extdgfialglem2 - Mathbox for Thierry Arnoux

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Theorem extdgfialglem2 34106

Description: Lemma for extdgfialg 34107. (Contributed by Thierry Arnoux, 10-Jan-2026.)

**Hypotheses**
Ref	Expression
extdgfialg.b	⊢ 𝐵 = (Base‘𝐸)
extdgfialg.d	⊢ 𝐷 = (dim‘((subringAlg ‘𝐸)‘𝐹))
extdgfialg.e	⊢ (𝜑 → 𝐸 ∈ Field)
extdgfialg.f	⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))
extdgfialg.1	⊢ (𝜑 → 𝐷 ∈ ℕ₀)
extdgfialglem1.2	⊢ 𝑍 = (0_g‘𝐸)
extdgfialglem1.3	⊢ · = (._r‘𝐸)
extdgfialglem1.r	⊢ 𝐺 = (𝑛 ∈ (0...𝐷) ↦ (𝑛(._g‘(mulGrp‘((subringAlg ‘𝐸)‘𝐹)))𝑋))
extdgfialglem1.4	⊢ (𝜑 → 𝑋 ∈ 𝐵)
extdgfialglem2.1	⊢ (𝜑 → 𝐴:(0...𝐷)⟶𝐹)
extdgfialglem2.2	⊢ (𝜑 → 𝐴 finSupp 𝑍)
extdgfialglem2.3	⊢ (𝜑 → (𝐸 Σ_g (𝐴 ∘_f · 𝐺)) = 𝑍)
extdgfialglem2.4	⊢ (𝜑 → 𝐴 ≠ ((0...𝐷) × {𝑍}))

**Assertion**
Ref	Expression
extdgfialglem2	⊢ (𝜑 → 𝑋 ∈ (𝐸 IntgRing 𝐹))

Distinct variable groups: · ,𝑛 𝐴,𝑛 𝐵,𝑛 𝐷,𝑛 𝑛,𝐸 𝑛,𝐹 𝑛,𝐺 𝑛,𝑋 𝑛,𝑍 𝜑,𝑛

Proof of Theorem extdgfialglem2 Dummy variables 𝑚 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2765	. 2 ⊢ (𝐸 evalSub₁ 𝐹) = (𝐸 evalSub₁ 𝐹)
2		eqid 2765	. 2 ⊢ (0_g‘(Poly₁‘𝐸)) = (0_g‘(Poly₁‘𝐸))
3		extdgfialglem1.2	. 2 ⊢ 𝑍 = (0_g‘𝐸)
4		extdgfialg.e	. 2 ⊢ (𝜑 → 𝐸 ∈ Field)
5		extdgfialg.f	. 2 ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))
6		extdgfialg.b	. 2 ⊢ 𝐵 = (Base‘𝐸)
7		eqid 2765	. . . 4 ⊢ (Base‘(Poly₁‘(𝐸 ↾_s 𝐹))) = (Base‘(Poly₁‘(𝐸 ↾_s 𝐹)))
8		eqid 2765	. . . 4 ⊢ (0_g‘(Poly₁‘(𝐸 ↾_s 𝐹))) = (0_g‘(Poly₁‘(𝐸 ↾_s 𝐹)))
9		sdrgsubrg 20923	. . . . . . . 8 ⊢ (𝐹 ∈ (SubDRing‘𝐸) → 𝐹 ∈ (SubRing‘𝐸))
10	5, 9	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ (SubRing‘𝐸))
11		eqid 2765	. . . . . . . 8 ⊢ (𝐸 ↾_s 𝐹) = (𝐸 ↾_s 𝐹)
12	11	subrgring 20702	. . . . . . 7 ⊢ (𝐹 ∈ (SubRing‘𝐸) → (𝐸 ↾_s 𝐹) ∈ Ring)
13	10, 12	syl 18	. . . . . 6 ⊢ (𝜑 → (𝐸 ↾_s 𝐹) ∈ Ring)
14		eqid 2765	. . . . . . 7 ⊢ (Poly₁‘(𝐸 ↾_s 𝐹)) = (Poly₁‘(𝐸 ↾_s 𝐹))
15	14	ply1ring 22436	. . . . . 6 ⊢ ((𝐸 ↾_s 𝐹) ∈ Ring → (Poly₁‘(𝐸 ↾_s 𝐹)) ∈ Ring)
16	13, 15	syl 18	. . . . 5 ⊢ (𝜑 → (Poly₁‘(𝐸 ↾_s 𝐹)) ∈ Ring)
17	16	ringcmnd 20391	. . . 4 ⊢ (𝜑 → (Poly₁‘(𝐸 ↾_s 𝐹)) ∈ CMnd)
18		fzfid 14022	. . . 4 ⊢ (𝜑 → (0...𝐷) ∈ Fin)
19		eqid 2765	. . . . . 6 ⊢ (Scalar‘(Poly₁‘(𝐸 ↾_s 𝐹))) = (Scalar‘(Poly₁‘(𝐸 ↾_s 𝐹)))
20		eqid 2765	. . . . . 6 ⊢ ( ·_𝑠 ‘(Poly₁‘(𝐸 ↾_s 𝐹))) = ( ·_𝑠 ‘(Poly₁‘(𝐸 ↾_s 𝐹)))
21		eqid 2765	. . . . . 6 ⊢ (Base‘(Scalar‘(Poly₁‘(𝐸 ↾_s 𝐹)))) = (Base‘(Scalar‘(Poly₁‘(𝐸 ↾_s 𝐹))))
22	14	ply1lmod 22440	. . . . . . . 8 ⊢ ((𝐸 ↾_s 𝐹) ∈ Ring → (Poly₁‘(𝐸 ↾_s 𝐹)) ∈ LMod)
23	13, 22	syl 18	. . . . . . 7 ⊢ (𝜑 → (Poly₁‘(𝐸 ↾_s 𝐹)) ∈ LMod)
24	23	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...𝐷)) → (Poly₁‘(𝐸 ↾_s 𝐹)) ∈ LMod)
25		extdgfialglem2.1	. . . . . . . 8 ⊢ (𝜑 → 𝐴:(0...𝐷)⟶𝐹)
26	25	ffvelcdmda 7083	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...𝐷)) → (𝐴‘𝑛) ∈ 𝐹)
27	6	sdrgss 20925	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (SubDRing‘𝐸) → 𝐹 ⊆ 𝐵)
28	5, 27	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ⊆ 𝐵)
29	11, 6	ressbas2 17315	. . . . . . . . . 10 ⊢ (𝐹 ⊆ 𝐵 → 𝐹 = (Base‘(𝐸 ↾_s 𝐹)))
30	28, 29	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 = (Base‘(𝐸 ↾_s 𝐹)))
31		ovex 7449	. . . . . . . . . . 11 ⊢ (𝐸 ↾_s 𝐹) ∈ V
32	14	ply1sca 22441	. . . . . . . . . . 11 ⊢ ((𝐸 ↾_s 𝐹) ∈ V → (𝐸 ↾_s 𝐹) = (Scalar‘(Poly₁‘(𝐸 ↾_s 𝐹))))
33	31, 32	ax-mp 5	. . . . . . . . . 10 ⊢ (𝐸 ↾_s 𝐹) = (Scalar‘(Poly₁‘(𝐸 ↾_s 𝐹)))
34	33	fveq2i 6888	. . . . . . . . 9 ⊢ (Base‘(𝐸 ↾_s 𝐹)) = (Base‘(Scalar‘(Poly₁‘(𝐸 ↾_s 𝐹))))
35	30, 34	eqtr2di 2817	. . . . . . . 8 ⊢ (𝜑 → (Base‘(Scalar‘(Poly₁‘(𝐸 ↾_s 𝐹)))) = 𝐹)
36	35	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...𝐷)) → (Base‘(Scalar‘(Poly₁‘(𝐸 ↾_s 𝐹)))) = 𝐹)
37	26, 36	eleqtrrd 2868	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...𝐷)) → (𝐴‘𝑛) ∈ (Base‘(Scalar‘(Poly₁‘(𝐸 ↾_s 𝐹)))))
38		eqid 2765	. . . . . . . 8 ⊢ (mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))) = (mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹)))
39	38, 7	mgpbas 20244	. . . . . . 7 ⊢ (Base‘(Poly₁‘(𝐸 ↾_s 𝐹))) = (Base‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))
40		eqid 2765	. . . . . . 7 ⊢ (._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹)))) = (._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))
