Theorem evls1fldgencl 33833

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 2737	. . . . . . . . 9 ⊢ (𝐸 ↾s 𝐹) = (𝐸 ↾s 𝐹)
5		evls1fldgencl.4	. . . . . . . . 9 ⊢ 𝑈 = (Base‘𝑃)
6		evls1fldgencl.5	. . . . . . . . . 10 ⊢ (𝜑 → 𝐸 ∈ Field)
7	6	fldcrngd 20713	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ CRing)
8		evls1fldgencl.6	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))
9		sdrgsubrg 20762	. . . . . . . . . 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 2737	. . . . . . . . 9 ⊢ (.r‘𝐸) = (.r‘𝐸)
13		eqid 2737	. . . . . . . . 9 ⊢ (.g‘(mulGrp‘𝐸)) = (.g‘(mulGrp‘𝐸))
14		eqid 2737	. . . . . . . . 9 ⊢ (coe1‘𝐺) = (coe1‘𝐺)
15	1, 2, 3, 4, 5, 7, 10, 11, 12, 13, 14	evls1fpws 22347	. . . . . . . 8 ⊢ (𝜑 → (𝑂‘𝐺) = (𝑥 ∈ 𝐵 ↦ (𝐸 Σg (𝑘 ∈ ℕ0 ↦ (((coe1‘𝐺)‘𝑘)(.r‘𝐸)(𝑘(.g‘(mulGrp‘𝐸))𝑥))))))
16		oveq2 7369	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐴 → (𝑘(.g‘(mulGrp‘𝐸))𝑥) = (𝑘(.g‘(mulGrp‘𝐸))𝐴))
17	16	oveq2d 7377	. . . . . . . . . . 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 5180	. . . . . . . . . 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 7377	. . . . . . . . 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 7396	. . . . . . . 8 ⊢ (𝜑 → (𝐸 Σg (𝑘 ∈ ℕ0 ↦ (((coe1‘𝐺)‘𝑘)(.r‘𝐸)(𝑘(.g‘(mulGrp‘𝐸))𝐴)))) ∈ V)
23	15, 20, 21, 22	fvmptd 6950	. . . . . . 7 ⊢ (𝜑 → ((𝑂‘𝐺)‘𝐴) = (𝐸 Σg (𝑘 ∈ ℕ0 ↦ (((coe1‘𝐺)‘𝑘)(.r‘𝐸)(𝑘(.g‘(mulGrp‘𝐸))𝐴)))))
24	23	ad2antrr 727	. . . . . 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 2737	. . . . . . 7 ⊢ (0g‘𝐸) = (0g‘𝐸)
26	7	crngringd 20221	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ Ring)
27	26	ringabld 20258	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ Abel)
28	27	ad2antrr 727	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) → 𝐸 ∈ Abel)
29		nn0ex 12437	. . . . . . . 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 20762	. . . . . . . 8 ⊢ (𝑎 ∈ (SubDRing‘𝐸) → 𝑎 ∈ (SubRing‘𝐸))
33		subrgsubg 20548	. . . . . . . 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 732	. . . . . . . . 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 4134	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑘 ∈ ℕ0) → 𝐹 ⊆ 𝑎)
38	11	ad3antrrr 731	. . . . . . . . . . . 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 2737	. . . . . . . . . . . . 13 ⊢ (Base‘(𝐸 ↾s 𝐹)) = (Base‘(𝐸 ↾s 𝐹))
41	14, 5, 3, 40	coe1fvalcl 22189	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ 𝑈 ∧ 𝑘 ∈ ℕ0) → ((coe1‘𝐺)‘𝑘) ∈ (Base‘(𝐸 ↾s 𝐹)))
42	38, 39, 41	syl2anc 585	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑘 ∈ ℕ0) → ((coe1‘𝐺)‘𝑘) ∈ (Base‘(𝐸 ↾s 𝐹)))
43	2	sdrgss 20764	. . . . . . . . . . . . . 14 ⊢ (𝐹 ∈ (SubDRing‘𝐸) → 𝐹 ⊆ 𝐵)
44	8, 43	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹 ⊆ 𝐵)
45	4, 2	ressbas2 17202	. . . . . . . . . . . . 13 ⊢ (𝐹 ⊆ 𝐵 → 𝐹 = (Base‘(𝐸 ↾s 𝐹)))
46	44, 45	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 = (Base‘(𝐸 ↾s 𝐹)))
47	46	ad3antrrr 731	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑘 ∈ ℕ0) → 𝐹 = (Base‘(𝐸 ↾s 𝐹)))
48	42, 47	eleqtrrd 2840	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑘 ∈ ℕ0) → ((coe1‘𝐺)‘𝑘) ∈ 𝐹)
49	37, 48	sseldd 3923	. . . . . . . . 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 731	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑘 ∈ ℕ0) → 𝐴 ∈ 𝐵)
