl.4	⊢ 𝑈 = (Base‘𝑃)
evls1fldgencl.5	⊢ (𝜑 → 𝐸 ∈ Field)
evls1fldgencl.6	⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))
evls1fldgencl.7	⊢ (𝜑 → 𝐴 ∈ 𝐵)
evls1fldgencl.8	⊢ (𝜑 → 𝐺 ∈ 𝑈)

Assertion

Ref	Expression
evls1fldgencl	⊢ (𝜑 → ((𝑂‘𝐺)‘𝐴) ∈ (𝐸 fldGen (𝐹 ∪ {𝐴})))

Proof of Theorem evls1fldgencl

Dummy variables 𝑎 𝑘 𝑥 𝑦 𝑖 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		evls1fldgencl.2	. . . . . . . . 9 ⊢ 𝑂 = (𝐸 evalSub1 𝐹)
2		evls1fldgencl.1	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝐸)
3		evls1fldgencl.3	. . . . . . . . 9 ⊢ 𝑃 = (Poly1‘(𝐸 ↾s 𝐹))
4		eqid 2763	. . . . . . . . 9 ⊢ (𝐸 ↾s 𝐹) = (𝐸 ↾s 𝐹)
5		evls1fldgencl.4	. . . . . . . . 9 ⊢ 𝑈 = (Base‘𝑃)
6		evls1fldgencl.5	. . . . . . . . . 10 ⊢ (𝜑 → 𝐸 ∈ Field)
7	6	fldcrngd 21046	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ CRing)
8		evls1fldgencl.6	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))
9		sdrgsubrg 21099	. . . . . . . . . 10 ⊢ (𝐹 ∈ (SubDRing‘𝐸) → 𝐹 ∈ (SubRing‘𝐸))
10	8, 9	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ (SubRing‘𝐸))
11		evls1fldgencl.8	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ 𝑈)
12		eqid 2763	. . . . . . . . 9 ⊢ (.r‘𝐸) = (.r‘𝐸)
13		eqid 2763	. . . . . . . . 9 ⊢ (.g‘(mulGrp‘𝐸)) = (.g‘(mulGrp‘𝐸))
14		eqid 2763	. . . . . . . . 9 ⊢ (coe1‘𝐺) = (coe1‘𝐺)
15	1, 2, 3, 4, 5, 7, 10, 11, 12, 13, 14	evls1fpws 22742	. . . . . . . 8 ⊢ (𝜑 → (𝑂‘𝐺) = (𝑥 ∈ 𝐵 ↦ (𝐸 Σg (𝑘 ∈ ℕ0 ↦ (((coe1‘𝐺)‘𝑘)(.r‘𝐸)(𝑘(.g‘(mulGrp‘𝐸))𝑥))))))
16		oveq2 7459	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐴 → (𝑘(.g‘(mulGrp‘𝐸))𝑥) = (𝑘(.g‘(mulGrp‘𝐸))𝐴))
17	16	oveq2d 7467	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → (((coe1‘𝐺)‘𝑘)(.r‘𝐸)(𝑘(.g‘(mulGrp‘𝐸))𝑥)) = (((coe1‘𝐺)‘𝑘)(.r‘𝐸)(𝑘(.g‘(mulGrp‘𝐸))𝐴)))
18	17	mpteq2dv 5205	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → (𝑘 ∈ ℕ0 ↦ (((coe1‘𝐺)‘𝑘)(.r‘𝐸)(𝑘(.g‘(mulGrp‘𝐸))𝑥))) = (𝑘 ∈ ℕ0 ↦ (((coe1‘𝐺)‘𝑘)(.r‘𝐸)(𝑘(.g‘(mulGrp‘𝐸))𝐴))))
19	18	oveq2d 7467	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (𝐸 Σg (𝑘 ∈ ℕ0 ↦ (((coe1‘𝐺)‘𝑘)(.r‘𝐸)(𝑘(.g‘(mulGrp‘𝐸))𝑥)))) = (𝐸 Σg (𝑘 ∈ ℕ0 ↦ (((coe1‘𝐺)‘𝑘)(.r‘𝐸)(𝑘(.g‘(mulGrp‘𝐸))𝐴)))))