41	38	ringmgp 20344	. . . . . . . . 9 ⊢ ((Poly₁‘(𝐸 ↾_s 𝐹)) ∈ Ring → (mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))) ∈ Mnd)
42	16, 41	syl 18	. . . . . . . 8 ⊢ (𝜑 → (mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))) ∈ Mnd)
43	42	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...𝐷)) → (mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))) ∈ Mnd)
44		fz0ssnn0 13662	. . . . . . . . 9 ⊢ (0...𝐷) ⊆ ℕ₀
45	44	a1i 11	. . . . . . . 8 ⊢ (𝜑 → (0...𝐷) ⊆ ℕ₀)
46	45	sselda 3938	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...𝐷)) → 𝑛 ∈ ℕ₀)
47		eqid 2765	. . . . . . . . . 10 ⊢ (var₁‘(𝐸 ↾_s 𝐹)) = (var₁‘(𝐸 ↾_s 𝐹))
48	47, 14, 7	vr1cl 22406	. . . . . . . . 9 ⊢ ((𝐸 ↾_s 𝐹) ∈ Ring → (var₁‘(𝐸 ↾_s 𝐹)) ∈ (Base‘(Poly₁‘(𝐸 ↾_s 𝐹))))
49	13, 48	syl 18	. . . . . . . 8 ⊢ (𝜑 → (var₁‘(𝐸 ↾_s 𝐹)) ∈ (Base‘(Poly₁‘(𝐸 ↾_s 𝐹))))
50	49	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...𝐷)) → (var₁‘(𝐸 ↾_s 𝐹)) ∈ (Base‘(Poly₁‘(𝐸 ↾_s 𝐹))))
51	39, 40, 43, 46, 50	mulgnn0cld 19184	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...𝐷)) → (𝑛(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹))) ∈ (Base‘(Poly₁‘(𝐸 ↾_s 𝐹))))
52	7, 19, 20, 21, 24, 37, 51	lmodvscld 21029	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...𝐷)) → ((𝐴‘𝑛)( ·_𝑠 ‘(Poly₁‘(𝐸 ↾_s 𝐹)))(𝑛(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹)))) ∈ (Base‘(Poly₁‘(𝐸 ↾_s 𝐹))))
53	52	fmpttd 7114	. . . 4 ⊢ (𝜑 → (𝑛 ∈ (0...𝐷) ↦ ((𝐴‘𝑛)( ·_𝑠 ‘(Poly₁‘(𝐸 ↾_s 𝐹)))(𝑛(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹))))):(0...𝐷)⟶(Base‘(Poly₁‘(𝐸 ↾_s 𝐹))))
54		eqid 2765	. . . . 5 ⊢ (𝑛 ∈ (0...𝐷) ↦ ((𝐴‘𝑛)( ·_𝑠 ‘(Poly₁‘(𝐸 ↾_s 𝐹)))(𝑛(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹))))) = (𝑛 ∈ (0...𝐷) ↦ ((𝐴‘𝑛)( ·_𝑠 ‘(Poly₁‘(𝐸 ↾_s 𝐹)))(𝑛(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹)))))
55		fvexd 6900	. . . . 5 ⊢ (𝜑 → (0_g‘(Poly₁‘(𝐸 ↾_s 𝐹))) ∈ V)
56	54, 18, 52, 55	fsuppmptdm 9339	. . . 4 ⊢ (𝜑 → (𝑛 ∈ (0...𝐷) ↦ ((𝐴‘𝑛)( ·_𝑠 ‘(Poly₁‘(𝐸 ↾_s 𝐹)))(𝑛(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹))))) finSupp (0_g‘(Poly₁‘(𝐸 ↾_s 𝐹))))
57	7, 8, 17, 18, 53, 56	gsumcl 20008	. . 3 ⊢ (𝜑 → ((Poly₁‘(𝐸 ↾_s 𝐹)) Σ_g (𝑛 ∈ (0...𝐷) ↦ ((𝐴‘𝑛)( ·_𝑠 ‘(Poly₁‘(𝐸 ↾_s 𝐹)))(𝑛(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹)))))) ∈ (Base‘(Poly₁‘(𝐸 ↾_s 𝐹))))
58	4	fldcrngd 20871	. . . 4 ⊢ (𝜑 → 𝐸 ∈ CRing)
59	1, 14, 7, 58, 10	evls1dm 33874	. . 3 ⊢ (𝜑 → dom (𝐸 evalSub₁ 𝐹) = (Base‘(Poly₁‘(𝐸 ↾_s 𝐹))))
60	57, 59	eleqtrrd 2868	. 2 ⊢ (𝜑 → ((Poly₁‘(𝐸 ↾_s 𝐹)) Σ_g (𝑛 ∈ (0...𝐷) ↦ ((𝐴‘𝑛)( ·_𝑠 ‘(Poly₁‘(𝐸 ↾_s 𝐹)))(𝑛(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹)))))) ∈ dom (𝐸 evalSub₁ 𝐹))
61		extdgfialglem2.4	. . 3 ⊢ (𝜑 → 𝐴 ≠ ((0...𝐷) × {𝑍}))
62		eqid 2765	. . . . . 6 ⊢ (Base‘(𝐸 ↾_s 𝐹)) = (Base‘(𝐸 ↾_s 𝐹))
63		eqid 2765	. . . . . 6 ⊢ (0_g‘(𝐸 ↾_s 𝐹)) = (0_g‘(𝐸 ↾_s 𝐹))
64	25	ffvelcdmda 7083	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...𝐷)) → (𝐴‘𝑚) ∈ 𝐹)
65	64	adantlr 728	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ₀) ∧ 𝑚 ∈ (0...𝐷)) → (𝐴‘𝑚) ∈ 𝐹)
66	30	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ₀) ∧ 𝑚 ∈ (0...𝐷)) → 𝐹 = (Base‘(𝐸 ↾_s 𝐹)))
67	65, 66	eleqtrd 2867	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ₀) ∧ 𝑚 ∈ (0...𝐷)) → (𝐴‘𝑚) ∈ (Base‘(𝐸 ↾_s 𝐹)))
68		subrgsubg 20705	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (SubRing‘𝐸) → 𝐹 ∈ (SubGrp‘𝐸))
69	10, 68	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ (SubGrp‘𝐸))
70	3	subg0cl 19223	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (SubGrp‘𝐸) → 𝑍 ∈ 𝐹)
71	69, 70	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝑍 ∈ 𝐹)
72	71, 30	eleqtrd 2867	. . . . . . . . 9 ⊢ (𝜑 → 𝑍 ∈ (Base‘(𝐸 ↾_s 𝐹)))
73	72	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ₀) ∧ ¬ 𝑚 ∈ (0...𝐷)) → 𝑍 ∈ (Base‘(𝐸 ↾_s 𝐹)))
74	67, 73	ifclda 4525	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → if(𝑚 ∈ (0...𝐷), (𝐴‘𝑚), 𝑍) ∈ (Base‘(𝐸 ↾_s 𝐹)))
75	74	ralrimiva 3159	. . . . . 6 ⊢ (𝜑 → ∀𝑚 ∈ ℕ₀ if(𝑚 ∈ (0...𝐷), (𝐴‘𝑚), 𝑍) ∈ (Base‘(𝐸 ↾_s 𝐹)))
76		eqid 2765	. . . . . . . 8 ⊢ (𝑚 ∈ ℕ₀ ↦ if(𝑚 ∈ (0...𝐷), (𝐴‘𝑚), 𝑍)) = (𝑚 ∈ ℕ₀ ↦ if(𝑚 ∈ (0...𝐷), (𝐴‘𝑚), 𝑍))
77		nn0ex 12521	. . . . . . . . 9 ⊢ ℕ₀ ∈ V
78	77	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ℕ₀ ∈ V)
79	76, 78, 18, 64, 71	mptiffisupp 33067	. . . . . . 7 ⊢ (𝜑 → (𝑚 ∈ ℕ₀ ↦ if(𝑚 ∈ (0...𝐷), (𝐴‘𝑚), 𝑍)) finSupp 𝑍)
80	58	crngringd 20351	. . . . . . . . . 10 ⊢ (𝜑 → 𝐸 ∈ Ring)
