selvvvval - Mathbox for Steven Nguyen

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Theorem selvvvval 43032

Description: Recover the original polynomial from a selectVars application. (Contributed by SN, 15-Mar-2025.)

**Hypotheses**
Ref	Expression
selvvvval.d	⊢ 𝐷 = {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}
selvvvval.p	⊢ 𝑃 = (𝐼 mPoly 𝑅)
selvvvval.b	⊢ 𝐵 = (Base‘𝑃)
selvvvval.r	⊢ (𝜑 → 𝑅 ∈ CRing)
selvvvval.j	⊢ (𝜑 → 𝐽 ⊆ 𝐼)
selvvvval.f	⊢ (𝜑 → 𝐹 ∈ 𝐵)
selvvvval.y	⊢ (𝜑 → 𝑌 ∈ 𝐷)

**Assertion**
Ref	Expression
selvvvval	⊢ (𝜑 → (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑌 ↾ 𝐽))‘(𝑌 ↾ (𝐼 ∖ 𝐽))) = (𝐹‘𝑌))

Distinct variable groups: ℎ,𝐼 ℎ,𝐽 𝑅,ℎ ℎ,𝑌 Allowed substitution hints: 𝜑(ℎ) 𝐵(ℎ) 𝐷(ℎ) 𝑃(ℎ) 𝐹(ℎ)

Proof of Theorem selvvvval Dummy variables 𝑒 𝑔 𝑖 𝑗 𝑘 𝑡 𝑢 𝑣 𝑤 𝑓 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		selvvvval.p	. . . . . 6 ⊢ 𝑃 = (𝐼 mPoly 𝑅)
2		selvvvval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑃)
3		eqid 2737	. . . . . 6 ⊢ ((𝐼 ∖ 𝐽) mPoly 𝑅) = ((𝐼 ∖ 𝐽) mPoly 𝑅)
4		eqid 2737	. . . . . 6 ⊢ (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)) = (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))
5		eqid 2737	. . . . . 6 ⊢ (algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) = (algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))
6		eqid 2737	. . . . . 6 ⊢ ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∘ (algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))) = ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∘ (algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅)))
7		selvvvval.r	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ CRing)
8		selvvvval.j	. . . . . 6 ⊢ (𝜑 → 𝐽 ⊆ 𝐼)
9		selvvvval.f	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ 𝐵)
10	1, 2, 3, 4, 5, 6, 7, 8, 9	selvval2 43031	. . . . 5 ⊢ (𝜑 → (((𝐼 selectVars 𝑅)‘𝐽)‘𝐹) = (((𝐼 eval (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∘ (algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))) ∘ 𝐹))‘(𝑧 ∈ 𝐼 ↦ if(𝑧 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧))))))
11		eqid 2737	. . . . . 6 ⊢ (𝐼 eval (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) = (𝐼 eval (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))
12		eqid 2737	. . . . . 6 ⊢ (𝐼 mPoly (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) = (𝐼 mPoly (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))
13		eqid 2737	. . . . . 6 ⊢ (Base‘(𝐼 mPoly (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) = (Base‘(𝐼 mPoly (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
14		selvvvval.d	. . . . . 6 ⊢ 𝐷 = {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}
15		eqid 2737	. . . . . 6 ⊢ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) = (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))
16		eqid 2737	. . . . . 6 ⊢ (mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) = (mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))
17		eqid 2737	. . . . . 6 ⊢ (._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) = (._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
18		eqid 2737	. . . . . 6 ⊢ (._r‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) = (._r‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))
19	1, 2	mplrcl 21982	. . . . . . 7 ⊢ (𝐹 ∈ 𝐵 → 𝐼 ∈ V)
20	9, 19	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ V)
21	20, 8	ssexd 5261	. . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ V)
22	20	difexd 5268	. . . . . . . 8 ⊢ (𝜑 → (𝐼 ∖ 𝐽) ∈ V)
23	3, 22, 7	mplcrngd 43004	. . . . . . 7 ⊢ (𝜑 → ((𝐼 ∖ 𝐽) mPoly 𝑅) ∈ CRing)
24	4, 21, 23	mplcrngd 43004	. . . . . 6 ⊢ (𝜑 → (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)) ∈ CRing)
25	4	mplassa 22010	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ V ∧ ((𝐼 ∖ 𝐽) mPoly 𝑅) ∈ CRing) → (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)) ∈ AssAlg)
26	21, 23, 25	syl2anc 585	. . . . . . . . . 10 ⊢ (𝜑 → (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)) ∈ AssAlg)
27		eqid 2737	. . . . . . . . . . 11 ⊢ (Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) = (Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))
28	5, 27	asclrhm 21880	. . . . . . . . . 10 ⊢ ((𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)) ∈ AssAlg → (algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∈ ((Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) RingHom (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
29	26, 28	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∈ ((Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) RingHom (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
30	3	mplassa 22010	. . . . . . . . . . . 12 ⊢ (((𝐼 ∖ 𝐽) ∈ V ∧ 𝑅 ∈ CRing) → ((𝐼 ∖ 𝐽) mPoly 𝑅) ∈ AssAlg)
31	22, 7, 30	syl2anc 585	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐼 ∖ 𝐽) mPoly 𝑅) ∈ AssAlg)
32		eqid 2737	. . . . . . . . . . . 12 ⊢ (algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅)) = (algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))
33		eqid 2737	. . . . . . . . . . . 12 ⊢ (Scalar‘((𝐼 ∖ 𝐽) mPoly 𝑅)) = (Scalar‘((𝐼 ∖ 𝐽) mPoly 𝑅))
34	32, 33	asclrhm 21880	. . . . . . . . . . 11 ⊢ (((𝐼 ∖ 𝐽) mPoly 𝑅) ∈ AssAlg → (algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅)) ∈ ((Scalar‘((𝐼 ∖ 𝐽) mPoly 𝑅)) RingHom ((𝐼 ∖ 𝐽) mPoly 𝑅)))
35	31, 34	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅)) ∈ ((Scalar‘((𝐼 ∖ 𝐽) mPoly 𝑅)) RingHom ((𝐼 ∖ 𝐽) mPoly 𝑅)))
36	3, 22, 7	mplsca 22001	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑅 = (Scalar‘((𝐼 ∖ 𝐽) mPoly 𝑅)))
37	36	eqcomd 2743	. . . . . . . . . . 11 ⊢ (𝜑 → (Scalar‘((𝐼 ∖ 𝐽) mPoly 𝑅)) = 𝑅)
38	4, 21, 23	mplsca 22001	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐼 ∖ 𝐽) mPoly 𝑅) = (Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
39	37, 38	oveq12d 7378	. . . . . . . . . 10 ⊢ (𝜑 → ((Scalar‘((𝐼 ∖ 𝐽) mPoly 𝑅)) RingHom ((𝐼 ∖ 𝐽) mPoly 𝑅)) = (𝑅 RingHom (Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))
40	35, 39	eleqtrd 2839	. . . . . . . . 9 ⊢ (𝜑 → (algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅)) ∈ (𝑅 RingHom (Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))
41		rhmco 20469	. . . . . . . . 9 ⊢ (((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∈ ((Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) RingHom (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∧ (algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅)) ∈ (𝑅 RingHom (Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))) → ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∘ (algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))) ∈ (𝑅 RingHom (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
42	29, 40, 41	syl2anc 585	. . . . . . . 8 ⊢ (𝜑 → ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∘ (algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))) ∈ (𝑅 RingHom (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
43		rhmghm 20454	. . . . . . . 8 ⊢ (((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∘ (algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))) ∈ (𝑅 RingHom (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) → ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∘ (algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))) ∈ (𝑅 GrpHom (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
44		ghmmhm 19192	. . . . . . . 8 ⊢ (((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∘ (algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))) ∈ (𝑅 GrpHom (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) → ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∘ (algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))) ∈ (𝑅 MndHom (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
45	42, 43, 44	3syl 18	. . . . . . 7 ⊢ (𝜑 → ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∘ (algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))) ∈ (𝑅 MndHom (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
46	1, 12, 2, 13, 45, 9	mhmcompl 22355	. . . . . 6 ⊢ (𝜑 → (((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∘ (algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))) ∘ 𝐹) ∈ (Base‘(𝐼 mPoly (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))
47		fvexd 6849	. . . . . . 7 ⊢ (𝜑 → (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∈ V)
48		eqid 2737	. . . . . . . . . . . 12 ⊢ (𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅)) = (𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))
49	23	crngringd 20218	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐼 ∖ 𝐽) mPoly 𝑅) ∈ Ring)
50	4, 48, 15, 21, 49	mvrf2 21981	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅)):𝐽⟶(Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
51	50	ffvelcdmda 7030	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐽) → ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧) ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
52	51	adantlr 716	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐼) ∧ 𝑧 ∈ 𝐽) → ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧) ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
53		eldif 3900	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (𝐼 ∖ 𝐽) ↔ (𝑧 ∈ 𝐼 ∧ ¬ 𝑧 ∈ 𝐽))
54		eqid 2737	. . . . . . . . . . . . . 14 ⊢ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)) = (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅))
55	4, 15, 54, 5, 21, 49	mplasclf 22053	. . . . . . . . . . . . 13 ⊢ (𝜑 → (algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))):(Base‘((𝐼 ∖ 𝐽) mPoly 𝑅))⟶(Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
56	55	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐼 ∖ 𝐽)) → (algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))):(Base‘((𝐼 ∖ 𝐽) mPoly 𝑅))⟶(Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
