Theorem evls1fldgencl 33665

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 2729	. . . . . . . . 9 ⊢ (𝐸 ↾s 𝐹) = (𝐸 ↾s 𝐹)
5		evls1fldgencl.4	. . . . . . . . 9 ⊢ 𝑈 = (Base‘𝑃)
6		evls1fldgencl.5	. . . . . . . . . 10 ⊢ (𝜑 → 𝐸 ∈ Field)
7	6	fldcrngd 20651	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ CRing)
8		evls1fldgencl.6	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))
9		sdrgsubrg 20700	. . . . . . . . . 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 2729	. . . . . . . . 9 ⊢ (.r‘𝐸) = (.r‘𝐸)
13		eqid 2729	. . . . . . . . 9 ⊢ (.g‘(mulGrp‘𝐸)) = (.g‘(mulGrp‘𝐸))
14		eqid 2729	. . . . . . . . 9 ⊢ (coe1‘𝐺) = (coe1‘𝐺)
15	1, 2, 3, 4, 5, 7, 10, 11, 12, 13, 14	evls1fpws 22256	. . . . . . . 8 ⊢ (𝜑 → (𝑂‘𝐺) = (𝑥 ∈ 𝐵 ↦ (𝐸 Σg (𝑘 ∈ ℕ0 ↦ (((coe1‘𝐺)‘𝑘)(.r‘𝐸)(𝑘(.g‘(mulGrp‘𝐸))𝑥))))))
16		oveq2 7395	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐴 → (𝑘(.g‘(mulGrp‘𝐸))𝑥) = (𝑘(.g‘(mulGrp‘𝐸))𝐴))
17	16	oveq2d 7403	. . . . . . . . . . 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 5201	. . . . . . . . . 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 7403	. . . . . . . . 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 7422	. . . . . . . 8 ⊢ (𝜑 → (𝐸 Σg (𝑘 ∈ ℕ0 ↦ (((coe1‘𝐺)‘𝑘)(.r‘𝐸)(𝑘(.g‘(mulGrp‘𝐸))𝐴)))) ∈ V)
23	15, 20, 21, 22	fvmptd 6975	. . . . . . 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 2729	. . . . . . 7 ⊢ (0g‘𝐸) = (0g‘𝐸)
26	7	crngringd 20155	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ Ring)
27	26	ringabld 20192	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ Abel)
28	27	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) → 𝐸 ∈ Abel)
29		nn0ex 12448	. . . . . . . 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 20700	. . . . . . . 8 ⊢ (𝑎 ∈ (SubDRing‘𝐸) → 𝑎 ∈ (SubRing‘𝐸))
33		subrgsubg 20486	. . . . . . . 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 4156	. . . . . . . . . 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 2729	. . . . . . . . . . . . 13 ⊢ (Base‘(𝐸 ↾s 𝐹)) = (Base‘(𝐸 ↾s 𝐹))
41	14, 5, 3, 40	coe1fvalcl 22097	. . . . . . . . . . . 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 20702	. . . . . . . . . . . . . 14 ⊢ (𝐹 ∈ (SubDRing‘𝐸) → 𝐹 ⊆ 𝐵)
44	8, 43	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹 ⊆ 𝐵)
45	4, 2	ressbas2 17208	. . . . . . . . . . . . 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 2831	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑘 ∈ ℕ0) → ((coe1‘𝐺)‘𝑘) ∈ 𝐹)
49	37, 48	sseldd 3947	. . . . . . . . 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 4157	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑘 ∈ ℕ0) → {𝐴} ⊆ 𝑎)
53		snssg 4747	. . . . . . . . . . . 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 2729	. . . . . . . . . . . 12 ⊢ (mulGrp‘𝐸) = (mulGrp‘𝐸)
57	56, 2	mgpbas 20054	. . . . . . . . . . 11 ⊢ 𝐵 = (Base‘(mulGrp‘𝐸))
58	56, 12	mgpplusg 20053	. . . . . . . . . . 11 ⊢ (.r‘𝐸) = (+g‘(mulGrp‘𝐸))
59		fvexd 6873	. . . . . . . . . . 11 ⊢ (𝑎 ∈ (SubDRing‘𝐸) → (mulGrp‘𝐸) ∈ V)
60	2	sdrgss 20702	. . . . . . . . . . 11 ⊢ (𝑎 ∈ (SubDRing‘𝐸) → 𝑎 ⊆ 𝐵)
61	12	subrgmcl 20493	. . . . . . . . . . . 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 2729	. . . . . . . . . . 11 ⊢ (0g‘(mulGrp‘𝐸)) = (0g‘(mulGrp‘𝐸))
64		eqid 2729	. . . . . . . . . . . . . . 15 ⊢ (1r‘𝐸) = (1r‘𝐸)
