Theorem evls1fldgencl 33672

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 2730	. . . . . . . . 9 ⊢ (𝐸 ↾s 𝐹) = (𝐸 ↾s 𝐹)
5		evls1fldgencl.4	. . . . . . . . 9 ⊢ 𝑈 = (Base‘𝑃)
6		evls1fldgencl.5	. . . . . . . . . 10 ⊢ (𝜑 → 𝐸 ∈ Field)
7	6	fldcrngd 20658	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ CRing)
8		evls1fldgencl.6	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))
9		sdrgsubrg 20707	. . . . . . . . . 10 ⊢ (𝐹 ∈ (SubDRing‘𝐸) → 𝐹 ∈ (SubRing‘𝐸))
10	8, 9	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ (SubRing‘𝐸))
11		evls1fldgencl.8	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ 𝑈)
12		eqid 2730	. . . . . . . . 9 ⊢ (.r‘𝐸) = (.r‘𝐸)
13		eqid 2730	. . . . . . . . 9 ⊢ (.g‘(mulGrp‘𝐸)) = (.g‘(mulGrp‘𝐸))
14		eqid 2730	. . . . . . . . 9 ⊢ (coe1‘𝐺) = (coe1‘𝐺)
15	1, 2, 3, 4, 5, 7, 10, 11, 12, 13, 14	evls1fpws 22263	. . . . . . . 8 ⊢ (𝜑 → (𝑂‘𝐺) = (𝑥 ∈ 𝐵 ↦ (𝐸 Σg (𝑘 ∈ ℕ0 ↦ (((coe1‘𝐺)‘𝑘)(.r‘𝐸)(𝑘(.g‘(mulGrp‘𝐸))𝑥))))))
16		oveq2 7398	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐴 → (𝑘(.g‘(mulGrp‘𝐸))𝑥) = (𝑘(.g‘(mulGrp‘𝐸))𝐴))
17	16	oveq2d 7406	. . . . . . . . . . 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 5204	. . . . . . . . . 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 7406	. . . . . . . . 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 481	. . . . . . . 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 7425	. . . . . . . 8 ⊢ (𝜑 → (𝐸 Σg (𝑘 ∈ ℕ0 ↦ (((coe1‘𝐺)‘𝑘)(.r‘𝐸)(𝑘(.g‘(mulGrp‘𝐸))𝐴)))) ∈ V)
23	15, 20, 21, 22	fvmptd 6978	. . . . . . 7 ⊢ (𝜑 → ((𝑂‘𝐺)‘𝐴) = (𝐸 Σg (𝑘 ∈ ℕ0 ↦ (((coe1‘𝐺)‘𝑘)(.r‘𝐸)(𝑘(.g‘(mulGrp‘𝐸))𝐴)))))
24	23	ad2antrr 726	. . . . . 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 2730	. . . . . . 7 ⊢ (0g‘𝐸) = (0g‘𝐸)
26	7	crngringd 20162	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ Ring)
27	26	ringabld 20199	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ Abel)
28	27	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) → 𝐸 ∈ Abel)
29		nn0ex 12455	. . . . . . . 8 ⊢ ℕ0 ∈ V
30	29	a1i 11	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) → ℕ0 ∈ V)
31		simplr 768	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) → 𝑎 ∈ (SubDRing‘𝐸))
32		sdrgsubrg 20707	. . . . . . . 8 ⊢ (𝑎 ∈ (SubDRing‘𝐸) → 𝑎 ∈ (SubRing‘𝐸))
33		subrgsubg 20493	. . . . . . . 8 ⊢ (𝑎 ∈ (SubRing‘𝐸) → 𝑎 ∈ (SubGrp‘𝐸))
34	31, 32, 33	3syl 18	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) → 𝑎 ∈ (SubGrp‘𝐸))
35	32	ad3antlr 731	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑘 ∈ ℕ0) → 𝑎 ∈ (SubRing‘𝐸))
36		simplr 768	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑘 ∈ ℕ0) → (𝐹 ∪ {𝐴}) ⊆ 𝑎)
37	36	unssad 4159	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑘 ∈ ℕ0) → 𝐹 ⊆ 𝑎)
38	11	ad3antrrr 730	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑘 ∈ ℕ0) → 𝐺 ∈ 𝑈)
