evls1fldgencl - Mathbox for Thierry Arnoux

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Theorem evls1fldgencl 33224

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 2724	. . . . . . . . 9 ⊢ (𝐸 ↾_s 𝐹) = (𝐸 ↾_s 𝐹)
5		evls1fldgencl.4	. . . . . . . . 9 ⊢ 𝑈 = (Base‘𝑃)
6		evls1fldgencl.5	. . . . . . . . . 10 ⊢ (𝜑 → 𝐸 ∈ Field)
7	6	fldcrngd 20590	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ CRing)
8		evls1fldgencl.6	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))
9		sdrgsubrg 20632	. . . . . . . . . 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 2724	. . . . . . . . 9 ⊢ (._r‘𝐸) = (._r‘𝐸)
13		eqid 2724	. . . . . . . . 9 ⊢ (._g‘(mulGrp‘𝐸)) = (._g‘(mulGrp‘𝐸))
14		eqid 2724	. . . . . . . . 9 ⊢ (coe₁‘𝐺) = (coe₁‘𝐺)
15	1, 2, 3, 4, 5, 7, 10, 11, 12, 13, 14	evls1fpws 33113	. . . . . . . 8 ⊢ (𝜑 → (𝑂‘𝐺) = (𝑥 ∈ 𝐵 ↦ (𝐸 Σ_g (𝑘 ∈ ℕ₀ ↦ (((coe₁‘𝐺)‘𝑘)(._r‘𝐸)(𝑘(._g‘(mulGrp‘𝐸))𝑥))))))
16		oveq2 7409	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐴 → (𝑘(._g‘(mulGrp‘𝐸))𝑥) = (𝑘(._g‘(mulGrp‘𝐸))𝐴))
17	16	oveq2d 7417	. . . . . . . . . . 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 5240	. . . . . . . . . 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 7417	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (𝐸 Σ_g (𝑘 ∈ ℕ₀ ↦ (((coe₁‘𝐺)‘𝑘)(._r‘𝐸)(𝑘(._g‘(mulGrp‘𝐸))𝑥)))) = (𝐸 Σ_g (𝑘 ∈ ℕ₀ ↦ (((coe₁‘𝐺)‘𝑘)(._r‘𝐸)(𝑘(._g‘(mulGrp‘𝐸))𝐴)))))
20	19	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝐸 Σ_g (𝑘 ∈ ℕ₀ ↦ (((coe₁‘𝐺)‘𝑘)(._r‘𝐸)(𝑘(._g‘(mulGrp‘𝐸))𝑥)))) = (𝐸 Σ_g (𝑘 ∈ ℕ₀ ↦ (((coe₁‘𝐺)‘𝑘)(._r‘𝐸)(𝑘(._g‘(mulGrp‘𝐸))𝐴)))))
21		evls1fldgencl.7	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
22		ovexd 7436	. . . . . . . 8 ⊢ (𝜑 → (𝐸 Σ_g (𝑘 ∈ ℕ₀ ↦ (((coe₁‘𝐺)‘𝑘)(._r‘𝐸)(𝑘(._g‘(mulGrp‘𝐸))𝐴)))) ∈ V)
23	15, 20, 21, 22	fvmptd 6995	. . . . . . 7 ⊢ (𝜑 → ((𝑂‘𝐺)‘𝐴) = (𝐸 Σ_g (𝑘 ∈ ℕ₀ ↦ (((coe₁‘𝐺)‘𝑘)(._r‘𝐸)(𝑘(._g‘(mulGrp‘𝐸))𝐴)))))
24	23	ad2antrr 723	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) → ((𝑂‘𝐺)‘𝐴) = (𝐸 Σ_g (𝑘 ∈ ℕ₀ ↦ (((coe₁‘𝐺)‘𝑘)(._r‘𝐸)(𝑘(._g‘(mulGrp‘𝐸))𝐴)))))
25		eqid 2724	. . . . . . 7 ⊢ (0_g‘𝐸) = (0_g‘𝐸)
26	7	crngringd 20141	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ Ring)
27	26	ringabld 20172	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ Abel)