20	19	adantl 487	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝐸 Σg (𝑘 ∈ ℕ0 ↦ (((coe1‘𝐺)‘𝑘)(.r‘𝐸)(𝑘(.g‘(mulGrp‘𝐸))𝑥)))) = (𝐸 Σg (𝑘 ∈ ℕ0 ↦ (((coe1‘𝐺)‘𝑘)(.r‘𝐸)(𝑘(.g‘(mulGrp‘𝐸))𝐴)))))
21		evls1fldgencl.7	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
22		ovexd 7486	. . . . . . . 8 ⊢ (𝜑 → (𝐸 Σg (𝑘 ∈ ℕ0 ↦ (((coe1‘𝐺)‘𝑘)(.r‘𝐸)(𝑘(.g‘(mulGrp‘𝐸))𝐴)))) ∈ V)
23	15, 20, 21, 22	fvmptd 7032	. . . . . . 7 ⊢ (𝜑 → ((𝑂‘𝐺)‘𝐴) = (𝐸 Σg (𝑘 ∈ ℕ0 ↦ (((coe1‘𝐺)‘𝑘)(.r‘𝐸)(𝑘(.g‘(mulGrp‘𝐸))𝐴)))))
24	23	ad2antrr 739	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) → ((𝑂‘𝐺)‘𝐴) = (𝐸 Σg (𝑘 ∈ ℕ0 ↦ (((coe1‘𝐺)‘𝑘)(.r‘𝐸)(𝑘(.g‘(mulGrp‘𝐸))𝐴)))))
25		eqid 2763	. . . . . . 7 ⊢ (0g‘𝐸) = (0g‘𝐸)
26	7	crngringd 20524	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ Ring)
27	26	ringabld 20563	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ Abel)
28	27	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) → 𝐸 ∈ Abel)
29		nn0ex 12652	. . . . . . . 8 ⊢ ℕ0 ∈ V
30	29	a1i 11	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) → ℕ0 ∈ V)
31		simplr 781	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) → 𝑎 ∈ (SubDRing‘𝐸))
32		sdrgsubrg 21099	. . . . . . . 8 ⊢ (𝑎 ∈ (SubDRing‘𝐸) → 𝑎 ∈ (SubRing‘𝐸))
33		subrgsubg 20880	. . . . . . . 8 ⊢ (𝑎 ∈ (SubRing‘𝐸) → 𝑎 ∈ (SubGrp‘𝐸))
34	31, 32, 33	3syl 19	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) → 𝑎 ∈ (SubGrp‘𝐸))
35	32	ad3antlr 744	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑘 ∈ ℕ0) → 𝑎 ∈ (SubRing‘𝐸))
36		simplr 781	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑘 ∈ ℕ0) → (𝐹 ∪ {𝐴}) ⊆ 𝑎)
37	36	unssad 4194	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑘 ∈ ℕ0) → 𝐹 ⊆ 𝑎)
38	11	ad3antrrr 743	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑘 ∈ ℕ0) → 𝐺 ∈ 𝑈)
39		simpr 490	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0)
40		eqid 2763	. . . . . . . . . . . . 13 ⊢ (Base‘(𝐸 ↾s 𝐹)) = (Base‘(𝐸 ↾s 𝐹))
41	14, 5, 3, 40	coe1fvalcl 22585	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ 𝑈 ∧ 𝑘 ∈ ℕ0) → ((coe1‘𝐺)‘𝑘) ∈ (Base‘(𝐸 ↾s 𝐹)))
42	38, 39, 41	syl2anc 596	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑘 ∈ ℕ0) → ((coe1‘𝐺)‘𝑘) ∈ (Base‘(𝐸 ↾s 𝐹)))