65	56, 64	ringidval 20092	. . . . . . . . . . . . . 14 ⊢ (1r‘𝐸) = (0g‘(mulGrp‘𝐸))
66	65	eqcomi 2738	. . . . . . . . . . . . 13 ⊢ (0g‘(mulGrp‘𝐸)) = (1r‘𝐸)
67	66	subrg1cl 20489	. . . . . . . . . . . 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 19019	. . . . . . . . . 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 20493	. . . . . . . . 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 7087	. . . . . . 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 7198	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) → (𝑘 ∈ ℕ0 ↦ (((coe1‘𝐺)‘𝑘)(.r‘𝐸)(𝑘(.g‘(mulGrp‘𝐸))𝐴))) ∈ V)
75	73	ffund 6692	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) → Fun (𝑘 ∈ ℕ0 ↦ (((coe1‘𝐺)‘𝑘)(.r‘𝐸)(𝑘(.g‘(mulGrp‘𝐸))𝐴))))
76		fvexd 6873	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) → (0g‘𝐸) ∈ V)
77	4	subrgring 20483	. . . . . . . . . . . . 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 2729	. . . . . . . . . . . 12 ⊢ (0g‘(𝐸 ↾s 𝐹)) = (0g‘(𝐸 ↾s 𝐹))
82	3, 5, 81	mptcoe1fsupp 22100	. . . . . . . . . . 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 20152	. . . . . . . . . . . . 13 ⊢ (𝐸 ∈ Ring → 𝐸 ∈ Mnd)
85	26, 84	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐸 ∈ Mnd)
86		subrgsubg 20486	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ (SubRing‘𝐸) → 𝐹 ∈ (SubGrp‘𝐸))
87		subgsubm 19080	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ (SubGrp‘𝐸) → 𝐹 ∈ (SubMnd‘𝐸))
88	25	subm0cl 18738	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ (SubMnd‘𝐸) → (0g‘𝐸) ∈ 𝐹)
89	10, 86, 87, 88	4syl 19	. . . . . . . . . . . 12 ⊢ (𝜑 → (0g‘𝐸) ∈ 𝐹)
90	4, 2, 25	ress0g 18689	. . . . . . . . . . . 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 5135	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) → (𝑘 ∈ ℕ0 ↦ ((coe1‘𝐺)‘𝑘)) finSupp (0g‘𝐸))
94	93	fsuppimpd 9320	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) → ((𝑘 ∈ ℕ0 ↦ ((coe1‘𝐺)‘𝑘)) supp (0g‘𝐸)) ∈ Fin)
95		fveq2 6858	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑖 → ((coe1‘𝐺)‘𝑘) = ((coe1‘𝐺)‘𝑖))
96		oveq1 7394	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑖 → (𝑘(.g‘(mulGrp‘𝐸))𝐴) = (𝑖(.g‘(mulGrp‘𝐸))𝐴))
97	95, 96	oveq12d 7405	. . . . . . . . . 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 5211	. . . . . . . . 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 2729	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ0 ↦ ((coe1‘𝐺)‘𝑘)) = (𝑘 ∈ ℕ0 ↦ ((coe1‘𝐺)‘𝑘))
101	99, 42, 100	fnmptd 6659	. . . . . . . . 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 6873	. . . . . . . . . . . . . 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 6991	. . . . . . . . . . . . 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 2766	. . . . . . . . . . . 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 7402	. . . . . . . . . . 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 20148	. . . . . . . . . . . . . . 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 19027	. . . . . . . . . . . 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 20204	. . . . . . . . . . 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 2764	. . . . . . . . . 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 32647	. . . . . . . 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 9331	. . . . . . . 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 19850	. . . . . 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 2828	. . . . 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 3125	. . 3 ⊢ (𝜑 → ∀𝑎 ∈ (SubDRing‘𝐸)((𝐹 ∪ {𝐴}) ⊆ 𝑎 → ((𝑂‘𝐺)‘𝐴) ∈ 𝑎))