28	27	ad2antrr 723	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) → 𝐸 ∈ Abel)
29		nn0ex 12475	. . . . . . . 8 ⊢ ℕ₀ ∈ V
30	29	a1i 11	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) → ℕ₀ ∈ V)
31		simplr 766	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) → 𝑎 ∈ (SubDRing‘𝐸))
32		sdrgsubrg 20632	. . . . . . . 8 ⊢ (𝑎 ∈ (SubDRing‘𝐸) → 𝑎 ∈ (SubRing‘𝐸))
33		subrgsubg 20469	. . . . . . . 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 728	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑘 ∈ ℕ₀) → 𝑎 ∈ (SubRing‘𝐸))
36		simplr 766	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑘 ∈ ℕ₀) → (𝐹 ∪ {𝐴}) ⊆ 𝑎)
37	36	unssad 4179	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑘 ∈ ℕ₀) → 𝐹 ⊆ 𝑎)
38	11	ad3antrrr 727	. . . . . . . . . . . 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 2724	. . . . . . . . . . . . 13 ⊢ (Base‘(𝐸 ↾_s 𝐹)) = (Base‘(𝐸 ↾_s 𝐹))
41	14, 5, 3, 40	coe1fvalcl 22054	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ 𝑈 ∧ 𝑘 ∈ ℕ₀) → ((coe₁‘𝐺)‘𝑘) ∈ (Base‘(𝐸 ↾_s 𝐹)))
42	38, 39, 41	syl2anc 583	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑘 ∈ ℕ₀) → ((coe₁‘𝐺)‘𝑘) ∈ (Base‘(𝐸 ↾_s 𝐹)))
43	2	sdrgss 20634	. . . . . . . . . . . . . 14 ⊢ (𝐹 ∈ (SubDRing‘𝐸) → 𝐹 ⊆ 𝐵)
44	8, 43	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹 ⊆ 𝐵)
45	4, 2	ressbas2 17181	. . . . . . . . . . . . 13 ⊢ (𝐹 ⊆ 𝐵 → 𝐹 = (Base‘(𝐸 ↾_s 𝐹)))
46	44, 45	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 = (Base‘(𝐸 ↾_s 𝐹)))
47	46	ad3antrrr 727	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑘 ∈ ℕ₀) → 𝐹 = (Base‘(𝐸 ↾_s 𝐹)))
48	42, 47	eleqtrrd 2828	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑘 ∈ ℕ₀) → ((coe₁‘𝐺)‘𝑘) ∈ 𝐹)
49	37, 48	sseldd 3975	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑘 ∈ ℕ₀) → ((coe₁‘𝐺)‘𝑘) ∈ 𝑎)
50		simpllr 773	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑘 ∈ ℕ₀) → 𝑎 ∈ (SubDRing‘𝐸))
51	21	ad3antrrr 727	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑘 ∈ ℕ₀) → 𝐴 ∈ 𝐵)
52	36	unssbd 4180	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑘 ∈ ℕ₀) → {𝐴} ⊆ 𝑎)
53		snssg 4779	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ 𝑎 ↔ {𝐴} ⊆ 𝑎))
54	53	biimpar 477	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝐵 ∧ {𝐴} ⊆ 𝑎) → 𝐴 ∈ 𝑎)