81	80	ringcmnd 20391	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ CMnd)
82	81	cmnmndd 19897	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ Mnd)
83	11, 6, 3	ress0g 18841	. . . . . . . 8 ⊢ ((𝐸 ∈ Mnd ∧ 𝑍 ∈ 𝐹 ∧ 𝐹 ⊆ 𝐵) → 𝑍 = (0_g‘(𝐸 ↾_s 𝐹)))
84	82, 71, 28, 83	syl3anc 1398	. . . . . . 7 ⊢ (𝜑 → 𝑍 = (0_g‘(𝐸 ↾_s 𝐹)))
85	79, 84	breqtrd 5139	. . . . . 6 ⊢ (𝜑 → (𝑚 ∈ ℕ₀ ↦ if(𝑚 ∈ (0...𝐷), (𝐴‘𝑚), 𝑍)) finSupp (0_g‘(𝐸 ↾_s 𝐹)))
86	72	ralrimivw 3163	. . . . . 6 ⊢ (𝜑 → ∀𝑚 ∈ ℕ₀ 𝑍 ∈ (Base‘(𝐸 ↾_s 𝐹)))
87		fconstmpt 5725	. . . . . . . 8 ⊢ (ℕ₀ × {𝑍}) = (𝑚 ∈ ℕ₀ ↦ 𝑍)
88	78, 71	fczfsuppd 9349	. . . . . . . 8 ⊢ (𝜑 → (ℕ₀ × {𝑍}) finSupp 𝑍)
89	87, 88	eqbrtrrid 5149	. . . . . . 7 ⊢ (𝜑 → (𝑚 ∈ ℕ₀ ↦ 𝑍) finSupp 𝑍)
90	89, 84	breqtrd 5139	. . . . . 6 ⊢ (𝜑 → (𝑚 ∈ ℕ₀ ↦ 𝑍) finSupp (0_g‘(𝐸 ↾_s 𝐹)))
91		simpr 490	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℕ₀ ∖ (0...𝐷))) → 𝑚 ∈ (ℕ₀ ∖ (0...𝐷)))
92	91	eldifbd 3919	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℕ₀ ∖ (0...𝐷))) → ¬ 𝑚 ∈ (0...𝐷))
93	92	iffalsed 4500	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℕ₀ ∖ (0...𝐷))) → if(𝑚 ∈ (0...𝐷), (𝐴‘𝑚), 𝑍) = 𝑍)
94	84	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℕ₀ ∖ (0...𝐷))) → 𝑍 = (0_g‘(𝐸 ↾_s 𝐹)))
95	93, 94	eqtrd 2800	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℕ₀ ∖ (0...𝐷))) → if(𝑚 ∈ (0...𝐷), (𝐴‘𝑚), 𝑍) = (0_g‘(𝐸 ↾_s 𝐹)))
96	95	oveq1d 7431	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℕ₀ ∖ (0...𝐷))) → (if(𝑚 ∈ (0...𝐷), (𝐴‘𝑚), 𝑍)( ·_𝑠 ‘(Poly₁‘(𝐸 ↾_s 𝐹)))(𝑚(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹)))) = ((0_g‘(𝐸 ↾_s 𝐹))( ·_𝑠 ‘(Poly₁‘(𝐸 ↾_s 𝐹)))(𝑚(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹)))))
97	23	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℕ₀ ∖ (0...𝐷))) → (Poly₁‘(𝐸 ↾_s 𝐹)) ∈ LMod)
98	42	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℕ₀ ∖ (0...𝐷))) → (mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))) ∈ Mnd)
99	91	eldifad 3918	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℕ₀ ∖ (0...𝐷))) → 𝑚 ∈ ℕ₀)
100	49	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℕ₀ ∖ (0...𝐷))) → (var₁‘(𝐸 ↾_s 𝐹)) ∈ (Base‘(Poly₁‘(𝐸 ↾_s 𝐹))))
101	39, 40, 98, 99, 100	mulgnn0cld 19184	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℕ₀ ∖ (0...𝐷))) → (𝑚(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹))) ∈ (Base‘(Poly₁‘(𝐸 ↾_s 𝐹))))
102	7, 33, 20, 63, 8	lmod0vs 21045	. . . . . . . . . 10 ⊢ (((Poly₁‘(𝐸 ↾_s 𝐹)) ∈ LMod ∧ (𝑚(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹))) ∈ (Base‘(Poly₁‘(𝐸 ↾_s 𝐹)))) → ((0_g‘(𝐸 ↾_s 𝐹))( ·_𝑠 ‘(Poly₁‘(𝐸 ↾_s 𝐹)))(𝑚(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹)))) = (0_g‘(Poly₁‘(𝐸 ↾_s 𝐹))))
103	97, 101, 102	syl2anc 596	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℕ₀ ∖ (0...𝐷))) → ((0_g‘(𝐸 ↾_s 𝐹))( ·_𝑠 ‘(Poly₁‘(𝐸 ↾_s 𝐹)))(𝑚(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹)))) = (0_g‘(Poly₁‘(𝐸 ↾_s 𝐹))))
104	96, 103	eqtrd 2800	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℕ₀ ∖ (0...𝐷))) → (if(𝑚 ∈ (0...𝐷), (𝐴‘𝑚), 𝑍)( ·_𝑠 ‘(Poly₁‘(𝐸 ↾_s 𝐹)))(𝑚(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹)))) = (0_g‘(Poly₁‘(𝐸 ↾_s 𝐹))))
105	23	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → (Poly₁‘(𝐸 ↾_s 𝐹)) ∈ LMod)
106	42	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → (mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))) ∈ Mnd)
107		simpr 490	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → 𝑚 ∈ ℕ₀)
108	49	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → (var₁‘(𝐸 ↾_s 𝐹)) ∈ (Base‘(Poly₁‘(𝐸 ↾_s 𝐹))))
109	39, 40, 106, 107, 108	mulgnn0cld 19184	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → (𝑚(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹))) ∈ (Base‘(Poly₁‘(𝐸 ↾_s 𝐹))))
110	7, 33, 20, 62, 105, 74, 109	lmodvscld 21029	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → (if(𝑚 ∈ (0...𝐷), (𝐴‘𝑚), 𝑍)( ·_𝑠 ‘(Poly₁‘(𝐸 ↾_s 𝐹)))(𝑚(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹)))) ∈ (Base‘(Poly₁‘(𝐸 ↾_s 𝐹))))
111	7, 8, 17, 78, 104, 18, 110, 45	gsummptres2 33396	. . . . . . 7 ⊢ (𝜑 → ((Poly₁‘(𝐸 ↾_s 𝐹)) Σ_g (𝑚 ∈ ℕ₀ ↦ (if(𝑚 ∈ (0...𝐷), (𝐴‘𝑚), 𝑍)( ·_𝑠 ‘(Poly₁‘(𝐸 ↾_s 𝐹)))(𝑚(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹)))))) = ((Poly₁‘(𝐸 ↾_s 𝐹)) Σ_g (𝑚 ∈ (0...𝐷) ↦ (if(𝑚 ∈ (0...𝐷), (𝐴‘𝑚), 𝑍)( ·_𝑠 ‘(Poly₁‘(𝐸 ↾_s 𝐹)))(𝑚(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹)))))))