57		eqid 2737	. . . . . . . . . . . . . 14 ⊢ ((𝐼 ∖ 𝐽) mVar 𝑅) = ((𝐼 ∖ 𝐽) mVar 𝑅)
58	7	crngringd 20218	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑅 ∈ Ring)
59	3, 57, 54, 22, 58	mvrf2 21981	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐼 ∖ 𝐽) mVar 𝑅):(𝐼 ∖ 𝐽)⟶(Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)))
60	59	ffvelcdmda 7030	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐼 ∖ 𝐽)) → (((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧) ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)))
61	56, 60	ffvelcdmd 7031	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐼 ∖ 𝐽)) → ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧)) ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
62	53, 61	sylan2br 596	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐼 ∧ ¬ 𝑧 ∈ 𝐽)) → ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧)) ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
63	62	anassrs 467	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐼) ∧ ¬ 𝑧 ∈ 𝐽) → ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧)) ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
64	52, 63	ifclda 4503	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐼) → if(𝑧 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧))) ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
65	64	fmpttd 7061	. . . . . . 7 ⊢ (𝜑 → (𝑧 ∈ 𝐼 ↦ if(𝑧 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧)))):𝐼⟶(Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
66	47, 20, 65	elmapdd 8781	. . . . . 6 ⊢ (𝜑 → (𝑧 ∈ 𝐼 ↦ if(𝑧 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧)))) ∈ ((Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ↑_m 𝐼))
67	11, 12, 13, 14, 15, 16, 17, 18, 20, 24, 46, 66	evlvvval 43022	. . . . 5 ⊢ (𝜑 → (((𝐼 eval (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∘ (algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))) ∘ 𝐹))‘(𝑧 ∈ 𝐼 ↦ if(𝑧 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧))))) = ((𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑔 ∈ 𝐷 ↦ (((((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∘ (algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))) ∘ 𝐹)‘𝑔)(._r‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐼 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝑧 ∈ 𝐼 ↦ if(𝑧 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧))))‘𝑘))))))))
68		eqid 2737	. . . . . . . . . . . . 13 ⊢ (Base‘𝑅) = (Base‘𝑅)
69	1, 68, 2, 14, 9	mplelf 21986	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹:𝐷⟶(Base‘𝑅))
70	69	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → 𝐹:𝐷⟶(Base‘𝑅))
71		simpr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → 𝑔 ∈ 𝐷)
72	70, 71	fvco3d 6934	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∘ (algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))) ∘ 𝐹)‘𝑔) = (((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∘ (algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(𝐹‘𝑔)))
73	3, 54, 68, 32, 22, 58	mplasclf 22053	. . . . . . . . . . . 12 ⊢ (𝜑 → (algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅)):(Base‘𝑅)⟶(Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)))
74	73	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅)):(Base‘𝑅)⟶(Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)))
75	69	ffvelcdmda 7030	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (𝐹‘𝑔) ∈ (Base‘𝑅))
76	74, 75	fvco3d 6934	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∘ (algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(𝐹‘𝑔)) = ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))))
77	72, 76	eqtrd 2772	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∘ (algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))) ∘ 𝐹)‘𝑔) = ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))))
78	16, 15	mgpbas 20117	. . . . . . . . . . 11 ⊢ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) = (Base‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
79		eqid 2737	. . . . . . . . . . 11 ⊢ (0_g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) = (0_g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
80	16, 18	mgpplusg 20116	. . . . . . . . . . 11 ⊢ (._r‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) = (+_g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
81	16	crngmgp 20213	. . . . . . . . . . . . 13 ⊢ ((𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)) ∈ CRing → (mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∈ CMnd)
82	24, 81	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∈ CMnd)
83	82	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∈ CMnd)
84	20	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → 𝐼 ∈ V)
85	82	cmnmndd 19770	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∈ Mnd)
86	85	ad2antrr 727	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ 𝐼) → (mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∈ Mnd)
87	14	psrbagf 21908	. . . . . . . . . . . . . . 15 ⊢ (𝑔 ∈ 𝐷 → 𝑔:𝐼⟶ℕ₀)
88	87	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → 𝑔:𝐼⟶ℕ₀)
89	88	ffvelcdmda 7030	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ 𝐼) → (𝑔‘𝑘) ∈ ℕ₀)
90		eqid 2737	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ 𝐼 ↦ if(𝑧 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧)))) = (𝑧 ∈ 𝐼 ↦ if(𝑧 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧))))
91		eleq1w 2820	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑘 → (𝑧 ∈ 𝐽 ↔ 𝑘 ∈ 𝐽))
92		fveq2 6834	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑘 → ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧) = ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))
93		fveq2 6834	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = 𝑘 → (((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧) = (((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))
94	93	fveq2d 6838	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑘 → ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧)) = ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘)))
95	91, 92, 94	ifbieq12d 4496	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑘 → if(𝑧 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧))) = if(𝑘 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))
96		simpr 484	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ 𝐼) → 𝑘 ∈ 𝐼)
97	50	ad2antrr 727	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ 𝐼) → (𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅)):𝐽⟶(Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
98	97	ffvelcdmda 7030	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ 𝐼) ∧ 𝑘 ∈ 𝐽) → ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘) ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
99		eldif 3900	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ (𝐼 ∖ 𝐽) ↔ (𝑘 ∈ 𝐼 ∧ ¬ 𝑘 ∈ 𝐽))
100	55	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → (algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))):(Base‘((𝐼 ∖ 𝐽) mPoly 𝑅))⟶(Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
101	59	ffvelcdmda 7030	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → (((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘) ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)))
102	100, 101	ffvelcdmd 7031	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘)) ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
103	99, 102	sylan2br 596	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐼 ∧ ¬ 𝑘 ∈ 𝐽)) → ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘)) ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
104	103	anassrs 467	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ ¬ 𝑘 ∈ 𝐽) → ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘)) ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
105	104	adantllr 720	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ 𝐼) ∧ ¬ 𝑘 ∈ 𝐽) → ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘)) ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
106	98, 105	ifclda 4503	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ 𝐼) → if(𝑘 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))) ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
107	90, 95, 96, 106	fvmptd3 6965	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ 𝐼) → ((𝑧 ∈ 𝐼 ↦ if(𝑧 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧))))‘𝑘) = if(𝑘 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))
108	107, 106	eqeltrd 2837	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ 𝐼) → ((𝑧 ∈ 𝐼 ↦ if(𝑧 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧))))‘𝑘) ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
109	78, 17, 86, 89, 108	mulgnn0cld 19062	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ 𝐼) → ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝑧 ∈ 𝐼 ↦ if(𝑧 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧))))‘𝑘)) ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
110	109	fmpttd 7061	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (𝑘 ∈ 𝐼 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝑧 ∈ 𝐼 ↦ if(𝑧 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧))))‘𝑘))):𝐼⟶(Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
111	88	feqmptd 6902	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → 𝑔 = (𝑘 ∈ 𝐼 ↦ (𝑔‘𝑘)))
112	14	psrbagfsupp 21909	. . . . . . . . . . . . . 14 ⊢ (𝑔 ∈ 𝐷 → 𝑔 finSupp 0)
113	112	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → 𝑔 finSupp 0)
114	111, 113	eqbrtrrd 5110	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (𝑘 ∈ 𝐼 ↦ (𝑔‘𝑘)) finSupp 0)
115	78, 79, 17	mulg0 19041	. . . . . . . . . . . . 13 ⊢ (𝑡 ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) → (0(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))𝑡) = (0_g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))