52	36	unssbd 4135	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑘 ∈ ℕ0) → {𝐴} ⊆ 𝑎)
53		snssg 4728	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ 𝑎 ↔ {𝐴} ⊆ 𝑎))
54	53	biimpar 477	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝐵 ∧ {𝐴} ⊆ 𝑎) → 𝐴 ∈ 𝑎)
55	51, 52, 54	syl2anc 585	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑘 ∈ ℕ0) → 𝐴 ∈ 𝑎)
56		eqid 2737	. . . . . . . . . . . 12 ⊢ (mulGrp‘𝐸) = (mulGrp‘𝐸)
57	56, 2	mgpbas 20120	. . . . . . . . . . 11 ⊢ 𝐵 = (Base‘(mulGrp‘𝐸))
58	56, 12	mgpplusg 20119	. . . . . . . . . . 11 ⊢ (.r‘𝐸) = (+g‘(mulGrp‘𝐸))
59		fvexd 6850	. . . . . . . . . . 11 ⊢ (𝑎 ∈ (SubDRing‘𝐸) → (mulGrp‘𝐸) ∈ V)
60	2	sdrgss 20764	. . . . . . . . . . 11 ⊢ (𝑎 ∈ (SubDRing‘𝐸) → 𝑎 ⊆ 𝐵)
61	12	subrgmcl 20555	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ (SubRing‘𝐸) ∧ 𝑥 ∈ 𝑎 ∧ 𝑦 ∈ 𝑎) → (𝑥(.r‘𝐸)𝑦) ∈ 𝑎)
62	32, 61	syl3an1 1164	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ (SubDRing‘𝐸) ∧ 𝑥 ∈ 𝑎 ∧ 𝑦 ∈ 𝑎) → (𝑥(.r‘𝐸)𝑦) ∈ 𝑎)
63		eqid 2737	. . . . . . . . . . 11 ⊢ (0g‘(mulGrp‘𝐸)) = (0g‘(mulGrp‘𝐸))
64		eqid 2737	. . . . . . . . . . . . . . 15 ⊢ (1r‘𝐸) = (1r‘𝐸)
65	56, 64	ringidval 20158	. . . . . . . . . . . . . 14 ⊢ (1r‘𝐸) = (0g‘(mulGrp‘𝐸))
66	65	eqcomi 2746	. . . . . . . . . . . . 13 ⊢ (0g‘(mulGrp‘𝐸)) = (1r‘𝐸)
67	66	subrg1cl 20551	. . . . . . . . . . . 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 19057	. . . . . . . . . 10 ⊢ ((𝑎 ∈ (SubDRing‘𝐸) ∧ 𝑘 ∈ ℕ0 ∧ 𝐴 ∈ 𝑎) → (𝑘(.g‘(mulGrp‘𝐸))𝐴) ∈ 𝑎)
70	50, 39, 55, 69	syl3anc 1374	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑘 ∈ ℕ0) → (𝑘(.g‘(mulGrp‘𝐸))𝐴) ∈ 𝑎)
71	12	subrgmcl 20555	. . . . . . . . 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 1374	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑘 ∈ ℕ0) → (((coe1‘𝐺)‘𝑘)(.r‘𝐸)(𝑘(.g‘(mulGrp‘𝐸))𝐴)) ∈ 𝑎)
73	72	fmpttd 7062	. . . . . . 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 7173	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) → (𝑘 ∈ ℕ0 ↦ (((coe1‘𝐺)‘𝑘)(.r‘𝐸)(𝑘(.g‘(mulGrp‘𝐸))𝐴))) ∈ V)
75	73	ffund 6667	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) → Fun (𝑘 ∈ ℕ0 ↦ (((coe1‘𝐺)‘𝑘)(.r‘𝐸)(𝑘(.g‘(mulGrp‘𝐸))𝐴))))
76		fvexd 6850	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) → (0g‘𝐸) ∈ V)
77	4	subrgring 20545	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ (SubRing‘𝐸) → (𝐸 ↾s 𝐹) ∈ Ring)
78	10, 77	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐸 ↾s 𝐹) ∈ Ring)
79	78	ad2antrr 727	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) → (𝐸 ↾s 𝐹) ∈ Ring)
80	11	ad2antrr 727	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) → 𝐺 ∈ 𝑈)
81		eqid 2737	. . . . . . . . . . . 12 ⊢ (0g‘(𝐸 ↾s 𝐹)) = (0g‘(𝐸 ↾s 𝐹))
82	3, 5, 81	mptcoe1fsupp 22192	. . . . . . . . . . 11 ⊢ (((𝐸 ↾s 𝐹) ∈ Ring ∧ 𝐺 ∈ 𝑈) → (𝑘 ∈ ℕ0 ↦ ((coe1‘𝐺)‘𝑘)) finSupp (0g‘(𝐸 ↾s 𝐹)))
83	79, 80, 82	syl2anc 585	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) → (𝑘 ∈ ℕ0 ↦ ((coe1‘𝐺)‘𝑘)) finSupp (0g‘(𝐸 ↾s 𝐹)))
84		ringmnd 20218	. . . . . . . . . . . . 13 ⊢ (𝐸 ∈ Ring → 𝐸 ∈ Mnd)
85	26, 84	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐸 ∈ Mnd)
86		subrgsubg 20548	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ (SubRing‘𝐸) → 𝐹 ∈ (SubGrp‘𝐸))
87		subgsubm 19118	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ (SubGrp‘𝐸) → 𝐹 ∈ (SubMnd‘𝐸))