43	2	sdrgss 21101	. . . . . . . . . . . . . 14 ⊢ (𝐹 ∈ (SubDRing‘𝐸) → 𝐹 ⊆ 𝐵)
44	8, 43	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹 ⊆ 𝐵)
45	4, 2	ressbas2 17460	. . . . . . . . . . . . 13 ⊢ (𝐹 ⊆ 𝐵 → 𝐹 = (Base‘(𝐸 ↾s 𝐹)))
46	44, 45	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 = (Base‘(𝐸 ↾s 𝐹)))
47	46	ad3antrrr 743	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑘 ∈ ℕ0) → 𝐹 = (Base‘(𝐸 ↾s 𝐹)))
48	42, 47	eleqtrrd 2867	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑘 ∈ ℕ0) → ((coe1‘𝐺)‘𝑘) ∈ 𝐹)
49	37, 48	sseldd 3914	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑘 ∈ ℕ0) → ((coe1‘𝐺)‘𝑘) ∈ 𝑎)
50		simpllr 788	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑘 ∈ ℕ0) → 𝑎 ∈ (SubDRing‘𝐸))
51	21	ad3antrrr 743	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑘 ∈ ℕ0) → 𝐴 ∈ 𝐵)
52	36	unssbd 4195	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑘 ∈ ℕ0) → {𝐴} ⊆ 𝑎)
53		snssg 4746	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ 𝑎 ↔ {𝐴} ⊆ 𝑎))
54	53	biimpar 483	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝐵 ∧ {𝐴} ⊆ 𝑎) → 𝐴 ∈ 𝑎)
55	51, 52, 54	syl2anc 596	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑘 ∈ ℕ0) → 𝐴 ∈ 𝑎)
56		eqid 2763	. . . . . . . . . . . 12 ⊢ (mulGrp‘𝐸) = (mulGrp‘𝐸)
57	56, 2	mgpbas 20416	. . . . . . . . . . 11 ⊢ 𝐵 = (Base‘(mulGrp‘𝐸))
58	56, 12	mgpplusg 20415	. . . . . . . . . . 11 ⊢ (.r‘𝐸) = (+g‘(mulGrp‘𝐸))
59		fvexd 6931	. . . . . . . . . . 11 ⊢ (𝑎 ∈ (SubDRing‘𝐸) → (mulGrp‘𝐸) ∈ V)
60	2	sdrgss 21101	. . . . . . . . . . 11 ⊢ (𝑎 ∈ (SubDRing‘𝐸) → 𝑎 ⊆ 𝐵)
61	12	subrgmcl 20887	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ (SubRing‘𝐸) ∧ 𝑥 ∈ 𝑎 ∧ 𝑦 ∈ 𝑎) → (𝑥(.r‘𝐸)𝑦) ∈ 𝑎)
62	32, 61	syl3an1 1181	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ (SubDRing‘𝐸) ∧ 𝑥 ∈ 𝑎 ∧ 𝑦 ∈ 𝑎) → (𝑥(.r‘𝐸)𝑦) ∈ 𝑎)
63		eqid 2763	. . . . . . . . . . 11 ⊢ (0g‘(mulGrp‘𝐸)) = (0g‘(mulGrp‘𝐸))
64		eqid 2763	. . . . . . . . . . . . . . 15 ⊢ (1r‘𝐸) = (1r‘𝐸)
65	56, 64	ringidval 20460	. . . . . . . . . . . . . 14 ⊢ (1r‘𝐸) = (0g‘(mulGrp‘𝐸))
66	65	eqcomi 2772	. . . . . . . . . . . . 13 ⊢ (0g‘(mulGrp‘𝐸)) = (1r‘𝐸)
67	66	subrg1cl 20883	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ (SubRing‘𝐸) → (0g‘(mulGrp‘𝐸)) ∈ 𝑎)