55	51, 52, 54	syl2anc 583	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑘 ∈ ℕ₀) → 𝐴 ∈ 𝑎)
56		eqid 2724	. . . . . . . . . . . 12 ⊢ (mulGrp‘𝐸) = (mulGrp‘𝐸)
57	56, 2	mgpbas 20035	. . . . . . . . . . 11 ⊢ 𝐵 = (Base‘(mulGrp‘𝐸))
58	56, 12	mgpplusg 20033	. . . . . . . . . . 11 ⊢ (._r‘𝐸) = (+_g‘(mulGrp‘𝐸))
59		fvexd 6896	. . . . . . . . . . 11 ⊢ (𝑎 ∈ (SubDRing‘𝐸) → (mulGrp‘𝐸) ∈ V)
60	2	sdrgss 20634	. . . . . . . . . . 11 ⊢ (𝑎 ∈ (SubDRing‘𝐸) → 𝑎 ⊆ 𝐵)
61	12	subrgmcl 20476	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ (SubRing‘𝐸) ∧ 𝑥 ∈ 𝑎 ∧ 𝑦 ∈ 𝑎) → (𝑥(._r‘𝐸)𝑦) ∈ 𝑎)
62	32, 61	syl3an1 1160	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ (SubDRing‘𝐸) ∧ 𝑥 ∈ 𝑎 ∧ 𝑦 ∈ 𝑎) → (𝑥(._r‘𝐸)𝑦) ∈ 𝑎)
63		eqid 2724	. . . . . . . . . . 11 ⊢ (0_g‘(mulGrp‘𝐸)) = (0_g‘(mulGrp‘𝐸))
64		eqid 2724	. . . . . . . . . . . . . . 15 ⊢ (1_r‘𝐸) = (1_r‘𝐸)
65	56, 64	ringidval 20078	. . . . . . . . . . . . . 14 ⊢ (1_r‘𝐸) = (0_g‘(mulGrp‘𝐸))
66	65	eqcomi 2733	. . . . . . . . . . . . 13 ⊢ (0_g‘(mulGrp‘𝐸)) = (1_r‘𝐸)
67	66	subrg1cl 20472	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ (SubRing‘𝐸) → (0_g‘(mulGrp‘𝐸)) ∈ 𝑎)
68	32, 67	syl 17	. . . . . . . . . . 11 ⊢ (𝑎 ∈ (SubDRing‘𝐸) → (0_g‘(mulGrp‘𝐸)) ∈ 𝑎)
69	57, 13, 58, 59, 60, 62, 63, 68	mulgnn0subcl 19004	. . . . . . . . . 10 ⊢ ((𝑎 ∈ (SubDRing‘𝐸) ∧ 𝑘 ∈ ℕ₀ ∧ 𝐴 ∈ 𝑎) → (𝑘(._g‘(mulGrp‘𝐸))𝐴) ∈ 𝑎)
70	50, 39, 55, 69	syl3anc 1368	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑘 ∈ ℕ₀) → (𝑘(._g‘(mulGrp‘𝐸))𝐴) ∈ 𝑎)
71	12	subrgmcl 20476	. . . . . . . . 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 1368	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑘 ∈ ℕ₀) → (((coe₁‘𝐺)‘𝑘)(._r‘𝐸)(𝑘(._g‘(mulGrp‘𝐸))𝐴)) ∈ 𝑎)
73	72	fmpttd 7106	. . . . . . 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 7217	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) → (𝑘 ∈ ℕ₀ ↦ (((coe₁‘𝐺)‘𝑘)(._r‘𝐸)(𝑘(._g‘(mulGrp‘𝐸))𝐴))) ∈ V)
75	73	ffund 6711	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) → Fun (𝑘 ∈ ℕ₀ ↦ (((coe₁‘𝐺)‘𝑘)(._r‘𝐸)(𝑘(._g‘(mulGrp‘𝐸))𝐴))))
76		fvexd 6896	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) → (0_g‘𝐸) ∈ V)
77	4	subrgring 20466	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ (SubRing‘𝐸) → (𝐸 ↾_s 𝐹) ∈ Ring)