112		eleq1w 2848	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑛 → (𝑚 ∈ (0...𝐷) ↔ 𝑛 ∈ (0...𝐷)))
113		fveq2 6885	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑛 → (𝐴‘𝑚) = (𝐴‘𝑛))
114	112, 113	ifbieq1d 4514	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → if(𝑚 ∈ (0...𝐷), (𝐴‘𝑚), 𝑍) = if(𝑛 ∈ (0...𝐷), (𝐴‘𝑛), 𝑍))
115		oveq1 7423	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (𝑚(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹))) = (𝑛(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹))))
116	114, 115	oveq12d 7434	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → (if(𝑚 ∈ (0...𝐷), (𝐴‘𝑚), 𝑍)( ·_𝑠 ‘(Poly₁‘(𝐸 ↾_s 𝐹)))(𝑚(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹)))) = (if(𝑛 ∈ (0...𝐷), (𝐴‘𝑛), 𝑍)( ·_𝑠 ‘(Poly₁‘(𝐸 ↾_s 𝐹)))(𝑛(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹)))))
117	116	cbvmptv 5217	. . . . . . . . 9 ⊢ (𝑚 ∈ (0...𝐷) ↦ (if(𝑚 ∈ (0...𝐷), (𝐴‘𝑚), 𝑍)( ·_𝑠 ‘(Poly₁‘(𝐸 ↾_s 𝐹)))(𝑚(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹))))) = (𝑛 ∈ (0...𝐷) ↦ (if(𝑛 ∈ (0...𝐷), (𝐴‘𝑛), 𝑍)( ·_𝑠 ‘(Poly₁‘(𝐸 ↾_s 𝐹)))(𝑛(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹)))))
118		simpr 490	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...𝐷)) → 𝑛 ∈ (0...𝐷))
119	118	iftrued 4497	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...𝐷)) → if(𝑛 ∈ (0...𝐷), (𝐴‘𝑛), 𝑍) = (𝐴‘𝑛))
120	119	oveq1d 7431	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...𝐷)) → (if(𝑛 ∈ (0...𝐷), (𝐴‘𝑛), 𝑍)( ·_𝑠 ‘(Poly₁‘(𝐸 ↾_s 𝐹)))(𝑛(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹)))) = ((𝐴‘𝑛)( ·_𝑠 ‘(Poly₁‘(𝐸 ↾_s 𝐹)))(𝑛(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹)))))
121	120	mpteq2dva 5206	. . . . . . . . 9 ⊢ (𝜑 → (𝑛 ∈ (0...𝐷) ↦ (if(𝑛 ∈ (0...𝐷), (𝐴‘𝑛), 𝑍)( ·_𝑠 ‘(Poly₁‘(𝐸 ↾_s 𝐹)))(𝑛(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹))))) = (𝑛 ∈ (0...𝐷) ↦ ((𝐴‘𝑛)( ·_𝑠 ‘(Poly₁‘(𝐸 ↾_s 𝐹)))(𝑛(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹))))))
122	117, 121	eqtrid 2812	. . . . . . . 8 ⊢ (𝜑 → (𝑚 ∈ (0...𝐷) ↦ (if(𝑚 ∈ (0...𝐷), (𝐴‘𝑚), 𝑍)( ·_𝑠 ‘(Poly₁‘(𝐸 ↾_s 𝐹)))(𝑚(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹))))) = (𝑛 ∈ (0...𝐷) ↦ ((𝐴‘𝑛)( ·_𝑠 ‘(Poly₁‘(𝐸 ↾_s 𝐹)))(𝑛(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹))))))
123	122	oveq2d 7432	. . . . . . 7 ⊢ (𝜑 → ((Poly₁‘(𝐸 ↾_s 𝐹)) Σ_g (𝑚 ∈ (0...𝐷) ↦ (if(𝑚 ∈ (0...𝐷), (𝐴‘𝑚), 𝑍)( ·_𝑠 ‘(Poly₁‘(𝐸 ↾_s 𝐹)))(𝑚(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹)))))) = ((Poly₁‘(𝐸 ↾_s 𝐹)) Σ_g (𝑛 ∈ (0...𝐷) ↦ ((𝐴‘𝑛)( ·_𝑠 ‘(Poly₁‘(𝐸 ↾_s 𝐹)))(𝑛(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹)))))))
124	111, 123	eqtr2d 2801	. . . . . 6 ⊢ (𝜑 → ((Poly₁‘(𝐸 ↾_s 𝐹)) Σ_g (𝑛 ∈ (0...𝐷) ↦ ((𝐴‘𝑛)( ·_𝑠 ‘(Poly₁‘(𝐸 ↾_s 𝐹)))(𝑛(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹)))))) = ((Poly₁‘(𝐸 ↾_s 𝐹)) Σ_g (𝑚 ∈ ℕ₀ ↦ (if(𝑚 ∈ (0...𝐷), (𝐴‘𝑚), 𝑍)( ·_𝑠 ‘(Poly₁‘(𝐸 ↾_s 𝐹)))(𝑚(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹)))))))
125	17	cmnmndd 19897	. . . . . . . 8 ⊢ (𝜑 → (Poly₁‘(𝐸 ↾_s 𝐹)) ∈ Mnd)
126	8	gsumz 18918	. . . . . . . 8 ⊢ (((Poly₁‘(𝐸 ↾_s 𝐹)) ∈ Mnd ∧ ℕ₀ ∈ V) → ((Poly₁‘(𝐸 ↾_s 𝐹)) Σ_g (𝑚 ∈ ℕ₀ ↦ (0_g‘(Poly₁‘(𝐸 ↾_s 𝐹))))) = (0_g‘(Poly₁‘(𝐸 ↾_s 𝐹))))
127	125, 78, 126	syl2anc 596	. . . . . . 7 ⊢ (𝜑 → ((Poly₁‘(𝐸 ↾_s 𝐹)) Σ_g (𝑚 ∈ ℕ₀ ↦ (0_g‘(Poly₁‘(𝐸 ↾_s 𝐹))))) = (0_g‘(Poly₁‘(𝐸 ↾_s 𝐹))))
128	84	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → 𝑍 = (0_g‘(𝐸 ↾_s 𝐹)))
129	128	oveq1d 7431	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → (𝑍( ·_𝑠 ‘(Poly₁‘(𝐸 ↾_s 𝐹)))(𝑚(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹)))) = ((0_g‘(𝐸 ↾_s 𝐹))( ·_𝑠 ‘(Poly₁‘(𝐸 ↾_s 𝐹)))(𝑚(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹)))))
130	105, 109, 102	syl2anc 596	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → ((0_g‘(𝐸 ↾_s 𝐹))( ·_𝑠 ‘(Poly₁‘(𝐸 ↾_s 𝐹)))(𝑚(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹)))) = (0_g‘(Poly₁‘(𝐸 ↾_s 𝐹))))
131	129, 130	eqtrd 2800	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → (𝑍( ·_𝑠 ‘(Poly₁‘(𝐸 ↾_s 𝐹)))(𝑚(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹)))) = (0_g‘(Poly₁‘(𝐸 ↾_s 𝐹))))
132	131	mpteq2dva 5206	. . . . . . . 8 ⊢ (𝜑 → (𝑚 ∈ ℕ₀ ↦ (𝑍( ·_𝑠 ‘(Poly₁‘(𝐸 ↾_s 𝐹)))(𝑚(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹))))) = (𝑚 ∈ ℕ₀ ↦ (0_g‘(Poly₁‘(𝐸 ↾_s 𝐹)))))