68	32, 67	syl 18	. . . . . . . . . . 11 ⊢ (𝑎 ∈ (SubDRing‘𝐸) → (0g‘(mulGrp‘𝐸)) ∈ 𝑎)
69	57, 13, 58, 59, 60, 62, 63, 68	mulgnn0subcl 19347	. . . . . . . . . 10 ⊢ ((𝑎 ∈ (SubDRing‘𝐸) ∧ 𝑘 ∈ ℕ0 ∧ 𝐴 ∈ 𝑎) → (𝑘(.g‘(mulGrp‘𝐸))𝐴) ∈ 𝑎)
70	50, 39, 55, 69	syl3anc 1399	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑘 ∈ ℕ0) → (𝑘(.g‘(mulGrp‘𝐸))𝐴) ∈ 𝑎)
71	12	subrgmcl 20887	. . . . . . . . 9 ⊢ ((𝑎 ∈ (SubRing‘𝐸) ∧ ((coe1‘𝐺)‘𝑘) ∈ 𝑎 ∧ (𝑘(.g‘(mulGrp‘𝐸))𝐴) ∈ 𝑎) → (((coe1‘𝐺)‘𝑘)(.r‘𝐸)(𝑘(.g‘(mulGrp‘𝐸))𝐴)) ∈ 𝑎)
72	35, 49, 70, 71	syl3anc 1399	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑘 ∈ ℕ0) → (((coe1‘𝐺)‘𝑘)(.r‘𝐸)(𝑘(.g‘(mulGrp‘𝐸))𝐴)) ∈ 𝑎)
73	72	fmpttd 7147	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) → (𝑘 ∈ ℕ0 ↦ (((coe1‘𝐺)‘𝑘)(.r‘𝐸)(𝑘(.g‘(mulGrp‘𝐸))𝐴))):ℕ0⟶𝑎)
74	30	mptexd 7261	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) → (𝑘 ∈ ℕ0 ↦ (((coe1‘𝐺)‘𝑘)(.r‘𝐸)(𝑘(.g‘(mulGrp‘𝐸))𝐴))) ∈ V)
75	73	ffund 6744	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) → Fun (𝑘 ∈ ℕ0 ↦ (((coe1‘𝐺)‘𝑘)(.r‘𝐸)(𝑘(.g‘(mulGrp‘𝐸))𝐴))))
76		fvexd 6931	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) → (0g‘𝐸) ∈ V)
77	4	subrgring 20877	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ (SubRing‘𝐸) → (𝐸 ↾s 𝐹) ∈ Ring)
78	10, 77	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐸 ↾s 𝐹) ∈ Ring)
79	78	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) → (𝐸 ↾s 𝐹) ∈ Ring)
80	11	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) → 𝐺 ∈ 𝑈)
81		eqid 2763	. . . . . . . . . . . 12 ⊢ (0g‘(𝐸 ↾s 𝐹)) = (0g‘(𝐸 ↾s 𝐹))
82	3, 5, 81	mptcoe1fsupp 22588	. . . . . . . . . . 11 ⊢ (((𝐸 ↾s 𝐹) ∈ Ring ∧ 𝐺 ∈ 𝑈) → (𝑘 ∈ ℕ0 ↦ ((coe1‘𝐺)‘𝑘)) finSupp (0g‘(𝐸 ↾s 𝐹)))
83	79, 80, 82	syl2anc 596	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) → (𝑘 ∈ ℕ0 ↦ ((coe1‘𝐺)‘𝑘)) finSupp (0g‘(𝐸 ↾s 𝐹)))
84		ringmnd 20521	. . . . . . . . . . . . 13 ⊢ (𝐸 ∈ Ring → 𝐸 ∈ Mnd)
85	26, 84	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐸 ∈ Mnd)
86		subrgsubg 20880	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ (SubRing‘𝐸) → 𝐹 ∈ (SubGrp‘𝐸))
87		subgsubm 19409	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ (SubGrp‘𝐸) → 𝐹 ∈ (SubMnd‘𝐸))