116	115	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑡 ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) → (0(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))𝑡) = (0_g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))
117		fvexd 6849	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (0_g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) ∈ V)
118	114, 116, 89, 108, 117	fsuppssov1 9290	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (𝑘 ∈ 𝐼 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝑧 ∈ 𝐼 ↦ if(𝑧 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧))))‘𝑘))) finSupp (0_g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))
119		disjdifr 4414	. . . . . . . . . . . 12 ⊢ ((𝐼 ∖ 𝐽) ∩ 𝐽) = ∅
120	119	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((𝐼 ∖ 𝐽) ∩ 𝐽) = ∅)
121		undifr 4424	. . . . . . . . . . . . . 14 ⊢ (𝐽 ⊆ 𝐼 ↔ ((𝐼 ∖ 𝐽) ∪ 𝐽) = 𝐼)
122	8, 121	sylib 218	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐼 ∖ 𝐽) ∪ 𝐽) = 𝐼)
123	122	eqcomd 2743	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐼 = ((𝐼 ∖ 𝐽) ∪ 𝐽))
124	123	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → 𝐼 = ((𝐼 ∖ 𝐽) ∪ 𝐽))
125	78, 79, 80, 83, 84, 110, 118, 120, 124	gsumsplit 19894	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐼 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝑧 ∈ 𝐼 ↦ if(𝑧 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧))))‘𝑘)))) = (((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g ((𝑘 ∈ 𝐼 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝑧 ∈ 𝐼 ↦ if(𝑧 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧))))‘𝑘))) ↾ (𝐼 ∖ 𝐽)))(._r‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g ((𝑘 ∈ 𝐼 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝑧 ∈ 𝐼 ↦ if(𝑧 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧))))‘𝑘))) ↾ 𝐽))))
126		eldifi 4072	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ (𝐼 ∖ 𝐽) → 𝑘 ∈ 𝐼)
127	126	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → 𝑘 ∈ 𝐼)
128	126, 106	sylan2 594	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → if(𝑘 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))) ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
129	90, 95, 127, 128	fvmptd3 6965	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → ((𝑧 ∈ 𝐼 ↦ if(𝑧 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧))))‘𝑘) = if(𝑘 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))
130		eldifn 4073	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ (𝐼 ∖ 𝐽) → ¬ 𝑘 ∈ 𝐽)
131	130	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → ¬ 𝑘 ∈ 𝐽)
132	131	iffalsed 4478	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → if(𝑘 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))) = ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘)))
133	129, 132	eqtrd 2772	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → ((𝑧 ∈ 𝐼 ↦ if(𝑧 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧))))‘𝑘) = ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘)))
134	133	oveq2d 7376	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝑧 ∈ 𝐼 ↦ if(𝑧 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧))))‘𝑘)) = ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))
135		eqid 2737	. . . . . . . . . . . . . . . . . . . 20 ⊢ (mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) = (mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
136	135, 16	rhmmhm 20450	. . . . . . . . . . . . . . . . . . 19 ⊢ ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∈ ((Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) RingHom (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) → (algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∈ ((mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) MndHom (mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))
137	29, 136	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∈ ((mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) MndHom (mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))
138	137	ad2antrr 727	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → (algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∈ ((mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) MndHom (mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))
139	126, 89	sylan2 594	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → (𝑔‘𝑘) ∈ ℕ₀)
140	101	adantlr 716	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → (((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘) ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)))
141	38	fveq2d 6838	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)) = (Base‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))
142	141	ad2antrr 727	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)) = (Base‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))
143	140, 142	eleqtrd 2839	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → (((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘) ∈ (Base‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))
144		eqid 2737	. . . . . . . . . . . . . . . . . . 19 ⊢ (Base‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) = (Base‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
145	135, 144	mgpbas 20117	. . . . . . . . . . . . . . . . . 18 ⊢ (Base‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) = (Base‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))
146		eqid 2737	. . . . . . . . . . . . . . . . . 18 ⊢ (._g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))) = (._g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))
147	145, 146, 17	mhmmulg 19082	. . . . . . . . . . . . . . . . 17 ⊢ (((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∈ ((mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) MndHom (mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) ∧ (𝑔‘𝑘) ∈ ℕ₀ ∧ (((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘) ∈ (Base‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))) → ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘((𝑔‘𝑘)(._g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))) = ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))
148	138, 139, 143, 147	syl3anc 1374	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘((𝑔‘𝑘)(._g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))) = ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))
149	134, 148	eqtr4d 2775	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝑧 ∈ 𝐼 ↦ if(𝑧 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧))))‘𝑘)) = ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘((𝑔‘𝑘)(._g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))
150	149	mpteq2dva 5179	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝑧 ∈ 𝐼 ↦ if(𝑧 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧))))‘𝑘))) = (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘((𝑔‘𝑘)(._g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘)))))
151		difssd 4078	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (𝐼 ∖ 𝐽) ⊆ 𝐼)
152	151	resmptd 5999	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((𝑘 ∈ 𝐼 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝑧 ∈ 𝐼 ↦ if(𝑧 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧))))‘𝑘))) ↾ (𝐼 ∖ 𝐽)) = (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝑧 ∈ 𝐼 ↦ if(𝑧 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧))))‘𝑘))))
153	55	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))):(Base‘((𝐼 ∖ 𝐽) mPoly 𝑅))⟶(Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
154	38	fveq2d 6838	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) = (mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))
155	154	fveq2d 6838	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅))) = (._g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))))
156	155	ad2antrr 727	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → (._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅))) = (._g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))))
157	156	oveqd 7377	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘)) = ((𝑔‘𝑘)(._g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘)))
158		eqid 2737	. . . . . . . . . . . . . . . . . 18 ⊢ (mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) = (mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅))
159	158, 54	mgpbas 20117	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)) = (Base‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))
160		eqid 2737	. . . . . . . . . . . . . . . . 17 ⊢ (._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅))) = (._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))
161	158	crngmgp 20213	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐼 ∖ 𝐽) mPoly 𝑅) ∈ CRing → (mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) ∈ CMnd)
162	23, 161	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) ∈ CMnd)
163	162	cmnmndd 19770	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) ∈ Mnd)
164	163	ad2antrr 727	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → (mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) ∈ Mnd)
165	159, 160, 164, 139, 140	mulgnn0cld 19062	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘)) ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)))
166	157, 165	eqeltrrd 2838	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → ((𝑔‘𝑘)(._g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘)) ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)))
