Theorem evls1fldgencl 33695

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 2735	. . . . . . . . 9 ⊢ (𝐸 ↾s 𝐹) = (𝐸 ↾s 𝐹)
5		evls1fldgencl.4	. . . . . . . . 9 ⊢ 𝑈 = (Base‘𝑃)
6		evls1fldgencl.5	. . . . . . . . . 10 ⊢ (𝜑 → 𝐸 ∈ Field)
7	6	fldcrngd 20759	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ CRing)
8		evls1fldgencl.6	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))
9		sdrgsubrg 20809	. . . . . . . . . 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 2735	. . . . . . . . 9 ⊢ (.r‘𝐸) = (.r‘𝐸)
13		eqid 2735	. . . . . . . . 9 ⊢ (.g‘(mulGrp‘𝐸)) = (.g‘(mulGrp‘𝐸))
14		eqid 2735	. . . . . . . . 9 ⊢ (coe1‘𝐺) = (coe1‘𝐺)
15	1, 2, 3, 4, 5, 7, 10, 11, 12, 13, 14	evls1fpws 22389	. . . . . . . 8 ⊢ (𝜑 → (𝑂‘𝐺) = (𝑥 ∈ 𝐵 ↦ (𝐸 Σg (𝑘 ∈ ℕ0 ↦ (((coe1‘𝐺)‘𝑘)(.r‘𝐸)(𝑘(.g‘(mulGrp‘𝐸))𝑥))))))
16		oveq2 7439	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐴 → (𝑘(.g‘(mulGrp‘𝐸))𝑥) = (𝑘(.g‘(mulGrp‘𝐸))𝐴))
17	16	oveq2d 7447	. . . . . . . . . . 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 5250	. . . . . . . . . 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 7447	. . . . . . . . 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 7466	. . . . . . . 8 ⊢ (𝜑 → (𝐸 Σg (𝑘 ∈ ℕ0 ↦ (((coe1‘𝐺)‘𝑘)(.r‘𝐸)(𝑘(.g‘(mulGrp‘𝐸))𝐴)))) ∈ V)
23	15, 20, 21, 22	fvmptd 7023	. . . . . . 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 2735	. . . . . . 7 ⊢ (0g‘𝐸) = (0g‘𝐸)
26	7	crngringd 20264	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ Ring)
27	26	ringabld 20297	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ Abel)
28	27	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) → 𝐸 ∈ Abel)
29		nn0ex 12530	. . . . . . . 8 ⊢ ℕ0 ∈ V
30	29	a1i 11	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) → ℕ0 ∈ V)
31		simplr 769	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) → 𝑎 ∈ (SubDRing‘𝐸))
32		sdrgsubrg 20809	. . . . . . . 8 ⊢ (𝑎 ∈ (SubDRing‘𝐸) → 𝑎 ∈ (SubRing‘𝐸))
33		subrgsubg 20594	. . . . . . . 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 769	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑘 ∈ ℕ0) → (𝐹 ∪ {𝐴}) ⊆ 𝑎)
37	36	unssad 4203	. . . . . . . . . 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 2735	. . . . . . . . . . . . 13 ⊢ (Base‘(𝐸 ↾s 𝐹)) = (Base‘(𝐸 ↾s 𝐹))
41	14, 5, 3, 40	coe1fvalcl 22230	. . . . . . . . . . . 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 20811	. . . . . . . . . . . . . 14 ⊢ (𝐹 ∈ (SubDRing‘𝐸) → 𝐹 ⊆ 𝐵)
44	8, 43	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹 ⊆ 𝐵)
45	4, 2	ressbas2 17283	. . . . . . . . . . . . 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 2842	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑘 ∈ ℕ0) → ((coe1‘𝐺)‘𝑘) ∈ 𝐹)
49	37, 48	sseldd 3996	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑘 ∈ ℕ0) → ((coe1‘𝐺)‘𝑘) ∈ 𝑎)
50		simpllr 776	. . . . . . . . . 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 4204	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑘 ∈ ℕ0) → {𝐴} ⊆ 𝑎)
53		snssg 4788	. . . . . . . . . . . 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 2735	. . . . . . . . . . . 12 ⊢ (mulGrp‘𝐸) = (mulGrp‘𝐸)
57	56, 2	mgpbas 20158	. . . . . . . . . . 11 ⊢ 𝐵 = (Base‘(mulGrp‘𝐸))