88	25	subm0cl 19056	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ (SubMnd‘𝐸) → (0g‘𝐸) ∈ 𝐹)
89	10, 86, 87, 88	4syl 20	. . . . . . . . . . . 12 ⊢ (𝜑 → (0g‘𝐸) ∈ 𝐹)
90	4, 2, 25	ress0g 19004	. . . . . . . . . . . 12 ⊢ ((𝐸 ∈ Mnd ∧ (0g‘𝐸) ∈ 𝐹 ∧ 𝐹 ⊆ 𝐵) → (0g‘𝐸) = (0g‘(𝐸 ↾s 𝐹)))
91	85, 89, 44, 90	syl3anc 1399	. . . . . . . . . . 11 ⊢ (𝜑 → (0g‘𝐸) = (0g‘(𝐸 ↾s 𝐹)))
92	91	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) → (0g‘𝐸) = (0g‘(𝐸 ↾s 𝐹)))
93	83, 92	breqtrrd 5139	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) → (𝑘 ∈ ℕ0 ↦ ((coe1‘𝐺)‘𝑘)) finSupp (0g‘𝐸))
94	93	fsuppimpd 9399	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) → ((𝑘 ∈ ℕ0 ↦ ((coe1‘𝐺)‘𝑘)) supp (0g‘𝐸)) ∈ Fin)
95		fveq2 6916	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑖 → ((coe1‘𝐺)‘𝑘) = ((coe1‘𝐺)‘𝑖))
96		oveq1 7458	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑖 → (𝑘(.g‘(mulGrp‘𝐸))𝐴) = (𝑖(.g‘(mulGrp‘𝐸))𝐴))
97	95, 96	oveq12d 7469	. . . . . . . . . 10 ⊢ (𝑘 = 𝑖 → (((coe1‘𝐺)‘𝑘)(.r‘𝐸)(𝑘(.g‘(mulGrp‘𝐸))𝐴)) = (((coe1‘𝐺)‘𝑖)(.r‘𝐸)(𝑖(.g‘(mulGrp‘𝐸))𝐴)))
98	97	cbvmptv 5215	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ0 ↦ (((coe1‘𝐺)‘𝑘)(.r‘𝐸)(𝑘(.g‘(mulGrp‘𝐸))𝐴))) = (𝑖 ∈ ℕ0 ↦ (((coe1‘𝐺)‘𝑖)(.r‘𝐸)(𝑖(.g‘(mulGrp‘𝐸))𝐴)))
99		nfv 1949	. . . . . . . . . 10 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎)
100		eqid 2763	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ0 ↦ ((coe1‘𝐺)‘𝑘)) = (𝑘 ∈ ℕ0 ↦ ((coe1‘𝐺)‘𝑘))
101	99, 42, 100	fnmptd 6710	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) → (𝑘 ∈ ℕ0 ↦ ((coe1‘𝐺)‘𝑘)) Fn ℕ0)
102		simplr 781	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑖 ∈ ℕ0) ∧ ((𝑘 ∈ ℕ0 ↦ ((coe1‘𝐺)‘𝑘))‘𝑖) = (0g‘𝐸)) → 𝑖 ∈ ℕ0)
103		fvexd 6931	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑖 ∈ ℕ0) ∧ ((𝑘 ∈ ℕ0 ↦ ((coe1‘𝐺)‘𝑘))‘𝑖) = (0g‘𝐸)) → ((coe1‘𝐺)‘𝑖) ∈ V)
104	100, 95, 102, 103	fvmptd3 7048	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑖 ∈ ℕ0) ∧ ((𝑘 ∈ ℕ0 ↦ ((coe1‘𝐺)‘𝑘))‘𝑖) = (0g‘𝐸)) → ((𝑘 ∈ ℕ0 ↦ ((coe1‘𝐺)‘𝑘))‘𝑖) = ((coe1‘𝐺)‘𝑖))
105		simpr 490	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑖 ∈ ℕ0) ∧ ((𝑘 ∈ ℕ0 ↦ ((coe1‘𝐺)‘𝑘))‘𝑖) = (0g‘𝐸)) → ((𝑘 ∈ ℕ0 ↦ ((coe1‘𝐺)‘𝑘))‘𝑖) = (0g‘𝐸))