78	10, 77	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐸 ↾_s 𝐹) ∈ Ring)
79	78	ad2antrr 723	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) → (𝐸 ↾_s 𝐹) ∈ Ring)
80	11	ad2antrr 723	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) → 𝐺 ∈ 𝑈)
81		eqid 2724	. . . . . . . . . . . 12 ⊢ (0_g‘(𝐸 ↾_s 𝐹)) = (0_g‘(𝐸 ↾_s 𝐹))
82	3, 5, 81	mptcoe1fsupp 22057	. . . . . . . . . . 11 ⊢ (((𝐸 ↾_s 𝐹) ∈ Ring ∧ 𝐺 ∈ 𝑈) → (𝑘 ∈ ℕ₀ ↦ ((coe₁‘𝐺)‘𝑘)) finSupp (0_g‘(𝐸 ↾_s 𝐹)))
83	79, 80, 82	syl2anc 583	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) → (𝑘 ∈ ℕ₀ ↦ ((coe₁‘𝐺)‘𝑘)) finSupp (0_g‘(𝐸 ↾_s 𝐹)))
84		ringmnd 20138	. . . . . . . . . . . . 13 ⊢ (𝐸 ∈ Ring → 𝐸 ∈ Mnd)
85	26, 84	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐸 ∈ Mnd)
86		subrgsubg 20469	. . . . . . . . . . . . . 14 ⊢ (𝐹 ∈ (SubRing‘𝐸) → 𝐹 ∈ (SubGrp‘𝐸))
87		subgsubm 19065	. . . . . . . . . . . . . 14 ⊢ (𝐹 ∈ (SubGrp‘𝐸) → 𝐹 ∈ (SubMnd‘𝐸))
88	10, 86, 87	3syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹 ∈ (SubMnd‘𝐸))
89	25	subm0cl 18726	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ (SubMnd‘𝐸) → (0_g‘𝐸) ∈ 𝐹)
90	88, 89	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (0_g‘𝐸) ∈ 𝐹)
91	4, 2, 25	ress0g 18685	. . . . . . . . . . . 12 ⊢ ((𝐸 ∈ Mnd ∧ (0_g‘𝐸) ∈ 𝐹 ∧ 𝐹 ⊆ 𝐵) → (0_g‘𝐸) = (0_g‘(𝐸 ↾_s 𝐹)))
92	85, 90, 44, 91	syl3anc 1368	. . . . . . . . . . 11 ⊢ (𝜑 → (0_g‘𝐸) = (0_g‘(𝐸 ↾_s 𝐹)))
93	92	ad2antrr 723	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) → (0_g‘𝐸) = (0_g‘(𝐸 ↾_s 𝐹)))
94	83, 93	breqtrrd 5166	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) → (𝑘 ∈ ℕ₀ ↦ ((coe₁‘𝐺)‘𝑘)) finSupp (0_g‘𝐸))
95	94	fsuppimpd 9365	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) → ((𝑘 ∈ ℕ₀ ↦ ((coe₁‘𝐺)‘𝑘)) supp (0_g‘𝐸)) ∈ Fin)
96		fveq2 6881	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑖 → ((coe₁‘𝐺)‘𝑘) = ((coe₁‘𝐺)‘𝑖))
97		oveq1 7408	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑖 → (𝑘(._g‘(mulGrp‘𝐸))𝐴) = (𝑖(._g‘(mulGrp‘𝐸))𝐴))
98	96, 97	oveq12d 7419	. . . . . . . . . 10 ⊢ (𝑘 = 𝑖 → (((coe₁‘𝐺)‘𝑘)(._r‘𝐸)(𝑘(._g‘(mulGrp‘𝐸))𝐴)) = (((coe₁‘𝐺)‘𝑖)(._r‘𝐸)(𝑖(._g‘(mulGrp‘𝐸))𝐴)))
99	98	cbvmptv 5251	. . . . . . . . 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)))