58	56, 12	mgpplusg 20156	. . . . . . . . . . 11 ⊢ (.r‘𝐸) = (+g‘(mulGrp‘𝐸))
59		fvexd 6922	. . . . . . . . . . 11 ⊢ (𝑎 ∈ (SubDRing‘𝐸) → (mulGrp‘𝐸) ∈ V)
60	2	sdrgss 20811	. . . . . . . . . . 11 ⊢ (𝑎 ∈ (SubDRing‘𝐸) → 𝑎 ⊆ 𝐵)
61	12	subrgmcl 20601	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ (SubRing‘𝐸) ∧ 𝑥 ∈ 𝑎 ∧ 𝑦 ∈ 𝑎) → (𝑥(.r‘𝐸)𝑦) ∈ 𝑎)
62	32, 61	syl3an1 1162	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ (SubDRing‘𝐸) ∧ 𝑥 ∈ 𝑎 ∧ 𝑦 ∈ 𝑎) → (𝑥(.r‘𝐸)𝑦) ∈ 𝑎)
63		eqid 2735	. . . . . . . . . . 11 ⊢ (0g‘(mulGrp‘𝐸)) = (0g‘(mulGrp‘𝐸))
64		eqid 2735	. . . . . . . . . . . . . . 15 ⊢ (1r‘𝐸) = (1r‘𝐸)
65	56, 64	ringidval 20201	. . . . . . . . . . . . . 14 ⊢ (1r‘𝐸) = (0g‘(mulGrp‘𝐸))
66	65	eqcomi 2744	. . . . . . . . . . . . 13 ⊢ (0g‘(mulGrp‘𝐸)) = (1r‘𝐸)
67	66	subrg1cl 20597	. . . . . . . . . . . 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 19118	. . . . . . . . . 10 ⊢ ((𝑎 ∈ (SubDRing‘𝐸) ∧ 𝑘 ∈ ℕ0 ∧ 𝐴 ∈ 𝑎) → (𝑘(.g‘(mulGrp‘𝐸))𝐴) ∈ 𝑎)
70	50, 39, 55, 69	syl3anc 1370	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑘 ∈ ℕ0) → (𝑘(.g‘(mulGrp‘𝐸))𝐴) ∈ 𝑎)
71	12	subrgmcl 20601	. . . . . . . . 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 1370	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑘 ∈ ℕ0) → (((coe1‘𝐺)‘𝑘)(.r‘𝐸)(𝑘(.g‘(mulGrp‘𝐸))𝐴)) ∈ 𝑎)
73	72	fmpttd 7135	. . . . . . 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 7244	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) → (𝑘 ∈ ℕ0 ↦ (((coe1‘𝐺)‘𝑘)(.r‘𝐸)(𝑘(.g‘(mulGrp‘𝐸))𝐴))) ∈ V)
75	73	ffund 6741	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) → Fun (𝑘 ∈ ℕ0 ↦ (((coe1‘𝐺)‘𝑘)(.r‘𝐸)(𝑘(.g‘(mulGrp‘𝐸))𝐴))))
76		fvexd 6922	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) → (0g‘𝐸) ∈ V)
77	4	subrgring 20591	. . . . . . . . . . . . 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 2735	. . . . . . . . . . . 12 ⊢ (0g‘(𝐸 ↾s 𝐹)) = (0g‘(𝐸 ↾s 𝐹))
82	3, 5, 81	mptcoe1fsupp 22233	. . . . . . . . . . 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 20261	. . . . . . . . . . . . 13 ⊢ (𝐸 ∈ Ring → 𝐸 ∈ Mnd)
85	26, 84	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐸 ∈ Mnd)
86		subrgsubg 20594	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ (SubRing‘𝐸) → 𝐹 ∈ (SubGrp‘𝐸))
87		subgsubm 19179	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ (SubGrp‘𝐸) → 𝐹 ∈ (SubMnd‘𝐸))
88	25	subm0cl 18837	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ (SubMnd‘𝐸) → (0g‘𝐸) ∈ 𝐹)
89	10, 86, 87, 88	4syl 19	. . . . . . . . . . . 12 ⊢ (𝜑 → (0g‘𝐸) ∈ 𝐹)
90	4, 2, 25	ress0g 18788	. . . . . . . . . . . 12 ⊢ ((𝐸 ∈ Mnd ∧ (0g‘𝐸) ∈ 𝐹 ∧ 𝐹 ⊆ 𝐵) → (0g‘𝐸) = (0g‘(𝐸 ↾s 𝐹)))
91	85, 89, 44, 90	syl3anc 1370	. . . . . . . . . . 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 5176	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) → (𝑘 ∈ ℕ0 ↦ ((coe1‘𝐺)‘𝑘)) finSupp (0g‘𝐸))
94	93	fsuppimpd 9407	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) → ((𝑘 ∈ ℕ0 ↦ ((coe1‘𝐺)‘𝑘)) supp (0g‘𝐸)) ∈ Fin)
95		fveq2 6907	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑖 → ((coe1‘𝐺)‘𝑘) = ((coe1‘𝐺)‘𝑖))
96		oveq1 7438	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑖 → (𝑘(.g‘(mulGrp‘𝐸))𝐴) = (𝑖(.g‘(mulGrp‘𝐸))𝐴))