106	104, 105	eqtr3d 2800	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑖 ∈ ℕ0) ∧ ((𝑘 ∈ ℕ0 ↦ ((coe1‘𝐺)‘𝑘))‘𝑖) = (0g‘𝐸)) → ((coe1‘𝐺)‘𝑖) = (0g‘𝐸))
107	106	oveq1d 7466	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑖 ∈ ℕ0) ∧ ((𝑘 ∈ ℕ0 ↦ ((coe1‘𝐺)‘𝑘))‘𝑖) = (0g‘𝐸)) → (((coe1‘𝐺)‘𝑖)(.r‘𝐸)(𝑖(.g‘(mulGrp‘𝐸))𝐴)) = ((0g‘𝐸)(.r‘𝐸)(𝑖(.g‘(mulGrp‘𝐸))𝐴)))
108	26	ad4antr 745	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑖 ∈ ℕ0) ∧ ((𝑘 ∈ ℕ0 ↦ ((coe1‘𝐺)‘𝑘))‘𝑖) = (0g‘𝐸)) → 𝐸 ∈ Ring)
109	56	ringmgp 20516	. . . . . . . . . . . . . . 15 ⊢ (𝐸 ∈ Ring → (mulGrp‘𝐸) ∈ Mnd)
110	26, 109	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (mulGrp‘𝐸) ∈ Mnd)
111	110	ad4antr 745	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑖 ∈ ℕ0) ∧ ((𝑘 ∈ ℕ0 ↦ ((coe1‘𝐺)‘𝑘))‘𝑖) = (0g‘𝐸)) → (mulGrp‘𝐸) ∈ Mnd)
112	21	ad4antr 745	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑖 ∈ ℕ0) ∧ ((𝑘 ∈ ℕ0 ↦ ((coe1‘𝐺)‘𝑘))‘𝑖) = (0g‘𝐸)) → 𝐴 ∈ 𝐵)
113	57, 13, 111, 102, 112	mulgnn0cld 19355	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑖 ∈ ℕ0) ∧ ((𝑘 ∈ ℕ0 ↦ ((coe1‘𝐺)‘𝑘))‘𝑖) = (0g‘𝐸)) → (𝑖(.g‘(mulGrp‘𝐸))𝐴) ∈ 𝐵)
114	2, 12, 25, 108, 113	ringlzd 20577	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑖 ∈ ℕ0) ∧ ((𝑘 ∈ ℕ0 ↦ ((coe1‘𝐺)‘𝑘))‘𝑖) = (0g‘𝐸)) → ((0g‘𝐸)(.r‘𝐸)(𝑖(.g‘(mulGrp‘𝐸))𝐴)) = (0g‘𝐸))
115	107, 114	eqtrd 2798	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑖 ∈ ℕ0) ∧ ((𝑘 ∈ ℕ0 ↦ ((coe1‘𝐺)‘𝑘))‘𝑖) = (0g‘𝐸)) → (((coe1‘𝐺)‘𝑖)(.r‘𝐸)(𝑖(.g‘(mulGrp‘𝐸))𝐴)) = (0g‘𝐸))
116	115	3impa 1127	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑖 ∈ ℕ0 ∧ ((𝑘 ∈ ℕ0 ↦ ((coe1‘𝐺)‘𝑘))‘𝑖) = (0g‘𝐸)) → (((coe1‘𝐺)‘𝑖)(.r‘𝐸)(𝑖(.g‘(mulGrp‘𝐸))𝐴)) = (0g‘𝐸))
117	98, 30, 76, 101, 116	suppss3 33411	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) → ((𝑘 ∈ ℕ0 ↦ (((coe1‘𝐺)‘𝑘)(.r‘𝐸)(𝑘(.g‘(mulGrp‘𝐸))𝐴))) supp (0g‘𝐸)) ⊆ ((𝑘 ∈ ℕ0 ↦ ((coe1‘𝐺)‘𝑘)) supp (0g‘𝐸)))
118		suppssfifsupp 9409	. . . . . . . 8 ⊢ ((((𝑘 ∈ ℕ0 ↦ (((coe1‘𝐺)‘𝑘)(.r‘𝐸)(𝑘(.g‘(mulGrp‘𝐸))𝐴))) ∈ V ∧ Fun (𝑘 ∈ ℕ0 ↦ (((coe1‘𝐺)‘𝑘)(.r‘𝐸)(𝑘(.g‘(mulGrp‘𝐸))𝐴))) ∧ (0g‘𝐸) ∈ V) ∧ (((𝑘 ∈ ℕ0 ↦ ((coe1‘𝐺)‘𝑘)) supp (0g‘𝐸)) ∈ Fin ∧ ((𝑘 ∈ ℕ0 ↦ (((coe1‘𝐺)‘𝑘)(.r‘𝐸)(𝑘(.g‘(mulGrp‘𝐸))𝐴))) supp (0g‘𝐸)) ⊆ ((𝑘 ∈ ℕ0 ↦ ((coe1‘𝐺)‘𝑘)) supp (0g‘𝐸)))) → (𝑘 ∈ ℕ0 ↦ (((coe1‘𝐺)‘𝑘)(.r‘𝐸)(𝑘(.g‘(mulGrp‘𝐸))𝐴))) finSupp (0g‘𝐸))