88	25	subm0cl 18773	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ (SubMnd‘𝐸) → (0g‘𝐸) ∈ 𝐹)
89	10, 86, 87, 88	4syl 19	. . . . . . . . . . . 12 ⊢ (𝜑 → (0g‘𝐸) ∈ 𝐹)
90	4, 2, 25	ress0g 18724	. . . . . . . . . . . 12 ⊢ ((𝐸 ∈ Mnd ∧ (0g‘𝐸) ∈ 𝐹 ∧ 𝐹 ⊆ 𝐵) → (0g‘𝐸) = (0g‘(𝐸 ↾s 𝐹)))
91	85, 89, 44, 90	syl3anc 1374	. . . . . . . . . . 11 ⊢ (𝜑 → (0g‘𝐸) = (0g‘(𝐸 ↾s 𝐹)))
92	91	ad2antrr 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) → (0g‘𝐸) = (0g‘(𝐸 ↾s 𝐹)))
93	83, 92	breqtrrd 5114	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) → (𝑘 ∈ ℕ0 ↦ ((coe1‘𝐺)‘𝑘)) finSupp (0g‘𝐸))
94	93	fsuppimpd 9276	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) → ((𝑘 ∈ ℕ0 ↦ ((coe1‘𝐺)‘𝑘)) supp (0g‘𝐸)) ∈ Fin)
95		fveq2 6835	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑖 → ((coe1‘𝐺)‘𝑘) = ((coe1‘𝐺)‘𝑖))
96		oveq1 7368	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑖 → (𝑘(.g‘(mulGrp‘𝐸))𝐴) = (𝑖(.g‘(mulGrp‘𝐸))𝐴))
97	95, 96	oveq12d 7379	. . . . . . . . . 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 5190	. . . . . . . . 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 1916	. . . . . . . . . 10 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎)
100		eqid 2737	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ0 ↦ ((coe1‘𝐺)‘𝑘)) = (𝑘 ∈ ℕ0 ↦ ((coe1‘𝐺)‘𝑘))
101	99, 42, 100	fnmptd 6634	. . . . . . . . 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 6850	. . . . . . . . . . . . . 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 6966	. . . . . . . . . . . . 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 2774	. . . . . . . . . . . 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 7376	. . . . . . . . . . 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 733	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑖 ∈ ℕ0) ∧ ((𝑘 ∈ ℕ0 ↦ ((coe1‘𝐺)‘𝑘))‘𝑖) = (0g‘𝐸)) → 𝐸 ∈ Ring)
109	56	ringmgp 20214	. . . . . . . . . . . . . . 15 ⊢ (𝐸 ∈ Ring → (mulGrp‘𝐸) ∈ Mnd)
110	26, 109	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (mulGrp‘𝐸) ∈ Mnd)
111	110	ad4antr 733	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑖 ∈ ℕ0) ∧ ((𝑘 ∈ ℕ0 ↦ ((coe1‘𝐺)‘𝑘))‘𝑖) = (0g‘𝐸)) → (mulGrp‘𝐸) ∈ Mnd)
112	21	ad4antr 733	. . . . . . . . . . . . 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 19065	. . . . . . . . . . . 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 20270	. . . . . . . . . . 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 2772	. . . . . . . . . 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 1110	. . . . . . . . 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 32814	. . . . . . . 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 9287	. . . . . . . 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 1381	. . . . . . 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 19889	. . . . . 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 2837	. . . . 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 3130	. . 3 ⊢ (𝜑 → ∀𝑎 ∈ (SubDRing‘𝐸)((𝐹 ∪ {𝐴}) ⊆ 𝑎 → ((𝑂‘𝐺)‘𝐴) ∈ 𝑎))
124		fvex 6848	. . . 4 ⊢ ((𝑂‘𝐺)‘𝐴) ∈ V
125	124	elintrab 4903	. . 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 20712	. . 3 ⊢ (𝜑 → 𝐸 ∈ DivRing)
128	21	snssd 4753	. . . 4 ⊢ (𝜑 → {𝐴} ⊆ 𝐵)