100		nfv 1909	. . . . . . . . . 10 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎)
101		eqid 2724	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ₀ ↦ ((coe₁‘𝐺)‘𝑘)) = (𝑘 ∈ ℕ₀ ↦ ((coe₁‘𝐺)‘𝑘))
102	100, 42, 101	fnmptd 6681	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) → (𝑘 ∈ ℕ₀ ↦ ((coe₁‘𝐺)‘𝑘)) Fn ℕ₀)
103		simplr 766	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑖 ∈ ℕ₀) ∧ ((𝑘 ∈ ℕ₀ ↦ ((coe₁‘𝐺)‘𝑘))‘𝑖) = (0_g‘𝐸)) → 𝑖 ∈ ℕ₀)
104		fvexd 6896	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑖 ∈ ℕ₀) ∧ ((𝑘 ∈ ℕ₀ ↦ ((coe₁‘𝐺)‘𝑘))‘𝑖) = (0_g‘𝐸)) → ((coe₁‘𝐺)‘𝑖) ∈ V)
105	101, 96, 103, 104	fvmptd3 7011	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑖 ∈ ℕ₀) ∧ ((𝑘 ∈ ℕ₀ ↦ ((coe₁‘𝐺)‘𝑘))‘𝑖) = (0_g‘𝐸)) → ((𝑘 ∈ ℕ₀ ↦ ((coe₁‘𝐺)‘𝑘))‘𝑖) = ((coe₁‘𝐺)‘𝑖))
106		simpr 484	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑖 ∈ ℕ₀) ∧ ((𝑘 ∈ ℕ₀ ↦ ((coe₁‘𝐺)‘𝑘))‘𝑖) = (0_g‘𝐸)) → ((𝑘 ∈ ℕ₀ ↦ ((coe₁‘𝐺)‘𝑘))‘𝑖) = (0_g‘𝐸))
107	105, 106	eqtr3d 2766	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑖 ∈ ℕ₀) ∧ ((𝑘 ∈ ℕ₀ ↦ ((coe₁‘𝐺)‘𝑘))‘𝑖) = (0_g‘𝐸)) → ((coe₁‘𝐺)‘𝑖) = (0_g‘𝐸))
108	107	oveq1d 7416	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑖 ∈ ℕ₀) ∧ ((𝑘 ∈ ℕ₀ ↦ ((coe₁‘𝐺)‘𝑘))‘𝑖) = (0_g‘𝐸)) → (((coe₁‘𝐺)‘𝑖)(._r‘𝐸)(𝑖(._g‘(mulGrp‘𝐸))𝐴)) = ((0_g‘𝐸)(._r‘𝐸)(𝑖(._g‘(mulGrp‘𝐸))𝐴)))
109	26	ad4antr 729	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑖 ∈ ℕ₀) ∧ ((𝑘 ∈ ℕ₀ ↦ ((coe₁‘𝐺)‘𝑘))‘𝑖) = (0_g‘𝐸)) → 𝐸 ∈ Ring)
110	56	ringmgp 20134	. . . . . . . . . . . . . . 15 ⊢ (𝐸 ∈ Ring → (mulGrp‘𝐸) ∈ Mnd)
111	26, 110	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (mulGrp‘𝐸) ∈ Mnd)
112	111	ad4antr 729	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑖 ∈ ℕ₀) ∧ ((𝑘 ∈ ℕ₀ ↦ ((coe₁‘𝐺)‘𝑘))‘𝑖) = (0_g‘𝐸)) → (mulGrp‘𝐸) ∈ Mnd)
113	21	ad4antr 729	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑖 ∈ ℕ₀) ∧ ((𝑘 ∈ ℕ₀ ↦ ((coe₁‘𝐺)‘𝑘))‘𝑖) = (0_g‘𝐸)) → 𝐴 ∈ 𝐵)
114	57, 13, 112, 103, 113	mulgnn0cld 19012	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑖 ∈ ℕ₀) ∧ ((𝑘 ∈ ℕ₀ ↦ ((coe₁‘𝐺)‘𝑘))‘𝑖) = (0_g‘𝐸)) → (𝑖(._g‘(mulGrp‘𝐸))𝐴) ∈ 𝐵)
115	2, 12, 25, 109, 114	ringlzd 20184	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑖 ∈ ℕ₀) ∧ ((𝑘 ∈ ℕ₀ ↦ ((coe₁‘𝐺)‘𝑘))‘𝑖) = (0_g‘𝐸)) → ((0_g‘𝐸)(._r‘𝐸)(𝑖(._g‘(mulGrp‘𝐸))𝐴)) = (0_g‘𝐸))