119	74, 75, 76, 94, 117, 118	syl32anc 1406	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) → (𝑘 ∈ ℕ0 ↦ (((coe1‘𝐺)‘𝑘)(.r‘𝐸)(𝑘(.g‘(mulGrp‘𝐸))𝐴))) finSupp (0g‘𝐸))
120	25, 28, 30, 34, 73, 119	gsumsubgcl 20185	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) → (𝐸 Σg (𝑘 ∈ ℕ0 ↦ (((coe1‘𝐺)‘𝑘)(.r‘𝐸)(𝑘(.g‘(mulGrp‘𝐸))𝐴)))) ∈ 𝑎)
121	24, 120	eqeltrd 2864	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) → ((𝑂‘𝐺)‘𝐴) ∈ 𝑎)
122	121	ex 418	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) → ((𝐹 ∪ {𝐴}) ⊆ 𝑎 → ((𝑂‘𝐺)‘𝐴) ∈ 𝑎))
123	122	ralrimiva 3158	. . 3 ⊢ (𝜑 → ∀𝑎 ∈ (SubDRing‘𝐸)((𝐹 ∪ {𝐴}) ⊆ 𝑎 → ((𝑂‘𝐺)‘𝐴) ∈ 𝑎))
124		fvex 6929	. . . 4 ⊢ ((𝑂‘𝐺)‘𝐴) ∈ V
125	124	elintrab 4925	. . 3 ⊢ (((𝑂‘𝐺)‘𝐴) ∈ ∩ {𝑎 ∈ (SubDRing‘𝐸) ∣ (𝐹 ∪ {𝐴}) ⊆ 𝑎} ↔ ∀𝑎 ∈ (SubDRing‘𝐸)((𝐹 ∪ {𝐴}) ⊆ 𝑎 → ((𝑂‘𝐺)‘𝐴) ∈ 𝑎))
126	123, 125	sylibr 237	. 2 ⊢ (𝜑 → ((𝑂‘𝐺)‘𝐴) ∈ ∩ {𝑎 ∈ (SubDRing‘𝐸) ∣ (𝐹 ∪ {𝐴}) ⊆ 𝑎})
127	6	flddrngd 21045	. . 3 ⊢ (𝜑 → 𝐸 ∈ DivRing)
128	21	snssd 4749	. . . 4 ⊢ (𝜑 → {𝐴} ⊆ 𝐵)
129	44, 128	unssd 4193	. . 3 ⊢ (𝜑 → (𝐹 ∪ {𝐴}) ⊆ 𝐵)
130	2, 127, 129	fldgenval 33978	. 2 ⊢ (𝜑 → (𝐸 fldGen (𝐹 ∪ {𝐴})) = ∩ {𝑎 ∈ (SubDRing‘𝐸) ∣ (𝐹 ∪ {𝐴}) ⊆ 𝑎})
131	126, 130	eleqtrrd 2867	1 ⊢ (𝜑 → ((𝑂‘𝐺)‘𝐴) ∈ (𝐸 fldGen (𝐹 ∪ {𝐴})))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080  {crab 3416  Vcvv 3454   ⊆ wss 3900   ∪ cun 4152  {csn 4588  ∩ cint 4912   class class class wbr 5108   ↦ cmpt 5192  Fun wfun 6560  ‘cfv 6566  (class class class)co 7451   supp csupp 8214  Fincfn 9011   finSupp cfsupp 9391  ℕ₀cn0 12646  Basecbs 17431   ↾_s cress 17452  ._rcmulr 17473  0_gc0g 17660   Σ_g cgsu 17661  Mndcmnd 18973  SubMndcsubmnd 19027  ._gcmg 19327  SubGrpcsubg 19380  Abelcabl 20046  mulGrpcmgp 20411  1_rcur 20458  Ringcrg 20510  SubRingcsubrg 20872  Fieldcfield 21032  SubDRingcsdrg 21094  Poly₁cpl1 22550  coe₁cco1 22551   evalSub₁ ces1 22686   fldGen cfldgen 33976