167	153, 166	cofmpt 7079	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∘ (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘)))) = (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘((𝑔‘𝑘)(._g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘)))))
168	150, 152, 167	3eqtr4d 2782	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((𝑘 ∈ 𝐼 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝑧 ∈ 𝐼 ↦ if(𝑧 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧))))‘𝑘))) ↾ (𝐼 ∖ 𝐽)) = ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∘ (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘)))))
169	168	oveq2d 7376	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g ((𝑘 ∈ 𝐼 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝑧 ∈ 𝐼 ↦ if(𝑧 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧))))‘𝑘))) ↾ (𝐼 ∖ 𝐽))) = ((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∘ (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))))
170		eqid 2737	. . . . . . . . . . . . 13 ⊢ (0_g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))) = (0_g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))
171	38, 23	eqeltrrd 2838	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∈ CRing)
172	135	crngmgp 20213	. . . . . . . . . . . . . . 15 ⊢ ((Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∈ CRing → (mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) ∈ CMnd)
173	171, 172	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) ∈ CMnd)
174	173	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) ∈ CMnd)
175	85	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∈ Mnd)
176	22	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (𝐼 ∖ 𝐽) ∈ V)
177	137	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∈ ((mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) MndHom (mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))
178	166, 142	eleqtrd 2839	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → ((𝑔‘𝑘)(._g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘)) ∈ (Base‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))
179	178	fmpttd 7061	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))):(𝐼 ∖ 𝐽)⟶(Base‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))
180		0zd 12527	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → 0 ∈ ℤ)
181	114, 151, 180	fmptssfisupp 9300	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (𝑔‘𝑘)) finSupp 0)
182	141	eqimssd 3979	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)) ⊆ (Base‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))
183	182	sselda 3922	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑢 ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅))) → 𝑢 ∈ (Base‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))
184	183	adantlr 716	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑢 ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅))) → 𝑢 ∈ (Base‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))
185	145, 170, 146	mulg0 19041	. . . . . . . . . . . . . . 15 ⊢ (𝑢 ∈ (Base‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) → (0(._g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))𝑢) = (0_g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))))
186	184, 185	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑢 ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅))) → (0(._g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))𝑢) = (0_g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))))
187		fvexd 6849	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (0_g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))) ∈ V)
188	181, 186, 139, 140, 187	fsuppssov1 9290	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))) finSupp (0_g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))))
189	145, 170, 174, 175, 176, 177, 179, 188	gsummhm 19904	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∘ (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))) = ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘((mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))))
190	169, 189	eqtrd 2772	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g ((𝑘 ∈ 𝐼 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝑧 ∈ 𝐼 ↦ if(𝑧 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧))))‘𝑘))) ↾ (𝐼 ∖ 𝐽))) = ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘((mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))))
191	8	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → 𝐽 ⊆ 𝐼)
192	191	resmptd 5999	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((𝑘 ∈ 𝐼 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝑧 ∈ 𝐼 ↦ if(𝑧 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧))))‘𝑘))) ↾ 𝐽) = (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝑧 ∈ 𝐼 ↦ if(𝑧 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧))))‘𝑘))))
193	191	sselda 3922	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ 𝐽) → 𝑘 ∈ 𝐼)
194	193, 106	syldan 592	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ 𝐽) → if(𝑘 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))) ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
195	90, 95, 193, 194	fvmptd3 6965	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ 𝐽) → ((𝑧 ∈ 𝐼 ↦ if(𝑧 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧))))‘𝑘) = if(𝑘 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))
196		iftrue 4473	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ 𝐽 → if(𝑘 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))) = ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))
197	196	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ 𝐽) → if(𝑘 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))) = ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))
198	195, 197	eqtrd 2772	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ 𝐽) → ((𝑧 ∈ 𝐼 ↦ if(𝑧 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧))))‘𝑘) = ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))
199	198	oveq2d 7376	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ 𝐽) → ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝑧 ∈ 𝐼 ↦ if(𝑧 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧))))‘𝑘)) = ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘)))
200	199	mpteq2dva 5179	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝑧 ∈ 𝐼 ↦ if(𝑧 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧))))‘𝑘))) = (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))
201	192, 200	eqtrd 2772	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((𝑘 ∈ 𝐼 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝑧 ∈ 𝐼 ↦ if(𝑧 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧))))‘𝑘))) ↾ 𝐽) = (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))
202	201	oveq2d 7376	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g ((𝑘 ∈ 𝐼 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝑧 ∈ 𝐼 ↦ if(𝑧 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧))))‘𝑘))) ↾ 𝐽)) = ((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘)))))
203	190, 202	oveq12d 7378	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g ((𝑘 ∈ 𝐼 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝑧 ∈ 𝐼 ↦ if(𝑧 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧))))‘𝑘))) ↾ (𝐼 ∖ 𝐽)))(._r‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g ((𝑘 ∈ 𝐼 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝑧 ∈ 𝐼 ↦ if(𝑧 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧))))‘𝑘))) ↾ 𝐽))) = (((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘((mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘)))))(._r‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))))
204	26	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)) ∈ AssAlg)
205	145, 170, 174, 176, 179, 188	gsumcl 19881	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘)))) ∈ (Base‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))
206	21	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → 𝐽 ∈ V)
207	85	ad2antrr 727	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ 𝐽) → (mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∈ Mnd)
208	193, 89	syldan 592	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ 𝐽) → (𝑔‘𝑘) ∈ ℕ₀)
209	50	ffvelcdmda 7030	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐽) → ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘) ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
210	209	adantlr 716	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ 𝐽) → ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘) ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
211	78, 17, 207, 208, 210	mulgnn0cld 19062	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ 𝐽) → ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘)) ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
212	211	fmpttd 7061	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))):𝐽⟶(Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
213	114, 191, 180	fmptssfisupp 9300	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (𝑘 ∈ 𝐽 ↦ (𝑔‘𝑘)) finSupp 0)
214	213, 116, 208, 210, 117	fsuppssov1 9290	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))) finSupp (0_g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))
215	78, 79, 83, 206, 212, 214	gsumcl 19881	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘)))) ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
216		eqid 2737	. . . . . . . . . . . . 13 ⊢ ( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) = ( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))