39		simpr 484	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0)
40		eqid 2730	. . . . . . . . . . . . 13 ⊢ (Base‘(𝐸 ↾s 𝐹)) = (Base‘(𝐸 ↾s 𝐹))
41	14, 5, 3, 40	coe1fvalcl 22104	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ 𝑈 ∧ 𝑘 ∈ ℕ0) → ((coe1‘𝐺)‘𝑘) ∈ (Base‘(𝐸 ↾s 𝐹)))
42	38, 39, 41	syl2anc 584	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑘 ∈ ℕ0) → ((coe1‘𝐺)‘𝑘) ∈ (Base‘(𝐸 ↾s 𝐹)))
43	2	sdrgss 20709	. . . . . . . . . . . . . 14 ⊢ (𝐹 ∈ (SubDRing‘𝐸) → 𝐹 ⊆ 𝐵)
44	8, 43	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹 ⊆ 𝐵)
45	4, 2	ressbas2 17215	. . . . . . . . . . . . 13 ⊢ (𝐹 ⊆ 𝐵 → 𝐹 = (Base‘(𝐸 ↾s 𝐹)))
46	44, 45	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 = (Base‘(𝐸 ↾s 𝐹)))
47	46	ad3antrrr 730	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑘 ∈ ℕ0) → 𝐹 = (Base‘(𝐸 ↾s 𝐹)))
48	42, 47	eleqtrrd 2832	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑘 ∈ ℕ0) → ((coe1‘𝐺)‘𝑘) ∈ 𝐹)
49	37, 48	sseldd 3950	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑘 ∈ ℕ0) → ((coe1‘𝐺)‘𝑘) ∈ 𝑎)
50		simpllr 775	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑘 ∈ ℕ0) → 𝑎 ∈ (SubDRing‘𝐸))
51	21	ad3antrrr 730	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑘 ∈ ℕ0) → 𝐴 ∈ 𝐵)
52	36	unssbd 4160	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑘 ∈ ℕ0) → {𝐴} ⊆ 𝑎)
53		snssg 4750	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ 𝑎 ↔ {𝐴} ⊆ 𝑎))
54	53	biimpar 477	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝐵 ∧ {𝐴} ⊆ 𝑎) → 𝐴 ∈ 𝑎)
55	51, 52, 54	syl2anc 584	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑘 ∈ ℕ0) → 𝐴 ∈ 𝑎)
56		eqid 2730	. . . . . . . . . . . 12 ⊢ (mulGrp‘𝐸) = (mulGrp‘𝐸)
57	56, 2	mgpbas 20061	. . . . . . . . . . 11 ⊢ 𝐵 = (Base‘(mulGrp‘𝐸))
58	56, 12	mgpplusg 20060	. . . . . . . . . . 11 ⊢ (.r‘𝐸) = (+g‘(mulGrp‘𝐸))
59		fvexd 6876	. . . . . . . . . . 11 ⊢ (𝑎 ∈ (SubDRing‘𝐸) → (mulGrp‘𝐸) ∈ V)
60	2	sdrgss 20709	. . . . . . . . . . 11 ⊢ (𝑎 ∈ (SubDRing‘𝐸) → 𝑎 ⊆ 𝐵)
61	12	subrgmcl 20500	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ (SubRing‘𝐸) ∧ 𝑥 ∈ 𝑎 ∧ 𝑦 ∈ 𝑎) → (𝑥(.r‘𝐸)𝑦) ∈ 𝑎)
62	32, 61	syl3an1 1163	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ (SubDRing‘𝐸) ∧ 𝑥 ∈ 𝑎 ∧ 𝑦 ∈ 𝑎) → (𝑥(.r‘𝐸)𝑦) ∈ 𝑎)
63		eqid 2730	. . . . . . . . . . 11 ⊢ (0g‘(mulGrp‘𝐸)) = (0g‘(mulGrp‘𝐸))
64		eqid 2730	. . . . . . . . . . . . . . 15 ⊢ (1r‘𝐸) = (1r‘𝐸)
65	56, 64	ringidval 20099	. . . . . . . . . . . . . 14 ⊢ (1r‘𝐸) = (0g‘(mulGrp‘𝐸))
66	65	eqcomi 2739	. . . . . . . . . . . . 13 ⊢ (0g‘(mulGrp‘𝐸)) = (1r‘𝐸)
67	66	subrg1cl 20496	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ (SubRing‘𝐸) → (0g‘(mulGrp‘𝐸)) ∈ 𝑎)
68	32, 67	syl 17	. . . . . . . . . . 11 ⊢ (𝑎 ∈ (SubDRing‘𝐸) → (0g‘(mulGrp‘𝐸)) ∈ 𝑎)
69	57, 13, 58, 59, 60, 62, 63, 68	mulgnn0subcl 19026	. . . . . . . . . 10 ⊢ ((𝑎 ∈ (SubDRing‘𝐸) ∧ 𝑘 ∈ ℕ0 ∧ 𝐴 ∈ 𝑎) → (𝑘(.g‘(mulGrp‘𝐸))𝐴) ∈ 𝑎)