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-cnex 11295  ax-resscn 11296  ax-1cn 11297  ax-icn 11298  ax-addcl 11299  ax-addrcl 11300  ax-mulcl 11301  ax-mulrcl 11302  ax-mulcom 11303  ax-addass 11304  ax-mulass 11305  ax-distr 11306  ax-i2m1 11307  ax-1ne0 11308  ax-1rid 11309  ax-rnegex 11310  ax-rrecex 11311  ax-cnre 11312  ax-pre-lttri 11313  ax-pre-lttrn 11314  ax-pre-ltadd 11315  ax-pre-mulgt0 11316

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4841  df-uni 4873  df-int 4913  df-iun 4958  df-iin 4959  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-isom 6575  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-of 7725  df-ofr 7726  df-om 7912  df-1st 8035  df-2nd 8036  df-supp 8215  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-1o 8514  df-2o 8515  df-er 8755  df-map 8887  df-pm 8888  df-ixp 8964  df-en 9012  df-dom 9013  df-sdom 9014  df-fin 9015  df-fsupp 9392  df-sup 9471  df-oi 9541  df-card 10057  df-pnf 11384  df-mnf 11385  df-xr 11386  df-ltxr 11387  df-le 11388  df-sub 11582  df-neg 11583  df-nn 12376  df-2 12445  df-3 12446  df-4 12447  df-5 12448  df-6 12449  df-7 12450  df-8 12451  df-9 12452  df-n0 12647  df-z 12734  df-dec 12855  df-uz 13006  df-fz 13680  df-fzo 13829  df-seq 14185  df-hash 14516  df-struct 17369  df-sets 17386  df-slot 17404  df-ndx 17416  df-base 17432  df-ress 17453  df-plusg 17486  df-mulr 17487  df-sca 17489  df-vsca 17490  df-ip 17491  df-tset 17492  df-ple 17493  df-ds 17495  df-hom 17497  df-cco 17498  df-0g 17662  df-gsum 17663  df-prds 17668  df-pws 17670  df-mre 17806  df-mrc 17807  df-acs 17809  df-mgm 18866  df-sgrp 18958  df-mnd 18974  df-mhm 19028  df-submnd 19029  df-grp 19197  df-minusg 19198  df-sbg 19199  df-mulg 19328  df-subg 19383  df-ghm 19478  df-cntz 19582  df-cmn 20047  df-abl 20048  df-mgp 20412  df-rng 20426  df-ur 20459  df-srg 20464  df-ring 20512  df-cring 20513  df-rhm 20753  df-subrng 20849  df-subrg 20873  df-drng 21033  df-field 21034  df-sdrg 21095  df-lmod 21188  df-lss 21258  df-lsp 21298  df-assa 22216  df-asp 22217  df-ascl 22218  df-psr 22272  df-mvr 22273  df-mpl 22274  df-opsr 22276  df-evls 22438  df-evl 22439  df-psr1 22553  df-vr1 22554  df-ply1 22555  df-coe1 22556  df-evls1 22688  df-evl1 22689  df-fldgen 33977

This theorem is used by:  algextdeglem2  34461  algextdeglem4  34463