129	44, 128	unssd 4133	. . 3 ⊢ (𝜑 → (𝐹 ∪ {𝐴}) ⊆ 𝐵)
130	2, 127, 129	fldgenval 33391	. 2 ⊢ (𝜑 → (𝐸 fldGen (𝐹 ∪ {𝐴})) = ∩ {𝑎 ∈ (SubDRing‘𝐸) ∣ (𝐹 ∪ {𝐴}) ⊆ 𝑎})
131	126, 130	eleqtrrd 2840	1 ⊢ (𝜑 → ((𝑂‘𝐺)‘𝐴) ∈ (𝐸 fldGen (𝐹 ∪ {𝐴})))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  {crab 3390  Vcvv 3430   ∪ cun 3888   ⊆ wss 3890  {csn 4568  ∩ cint 4890   class class class wbr 5086   ↦ cmpt 5167  Fun wfun 6487  ‘cfv 6493  (class class class)co 7361   supp csupp 8104  Fincfn 8887   finSupp cfsupp 9268  ℕ₀cn0 12431  Basecbs 17173   ↾_s cress 17194  ._rcmulr 17215  0_gc0g 17396   Σ_g cgsu 17397  Mndcmnd 18696  SubMndcsubmnd 18744  ._gcmg 19037  SubGrpcsubg 19090  Abelcabl 19750  mulGrpcmgp 20115  1_rcur 20156  Ringcrg 20208  SubRingcsubrg 20540  Fieldcfield 20701  SubDRingcsdrg 20757  Poly₁cpl1 22153  coe₁cco1 22154   evalSub₁ ces1 22291   fldGen cfldgen 33389

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683  ax-cnex 11088  ax-resscn 11089  ax-1cn 11090  ax-icn 11091  ax-addcl 11092  ax-addrcl 11093  ax-mulcl 11094  ax-mulrcl 11095  ax-mulcom 11096  ax-addass 11097  ax-mulass 11098  ax-distr 11099  ax-i2m1 11100  ax-1ne0 11101  ax-1rid 11102  ax-rnegex 11103  ax-rrecex 11104  ax-cnre 11105  ax-pre-lttri 11106  ax-pre-lttrn 11107  ax-pre-ltadd 11108  ax-pre-mulgt0 11109

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-iin 4937  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-se 5579  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-isom 6502  df-riota 7318  df-ov 7364  df-oprab 7365  df-mpo 7366  df-of 7625  df-ofr 7626  df-om 7812  df-1st 7936  df-2nd 7937  df-supp 8105  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-2o 8400  df-er 8637  df-map 8769  df-pm 8770  df-ixp 8840  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891  df-fsupp 9269  df-sup 9349  df-oi 9419  df-card 9857  df-pnf 11175  df-mnf 11176  df-xr 11177  df-ltxr 11178  df-le 11179  df-sub 11373  df-neg 11374  df-nn 12169  df-2 12238  df-3 12239  df-4 12240  df-5 12241  df-6 12242  df-7 12243  df-8 12244  df-9 12245  df-n0 12432  df-z 12519  df-dec 12639  df-uz 12783  df-fz 13456  df-fzo 13603  df-seq 13958  df-hash 14287  df-struct 17111  df-sets 17128  df-slot 17146  df-ndx 17158  df-base 17174  df-ress 17195  df-plusg 17227  df-mulr 17228  df-sca 17230  df-vsca 17231  df-ip 17232  df-tset 17233  df-ple 17234  df-ds 17236  df-hom 17238  df-cco 17239  df-0g 17398  df-gsum 17399  df-prds 17404  df-pws 17406  df-mre 17542  df-mrc 17543  df-acs 17545  df-mgm 18602  df-sgrp 18681  df-mnd 18697  df-mhm 18745  df-submnd 18746  df-grp 18906  df-minusg 18907  df-sbg 18908  df-mulg 19038  df-subg 19093  df-ghm 19182  df-cntz 19286  df-cmn 19751  df-abl 19752  df-mgp 20116  df-rng 20128  df-ur 20157  df-srg 20162  df-ring 20210  df-cring 20211  df-rhm 20446  df-subrng 20517  df-subrg 20541  df-drng 20702  df-field 20703  df-sdrg 20758  df-lmod 20851  df-lss 20921  df-lsp 20961  df-assa 21846  df-asp 21847  df-ascl 21848  df-psr 21902  df-mvr 21903  df-mpl 21904  df-opsr 21906  df-evls 22065  df-evl 22066  df-psr1 22156  df-vr1 22157  df-ply1 22158  df-coe1 22159  df-evls1 22293  df-evl1 22294  df-fldgen 33390

This theorem is referenced by:  algextdeglem2  33881  algextdeglem4  33883