97	95, 96	oveq12d 7449	. . . . . . . . . 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 5261	. . . . . . . . 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 1912	. . . . . . . . . 10 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎)
100		eqid 2735	. . . . . . . . . 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 769	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑖 ∈ ℕ0) ∧ ((𝑘 ∈ ℕ0 ↦ ((coe1‘𝐺)‘𝑘))‘𝑖) = (0g‘𝐸)) → 𝑖 ∈ ℕ0)
103		fvexd 6922	. . . . . . . . . . . . . 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 7039	. . . . . . . . . . . . 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 2777	. . . . . . . . . . . 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 7446	. . . . . . . . . . 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 20257	. . . . . . . . . . . . . . 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 19126	. . . . . . . . . . . 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 20309	. . . . . . . . . . 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 2775	. . . . . . . . . 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 32742	. . . . . . . 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 9418	. . . . . . . 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 1377	. . . . . . 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 19953	. . . . . 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 2839	. . . . 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 3144	. . 3 ⊢ (𝜑 → ∀𝑎 ∈ (SubDRing‘𝐸)((𝐹 ∪ {𝐴}) ⊆ 𝑎 → ((𝑂‘𝐺)‘𝐴) ∈ 𝑎))
124		fvex 6920	. . . 4 ⊢ ((𝑂‘𝐺)‘𝐴) ∈ V
125	124	elintrab 4965	. . 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 20758	. . 3 ⊢ (𝜑 → 𝐸 ∈ DivRing)
128	21	snssd 4814	. . . 4 ⊢ (𝜑 → {𝐴} ⊆ 𝐵)
129	44, 128	unssd 4202	. . 3 ⊢ (𝜑 → (𝐹 ∪ {𝐴}) ⊆ 𝐵)
130	2, 127, 129	fldgenval 33294	. 2 ⊢ (𝜑 → (𝐸 fldGen (𝐹 ∪ {𝐴})) = ∩ {𝑎 ∈ (SubDRing‘𝐸) ∣ (𝐹 ∪ {𝐴}) ⊆ 𝑎})
131	126, 130	eleqtrrd 2842	1 ⊢ (𝜑 → ((𝑂‘𝐺)‘𝐴) ∈ (𝐸 fldGen (𝐹 ∪ {𝐴})))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2106  ∀wral 3059  {crab 3433  Vcvv 3478   ∪ cun 3961   ⊆ wss 3963  {csn 4631  ∩ cint 4951   class class class wbr 5148   ↦ cmpt 5231  Fun wfun 6557  ‘cfv 6563  (class class class)co 7431   supp csupp 8184  Fincfn 8984   finSupp cfsupp 9399  ℕ₀cn0 12524  Basecbs 17245   ↾_s cress 17274  ._rcmulr 17299  0_gc0g 17486   Σ_g cgsu 17487  Mndcmnd 18760  SubMndcsubmnd 18808  ._gcmg 19098  SubGrpcsubg 19151  Abelcabl 19814  mulGrpcmgp 20152  1_rcur 20199  Ringcrg 20251  SubRingcsubrg 20586  Fieldcfield 20747  SubDRingcsdrg 20804  Poly₁cpl1 22194  coe₁cco1 22195   evalSub₁ ces1 22333   fldGen cfldgen 33292

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

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-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-sca 17314  df-vsca 17315  df-ip 17316  df-tset 17317  df-ple 17318  df-ds 17320  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-rhm 20489  df-subrng 20563  df-subrg 20587  df-drng 20748  df-field 20749  df-sdrg 20805  df-lmod 20877  df-lss 20948  df-lsp 20988  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-fldgen 33293

This theorem is referenced by:  algextdeglem2  33724  algextdeglem4  33726