116	108, 115	eqtrd 2764	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑖 ∈ ℕ₀) ∧ ((𝑘 ∈ ℕ₀ ↦ ((coe₁‘𝐺)‘𝑘))‘𝑖) = (0_g‘𝐸)) → (((coe₁‘𝐺)‘𝑖)(._r‘𝐸)(𝑖(._g‘(mulGrp‘𝐸))𝐴)) = (0_g‘𝐸))
117	116	3impa 1107	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) ∧ 𝑖 ∈ ℕ₀ ∧ ((𝑘 ∈ ℕ₀ ↦ ((coe₁‘𝐺)‘𝑘))‘𝑖) = (0_g‘𝐸)) → (((coe₁‘𝐺)‘𝑖)(._r‘𝐸)(𝑖(._g‘(mulGrp‘𝐸))𝐴)) = (0_g‘𝐸))
118	99, 30, 76, 102, 117	suppss3 32418	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) → ((𝑘 ∈ ℕ₀ ↦ (((coe₁‘𝐺)‘𝑘)(._r‘𝐸)(𝑘(._g‘(mulGrp‘𝐸))𝐴))) supp (0_g‘𝐸)) ⊆ ((𝑘 ∈ ℕ₀ ↦ ((coe₁‘𝐺)‘𝑘)) supp (0_g‘𝐸)))
119		suppssfifsupp 9374	. . . . . . . 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))) ∧ (0_g‘𝐸) ∈ V) ∧ (((𝑘 ∈ ℕ₀ ↦ ((coe₁‘𝐺)‘𝑘)) supp (0_g‘𝐸)) ∈ Fin ∧ ((𝑘 ∈ ℕ₀ ↦ (((coe₁‘𝐺)‘𝑘)(._r‘𝐸)(𝑘(._g‘(mulGrp‘𝐸))𝐴))) supp (0_g‘𝐸)) ⊆ ((𝑘 ∈ ℕ₀ ↦ ((coe₁‘𝐺)‘𝑘)) supp (0_g‘𝐸)))) → (𝑘 ∈ ℕ₀ ↦ (((coe₁‘𝐺)‘𝑘)(._r‘𝐸)(𝑘(._g‘(mulGrp‘𝐸))𝐴))) finSupp (0_g‘𝐸))
120	74, 75, 76, 95, 118, 119	syl32anc 1375	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) → (𝑘 ∈ ℕ₀ ↦ (((coe₁‘𝐺)‘𝑘)(._r‘𝐸)(𝑘(._g‘(mulGrp‘𝐸))𝐴))) finSupp (0_g‘𝐸))
121	25, 28, 30, 34, 73, 120	gsumsubgcl 19830	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) → (𝐸 Σ_g (𝑘 ∈ ℕ₀ ↦ (((coe₁‘𝐺)‘𝑘)(._r‘𝐸)(𝑘(._g‘(mulGrp‘𝐸))𝐴)))) ∈ 𝑎)
122	24, 121	eqeltrd 2825	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) ∧ (𝐹 ∪ {𝐴}) ⊆ 𝑎) → ((𝑂‘𝐺)‘𝐴) ∈ 𝑎)
123	122	ex 412	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐸)) → ((𝐹 ∪ {𝐴}) ⊆ 𝑎 → ((𝑂‘𝐺)‘𝐴) ∈ 𝑎))
124	123	ralrimiva 3138	. . 3 ⊢ (𝜑 → ∀𝑎 ∈ (SubDRing‘𝐸)((𝐹 ∪ {𝐴}) ⊆ 𝑎 → ((𝑂‘𝐺)‘𝐴) ∈ 𝑎))
125		fvex 6894	. . . 4 ⊢ ((𝑂‘𝐺)‘𝐴) ∈ V
126	125	elintrab 4954	. . 3 ⊢ (((𝑂‘𝐺)‘𝐴) ∈ ∩ {𝑎 ∈ (SubDRing‘𝐸) ∣ (𝐹 ∪ {𝐴}) ⊆ 𝑎} ↔ ∀𝑎 ∈ (SubDRing‘𝐸)((𝐹 ∪ {𝐴}) ⊆ 𝑎 → ((𝑂‘𝐺)‘𝐴) ∈ 𝑎))
127	124, 126	sylibr 233	. 2 ⊢ (𝜑 → ((𝑂‘𝐺)‘𝐴) ∈ ∩ {𝑎 ∈ (SubDRing‘𝐸) ∣ (𝐹 ∪ {𝐴}) ⊆ 𝑎})
128	6	flddrngd 20589	. . 3 ⊢ (𝜑 → 𝐸 ∈ DivRing)
129	21	snssd 4804	. . . 4 ⊢ (𝜑 → {𝐴} ⊆ 𝐵)
130	44, 129	unssd 4178	. . 3 ⊢ (𝜑 → (𝐹 ∪ {𝐴}) ⊆ 𝐵)
131	2, 128, 130	fldgenval 32868	. 2 ⊢ (𝜑 → (𝐸 fldGen (𝐹 ∪ {𝐴})) = ∩ {𝑎 ∈ (SubDRing‘𝐸) ∣ (𝐹 ∪ {𝐴}) ⊆ 𝑎})