133	132	oveq2d 7432	. . . . . . 7 ⊢ (𝜑 → ((Poly₁‘(𝐸 ↾_s 𝐹)) Σ_g (𝑚 ∈ ℕ₀ ↦ (𝑍( ·_𝑠 ‘(Poly₁‘(𝐸 ↾_s 𝐹)))(𝑚(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹)))))) = ((Poly₁‘(𝐸 ↾_s 𝐹)) Σ_g (𝑚 ∈ ℕ₀ ↦ (0_g‘(Poly₁‘(𝐸 ↾_s 𝐹))))))
134		eqid 2765	. . . . . . . 8 ⊢ (Poly₁‘𝐸) = (Poly₁‘𝐸)
135	134, 11, 14, 7, 10, 2	ressply10g 33880	. . . . . . 7 ⊢ (𝜑 → (0_g‘(Poly₁‘𝐸)) = (0_g‘(Poly₁‘(𝐸 ↾_s 𝐹))))
136	127, 133, 135	3eqtr4rd 2811	. . . . . 6 ⊢ (𝜑 → (0_g‘(Poly₁‘𝐸)) = ((Poly₁‘(𝐸 ↾_s 𝐹)) Σ_g (𝑚 ∈ ℕ₀ ↦ (𝑍( ·_𝑠 ‘(Poly₁‘(𝐸 ↾_s 𝐹)))(𝑚(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹)))))))
137	14, 47, 40, 13, 62, 20, 63, 75, 85, 86, 90, 124, 136	gsumply1eq 22498	. . . . 5 ⊢ (𝜑 → (((Poly₁‘(𝐸 ↾_s 𝐹)) Σ_g (𝑛 ∈ (0...𝐷) ↦ ((𝐴‘𝑛)( ·_𝑠 ‘(Poly₁‘(𝐸 ↾_s 𝐹)))(𝑛(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹)))))) = (0_g‘(Poly₁‘𝐸)) ↔ ∀𝑚 ∈ ℕ₀ if(𝑚 ∈ (0...𝐷), (𝐴‘𝑚), 𝑍) = 𝑍))
138	25	ffnd 6710	. . . . . . . 8 ⊢ (𝜑 → 𝐴 Fn (0...𝐷))
139	138	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ ∀𝑚 ∈ ℕ₀ if(𝑚 ∈ (0...𝐷), (𝐴‘𝑚), 𝑍) = 𝑍) → 𝐴 Fn (0...𝐷))
140	119	adantlr 728	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ₀ if(𝑚 ∈ (0...𝐷), (𝐴‘𝑚), 𝑍) = 𝑍) ∧ 𝑛 ∈ (0...𝐷)) → if(𝑛 ∈ (0...𝐷), (𝐴‘𝑛), 𝑍) = (𝐴‘𝑛))
141	114	eqeq1d 2767	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → (if(𝑚 ∈ (0...𝐷), (𝐴‘𝑚), 𝑍) = 𝑍 ↔ if(𝑛 ∈ (0...𝐷), (𝐴‘𝑛), 𝑍) = 𝑍))
142		simplr 781	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ₀ if(𝑚 ∈ (0...𝐷), (𝐴‘𝑚), 𝑍) = 𝑍) ∧ 𝑛 ∈ (0...𝐷)) → ∀𝑚 ∈ ℕ₀ if(𝑚 ∈ (0...𝐷), (𝐴‘𝑚), 𝑍) = 𝑍)
143	44	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ∀𝑚 ∈ ℕ₀ if(𝑚 ∈ (0...𝐷), (𝐴‘𝑚), 𝑍) = 𝑍) → (0...𝐷) ⊆ ℕ₀)
144	143	sselda 3938	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ₀ if(𝑚 ∈ (0...𝐷), (𝐴‘𝑚), 𝑍) = 𝑍) ∧ 𝑛 ∈ (0...𝐷)) → 𝑛 ∈ ℕ₀)
145	141, 142, 144	rspcdva 3584	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ₀ if(𝑚 ∈ (0...𝐷), (𝐴‘𝑚), 𝑍) = 𝑍) ∧ 𝑛 ∈ (0...𝐷)) → if(𝑛 ∈ (0...𝐷), (𝐴‘𝑛), 𝑍) = 𝑍)
146	140, 145	eqtr3d 2802	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ₀ if(𝑚 ∈ (0...𝐷), (𝐴‘𝑚), 𝑍) = 𝑍) ∧ 𝑛 ∈ (0...𝐷)) → (𝐴‘𝑛) = 𝑍)
147	139, 146	fconst7v 32994	. . . . . 6 ⊢ ((𝜑 ∧ ∀𝑚 ∈ ℕ₀ if(𝑚 ∈ (0...𝐷), (𝐴‘𝑚), 𝑍) = 𝑍) → 𝐴 = ((0...𝐷) × {𝑍}))
148	147	ex 418	. . . . 5 ⊢ (𝜑 → (∀𝑚 ∈ ℕ₀ if(𝑚 ∈ (0...𝐷), (𝐴‘𝑚), 𝑍) = 𝑍 → 𝐴 = ((0...𝐷) × {𝑍})))
149	137, 148	sylbid 243	. . . 4 ⊢ (𝜑 → (((Poly₁‘(𝐸 ↾_s 𝐹)) Σ_g (𝑛 ∈ (0...𝐷) ↦ ((𝐴‘𝑛)( ·_𝑠 ‘(Poly₁‘(𝐸 ↾_s 𝐹)))(𝑛(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹)))))) = (0_g‘(Poly₁‘𝐸)) → 𝐴 = ((0...𝐷) × {𝑍})))
150	149	necon3d 2981	. . 3 ⊢ (𝜑 → (𝐴 ≠ ((0...𝐷) × {𝑍}) → ((Poly₁‘(𝐸 ↾_s 𝐹)) Σ_g (𝑛 ∈ (0...𝐷) ↦ ((𝐴‘𝑛)( ·_𝑠 ‘(Poly₁‘(𝐸 ↾_s 𝐹)))(𝑛(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹)))))) ≠ (0_g‘(Poly₁‘𝐸))))
151	61, 150	mpd 16	. 2 ⊢ (𝜑 → ((Poly₁‘(𝐸 ↾_s 𝐹)) Σ_g (𝑛 ∈ (0...𝐷) ↦ ((𝐴‘𝑛)( ·_𝑠 ‘(Poly₁‘(𝐸 ↾_s 𝐹)))(𝑛(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹)))))) ≠ (0_g‘(Poly₁‘𝐸)))
152		eqid 2765	. . . . 5 ⊢ (𝐸 ↑_s 𝐵) = (𝐸 ↑_s 𝐵)
153	1, 6, 14, 8, 11, 152, 7, 58, 10, 52, 45, 56	evls1gsumadd 22513	. . . 4 ⊢ (𝜑 → ((𝐸 evalSub₁ 𝐹)‘((Poly₁‘(𝐸 ↾_s 𝐹)) Σ_g (𝑛 ∈ (0...𝐷) ↦ ((𝐴‘𝑛)( ·_𝑠 ‘(Poly₁‘(𝐸 ↾_s 𝐹)))(𝑛(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹))))))) = ((𝐸 ↑_s 𝐵) Σ_g (𝑛 ∈ (0...𝐷) ↦ ((𝐸 evalSub₁ 𝐹)‘((𝐴‘𝑛)( ·_𝑠 ‘(Poly₁‘(𝐸 ↾_s 𝐹)))(𝑛(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹))))))))
154	153	fveq1d 6887	. . 3 ⊢ (𝜑 → (((𝐸 evalSub₁ 𝐹)‘((Poly₁‘(𝐸 ↾_s 𝐹)) Σ_g (𝑛 ∈ (0...𝐷) ↦ ((𝐴‘𝑛)( ·_𝑠 ‘(Poly₁‘(𝐸 ↾_s 𝐹)))(𝑛(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹)))))))‘𝑋) = (((𝐸 ↑_s 𝐵) Σ_g (𝑛 ∈ (0...𝐷) ↦ ((𝐸 evalSub₁ 𝐹)‘((𝐴‘𝑛)( ·_𝑠 ‘(Poly₁‘(𝐸 ↾_s 𝐹)))(𝑛(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹)))))))‘𝑋))
155	58	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...𝐷)) → 𝐸 ∈ CRing)
156	10	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...𝐷)) → 𝐹 ∈ (SubRing‘𝐸))
157	1, 14, 7, 155, 156, 6, 52	evls1fvf 33875	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...𝐷)) → ((𝐸 evalSub₁ 𝐹)‘((𝐴‘𝑛)( ·_𝑠 ‘(Poly₁‘(𝐸 ↾_s 𝐹)))(𝑛(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹))))):𝐵⟶𝐵)
158	157	feqmptd 6953	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...𝐷)) → ((𝐸 evalSub₁ 𝐹)‘((𝐴‘𝑛)( ·_𝑠 ‘(Poly₁‘(𝐸 ↾_s 𝐹)))(𝑛(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹))))) = (𝑥 ∈ 𝐵 ↦ (((𝐸 evalSub₁ 𝐹)‘((𝐴‘𝑛)( ·_𝑠 ‘(Poly₁‘(𝐸 ↾_s 𝐹)))(𝑛(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹)))))‘𝑥)))