217	5, 27, 144, 15, 18, 216	asclmul1 21876	. . . . . . . . . . . 12 ⊢ (((𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)) ∈ AssAlg ∧ ((mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘)))) ∈ (Base‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) ∧ ((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘)))) ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) → (((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘((mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘)))))(._r‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))) = (((mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))))
218	204, 205, 215, 217	syl3anc 1374	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘((mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘)))))(._r‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))) = (((mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))))
219	155	oveqd 7377	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘)) = ((𝑔‘𝑘)(._g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘)))
220	219	mpteq2dv 5180	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))) = (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))
221	154, 220	oveq12d 7378	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘)))) = ((mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘)))))
222	221	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘)))) = ((mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘)))))
223	222	oveq1d 7375	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))) = (((mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))))
224	218, 223	eqtr4d 2775	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘((mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘)))))(._r‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))) = (((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))))
225	125, 203, 224	3eqtrd 2776	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐼 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝑧 ∈ 𝐼 ↦ if(𝑧 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧))))‘𝑘)))) = (((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))))
226	77, 225	oveq12d 7378	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (((((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∘ (algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))) ∘ 𝐹)‘𝑔)(._r‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐼 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝑧 ∈ 𝐼 ↦ if(𝑧 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧))))‘𝑘))))) = (((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔)))(._r‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘)))))))
227	74, 75	ffvelcdmd 7031	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔)) ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)))
228	141	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)) = (Base‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))
229	227, 228	eleqtrd 2839	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔)) ∈ (Base‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))
230	4, 21, 49	mpllmodd 22012	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)) ∈ LMod)
231	230	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)) ∈ LMod)
232		eqid 2737	. . . . . . . . . . . 12 ⊢ (0_g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅))) = (0_g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))
233	162	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) ∈ CMnd)
234	165	fmpttd 7061	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))):(𝐼 ∖ 𝐽)⟶(Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)))
235	159, 232, 160	mulg0 19041	. . . . . . . . . . . . . 14 ⊢ (𝑒 ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)) → (0(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))𝑒) = (0_g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅))))
236	235	adantl 481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑒 ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅))) → (0(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))𝑒) = (0_g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅))))
237		fvexd 6849	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (0_g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅))) ∈ V)
238	181, 236, 139, 140, 237	fsuppssov1 9290	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))) finSupp (0_g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅))))
239	159, 232, 233, 176, 234, 238	gsumcl 19881	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘)))) ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)))
240	239, 228	eleqtrd 2839	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘)))) ∈ (Base‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))
241	15, 27, 216, 144, 231, 240, 215	lmodvscld 20865	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))) ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
242	5, 27, 144, 15, 18, 216	asclmul1 21876	. . . . . . . . 9 ⊢ (((𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)) ∈ AssAlg ∧ ((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔)) ∈ (Base‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) ∧ (((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))) ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) → (((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔)))(._r‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘)))))) = (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘)))))))
243	204, 229, 241, 242	syl3anc 1374	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔)))(._r‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘)))))) = (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘)))))))
244	226, 243	eqtrd 2772	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (((((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∘ (algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))) ∘ 𝐹)‘𝑔)(._r‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐼 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝑧 ∈ 𝐼 ↦ if(𝑧 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧))))‘𝑘))))) = (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘)))))))
245	244	mpteq2dva 5179	. . . . . 6 ⊢ (𝜑 → (𝑔 ∈ 𝐷 ↦ (((((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∘ (algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))) ∘ 𝐹)‘𝑔)(._r‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐼 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝑧 ∈ 𝐼 ↦ if(𝑧 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧))))‘𝑘)))))) = (𝑔 ∈ 𝐷 ↦ (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))))))
246	245	oveq2d 7376	. . . . 5 ⊢ (𝜑 → ((𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑔 ∈ 𝐷 ↦ (((((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∘ (algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))) ∘ 𝐹)‘𝑔)(._r‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐼 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝑧 ∈ 𝐼 ↦ if(𝑧 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧))))‘𝑘))))))) = ((𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑔 ∈ 𝐷 ↦ (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘)))))))))
247	10, 67, 246	3eqtrd 2776	. . . 4 ⊢ (𝜑 → (((𝐼 selectVars 𝑅)‘𝐽)‘𝐹) = ((𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑔 ∈ 𝐷 ↦ (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘)))))))))
248	247	fveq1d 6836	. . 3 ⊢ (𝜑 → ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑌 ↾ 𝐽)) = (((𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑔 ∈ 𝐷 ↦ (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))))))‘(𝑌 ↾ 𝐽)))
249	248	fveq1d 6836	. 2 ⊢ (𝜑 → (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑌 ↾ 𝐽))‘(𝑌 ↾ (𝐼 ∖ 𝐽))) = ((((𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑔 ∈ 𝐷 ↦ (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))))))‘(𝑌 ↾ 𝐽))‘(𝑌 ↾ (𝐼 ∖ 𝐽))))
250		eqid 2737	. . . 4 ⊢ (0_g‘((𝐼 ∖ 𝐽) mPoly 𝑅)) = (0_g‘((𝐼 ∖ 𝐽) mPoly 𝑅))
251	49	ringcmnd 20256	. . . 4 ⊢ (𝜑 → ((𝐼 ∖ 𝐽) mPoly 𝑅) ∈ CMnd)
252	7	crnggrpd 20219	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ Grp)
253	252	grpmndd 18913	. . . 4 ⊢ (𝜑 → 𝑅 ∈ Mnd)
254		ovex 7393	. . . . . 6 ⊢ (ℕ₀ ↑_m 𝐼) ∈ V
255	14, 254	rabex2 5278	. . . . 5 ⊢ 𝐷 ∈ V
256	255	a1i 11	. . . 4 ⊢ (𝜑 → 𝐷 ∈ V)
257		eqid 2737	. . . . . 6 ⊢ {𝑦 ∈ (ℕ₀ ↑_m (𝐼 ∖ 𝐽)) ∣ (^◡𝑦 “ ℕ) ∈ Fin} = {𝑦 ∈ (ℕ₀ ↑_m (𝐼 ∖ 𝐽)) ∣ (^◡𝑦 “ ℕ) ∈ Fin}
258		eqid 2737	. . . . . 6 ⊢ (𝑣 ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)) ↦ (𝑣‘(𝑌 ↾ (𝐼 ∖ 𝐽)))) = (𝑣 ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)) ↦ (𝑣‘(𝑌 ↾ (𝐼 ∖ 𝐽))))
259		difssd 4078	. . . . . . 7 ⊢ (𝜑 → (𝐼 ∖ 𝐽) ⊆ 𝐼)
260		selvvvval.y	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ 𝐷)
261	14, 257, 20, 259, 260	psrbagres 43003	. . . . . 6 ⊢ (𝜑 → (𝑌 ↾ (𝐼 ∖ 𝐽)) ∈ {𝑦 ∈ (ℕ₀ ↑_m (𝐼 ∖ 𝐽)) ∣ (^◡𝑦 “ ℕ) ∈ Fin})
262	3, 54, 257, 258, 22, 252, 261	mplmapghm 43011	. . . . 5 ⊢ (𝜑 → (𝑣 ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)) ↦ (𝑣‘(𝑌 ↾ (𝐼 ∖ 𝐽)))) ∈ (((𝐼 ∖ 𝐽) mPoly 𝑅) GrpHom 𝑅))
263		ghmmhm 19192	. . . . 5 ⊢ ((𝑣 ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)) ↦ (𝑣‘(𝑌 ↾ (𝐼 ∖ 𝐽)))) ∈ (((𝐼 ∖ 𝐽) mPoly 𝑅) GrpHom 𝑅) → (𝑣 ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)) ↦ (𝑣‘(𝑌 ↾ (𝐼 ∖ 𝐽)))) ∈ (((𝐼 ∖ 𝐽) mPoly 𝑅) MndHom 𝑅))
264	262, 263	syl 17	. . . 4 ⊢ (𝜑 → (𝑣 ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)) ↦ (𝑣‘(𝑌 ↾ (𝐼 ∖ 𝐽)))) ∈ (((𝐼 ∖ 𝐽) mPoly 𝑅) MndHom 𝑅))
265		eqid 2737	. . . . . . . 8 ⊢ {𝑥 ∈ (ℕ₀ ↑_m 𝐽) ∣ (^◡𝑥 “ ℕ) ∈ Fin} = {𝑥 ∈ (ℕ₀ ↑_m 𝐽) ∣ (^◡𝑥 “ ℕ) ∈ Fin}
266		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) → 𝑤 ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
267	4, 54, 15, 265, 266	mplelf 21986	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) → 𝑤:{𝑥 ∈ (ℕ₀ ↑_m 𝐽) ∣ (^◡𝑥 “ ℕ) ∈ Fin}⟶(Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)))
268	14, 265, 20, 8, 260	psrbagres 43003	. . . . . . . 8 ⊢ (𝜑 → (𝑌 ↾ 𝐽) ∈ {𝑥 ∈ (ℕ₀ ↑_m 𝐽) ∣ (^◡𝑥 “ ℕ) ∈ Fin})