70	50, 39, 55, 69	syl3anc 1373	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑘 ∈ ℕ0) → (𝑘(.g‘(mulGrp‘𝐸))𝐴) ∈ 𝑎)
71	12	subrgmcl 20500	. . . . . . . . 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 1373	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑘 ∈ ℕ0) → (((coe1‘𝐺)‘𝑘)(.r‘𝐸)(𝑘(.g‘(mulGrp‘𝐸))𝐴)) ∈ 𝑎)
73	72	fmpttd 7090	. . . . . . 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 7201	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) → (𝑘 ∈ ℕ0 ↦ (((coe1‘𝐺)‘𝑘)(.r‘𝐸)(𝑘(.g‘(mulGrp‘𝐸))𝐴))) ∈ V)
75	73	ffund 6695	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) → Fun (𝑘 ∈ ℕ0 ↦ (((coe1‘𝐺)‘𝑘)(.r‘𝐸)(𝑘(.g‘(mulGrp‘𝐸))𝐴))))
76		fvexd 6876	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) → (0g‘𝐸) ∈ V)
77	4	subrgring 20490	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ (SubRing‘𝐸) → (𝐸 ↾s 𝐹) ∈ Ring)
78	10, 77	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐸 ↾s 𝐹) ∈ Ring)
79	78	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) → (𝐸 ↾s 𝐹) ∈ Ring)
80	11	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) → 𝐺 ∈ 𝑈)
81		eqid 2730	. . . . . . . . . . . 12 ⊢ (0g‘(𝐸 ↾s 𝐹)) = (0g‘(𝐸 ↾s 𝐹))
82	3, 5, 81	mptcoe1fsupp 22107	. . . . . . . . . . 11 ⊢ (((𝐸 ↾s 𝐹) ∈ Ring ∧ 𝐺 ∈ 𝑈) → (𝑘 ∈ ℕ0 ↦ ((coe1‘𝐺)‘𝑘)) finSupp (0g‘(𝐸 ↾s 𝐹)))
83	79, 80, 82	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) → (𝑘 ∈ ℕ0 ↦ ((coe1‘𝐺)‘𝑘)) finSupp (0g‘(𝐸 ↾s 𝐹)))
84		ringmnd 20159	. . . . . . . . . . . . 13 ⊢ (𝐸 ∈ Ring → 𝐸 ∈ Mnd)
85	26, 84	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐸 ∈ Mnd)
86		subrgsubg 20493	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ (SubRing‘𝐸) → 𝐹 ∈ (SubGrp‘𝐸))
87		subgsubm 19087	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ (SubGrp‘𝐸) → 𝐹 ∈ (SubMnd‘𝐸))
88	25	subm0cl 18745	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ (SubMnd‘𝐸) → (0g‘𝐸) ∈ 𝐹)
89	10, 86, 87, 88	4syl 19	. . . . . . . . . . . 12 ⊢ (𝜑 → (0g‘𝐸) ∈ 𝐹)
90	4, 2, 25	ress0g 18696	. . . . . . . . . . . 12 ⊢ ((𝐸 ∈ Mnd ∧ (0g‘𝐸) ∈ 𝐹 ∧ 𝐹 ⊆ 𝐵) → (0g‘𝐸) = (0g‘(𝐸 ↾s 𝐹)))
91	85, 89, 44, 90	syl3anc 1373	. . . . . . . . . . 11 ⊢ (𝜑 → (0g‘𝐸) = (0g‘(𝐸 ↾s 𝐹)))
92	91	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) → (0g‘𝐸) = (0g‘(𝐸 ↾s 𝐹)))
93	83, 92	breqtrrd 5138	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) → (𝑘 ∈ ℕ0 ↦ ((coe1‘𝐺)‘𝑘)) finSupp (0g‘𝐸))
94	93	fsuppimpd 9327	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) → ((𝑘 ∈ ℕ0 ↦ ((coe1‘𝐺)‘𝑘)) supp (0g‘𝐸)) ∈ Fin)
95		fveq2 6861	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑖 → ((coe1‘𝐺)‘𝑘) = ((coe1‘𝐺)‘𝑖))
96		oveq1 7397	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑖 → (𝑘(.g‘(mulGrp‘𝐸))𝐴) = (𝑖(.g‘(mulGrp‘𝐸))𝐴))