159	158	mpteq2dva 5206	. . . . . . 7 ⊢ (𝜑 → (𝑛 ∈ (0...𝐷) ↦ ((𝐸 evalSub₁ 𝐹)‘((𝐴‘𝑛)( ·_𝑠 ‘(Poly₁‘(𝐸 ↾_s 𝐹)))(𝑛(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹)))))) = (𝑛 ∈ (0...𝐷) ↦ (𝑥 ∈ 𝐵 ↦ (((𝐸 evalSub₁ 𝐹)‘((𝐴‘𝑛)( ·_𝑠 ‘(Poly₁‘(𝐸 ↾_s 𝐹)))(𝑛(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹)))))‘𝑥))))
160	159	oveq2d 7432	. . . . . 6 ⊢ (𝜑 → ((𝐸 ↑_s 𝐵) Σ_g (𝑛 ∈ (0...𝐷) ↦ ((𝐸 evalSub₁ 𝐹)‘((𝐴‘𝑛)( ·_𝑠 ‘(Poly₁‘(𝐸 ↾_s 𝐹)))(𝑛(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹))))))) = ((𝐸 ↑_s 𝐵) Σ_g (𝑛 ∈ (0...𝐷) ↦ (𝑥 ∈ 𝐵 ↦ (((𝐸 evalSub₁ 𝐹)‘((𝐴‘𝑛)( ·_𝑠 ‘(Poly₁‘(𝐸 ↾_s 𝐹)))(𝑛(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹)))))‘𝑥)))))
161	160	fveq1d 6887	. . . . 5 ⊢ (𝜑 → (((𝐸 ↑_s 𝐵) Σ_g (𝑛 ∈ (0...𝐷) ↦ ((𝐸 evalSub₁ 𝐹)‘((𝐴‘𝑛)( ·_𝑠 ‘(Poly₁‘(𝐸 ↾_s 𝐹)))(𝑛(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹)))))))‘𝑋) = (((𝐸 ↑_s 𝐵) Σ_g (𝑛 ∈ (0...𝐷) ↦ (𝑥 ∈ 𝐵 ↦ (((𝐸 evalSub₁ 𝐹)‘((𝐴‘𝑛)( ·_𝑠 ‘(Poly₁‘(𝐸 ↾_s 𝐹)))(𝑛(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹)))))‘𝑥))))‘𝑋))
162		eqid 2765	. . . . . . 7 ⊢ (0_g‘(𝐸 ↑_s 𝐵)) = (0_g‘(𝐸 ↑_s 𝐵))
163	6	fvexi 6899	. . . . . . . 8 ⊢ 𝐵 ∈ V
164	163	a1i 11	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ V)
165	155	adantr 486	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ (0...𝐷)) ∧ 𝑥 ∈ 𝐵) → 𝐸 ∈ CRing)
166	156	adantr 486	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ (0...𝐷)) ∧ 𝑥 ∈ 𝐵) → 𝐹 ∈ (SubRing‘𝐸))
167		simpr 490	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ (0...𝐷)) ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
168	52	adantr 486	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ (0...𝐷)) ∧ 𝑥 ∈ 𝐵) → ((𝐴‘𝑛)( ·_𝑠 ‘(Poly₁‘(𝐸 ↾_s 𝐹)))(𝑛(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹)))) ∈ (Base‘(Poly₁‘(𝐸 ↾_s 𝐹))))
169	1, 14, 6, 7, 165, 166, 167, 168	evls1fvcl 22564	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ (0...𝐷)) ∧ 𝑥 ∈ 𝐵) → (((𝐸 evalSub₁ 𝐹)‘((𝐴‘𝑛)( ·_𝑠 ‘(Poly₁‘(𝐸 ↾_s 𝐹)))(𝑛(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹)))))‘𝑥) ∈ 𝐵)
170	169	an32s 665	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑛 ∈ (0...𝐷)) → (((𝐸 evalSub₁ 𝐹)‘((𝐴‘𝑛)( ·_𝑠 ‘(Poly₁‘(𝐸 ↾_s 𝐹)))(𝑛(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹)))))‘𝑥) ∈ 𝐵)
171	170	anasss 472	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑛 ∈ (0...𝐷))) → (((𝐸 evalSub₁ 𝐹)‘((𝐴‘𝑛)( ·_𝑠 ‘(Poly₁‘(𝐸 ↾_s 𝐹)))(𝑛(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹)))))‘𝑥) ∈ 𝐵)
172		eqid 2765	. . . . . . . 8 ⊢ (𝑛 ∈ (0...𝐷) ↦ (𝑥 ∈ 𝐵 ↦ (((𝐸 evalSub₁ 𝐹)‘((𝐴‘𝑛)( ·_𝑠 ‘(Poly₁‘(𝐸 ↾_s 𝐹)))(𝑛(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹)))))‘𝑥))) = (𝑛 ∈ (0...𝐷) ↦ (𝑥 ∈ 𝐵 ↦ (((𝐸 evalSub₁ 𝐹)‘((𝐴‘𝑛)( ·_𝑠 ‘(Poly₁‘(𝐸 ↾_s 𝐹)))(𝑛(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹)))))‘𝑥)))
173	163	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...𝐷)) → 𝐵 ∈ V)
174	173	mptexd 7226	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...𝐷)) → (𝑥 ∈ 𝐵 ↦ (((𝐸 evalSub₁ 𝐹)‘((𝐴‘𝑛)( ·_𝑠 ‘(Poly₁‘(𝐸 ↾_s 𝐹)))(𝑛(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹)))))‘𝑥)) ∈ V)
175		fvexd 6900	. . . . . . . 8 ⊢ (𝜑 → (0_g‘(𝐸 ↑_s 𝐵)) ∈ V)
176	172, 18, 174, 175	fsuppmptdm 9339	. . . . . . 7 ⊢ (𝜑 → (𝑛 ∈ (0...𝐷) ↦ (𝑥 ∈ 𝐵 ↦ (((𝐸 evalSub₁ 𝐹)‘((𝐴‘𝑛)( ·_𝑠 ‘(Poly₁‘(𝐸 ↾_s 𝐹)))(𝑛(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹)))))‘𝑥))) finSupp (0_g‘(𝐸 ↑_s 𝐵)))
177	152, 6, 162, 164, 18, 81, 171, 176	pwsgsum 20075	. . . . . 6 ⊢ (𝜑 → ((𝐸 ↑_s 𝐵) Σ_g (𝑛 ∈ (0...𝐷) ↦ (𝑥 ∈ 𝐵 ↦ (((𝐸 evalSub₁ 𝐹)‘((𝐴‘𝑛)( ·_𝑠 ‘(Poly₁‘(𝐸 ↾_s 𝐹)))(𝑛(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹)))))‘𝑥)))) = (𝑥 ∈ 𝐵 ↦ (𝐸 Σ_g (𝑛 ∈ (0...𝐷) ↦ (((𝐸 evalSub₁ 𝐹)‘((𝐴‘𝑛)( ·_𝑠 ‘(Poly₁‘(𝐸 ↾_s 𝐹)))(𝑛(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹)))))‘𝑥)))))
178	177	fveq1d 6887	. . . . 5 ⊢ (𝜑 → (((𝐸 ↑_s 𝐵) Σ_g (𝑛 ∈ (0...𝐷) ↦ (𝑥 ∈ 𝐵 ↦ (((𝐸 evalSub₁ 𝐹)‘((𝐴‘𝑛)( ·_𝑠 ‘(Poly₁‘(𝐸 ↾_s 𝐹)))(𝑛(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹)))))‘𝑥))))‘𝑋) = ((𝑥 ∈ 𝐵 ↦ (𝐸 Σ_g (𝑛 ∈ (0...𝐷) ↦ (((𝐸 evalSub₁ 𝐹)‘((𝐴‘𝑛)( ·_𝑠 ‘(Poly₁‘(𝐸 ↾_s 𝐹)))(𝑛(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹)))))‘𝑥))))‘𝑋))