269	268	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) → (𝑌 ↾ 𝐽) ∈ {𝑥 ∈ (ℕ₀ ↑_m 𝐽) ∣ (^◡𝑥 “ ℕ) ∈ Fin})
270	267, 269	ffvelcdmd 7031	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) → (𝑤‘(𝑌 ↾ 𝐽)) ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)))
271	270	fmpttd 7061	. . . . 5 ⊢ (𝜑 → (𝑤 ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ↦ (𝑤‘(𝑌 ↾ 𝐽))):(Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))⟶(Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)))
272	15, 27, 216, 144, 231, 229, 241	lmodvscld 20865	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘)))))) ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
273	272	fmpttd 7061	. . . . 5 ⊢ (𝜑 → (𝑔 ∈ 𝐷 ↦ (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))))):𝐷⟶(Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
274	271, 273	fcod 6687	. . . 4 ⊢ (𝜑 → ((𝑤 ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ↦ (𝑤‘(𝑌 ↾ 𝐽))) ∘ (𝑔 ∈ 𝐷 ↦ (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘)))))))):𝐷⟶(Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)))
275		fvexd 6849	. . . . 5 ⊢ (𝜑 → (0_g‘((𝐼 ∖ 𝐽) mPoly 𝑅)) ∈ V)
276	24	crngringd 20218	. . . . . 6 ⊢ (𝜑 → (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)) ∈ Ring)
277		eqid 2737	. . . . . . 7 ⊢ (0_g‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) = (0_g‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))
278	15, 277	ring0cl 20239	. . . . . 6 ⊢ ((𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)) ∈ Ring → (0_g‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
279	276, 278	syl 17	. . . . 5 ⊢ (𝜑 → (0_g‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
280		ssidd 3946	. . . . 5 ⊢ (𝜑 → (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ⊆ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
281	255	mptex 7171	. . . . . . . 8 ⊢ (𝑔 ∈ 𝐷 ↦ ((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))) ∈ V
282	281	a1i 11	. . . . . . 7 ⊢ (𝜑 → (𝑔 ∈ 𝐷 ↦ ((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))) ∈ V)
283		fvexd 6849	. . . . . . 7 ⊢ (𝜑 → (0_g‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) ∈ V)
284		funmpt 6530	. . . . . . . 8 ⊢ Fun (𝑔 ∈ 𝐷 ↦ ((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔)))
285	284	a1i 11	. . . . . . 7 ⊢ (𝜑 → Fun (𝑔 ∈ 𝐷 ↦ ((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))))
286		eqid 2737	. . . . . . . 8 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
287	1, 2, 286, 9	mplelsfi 21983	. . . . . . 7 ⊢ (𝜑 → 𝐹 finSupp (0_g‘𝑅))
288		ssidd 3946	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹 supp (0_g‘𝑅)) ⊆ (𝐹 supp (0_g‘𝑅)))
289		fvexd 6849	. . . . . . . . . . 11 ⊢ (𝜑 → (0_g‘𝑅) ∈ V)
290	69, 288, 9, 289	suppssrg 8139	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐷 ∖ (𝐹 supp (0_g‘𝑅)))) → (𝐹‘𝑔) = (0_g‘𝑅))
291	290	fveq2d 6838	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐷 ∖ (𝐹 supp (0_g‘𝑅)))) → ((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔)) = ((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(0_g‘𝑅)))
292	3, 32, 286, 250, 22, 58	mplascl0 22013	. . . . . . . . . . 11 ⊢ (𝜑 → ((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(0_g‘𝑅)) = (0_g‘((𝐼 ∖ 𝐽) mPoly 𝑅)))
293	38	fveq2d 6838	. . . . . . . . . . 11 ⊢ (𝜑 → (0_g‘((𝐼 ∖ 𝐽) mPoly 𝑅)) = (0_g‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))
294	292, 293	eqtrd 2772	. . . . . . . . . 10 ⊢ (𝜑 → ((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(0_g‘𝑅)) = (0_g‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))
295	294	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐷 ∖ (𝐹 supp (0_g‘𝑅)))) → ((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(0_g‘𝑅)) = (0_g‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))
296	291, 295	eqtrd 2772	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐷 ∖ (𝐹 supp (0_g‘𝑅)))) → ((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔)) = (0_g‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))
297	296, 256	suppss2 8143	. . . . . . 7 ⊢ (𝜑 → ((𝑔 ∈ 𝐷 ↦ ((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))) supp (0_g‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))) ⊆ (𝐹 supp (0_g‘𝑅)))
298	282, 283, 285, 287, 297	fsuppsssuppgd 9288	. . . . . 6 ⊢ (𝜑 → (𝑔 ∈ 𝐷 ↦ ((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))) finSupp (0_g‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))
299		eqid 2737	. . . . . . . 8 ⊢ (0_g‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) = (0_g‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
300	15, 27, 216, 299, 277	lmod0vs 20881	. . . . . . 7 ⊢ (((𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)) ∈ LMod ∧ 𝑓 ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) → ((0_g‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))𝑓) = (0_g‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
301	230, 300	sylan 581	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) → ((0_g‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))𝑓) = (0_g‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
302		fvexd 6849	. . . . . 6 ⊢ (𝜑 → (0_g‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∈ V)
303	298, 301, 227, 241, 302	fsuppssov1 9290	. . . . 5 ⊢ (𝜑 → (𝑔 ∈ 𝐷 ↦ (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))))) finSupp (0_g‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
304		eqid 2737	. . . . . . . 8 ⊢ (𝑤 ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ↦ (𝑤‘(𝑌 ↾ 𝐽))) = (𝑤 ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ↦ (𝑤‘(𝑌 ↾ 𝐽)))
305	23	crnggrpd 20219	. . . . . . . 8 ⊢ (𝜑 → ((𝐼 ∖ 𝐽) mPoly 𝑅) ∈ Grp)
306	4, 15, 265, 304, 21, 305, 268	mplmapghm 43011	. . . . . . 7 ⊢ (𝜑 → (𝑤 ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ↦ (𝑤‘(𝑌 ↾ 𝐽))) ∈ ((𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)) GrpHom ((𝐼 ∖ 𝐽) mPoly 𝑅)))
307		ghmmhm 19192	. . . . . . 7 ⊢ ((𝑤 ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ↦ (𝑤‘(𝑌 ↾ 𝐽))) ∈ ((𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)) GrpHom ((𝐼 ∖ 𝐽) mPoly 𝑅)) → (𝑤 ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ↦ (𝑤‘(𝑌 ↾ 𝐽))) ∈ ((𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)) MndHom ((𝐼 ∖ 𝐽) mPoly 𝑅)))
308	306, 307	syl 17	. . . . . 6 ⊢ (𝜑 → (𝑤 ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ↦ (𝑤‘(𝑌 ↾ 𝐽))) ∈ ((𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)) MndHom ((𝐼 ∖ 𝐽) mPoly 𝑅)))
309	277, 250	mhm0 18753	. . . . . 6 ⊢ ((𝑤 ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ↦ (𝑤‘(𝑌 ↾ 𝐽))) ∈ ((𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)) MndHom ((𝐼 ∖ 𝐽) mPoly 𝑅)) → ((𝑤 ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ↦ (𝑤‘(𝑌 ↾ 𝐽)))‘(0_g‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) = (0_g‘((𝐼 ∖ 𝐽) mPoly 𝑅)))
310	308, 309	syl 17	. . . . 5 ⊢ (𝜑 → ((𝑤 ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ↦ (𝑤‘(𝑌 ↾ 𝐽)))‘(0_g‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) = (0_g‘((𝐼 ∖ 𝐽) mPoly 𝑅)))
311	275, 279, 273, 271, 280, 256, 47, 303, 310	fsuppcor 9310	. . . 4 ⊢ (𝜑 → ((𝑤 ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ↦ (𝑤‘(𝑌 ↾ 𝐽))) ∘ (𝑔 ∈ 𝐷 ↦ (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘)))))))) finSupp (0_g‘((𝐼 ∖ 𝐽) mPoly 𝑅)))
312	54, 250, 251, 253, 256, 264, 274, 311	gsummhm 19904	. . 3 ⊢ (𝜑 → (𝑅 Σ_g ((𝑣 ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)) ↦ (𝑣‘(𝑌 ↾ (𝐼 ∖ 𝐽)))) ∘ ((𝑤 ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ↦ (𝑤‘(𝑌 ↾ 𝐽))) ∘ (𝑔 ∈ 𝐷 ↦ (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘)))))))))) = ((𝑣 ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)) ↦ (𝑣‘(𝑌 ↾ (𝐼 ∖ 𝐽))))‘(((𝐼 ∖ 𝐽) mPoly 𝑅) Σ_g ((𝑤 ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ↦ (𝑤‘(𝑌 ↾ 𝐽))) ∘ (𝑔 ∈ 𝐷 ↦ (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘)))))))))))
313		fveq1 6833	. . . 4 ⊢ (𝑣 = (((𝐼 ∖ 𝐽) mPoly 𝑅) Σ_g ((𝑤 ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ↦ (𝑤‘(𝑌 ↾ 𝐽))) ∘ (𝑔 ∈ 𝐷 ↦ (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))))))) → (𝑣‘(𝑌 ↾ (𝐼 ∖ 𝐽))) = ((((𝐼 ∖ 𝐽) mPoly 𝑅) Σ_g ((𝑤 ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ↦ (𝑤‘(𝑌 ↾ 𝐽))) ∘ (𝑔 ∈ 𝐷 ↦ (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘)))))))))‘(𝑌 ↾ (𝐼 ∖ 𝐽))))
314	54, 250, 251, 256, 274, 311	gsumcl 19881	. . . 4 ⊢ (𝜑 → (((𝐼 ∖ 𝐽) mPoly 𝑅) Σ_g ((𝑤 ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ↦ (𝑤‘(𝑌 ↾ 𝐽))) ∘ (𝑔 ∈ 𝐷 ↦ (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))))))) ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)))
315		fvexd 6849	. . . 4 ⊢ (𝜑 → ((((𝐼 ∖ 𝐽) mPoly 𝑅) Σ_g ((𝑤 ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ↦ (𝑤‘(𝑌 ↾ 𝐽))) ∘ (𝑔 ∈ 𝐷 ↦ (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘)))))))))‘(𝑌 ↾ (𝐼 ∖ 𝐽))) ∈ V)
316	258, 313, 314, 315	fvmptd3 6965	. . 3 ⊢ (𝜑 → ((𝑣 ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)) ↦ (𝑣‘(𝑌 ↾ (𝐼 ∖ 𝐽))))‘(((𝐼 ∖ 𝐽) mPoly 𝑅) Σ_g ((𝑤 ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ↦ (𝑤‘(𝑌 ↾ 𝐽))) ∘ (𝑔 ∈ 𝐷 ↦ (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘)))))))))) = ((((𝐼 ∖ 𝐽) mPoly 𝑅) Σ_g ((𝑤 ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ↦ (𝑤‘(𝑌 ↾ 𝐽))) ∘ (𝑔 ∈ 𝐷 ↦ (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘)))))))))‘(𝑌 ↾ (𝐼 ∖ 𝐽))))
317	276	ringcmnd 20256	. . . . . 6 ⊢ (𝜑 → (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)) ∈ CMnd)
318	305	grpmndd 18913	. . . . . 6 ⊢ (𝜑 → ((𝐼 ∖ 𝐽) mPoly 𝑅) ∈ Mnd)
319	15, 277, 317, 318, 256, 308, 273, 303	gsummhm 19904	. . . . 5 ⊢ (𝜑 → (((𝐼 ∖ 𝐽) mPoly 𝑅) Σ_g ((𝑤 ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ↦ (𝑤‘(𝑌 ↾ 𝐽))) ∘ (𝑔 ∈ 𝐷 ↦ (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))))))) = ((𝑤 ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ↦ (𝑤‘(𝑌 ↾ 𝐽)))‘((𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑔 ∈ 𝐷 ↦ (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))))))))