97	95, 96	oveq12d 7408	. . . . . . . . . 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 5214	. . . . . . . . 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 1914	. . . . . . . . . 10 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎)
100		eqid 2730	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ0 ↦ ((coe1‘𝐺)‘𝑘)) = (𝑘 ∈ ℕ0 ↦ ((coe1‘𝐺)‘𝑘))
101	99, 42, 100	fnmptd 6662	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) → (𝑘 ∈ ℕ0 ↦ ((coe1‘𝐺)‘𝑘)) Fn ℕ0)
102		simplr 768	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑖 ∈ ℕ0) ∧ ((𝑘 ∈ ℕ0 ↦ ((coe1‘𝐺)‘𝑘))‘𝑖) = (0g‘𝐸)) → 𝑖 ∈ ℕ0)
103		fvexd 6876	. . . . . . . . . . . . . 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 6994	. . . . . . . . . . . . 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 484	. . . . . . . . . . . . 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 2767	. . . . . . . . . . . 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 7405	. . . . . . . . . . 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 732	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑖 ∈ ℕ0) ∧ ((𝑘 ∈ ℕ0 ↦ ((coe1‘𝐺)‘𝑘))‘𝑖) = (0g‘𝐸)) → 𝐸 ∈ Ring)
109	56	ringmgp 20155	. . . . . . . . . . . . . . 15 ⊢ (𝐸 ∈ Ring → (mulGrp‘𝐸) ∈ Mnd)
110	26, 109	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (mulGrp‘𝐸) ∈ Mnd)
111	110	ad4antr 732	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑖 ∈ ℕ0) ∧ ((𝑘 ∈ ℕ0 ↦ ((coe1‘𝐺)‘𝑘))‘𝑖) = (0g‘𝐸)) → (mulGrp‘𝐸) ∈ Mnd)
112	21	ad4antr 732	. . . . . . . . . . . . 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 19034	. . . . . . . . . . . 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 20211	. . . . . . . . . . 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 2765	. . . . . . . . . 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 1109	. . . . . . . . 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 32654	. . . . . . . 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 9338	. . . . . . . 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 1380	. . . . . . 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 19857	. . . . . 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 2829	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) → ((𝑂‘𝐺)‘𝐴) ∈ 𝑎)
122	121	ex 412	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) → ((𝐹 ∪ {𝐴}) ⊆ 𝑎 → ((𝑂‘𝐺)‘𝐴) ∈ 𝑎))
123	122	ralrimiva 3126	. . 3 ⊢ (𝜑 → ∀𝑎 ∈ (SubDRing‘𝐸)((𝐹 ∪ {𝐴}) ⊆ 𝑎 → ((𝑂‘𝐺)‘𝐴) ∈ 𝑎))
124		fvex 6874	. . . 4 ⊢ ((𝑂‘𝐺)‘𝐴) ∈ V
125	124	elintrab 4927	. . 3 ⊢ (((𝑂‘𝐺)‘𝐴) ∈ ∩ {𝑎 ∈ (SubDRing‘𝐸) ∣ (𝐹 ∪ {𝐴}) ⊆ 𝑎} ↔ ∀𝑎 ∈ (SubDRing‘𝐸)((𝐹 ∪ {𝐴}) ⊆ 𝑎 → ((𝑂‘𝐺)‘𝐴) ∈ 𝑎))
126	123, 125	sylibr 234	. 2 ⊢ (𝜑 → ((𝑂‘𝐺)‘𝐴) ∈ ∩ {𝑎 ∈ (SubDRing‘𝐸) ∣ (𝐹 ∪ {𝐴}) ⊆ 𝑎})
127	6	flddrngd 20657	. . 3 ⊢ (𝜑 → 𝐸 ∈ DivRing)
128	21	snssd 4776	. . . 4 ⊢ (𝜑 → {𝐴} ⊆ 𝐵)
129	44, 128	unssd 4158	. . 3 ⊢ (𝜑 → (𝐹 ∪ {𝐴}) ⊆ 𝐵)