179	161, 178	eqtrd 2800	. . . 4 ⊢ (𝜑 → (((𝐸 ↑_s 𝐵) Σ_g (𝑛 ∈ (0...𝐷) ↦ ((𝐸 evalSub₁ 𝐹)‘((𝐴‘𝑛)( ·_𝑠 ‘(Poly₁‘(𝐸 ↾_s 𝐹)))(𝑛(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹)))))))‘𝑋) = ((𝑥 ∈ 𝐵 ↦ (𝐸 Σ_g (𝑛 ∈ (0...𝐷) ↦ (((𝐸 evalSub₁ 𝐹)‘((𝐴‘𝑛)( ·_𝑠 ‘(Poly₁‘(𝐸 ↾_s 𝐹)))(𝑛(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹)))))‘𝑥))))‘𝑋))
180		fveq2 6885	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (((𝐸 evalSub₁ 𝐹)‘((𝐴‘𝑛)( ·_𝑠 ‘(Poly₁‘(𝐸 ↾_s 𝐹)))(𝑛(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹)))))‘𝑥) = (((𝐸 evalSub₁ 𝐹)‘((𝐴‘𝑛)( ·_𝑠 ‘(Poly₁‘(𝐸 ↾_s 𝐹)))(𝑛(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹)))))‘𝑋))
181	180	mpteq2dv 5207	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑛 ∈ (0...𝐷) ↦ (((𝐸 evalSub₁ 𝐹)‘((𝐴‘𝑛)( ·_𝑠 ‘(Poly₁‘(𝐸 ↾_s 𝐹)))(𝑛(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹)))))‘𝑥)) = (𝑛 ∈ (0...𝐷) ↦ (((𝐸 evalSub₁ 𝐹)‘((𝐴‘𝑛)( ·_𝑠 ‘(Poly₁‘(𝐸 ↾_s 𝐹)))(𝑛(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹)))))‘𝑋)))
182	181	oveq2d 7432	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝐸 Σ_g (𝑛 ∈ (0...𝐷) ↦ (((𝐸 evalSub₁ 𝐹)‘((𝐴‘𝑛)( ·_𝑠 ‘(Poly₁‘(𝐸 ↾_s 𝐹)))(𝑛(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹)))))‘𝑥))) = (𝐸 Σ_g (𝑛 ∈ (0...𝐷) ↦ (((𝐸 evalSub₁ 𝐹)‘((𝐴‘𝑛)( ·_𝑠 ‘(Poly₁‘(𝐸 ↾_s 𝐹)))(𝑛(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹)))))‘𝑋))))
183		eqidd 2766	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ (𝐸 Σ_g (𝑛 ∈ (0...𝐷) ↦ (((𝐸 evalSub₁ 𝐹)‘((𝐴‘𝑛)( ·_𝑠 ‘(Poly₁‘(𝐸 ↾_s 𝐹)))(𝑛(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹)))))‘𝑥)))) = (𝑥 ∈ 𝐵 ↦ (𝐸 Σ_g (𝑛 ∈ (0...𝐷) ↦ (((𝐸 evalSub₁ 𝐹)‘((𝐴‘𝑛)( ·_𝑠 ‘(Poly₁‘(𝐸 ↾_s 𝐹)))(𝑛(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹)))))‘𝑥)))))
184		extdgfialglem1.4	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
185		ovexd 7451	. . . . 5 ⊢ (𝜑 → (𝐸 Σ_g (𝑛 ∈ (0...𝐷) ↦ (((𝐸 evalSub₁ 𝐹)‘((𝐴‘𝑛)( ·_𝑠 ‘(Poly₁‘(𝐸 ↾_s 𝐹)))(𝑛(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹)))))‘𝑋))) ∈ V)
186	182, 183, 184, 185	fvmptd4 7018	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ↦ (𝐸 Σ_g (𝑛 ∈ (0...𝐷) ↦ (((𝐸 evalSub₁ 𝐹)‘((𝐴‘𝑛)( ·_𝑠 ‘(Poly₁‘(𝐸 ↾_s 𝐹)))(𝑛(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹)))))‘𝑥))))‘𝑋) = (𝐸 Σ_g (𝑛 ∈ (0...𝐷) ↦ (((𝐸 evalSub₁ 𝐹)‘((𝐴‘𝑛)( ·_𝑠 ‘(Poly₁‘(𝐸 ↾_s 𝐹)))(𝑛(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹)))))‘𝑋))))
187		eqid 2765	. . . . . . . 8 ⊢ (._g‘(mulGrp‘𝐸)) = (._g‘(mulGrp‘𝐸))
188		extdgfialglem1.3	. . . . . . . 8 ⊢ · = (._r‘𝐸)
189	184	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...𝐷)) → 𝑋 ∈ 𝐵)
190	1, 6, 14, 11, 47, 40, 187, 20, 188, 155, 156, 26, 46, 189	evls1monply1 33892	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...𝐷)) → (((𝐸 evalSub₁ 𝐹)‘((𝐴‘𝑛)( ·_𝑠 ‘(Poly₁‘(𝐸 ↾_s 𝐹)))(𝑛(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹)))))‘𝑋) = ((𝐴‘𝑛) · (𝑛(._g‘(mulGrp‘𝐸))𝑋)))
191	190	mpteq2dva 5206	. . . . . 6 ⊢ (𝜑 → (𝑛 ∈ (0...𝐷) ↦ (((𝐸 evalSub₁ 𝐹)‘((𝐴‘𝑛)( ·_𝑠 ‘(Poly₁‘(𝐸 ↾_s 𝐹)))(𝑛(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹)))))‘𝑋)) = (𝑛 ∈ (0...𝐷) ↦ ((𝐴‘𝑛) · (𝑛(._g‘(mulGrp‘𝐸))𝑋))))
192		nfv 1947	. . . . . . . 8 ⊢ Ⅎ𝑛𝜑
193		ovexd 7451	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...𝐷)) → (𝑛(._g‘(mulGrp‘((subringAlg ‘𝐸)‘𝐹)))𝑋) ∈ V)
194		extdgfialglem1.r	. . . . . . . 8 ⊢ 𝐺 = (𝑛 ∈ (0...𝐷) ↦ (𝑛(._g‘(mulGrp‘((subringAlg ‘𝐸)‘𝐹)))𝑋))
195	192, 193, 194	fnmptd 6680	. . . . . . 7 ⊢ (𝜑 → 𝐺 Fn (0...𝐷))
196		inidm 4179	. . . . . . 7 ⊢ ((0...𝐷) ∩ (0...𝐷)) = (0...𝐷)
197		eqidd 2766	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...𝐷)) → (𝐴‘𝑛) = (𝐴‘𝑛))
198	194	fvmpt2 7005	. . . . . . . . 9 ⊢ ((𝑛 ∈ (0...𝐷) ∧ (𝑛(._g‘(mulGrp‘((subringAlg ‘𝐸)‘𝐹)))𝑋) ∈ V) → (𝐺‘𝑛) = (𝑛(._g‘(mulGrp‘((subringAlg ‘𝐸)‘𝐹)))𝑋))
199	118, 193, 198	syl2anc 596	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...𝐷)) → (𝐺‘𝑛) = (𝑛(._g‘(mulGrp‘((subringAlg ‘𝐸)‘𝐹)))𝑋))
200		eqid 2765	. . . . . . . . . . 11 ⊢ ((subringAlg ‘𝐸)‘𝐹) = ((subringAlg ‘𝐸)‘𝐹)
201	28, 6	sseqtrdi 3978	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ⊆ (Base‘𝐸))
202	200, 80, 201	srapwov 34002	. . . . . . . . . 10 ⊢ (𝜑 → (._g‘(mulGrp‘𝐸)) = (._g‘(mulGrp‘((subringAlg ‘𝐸)‘𝐹))))
203	202	oveqd 7433	. . . . . . . . 9 ⊢ (𝜑 → (𝑛(._g‘(mulGrp‘𝐸))𝑋) = (𝑛(._g‘(mulGrp‘((subringAlg ‘𝐸)‘𝐹)))𝑋))
204	203	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...𝐷)) → (𝑛(._g‘(mulGrp‘𝐸))𝑋) = (𝑛(._g‘(mulGrp‘((subringAlg ‘𝐸)‘𝐹)))𝑋))
205	199, 204	eqtr4d 2803	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...𝐷)) → (𝐺‘𝑛) = (𝑛(._g‘(mulGrp‘𝐸))𝑋))