124		fvex 6871	. . . 4 ⊢ ((𝑂‘𝐺)‘𝐴) ∈ V
125	124	elintrab 4924	. . 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 20650	. . 3 ⊢ (𝜑 → 𝐸 ∈ DivRing)
128	21	snssd 4773	. . . 4 ⊢ (𝜑 → {𝐴} ⊆ 𝐵)
129	44, 128	unssd 4155	. . 3 ⊢ (𝜑 → (𝐹 ∪ {𝐴}) ⊆ 𝐵)
130	2, 127, 129	fldgenval 33262	. 2 ⊢ (𝜑 → (𝐸 fldGen (𝐹 ∪ {𝐴})) = ∩ {𝑎 ∈ (SubDRing‘𝐸) ∣ (𝐹 ∪ {𝐴}) ⊆ 𝑎})
131	126, 130	eleqtrrd 2831	1 ⊢ (𝜑 → ((𝑂‘𝐺)‘𝐴) ∈ (𝐸 fldGen (𝐹 ∪ {𝐴})))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  {crab 3405  Vcvv 3447   ∪ cun 3912   ⊆ wss 3914  {csn 4589  ∩ cint 4910   class class class wbr 5107   ↦ cmpt 5188  Fun wfun 6505  ‘cfv 6511  (class class class)co 7387   supp csupp 8139  Fincfn 8918   finSupp cfsupp 9312  ℕ₀cn0 12442  Basecbs 17179   ↾_s cress 17200  ._rcmulr 17221  0_gc0g 17402   Σ_g cgsu 17403  Mndcmnd 18661  SubMndcsubmnd 18709  ._gcmg 18999  SubGrpcsubg 19052  Abelcabl 19711  mulGrpcmgp 20049  1_rcur 20090  Ringcrg 20142  SubRingcsubrg 20478  Fieldcfield 20639  SubDRingcsdrg 20695  Poly₁cpl1 22061  coe₁cco1 22062   evalSub₁ ces1 22200   fldGen cfldgen 33260

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 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-iin 4958  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-se 5592  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-isom 6520  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-of 7653  df-ofr 7654  df-om 7843  df-1st 7968  df-2nd 7969  df-supp 8140  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-2o 8435  df-er 8671  df-map 8801  df-pm 8802  df-ixp 8871  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-fsupp 9313  df-sup 9393  df-oi 9463  df-card 9892  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-nn 12187  df-2 12249  df-3 12250  df-4 12251  df-5 12252  df-6 12253  df-7 12254  df-8 12255  df-9 12256  df-n0 12443  df-z 12530  df-dec 12650  df-uz 12794  df-fz 13469  df-fzo 13616  df-seq 13967  df-hash 14296  df-struct 17117  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-ress 17201  df-plusg 17233  df-mulr 17234  df-sca 17236  df-vsca 17237  df-ip 17238  df-tset 17239  df-ple 17240  df-ds 17242  df-hom 17244  df-cco 17245  df-0g 17404  df-gsum 17405  df-prds 17410  df-pws 17412  df-mre 17547  df-mrc 17548  df-acs 17550  df-mgm 18567  df-sgrp 18646  df-mnd 18662  df-mhm 18710  df-submnd 18711  df-grp 18868  df-minusg 18869  df-sbg 18870  df-mulg 19000  df-subg 19055  df-ghm 19145  df-cntz 19249  df-cmn 19712  df-abl 19713  df-mgp 20050  df-rng 20062  df-ur 20091  df-srg 20096  df-ring 20144  df-cring 20145  df-rhm 20381  df-subrng 20455  df-subrg 20479  df-drng 20640  df-field 20641  df-sdrg 20696  df-lmod 20768  df-lss 20838  df-lsp 20878  df-assa 21762  df-asp 21763  df-ascl 21764  df-psr 21818  df-mvr 21819  df-mpl 21820  df-opsr 21822  df-evls 21981  df-evl 21982  df-psr1 22064  df-vr1 22065  df-ply1 22066  df-coe1 22067  df-evls1 22202  df-evl1 22203  df-fldgen 33261

This theorem is referenced by:  algextdeglem2  33708  algextdeglem4  33710