320		fveq1 6833	. . . . . 6 ⊢ (𝑤 = ((𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑔 ∈ 𝐷 ↦ (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘)))))))) → (𝑤‘(𝑌 ↾ 𝐽)) = (((𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑔 ∈ 𝐷 ↦ (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))))))‘(𝑌 ↾ 𝐽)))
321	15, 277, 317, 256, 273, 303	gsumcl 19881	. . . . . 6 ⊢ (𝜑 → ((𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑔 ∈ 𝐷 ↦ (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘)))))))) ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
322		fvexd 6849	. . . . . 6 ⊢ (𝜑 → (((𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑔 ∈ 𝐷 ↦ (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))))))‘(𝑌 ↾ 𝐽)) ∈ V)
323	304, 320, 321, 322	fvmptd3 6965	. . . . 5 ⊢ (𝜑 → ((𝑤 ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ↦ (𝑤‘(𝑌 ↾ 𝐽)))‘((𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑔 ∈ 𝐷 ↦ (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))))))) = (((𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑔 ∈ 𝐷 ↦ (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))))))‘(𝑌 ↾ 𝐽)))
324	319, 323	eqtrd 2772	. . . 4 ⊢ (𝜑 → (((𝐼 ∖ 𝐽) mPoly 𝑅) Σ_g ((𝑤 ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ↦ (𝑤‘(𝑌 ↾ 𝐽))) ∘ (𝑔 ∈ 𝐷 ↦ (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))))))) = (((𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑔 ∈ 𝐷 ↦ (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))))))‘(𝑌 ↾ 𝐽)))
325	324	fveq1d 6836	. . 3 ⊢ (𝜑 → ((((𝐼 ∖ 𝐽) mPoly 𝑅) Σ_g ((𝑤 ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ↦ (𝑤‘(𝑌 ↾ 𝐽))) ∘ (𝑔 ∈ 𝐷 ↦ (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘)))))))))‘(𝑌 ↾ (𝐼 ∖ 𝐽))) = ((((𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑔 ∈ 𝐷 ↦ (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))))))‘(𝑌 ↾ 𝐽))‘(𝑌 ↾ (𝐼 ∖ 𝐽))))
326	312, 316, 325	3eqtrrd 2777	. 2 ⊢ (𝜑 → ((((𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑔 ∈ 𝐷 ↦ (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))))))‘(𝑌 ↾ 𝐽))‘(𝑌 ↾ (𝐼 ∖ 𝐽))) = (𝑅 Σ_g ((𝑣 ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)) ↦ (𝑣‘(𝑌 ↾ (𝐼 ∖ 𝐽)))) ∘ ((𝑤 ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ↦ (𝑤‘(𝑌 ↾ 𝐽))) ∘ (𝑔 ∈ 𝐷 ↦ (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘)))))))))))
327	4, 54, 15, 265, 272	mplelf 21986	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘)))))):{𝑥 ∈ (ℕ₀ ↑_m 𝐽) ∣ (^◡𝑥 “ ℕ) ∈ Fin}⟶(Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)))
328	268	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (𝑌 ↾ 𝐽) ∈ {𝑥 ∈ (ℕ₀ ↑_m 𝐽) ∣ (^◡𝑥 “ ℕ) ∈ Fin})
329	327, 328	ffvelcdmd 7031	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))))‘(𝑌 ↾ 𝐽)) ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)))
330		eqidd 2738	. . . . . . 7 ⊢ (𝜑 → (𝑔 ∈ 𝐷 ↦ (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))))) = (𝑔 ∈ 𝐷 ↦ (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))))))
331		eqidd 2738	. . . . . . 7 ⊢ (𝜑 → (𝑤 ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ↦ (𝑤‘(𝑌 ↾ 𝐽))) = (𝑤 ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ↦ (𝑤‘(𝑌 ↾ 𝐽))))
332		fveq1 6833	. . . . . . 7 ⊢ (𝑤 = (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘)))))) → (𝑤‘(𝑌 ↾ 𝐽)) = ((((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))))‘(𝑌 ↾ 𝐽)))
333	272, 330, 331, 332	fmptco 7076	. . . . . 6 ⊢ (𝜑 → ((𝑤 ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ↦ (𝑤‘(𝑌 ↾ 𝐽))) ∘ (𝑔 ∈ 𝐷 ↦ (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘)))))))) = (𝑔 ∈ 𝐷 ↦ ((((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))))‘(𝑌 ↾ 𝐽))))
334		eqidd 2738	. . . . . 6 ⊢ (𝜑 → (𝑣 ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)) ↦ (𝑣‘(𝑌 ↾ (𝐼 ∖ 𝐽)))) = (𝑣 ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)) ↦ (𝑣‘(𝑌 ↾ (𝐼 ∖ 𝐽)))))
335		fveq1 6833	. . . . . 6 ⊢ (𝑣 = ((((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))))‘(𝑌 ↾ 𝐽)) → (𝑣‘(𝑌 ↾ (𝐼 ∖ 𝐽))) = (((((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))))‘(𝑌 ↾ 𝐽))‘(𝑌 ↾ (𝐼 ∖ 𝐽))))
336	329, 333, 334, 335	fmptco 7076	. . . . 5 ⊢ (𝜑 → ((𝑣 ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)) ↦ (𝑣‘(𝑌 ↾ (𝐼 ∖ 𝐽)))) ∘ ((𝑤 ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ↦ (𝑤‘(𝑌 ↾ 𝐽))) ∘ (𝑔 ∈ 𝐷 ↦ (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))))))) = (𝑔 ∈ 𝐷 ↦ (((((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))))‘(𝑌 ↾ 𝐽))‘(𝑌 ↾ (𝐼 ∖ 𝐽)))))
337		eqid 2737	. . . . . . . . . 10 ⊢ (._r‘((𝐼 ∖ 𝐽) mPoly 𝑅)) = (._r‘((𝐼 ∖ 𝐽) mPoly 𝑅))
338	4, 216, 54, 15, 337, 265, 227, 241, 328	mplvscaval 22004	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))))‘(𝑌 ↾ 𝐽)) = (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))(._r‘((𝐼 ∖ 𝐽) mPoly 𝑅))((((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘)))))‘(𝑌 ↾ 𝐽))))
339	4, 216, 54, 15, 337, 265, 239, 215, 328	mplvscaval 22004	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘)))))‘(𝑌 ↾ 𝐽)) = (((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))(._r‘((𝐼 ∖ 𝐽) mPoly 𝑅))(((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))‘(𝑌 ↾ 𝐽))))
340	339	oveq2d 7376	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))(._r‘((𝐼 ∖ 𝐽) mPoly 𝑅))((((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘)))))‘(𝑌 ↾ 𝐽))) = (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))(._r‘((𝐼 ∖ 𝐽) mPoly 𝑅))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))(._r‘((𝐼 ∖ 𝐽) mPoly 𝑅))(((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))‘(𝑌 ↾ 𝐽)))))
341	31	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((𝐼 ∖ 𝐽) mPoly 𝑅) ∈ AssAlg)
342	36	fveq2d 6838	. . . . . . . . . . . 12 ⊢ (𝜑 → (Base‘𝑅) = (Base‘(Scalar‘((𝐼 ∖ 𝐽) mPoly 𝑅))))
343	342	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (Base‘𝑅) = (Base‘(Scalar‘((𝐼 ∖ 𝐽) mPoly 𝑅))))
344	75, 343	eleqtrd 2839	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (𝐹‘𝑔) ∈ (Base‘(Scalar‘((𝐼 ∖ 𝐽) mPoly 𝑅))))
345	49	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((𝐼 ∖ 𝐽) mPoly 𝑅) ∈ Ring)
346	4, 54, 15, 265, 215	mplelf 21986	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘)))):{𝑥 ∈ (ℕ₀ ↑_m 𝐽) ∣ (^◡𝑥 “ ℕ) ∈ Fin}⟶(Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)))
347	346, 328	ffvelcdmd 7031	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))‘(𝑌 ↾ 𝐽)) ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)))
348	54, 337, 345, 239, 347	ringcld 20232	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))(._r‘((𝐼 ∖ 𝐽) mPoly 𝑅))(((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))‘(𝑌 ↾ 𝐽))) ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)))
349		eqid 2737	. . . . . . . . . . 11 ⊢ (Base‘(Scalar‘((𝐼 ∖ 𝐽) mPoly 𝑅))) = (Base‘(Scalar‘((𝐼 ∖ 𝐽) mPoly 𝑅)))
350		eqid 2737	. . . . . . . . . . 11 ⊢ ( ·_𝑠 ‘((𝐼 ∖ 𝐽) mPoly 𝑅)) = ( ·_𝑠 ‘((𝐼 ∖ 𝐽) mPoly 𝑅))
351	32, 33, 349, 54, 337, 350	asclmul1 21876	. . . . . . . . . 10 ⊢ ((((𝐼 ∖ 𝐽) mPoly 𝑅) ∈ AssAlg ∧ (𝐹‘𝑔) ∈ (Base‘(Scalar‘((𝐼 ∖ 𝐽) mPoly 𝑅))) ∧ (((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))(._r‘((𝐼 ∖ 𝐽) mPoly 𝑅))(((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))‘(𝑌 ↾ 𝐽))) ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅))) → (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))(._r‘((𝐼 ∖ 𝐽) mPoly 𝑅))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))(._r‘((𝐼 ∖ 𝐽) mPoly 𝑅))(((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))‘(𝑌 ↾ 𝐽)))) = ((𝐹‘𝑔)( ·_𝑠 ‘((𝐼 ∖ 𝐽) mPoly 𝑅))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))(._r‘((𝐼 ∖ 𝐽) mPoly 𝑅))(((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))‘(𝑌 ↾ 𝐽)))))
352	341, 344, 348, 351	syl3anc 1374	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))(._r‘((𝐼 ∖ 𝐽) mPoly 𝑅))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))(._r‘((𝐼 ∖ 𝐽) mPoly 𝑅))(((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))‘(𝑌 ↾ 𝐽)))) = ((𝐹‘𝑔)( ·_𝑠 ‘((𝐼 ∖ 𝐽) mPoly 𝑅))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))(._r‘((𝐼 ∖ 𝐽) mPoly 𝑅))(((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))‘(𝑌 ↾ 𝐽)))))
353	338, 340, 352	3eqtrd 2776	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))))‘(𝑌 ↾ 𝐽)) = ((𝐹‘𝑔)( ·_𝑠 ‘((𝐼 ∖ 𝐽) mPoly 𝑅))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))(._r‘((𝐼 ∖ 𝐽) mPoly 𝑅))(((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))‘(𝑌 ↾ 𝐽)))))
354	353	fveq1d 6836	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (((((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))))‘(𝑌 ↾ 𝐽))‘(𝑌 ↾ (𝐼 ∖ 𝐽))) = (((𝐹‘𝑔)( ·_𝑠 ‘((𝐼 ∖ 𝐽) mPoly 𝑅))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))(._r‘((𝐼 ∖ 𝐽) mPoly 𝑅))(((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))‘(𝑌 ↾ 𝐽))))‘(𝑌 ↾ (𝐼 ∖ 𝐽))))
355		eqid 2737	. . . . . . . 8 ⊢ (._r‘𝑅) = (._r‘𝑅)
356	261	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (𝑌 ↾ (𝐼 ∖ 𝐽)) ∈ {𝑦 ∈ (ℕ₀ ↑_m (𝐼 ∖ 𝐽)) ∣ (^◡𝑦 “ ℕ) ∈ Fin})
357	3, 350, 68, 54, 355, 257, 75, 348, 356	mplvscaval 22004	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (((𝐹‘𝑔)( ·_𝑠 ‘((𝐼 ∖ 𝐽) mPoly 𝑅))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))(._r‘((𝐼 ∖ 𝐽) mPoly 𝑅))(((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))‘(𝑌 ↾ 𝐽))))‘(𝑌 ↾ (𝐼 ∖ 𝐽))) = ((𝐹‘𝑔)(._r‘𝑅)((((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))(._r‘((𝐼 ∖ 𝐽) mPoly 𝑅))(((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))‘(𝑌 ↾ 𝐽)))‘(𝑌 ↾ (𝐼 ∖ 𝐽)))))
358		ovif2 7459	. . . . . . . . . . . . 13 ⊢ ((𝑖 ∈ {𝑦 ∈ (ℕ₀ ↑_m (𝐼 ∖ 𝐽)) ∣ (^◡𝑦 “ ℕ) ∈ Fin} ↦ if(𝑖 = (𝑔 ↾ (𝐼 ∖ 𝐽)), (1_r‘𝑅), (0_g‘𝑅)))(._r‘((𝐼 ∖ 𝐽) mPoly 𝑅))if((𝑌 ↾ 𝐽) = (𝑔 ↾ 𝐽), (1_r‘((𝐼 ∖ 𝐽) mPoly 𝑅)), (0_g‘((𝐼 ∖ 𝐽) mPoly 𝑅)))) = if((𝑌 ↾ 𝐽) = (𝑔 ↾ 𝐽), ((𝑖 ∈ {𝑦 ∈ (ℕ₀ ↑_m (𝐼 ∖ 𝐽)) ∣ (^◡𝑦 “ ℕ) ∈ Fin} ↦ if(𝑖 = (𝑔 ↾ (𝐼 ∖ 𝐽)), (1_r‘𝑅), (0_g‘𝑅)))(._r‘((𝐼 ∖ 𝐽) mPoly 𝑅))(1_r‘((𝐼 ∖ 𝐽) mPoly 𝑅))), ((𝑖 ∈ {𝑦 ∈ (ℕ₀ ↑_m (𝐼 ∖ 𝐽)) ∣ (^◡𝑦 “ ℕ) ∈ Fin} ↦ if(𝑖 = (𝑔 ↾ (𝐼 ∖ 𝐽)), (1_r‘𝑅), (0_g‘𝑅)))(._r‘((𝐼 ∖ 𝐽) mPoly 𝑅))(0_g‘((𝐼 ∖ 𝐽) mPoly 𝑅))))