206	138, 195, 18, 18, 196, 197, 205	offval 7689	. . . . . 6 ⊢ (𝜑 → (𝐴 ∘_f · 𝐺) = (𝑛 ∈ (0...𝐷) ↦ ((𝐴‘𝑛) · (𝑛(._g‘(mulGrp‘𝐸))𝑋))))
207	191, 206	eqtr4d 2803	. . . . 5 ⊢ (𝜑 → (𝑛 ∈ (0...𝐷) ↦ (((𝐸 evalSub₁ 𝐹)‘((𝐴‘𝑛)( ·_𝑠 ‘(Poly₁‘(𝐸 ↾_s 𝐹)))(𝑛(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹)))))‘𝑋)) = (𝐴 ∘_f · 𝐺))
208	207	oveq2d 7432	. . . 4 ⊢ (𝜑 → (𝐸 Σ_g (𝑛 ∈ (0...𝐷) ↦ (((𝐸 evalSub₁ 𝐹)‘((𝐴‘𝑛)( ·_𝑠 ‘(Poly₁‘(𝐸 ↾_s 𝐹)))(𝑛(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹)))))‘𝑋))) = (𝐸 Σ_g (𝐴 ∘_f · 𝐺)))
209	179, 186, 208	3eqtrd 2804	. . 3 ⊢ (𝜑 → (((𝐸 ↑_s 𝐵) Σ_g (𝑛 ∈ (0...𝐷) ↦ ((𝐸 evalSub₁ 𝐹)‘((𝐴‘𝑛)( ·_𝑠 ‘(Poly₁‘(𝐸 ↾_s 𝐹)))(𝑛(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹)))))))‘𝑋) = (𝐸 Σ_g (𝐴 ∘_f · 𝐺)))
210		extdgfialglem2.3	. . 3 ⊢ (𝜑 → (𝐸 Σ_g (𝐴 ∘_f · 𝐺)) = 𝑍)
211	154, 209, 210	3eqtrd 2804	. 2 ⊢ (𝜑 → (((𝐸 evalSub₁ 𝐹)‘((Poly₁‘(𝐸 ↾_s 𝐹)) Σ_g (𝑛 ∈ (0...𝐷) ↦ ((𝐴‘𝑛)( ·_𝑠 ‘(Poly₁‘(𝐸 ↾_s 𝐹)))(𝑛(._g‘(mulGrp‘(Poly₁‘(𝐸 ↾_s 𝐹))))(var₁‘(𝐸 ↾_s 𝐹)))))))‘𝑋) = 𝑍)
212	1, 2, 3, 4, 5, 6, 60, 151, 211, 184	irngnzply1lem 34103	1 ⊢ (𝜑 → 𝑋 ∈ (𝐸 IntgRing 𝐹))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2146 ≠ wne 2960 ∀wral 3081 Vcvv 3457 ∖ cdif 3903 ⊆ wss 3906 ifcif 4489 {csn 4591 class class class wbr 5111 ↦ cmpt 5194 × cxp 5661 dom cdm 5663 Fn wfn 6535 ⟶wf 6536 ‘cfv 6540 (class class class)co 7416 ∘_f cof 7678 Fincfn 8945 finSupp cfsupp 9324 0cc0 11111 ℕ₀cn0 12515 ...cfz 13546 Basecbs 17286 ↾_s cress 17307 ._rcmulr 17328 Scalarcsca 17330 ·_𝑠 cvsca 17331 0_gc0g 17509 Σ_g cgsu 17510 ↑_s cpws 17516 Mndcmnd 18813 ._gcmg 19156 SubGrpcsubg 19209 mulGrpcmgp 20239 Ringcrg 20338 CRingccrg 20339 SubRingcsubrg 20697 Fieldcfield 20857 SubDRingcsdrg 20918 LModclmod 21010 subringAlg csra 21321 var₁cv1 22365 Poly₁cpl1 22366 evalSub₁ ces1 22502 dimcldim 34012 IntgRing cirng 34096

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7738 ax-cnex 11167 ax-resscn 11168 ax-1cn 11169 ax-icn 11170 ax-addcl 11171 ax-addrcl 11172 ax-mulcl 11173 ax-mulrcl 11174 ax-mulcom 11175 ax-addass 11176 ax-mulass 11177 ax-distr 11178 ax-i2m1 11179 ax-1ne0 11180 ax-1rid 11181 ax-rnegex 11182 ax-rrecex 11183 ax-cnre 11184 ax-pre-lttri 11185 ax-pre-lttrn 11186 ax-pre-ltadd 11187 ax-pre-mulgt0 11188 ax-addf 11190

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4875 df-int 4915 df-iun 4960 df-iin 4961 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-se 5617 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7680 df-ofr 7681 df-om 7865 df-1st 7988 df-2nd 7989 df-supp 8159 df-tpos 8224 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-1o 8455 df-2o 8456 df-er 8696 df-map 8828 df-pm 8829 df-ixp 8898 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-fsupp 9325 df-sup 9405 df-oi 9475 df-card 9937 df-pnf 11256 df-mnf 11257 df-xr 11258 df-ltxr 11259 df-le 11260 df-sub 11454 df-neg 11455 df-nn 12245 df-2 12314 df-3 12315 df-4 12316 df-5 12317 df-6 12318 df-7 12319 df-8 12320 df-9 12321 df-n0 12516 df-z 12603 df-dec 12723 df-uz 12874 df-fz 13547 df-fzo 13695 df-seq 14051 df-hash 14380 df-struct 17224 df-sets 17241 df-slot 17259 df-ndx 17271 df-base 17287 df-ress 17308 df-plusg 17340 df-mulr 17341 df-starv 17342 df-sca 17343 df-vsca 17344 df-ip 17345 df-tset 17346 df-ple 17347 df-ds 17349 df-unif 17350 df-hom 17351 df-cco 17352 df-0g 17511 df-gsum 17512 df-prds 17517 df-pws 17519 df-mre 17655 df-mrc 17656 df-acs 17658 df-mgm 18715 df-sgrp 18798 df-mnd 18814 df-mhm 18864 df-submnd 18865 df-grp 19026 df-minusg 19027 df-sbg 19028 df-mulg 19157 df-subg 19212 df-ghm 19307 df-cntz 19410 df-cmn 19875 df-abl 19876 df-mgp 20240 df-rng 20254 df-ur 20287 df-srg 20292 df-ring 20340 df-cring 20341 df-oppr 20444 df-dvdsr 20464 df-unit 20465 df-invr 20495 df-rhm 20579 df-subrng 20674 df-subrg 20698 df-rlreg 20822 df-drng 20858 df-field 20859 df-sdrg 20919 df-lmod 21012 df-lss 21082 df-lsp 21122 df-sra 21323 df-cnfld 21552 df-assa 22032 df-asp 22033 df-ascl 22034 df-psr 22088 df-mvr 22089 df-mpl 22090 df-opsr 22092 df-evls 22254 df-evl 22255 df-psr1 22369 df-vr1 22370 df-ply1 22371 df-coe1 22372 df-evls1 22504 df-evl1 22505 df-mdeg 26241 df-deg1 26242 df-mon1 26317 df-uc1p 26318 df-irng 34097

This theorem is used by: extdgfialg 34107

W3C validator