130	2, 127, 129	fldgenval 33269	. 2 ⊢ (𝜑 → (𝐸 fldGen (𝐹 ∪ {𝐴})) = ∩ {𝑎 ∈ (SubDRing‘𝐸) ∣ (𝐹 ∪ {𝐴}) ⊆ 𝑎})
131	126, 130	eleqtrrd 2832	1 ⊢ (𝜑 → ((𝑂‘𝐺)‘𝐴) ∈ (𝐸 fldGen (𝐹 ∪ {𝐴})))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3045  {crab 3408  Vcvv 3450   ∪ cun 3915   ⊆ wss 3917  {csn 4592  ∩ cint 4913   class class class wbr 5110   ↦ cmpt 5191  Fun wfun 6508  ‘cfv 6514  (class class class)co 7390   supp csupp 8142  Fincfn 8921   finSupp cfsupp 9319  ℕ₀cn0 12449  Basecbs 17186   ↾_s cress 17207  ._rcmulr 17228  0_gc0g 17409   Σ_g cgsu 17410  Mndcmnd 18668  SubMndcsubmnd 18716  ._gcmg 19006  SubGrpcsubg 19059  Abelcabl 19718  mulGrpcmgp 20056  1_rcur 20097  Ringcrg 20149  SubRingcsubrg 20485  Fieldcfield 20646  SubDRingcsdrg 20702  Poly₁cpl1 22068  coe₁cco1 22069   evalSub₁ ces1 22207   fldGen cfldgen 33267

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-tp 4597  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-iin 4961  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-se 5595  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-isom 6523  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-of 7656  df-ofr 7657  df-om 7846  df-1st 7971  df-2nd 7972  df-supp 8143  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-2o 8438  df-er 8674  df-map 8804  df-pm 8805  df-ixp 8874  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-fsupp 9320  df-sup 9400  df-oi 9470  df-card 9899  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-nn 12194  df-2 12256  df-3 12257  df-4 12258  df-5 12259  df-6 12260  df-7 12261  df-8 12262  df-9 12263  df-n0 12450  df-z 12537  df-dec 12657  df-uz 12801  df-fz 13476  df-fzo 13623  df-seq 13974  df-hash 14303  df-struct 17124  df-sets 17141  df-slot 17159  df-ndx 17171  df-base 17187  df-ress 17208  df-plusg 17240  df-mulr 17241  df-sca 17243  df-vsca 17244  df-ip 17245  df-tset 17246  df-ple 17247  df-ds 17249  df-hom 17251  df-cco 17252  df-0g 17411  df-gsum 17412  df-prds 17417  df-pws 17419  df-mre 17554  df-mrc 17555  df-acs 17557  df-mgm 18574  df-sgrp 18653  df-mnd 18669  df-mhm 18717  df-submnd 18718  df-grp 18875  df-minusg 18876  df-sbg 18877  df-mulg 19007  df-subg 19062  df-ghm 19152  df-cntz 19256  df-cmn 19719  df-abl 19720  df-mgp 20057  df-rng 20069  df-ur 20098  df-srg 20103  df-ring 20151  df-cring 20152  df-rhm 20388  df-subrng 20462  df-subrg 20486  df-drng 20647  df-field 20648  df-sdrg 20703  df-lmod 20775  df-lss 20845  df-lsp 20885  df-assa 21769  df-asp 21770  df-ascl 21771  df-psr 21825  df-mvr 21826  df-mpl 21827  df-opsr 21829  df-evls 21988  df-evl 21989  df-psr1 22071  df-vr1 22072  df-ply1 22073  df-coe1 22074  df-evls1 22209  df-evl1 22210  df-fldgen 33268

This theorem is referenced by:  algextdeglem2  33715  algextdeglem4  33717