132	127, 131	eleqtrrd 2828	1 ⊢ (𝜑 → ((𝑂‘𝐺)‘𝐴) ∈ (𝐸 fldGen (𝐹 ∪ {𝐴})))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∀wral 3053 {crab 3424 Vcvv 3466 ∪ cun 3938 ⊆ wss 3940 {csn 4620 ∩ cint 4940 class class class wbr 5138 ↦ cmpt 5221 Fun wfun 6527 ‘cfv 6533 (class class class)co 7401 supp csupp 8140 Fincfn 8935 finSupp cfsupp 9357 ℕ₀cn0 12469 Basecbs 17143 ↾_s cress 17172 ._rcmulr 17197 0_gc0g 17384 Σ_g cgsu 17385 Mndcmnd 18657 SubMndcsubmnd 18702 ._gcmg 18985 SubGrpcsubg 19037 Abelcabl 19691 mulGrpcmgp 20029 1_rcur 20076 Ringcrg 20128 SubRingcsubrg 20459 Fieldcfield 20578 SubDRingcsdrg 20627 Poly₁cpl1 22019 coe₁cco1 22020 evalSub₁ ces1 22154 fldGen cfldgen 32866

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-tp 4625 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-iin 4990 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-se 5622 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-of 7663 df-ofr 7664 df-om 7849 df-1st 7968 df-2nd 7969 df-supp 8141 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8699 df-map 8818 df-pm 8819 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-sup 9433 df-oi 9501 df-card 9930 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-7 12277 df-8 12278 df-9 12279 df-n0 12470 df-z 12556 df-dec 12675 df-uz 12820 df-fz 13482 df-fzo 13625 df-seq 13964 df-hash 14288 df-struct 17079 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-ress 17173 df-plusg 17209 df-mulr 17210 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-hom 17220 df-cco 17221 df-0g 17386 df-gsum 17387 df-prds 17392 df-pws 17394 df-mre 17529 df-mrc 17530 df-acs 17532 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-mhm 18703 df-submnd 18704 df-grp 18856 df-minusg 18857 df-sbg 18858 df-mulg 18986 df-subg 19040 df-ghm 19129 df-cntz 19223 df-cmn 19692 df-abl 19693 df-mgp 20030 df-rng 20048 df-ur 20077 df-srg 20082 df-ring 20130 df-cring 20131 df-rhm 20364 df-subrng 20436 df-subrg 20461 df-drng 20579 df-field 20580 df-sdrg 20628 df-lmod 20698 df-lss 20769 df-lsp 20809 df-assa 21716 df-asp 21717 df-ascl 21718 df-psr 21771 df-mvr 21772 df-mpl 21773 df-opsr 21775 df-evls 21945 df-evl 21946 df-psr1 22022 df-vr1 22023 df-ply1 22024 df-coe1 22025 df-evls1 22156 df-evl1 22157 df-fldgen 32867

This theorem is referenced by: algextdeglem2 33254 algextdeglem4 33256

W3C validator