359	358	fveq1i 6835	. . . . . . . . . . . 12 ⊢ (((𝑖 ∈ {𝑦 ∈ (ℕ₀ ↑_m (𝐼 ∖ 𝐽)) ∣ (^◡𝑦 “ ℕ) ∈ Fin} ↦ if(𝑖 = (𝑔 ↾ (𝐼 ∖ 𝐽)), (1_r‘𝑅), (0_g‘𝑅)))(._r‘((𝐼 ∖ 𝐽) mPoly 𝑅))if((𝑌 ↾ 𝐽) = (𝑔 ↾ 𝐽), (1_r‘((𝐼 ∖ 𝐽) mPoly 𝑅)), (0_g‘((𝐼 ∖ 𝐽) mPoly 𝑅))))‘(𝑌 ↾ (𝐼 ∖ 𝐽))) = (if((𝑌 ↾ 𝐽) = (𝑔 ↾ 𝐽), ((𝑖 ∈ {𝑦 ∈ (ℕ₀ ↑_m (𝐼 ∖ 𝐽)) ∣ (^◡𝑦 “ ℕ) ∈ Fin} ↦ if(𝑖 = (𝑔 ↾ (𝐼 ∖ 𝐽)), (1_r‘𝑅), (0_g‘𝑅)))(._r‘((𝐼 ∖ 𝐽) mPoly 𝑅))(1_r‘((𝐼 ∖ 𝐽) mPoly 𝑅))), ((𝑖 ∈ {𝑦 ∈ (ℕ₀ ↑_m (𝐼 ∖ 𝐽)) ∣ (^◡𝑦 “ ℕ) ∈ Fin} ↦ if(𝑖 = (𝑔 ↾ (𝐼 ∖ 𝐽)), (1_r‘𝑅), (0_g‘𝑅)))(._r‘((𝐼 ∖ 𝐽) mPoly 𝑅))(0_g‘((𝐼 ∖ 𝐽) mPoly 𝑅))))‘(𝑌 ↾ (𝐼 ∖ 𝐽)))
360		iffv 6851	. . . . . . . . . . . 12 ⊢ (if((𝑌 ↾ 𝐽) = (𝑔 ↾ 𝐽), ((𝑖 ∈ {𝑦 ∈ (ℕ₀ ↑_m (𝐼 ∖ 𝐽)) ∣ (^◡𝑦 “ ℕ) ∈ Fin} ↦ if(𝑖 = (𝑔 ↾ (𝐼 ∖ 𝐽)), (1_r‘𝑅), (0_g‘𝑅)))(._r‘((𝐼 ∖ 𝐽) mPoly 𝑅))(1_r‘((𝐼 ∖ 𝐽) mPoly 𝑅))), ((𝑖 ∈ {𝑦 ∈ (ℕ₀ ↑_m (𝐼 ∖ 𝐽)) ∣ (^◡𝑦 “ ℕ) ∈ Fin} ↦ if(𝑖 = (𝑔 ↾ (𝐼 ∖ 𝐽)), (1_r‘𝑅), (0_g‘𝑅)))(._r‘((𝐼 ∖ 𝐽) mPoly 𝑅))(0_g‘((𝐼 ∖ 𝐽) mPoly 𝑅))))‘(𝑌 ↾ (𝐼 ∖ 𝐽))) = if((𝑌 ↾ 𝐽) = (𝑔 ↾ 𝐽), (((𝑖 ∈ {𝑦 ∈ (ℕ₀ ↑_m (𝐼 ∖ 𝐽)) ∣ (^◡𝑦 “ ℕ) ∈ Fin} ↦ if(𝑖 = (𝑔 ↾ (𝐼 ∖ 𝐽)), (1_r‘𝑅), (0_g‘𝑅)))(._r‘((𝐼 ∖ 𝐽) mPoly 𝑅))(1_r‘((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(𝑌 ↾ (𝐼 ∖ 𝐽))), (((𝑖 ∈ {𝑦 ∈ (ℕ₀ ↑_m (𝐼 ∖ 𝐽)) ∣ (^◡𝑦 “ ℕ) ∈ Fin} ↦ if(𝑖 = (𝑔 ↾ (𝐼 ∖ 𝐽)), (1_r‘𝑅), (0_g‘𝑅)))(._r‘((𝐼 ∖ 𝐽) mPoly 𝑅))(0_g‘((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(𝑌 ↾ (𝐼 ∖ 𝐽))))
361	359, 360	eqtri 2760	. . . . . . . . . . 11 ⊢ (((𝑖 ∈ {𝑦 ∈ (ℕ₀ ↑_m (𝐼 ∖ 𝐽)) ∣ (^◡𝑦 “ ℕ) ∈ Fin} ↦ if(𝑖 = (𝑔 ↾ (𝐼 ∖ 𝐽)), (1_r‘𝑅), (0_g‘𝑅)))(._r‘((𝐼 ∖ 𝐽) mPoly 𝑅))if((𝑌 ↾ 𝐽) = (𝑔 ↾ 𝐽), (1_r‘((𝐼 ∖ 𝐽) mPoly 𝑅)), (0_g‘((𝐼 ∖ 𝐽) mPoly 𝑅))))‘(𝑌 ↾ (𝐼 ∖ 𝐽))) = if((𝑌 ↾ 𝐽) = (𝑔 ↾ 𝐽), (((𝑖 ∈ {𝑦 ∈ (ℕ₀ ↑_m (𝐼 ∖ 𝐽)) ∣ (^◡𝑦 “ ℕ) ∈ Fin} ↦ if(𝑖 = (𝑔 ↾ (𝐼 ∖ 𝐽)), (1_r‘𝑅), (0_g‘𝑅)))(._r‘((𝐼 ∖ 𝐽) mPoly 𝑅))(1_r‘((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(𝑌 ↾ (𝐼 ∖ 𝐽))), (((𝑖 ∈ {𝑦 ∈ (ℕ₀ ↑_m (𝐼 ∖ 𝐽)) ∣ (^◡𝑦 “ ℕ) ∈ Fin} ↦ if(𝑖 = (𝑔 ↾ (𝐼 ∖ 𝐽)), (1_r‘𝑅), (0_g‘𝑅)))(._r‘((𝐼 ∖ 𝐽) mPoly 𝑅))(0_g‘((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(𝑌 ↾ (𝐼 ∖ 𝐽))))
362		eqeq1 2741	. . . . . . . . . . . . . 14 ⊢ (𝑖 = (𝑌 ↾ (𝐼 ∖ 𝐽)) → (𝑖 = (𝑔 ↾ (𝐼 ∖ 𝐽)) ↔ (𝑌 ↾ (𝐼 ∖ 𝐽)) = (𝑔 ↾ (𝐼 ∖ 𝐽))))
363	362	ifbid 4491	. . . . . . . . . . . . 13 ⊢ (𝑖 = (𝑌 ↾ (𝐼 ∖ 𝐽)) → if(𝑖 = (𝑔 ↾ (𝐼 ∖ 𝐽)), (1_r‘𝑅), (0_g‘𝑅)) = if((𝑌 ↾ (𝐼 ∖ 𝐽)) = (𝑔 ↾ (𝐼 ∖ 𝐽)), (1_r‘𝑅), (0_g‘𝑅)))
364		eqid 2737	. . . . . . . . . . . . . 14 ⊢ (1_r‘((𝐼 ∖ 𝐽) mPoly 𝑅)) = (1_r‘((𝐼 ∖ 𝐽) mPoly 𝑅))
365		eqid 2737	. . . . . . . . . . . . . . 15 ⊢ (1_r‘𝑅) = (1_r‘𝑅)
366	58	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → 𝑅 ∈ Ring)
367	14, 257, 84, 151, 71	psrbagres 43003	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (𝑔 ↾ (𝐼 ∖ 𝐽)) ∈ {𝑦 ∈ (ℕ₀ ↑_m (𝐼 ∖ 𝐽)) ∣ (^◡𝑦 “ ℕ) ∈ Fin})
368	3, 54, 286, 365, 257, 176, 366, 367	mplmon 22023	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (𝑖 ∈ {𝑦 ∈ (ℕ₀ ↑_m (𝐼 ∖ 𝐽)) ∣ (^◡𝑦 “ ℕ) ∈ Fin} ↦ if(𝑖 = (𝑔 ↾ (𝐼 ∖ 𝐽)), (1_r‘𝑅), (0_g‘𝑅))) ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)))
369	54, 337, 364, 345, 368	ringridmd 20245	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((𝑖 ∈ {𝑦 ∈ (ℕ₀ ↑_m (𝐼 ∖ 𝐽)) ∣ (^◡𝑦 “ ℕ) ∈ Fin} ↦ if(𝑖 = (𝑔 ↾ (𝐼 ∖ 𝐽)), (1_r‘𝑅), (0_g‘𝑅)))(._r‘((𝐼 ∖ 𝐽) mPoly 𝑅))(1_r‘((𝐼 ∖ 𝐽) mPoly 𝑅))) = (𝑖 ∈ {𝑦 ∈ (ℕ₀ ↑_m (𝐼 ∖ 𝐽)) ∣ (^◡𝑦 “ ℕ) ∈ Fin} ↦ if(𝑖 = (𝑔 ↾ (𝐼 ∖ 𝐽)), (1_r‘𝑅), (0_g‘𝑅))))
370		fvexd 6849	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (1_r‘𝑅) ∈ V)
371		fvexd 6849	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (0_g‘𝑅) ∈ V)
372	370, 371	ifcld 4514	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → if((𝑌 ↾ (𝐼 ∖ 𝐽)) = (𝑔 ↾ (𝐼 ∖ 𝐽)), (1_r‘𝑅), (0_g‘𝑅)) ∈ V)
373	363, 369, 356, 372	fvmptd4 6966	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (((𝑖 ∈ {𝑦 ∈ (ℕ₀ ↑_m (𝐼 ∖ 𝐽)) ∣ (^◡𝑦 “ ℕ) ∈ Fin} ↦ if(𝑖 = (𝑔 ↾ (𝐼 ∖ 𝐽)), (1_r‘𝑅), (0_g‘𝑅)))(._r‘((𝐼 ∖ 𝐽) mPoly 𝑅))(1_r‘((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(𝑌 ↾ (𝐼 ∖ 𝐽))) = if((𝑌 ↾ (𝐼 ∖ 𝐽)) = (𝑔 ↾ (𝐼 ∖ 𝐽)), (1_r‘𝑅), (0_g‘𝑅)))
374	54, 337, 250, 345, 368	ringrzd 20268	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((𝑖 ∈ {𝑦 ∈ (ℕ₀ ↑_m (𝐼 ∖ 𝐽)) ∣ (^◡𝑦 “ ℕ) ∈ Fin} ↦ if(𝑖 = (𝑔 ↾ (𝐼 ∖ 𝐽)), (1_r‘𝑅), (0_g‘𝑅)))(._r‘((𝐼 ∖ 𝐽) mPoly 𝑅))(0_g‘((𝐼 ∖ 𝐽) mPoly 𝑅))) = (0_g‘((𝐼 ∖ 𝐽) mPoly 𝑅)))
375	3, 257, 286, 250, 22, 252	mpl0 21994	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (0_g‘((𝐼 ∖ 𝐽) mPoly 𝑅)) = ({𝑦 ∈ (ℕ₀ ↑_m (𝐼 ∖ 𝐽)) ∣ (^◡𝑦 “ ℕ) ∈ Fin} × {(0_g‘𝑅)}))
376	375	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (0_g‘((𝐼 ∖ 𝐽) mPoly 𝑅)) = ({𝑦 ∈ (ℕ₀ ↑_m (𝐼 ∖ 𝐽)) ∣ (^◡𝑦 “ ℕ) ∈ Fin} × {(0_g‘𝑅)}))
377	374, 376	eqtrd 2772	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((𝑖 ∈ {𝑦 ∈ (ℕ₀ ↑_m (𝐼 ∖ 𝐽)) ∣ (^◡𝑦 “ ℕ) ∈ Fin} ↦ if(𝑖 = (𝑔 ↾ (𝐼 ∖ 𝐽)), (1_r‘𝑅), (0_g‘𝑅)))(._r‘((𝐼 ∖ 𝐽) mPoly 𝑅))(0_g‘((𝐼 ∖ 𝐽) mPoly 𝑅))) = ({𝑦 ∈ (ℕ₀ ↑_m (𝐼 ∖ 𝐽)) ∣ (^◡𝑦 “ ℕ) ∈ Fin} × {(0_g‘𝑅)}))
378	377	fveq1d 6836	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (((𝑖 ∈ {𝑦 ∈ (ℕ₀ ↑_m (𝐼 ∖ 𝐽)) ∣ (^◡𝑦 “ ℕ) ∈ Fin} ↦ if(𝑖 = (𝑔 ↾ (𝐼 ∖ 𝐽)), (1_r‘𝑅), (0_g‘𝑅)))(._r‘((𝐼 ∖ 𝐽) mPoly 𝑅))(0_g‘((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(𝑌 ↾ (𝐼 ∖ 𝐽))) = (({𝑦 ∈ (ℕ₀ ↑_m (𝐼 ∖ 𝐽)) ∣ (^◡𝑦 “ ℕ) ∈ Fin} × {(0_g‘𝑅)})‘(𝑌 ↾ (𝐼 ∖ 𝐽))))
379		fvex 6847	. . . . . . . . . . . . . . 15 ⊢ (0_g‘𝑅) ∈ V
380	379	fvconst2 7152	. . . . . . . . . . . . . 14 ⊢ ((𝑌 ↾ (𝐼 ∖ 𝐽)) ∈ {𝑦 ∈ (ℕ₀ ↑_m (𝐼 ∖ 𝐽)) ∣ (^◡𝑦 “ ℕ) ∈ Fin} → (({𝑦 ∈ (ℕ₀ ↑_m (𝐼 ∖ 𝐽)) ∣ (^◡𝑦 “ ℕ) ∈ Fin} × {(0_g‘𝑅)})‘(𝑌 ↾ (𝐼 ∖ 𝐽))) = (0_g‘𝑅))
381	356, 380	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (({𝑦 ∈ (ℕ₀ ↑_m (𝐼 ∖ 𝐽)) ∣ (^◡𝑦 “ ℕ) ∈ Fin} × {(0_g‘𝑅)})‘(𝑌 ↾ (𝐼 ∖ 𝐽))) = (0_g‘𝑅))
382	378, 381	eqtrd 2772	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (((𝑖 ∈ {𝑦 ∈ (ℕ₀ ↑_m (𝐼 ∖ 𝐽)) ∣ (^◡𝑦 “ ℕ) ∈ Fin} ↦ if(𝑖 = (𝑔 ↾ (𝐼 ∖ 𝐽)), (1_r‘𝑅), (0_g‘𝑅)))(._r‘((𝐼 ∖ 𝐽) mPoly 𝑅))(0_g‘((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(𝑌 ↾ (𝐼 ∖ 𝐽))) = (0_g‘𝑅))
383	373, 382	ifeq12d 4489	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → if((𝑌 ↾ 𝐽) = (𝑔 ↾ 𝐽), (((𝑖 ∈ {𝑦 ∈ (ℕ₀ ↑_m (𝐼 ∖ 𝐽)) ∣ (^◡𝑦 “ ℕ) ∈ Fin} ↦ if(𝑖 = (𝑔 ↾ (𝐼 ∖ 𝐽)), (1_r‘𝑅), (0_g‘𝑅)))(._r‘((𝐼 ∖ 𝐽) mPoly 𝑅))(1_r‘((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(𝑌 ↾ (𝐼 ∖ 𝐽))), (((𝑖 ∈ {𝑦 ∈ (ℕ₀ ↑_m (𝐼 ∖ 𝐽)) ∣ (^◡𝑦 “ ℕ) ∈ Fin} ↦ if(𝑖 = (𝑔 ↾ (𝐼 ∖ 𝐽)), (1_r‘𝑅), (0_g‘𝑅)))(._r‘((𝐼 ∖ 𝐽) mPoly 𝑅))(0_g‘((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(𝑌 ↾ (𝐼 ∖ 𝐽)))) = if((𝑌 ↾ 𝐽) = (𝑔 ↾ 𝐽), if((𝑌 ↾ (𝐼 ∖ 𝐽)) = (𝑔 ↾ (𝐼 ∖ 𝐽)), (1_r‘𝑅), (0_g‘𝑅)), (0_g‘𝑅)))
384	361, 383	eqtrid 2784	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (((𝑖 ∈ {𝑦 ∈ (ℕ₀ ↑_m (𝐼 ∖ 𝐽)) ∣ (^◡𝑦 “ ℕ) ∈ Fin} ↦ if(𝑖 = (𝑔 ↾ (𝐼 ∖ 𝐽)), (1_r‘𝑅), (0_g‘𝑅)))(._r‘((𝐼 ∖ 𝐽) mPoly 𝑅))if((𝑌 ↾ 𝐽) = (𝑔 ↾ 𝐽), (1_r‘((𝐼 ∖ 𝐽) mPoly 𝑅)), (0_g‘((𝐼 ∖ 𝐽) mPoly 𝑅))))‘(𝑌 ↾ (𝐼 ∖ 𝐽))) = if((𝑌 ↾ 𝐽) = (𝑔 ↾ 𝐽), if((𝑌 ↾ (𝐼 ∖ 𝐽)) = (𝑔 ↾ (𝐼 ∖ 𝐽)), (1_r‘𝑅), (0_g‘𝑅)), (0_g‘𝑅)))
385	384	oveq2d 7376	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((𝐹‘𝑔)(._r‘𝑅)(((𝑖 ∈ {𝑦 ∈ (ℕ₀ ↑_m (𝐼 ∖ 𝐽)) ∣ (^◡𝑦 “ ℕ) ∈ Fin} ↦ if(𝑖 = (𝑔 ↾ (𝐼 ∖ 𝐽)), (1_r‘𝑅), (0_g‘𝑅)))(._r‘((𝐼 ∖ 𝐽) mPoly 𝑅))if((𝑌 ↾ 𝐽) = (𝑔 ↾ 𝐽), (1_r‘((𝐼 ∖ 𝐽) mPoly 𝑅)), (0_g‘((𝐼 ∖ 𝐽) mPoly 𝑅))))‘(𝑌 ↾ (𝐼 ∖ 𝐽)))) = ((𝐹‘𝑔)(._r‘𝑅)if((𝑌 ↾ 𝐽) = (𝑔 ↾ 𝐽), if((𝑌 ↾ (𝐼 ∖ 𝐽)) = (𝑔 ↾ (𝐼 ∖ 𝐽)), (1_r‘𝑅), (0_g‘𝑅)), (0_g‘𝑅))))
386		ifan 4521	. . . . . . . . . . 11 ⊢ if(((𝑌 ↾ 𝐽) = (𝑔 ↾ 𝐽) ∧ (𝑌 ↾ (𝐼 ∖ 𝐽)) = (𝑔 ↾ (𝐼 ∖ 𝐽))), (1_r‘𝑅), (0_g‘𝑅)) = if((𝑌 ↾ 𝐽) = (𝑔 ↾ 𝐽), if((𝑌 ↾ (𝐼 ∖ 𝐽)) = (𝑔 ↾ (𝐼 ∖ 𝐽)), (1_r‘𝑅), (0_g‘𝑅)), (0_g‘𝑅))
387	386	oveq2i 7371	. . . . . . . . . 10 ⊢ ((𝐹‘𝑔)(._r‘𝑅)if(((𝑌 ↾ 𝐽) = (𝑔 ↾ 𝐽) ∧ (𝑌 ↾ (𝐼 ∖ 𝐽)) = (𝑔 ↾ (𝐼 ∖ 𝐽))), (1_r‘𝑅), (0_g‘𝑅))) = ((𝐹‘𝑔)(._r‘𝑅)if((𝑌 ↾ 𝐽) = (𝑔 ↾ 𝐽), if((𝑌 ↾ (𝐼 ∖ 𝐽)) = (𝑔 ↾ (𝐼 ∖ 𝐽)), (1_r‘𝑅), (0_g‘𝑅)), (0_g‘𝑅)))
388	14	psrbagf 21908	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑌 ∈ 𝐷 → 𝑌:𝐼⟶ℕ₀)
389	260, 388	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑌:𝐼⟶ℕ₀)
390	389	ffnd 6663	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑌 Fn 𝐼)
391	390	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → 𝑌 Fn 𝐼)
392		undif 4423	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐽 ⊆ 𝐼 ↔ (𝐽 ∪ (𝐼 ∖ 𝐽)) = 𝐼)
393	8, 392	sylib 218	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐽 ∪ (𝐼 ∖ 𝐽)) = 𝐼)
394	393	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (𝐽 ∪ (𝐼 ∖ 𝐽)) = 𝐼)
395	394	fneq2d 6586	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (𝑌 Fn (𝐽 ∪ (𝐼 ∖ 𝐽)) ↔ 𝑌 Fn 𝐼))
396	391, 395	mpbird 257	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → 𝑌 Fn (𝐽 ∪ (𝐼 ∖ 𝐽)))
397	88	ffnd 6663	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → 𝑔 Fn 𝐼)
398	394	fneq2d 6586	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (𝑔 Fn (𝐽 ∪ (𝐼 ∖ 𝐽)) ↔ 𝑔 Fn 𝐼))
399	397, 398	mpbird 257	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → 𝑔 Fn (𝐽 ∪ (𝐼 ∖ 𝐽)))
400		eqfnun 6983	. . . . . . . . . . . . . 14 ⊢ ((𝑌 Fn (𝐽 ∪ (𝐼 ∖ 𝐽)) ∧ 𝑔 Fn (𝐽 ∪ (𝐼 ∖ 𝐽))) → (𝑌 = 𝑔 ↔ ((𝑌 ↾ 𝐽) = (𝑔 ↾ 𝐽) ∧ (𝑌 ↾ (𝐼 ∖ 𝐽)) = (𝑔 ↾ (𝐼 ∖ 𝐽)))))
401	396, 399, 400	syl2anc 585	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (𝑌 = 𝑔 ↔ ((𝑌 ↾ 𝐽) = (𝑔 ↾ 𝐽) ∧ (𝑌 ↾ (𝐼 ∖ 𝐽)) = (𝑔 ↾ (𝐼 ∖ 𝐽)))))
402	401	ifbid 4491	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → if(𝑌 = 𝑔, (1_r‘𝑅), (0_g‘𝑅)) = if(((𝑌 ↾ 𝐽) = (𝑔 ↾ 𝐽) ∧ (𝑌 ↾ (𝐼 ∖ 𝐽)) = (𝑔 ↾ (𝐼 ∖ 𝐽))), (1_r‘𝑅), (0_g‘𝑅)))
403	402	oveq2d 7376	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((𝐹‘𝑔)(._r‘𝑅)if(𝑌 = 𝑔, (1_r‘𝑅), (0_g‘𝑅))) = ((𝐹‘𝑔)(._r‘𝑅)if(((𝑌 ↾ 𝐽) = (𝑔 ↾ 𝐽) ∧ (𝑌 ↾ (𝐼 ∖ 𝐽)) = (𝑔 ↾ (𝐼 ∖ 𝐽))), (1_r‘𝑅), (0_g‘𝑅))))
404		ovif2 7459	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑔)(._r‘𝑅)if(𝑌 = 𝑔, (1_r‘𝑅), (0_g‘𝑅))) = if(𝑌 = 𝑔, ((𝐹‘𝑔)(._r‘𝑅)(1_r‘𝑅)), ((𝐹‘𝑔)(._r‘𝑅)(0_g‘𝑅)))
405	403, 404	eqtr3di 2787	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((𝐹‘𝑔)(._r‘𝑅)if(((𝑌 ↾ 𝐽) = (𝑔 ↾ 𝐽) ∧ (𝑌 ↾ (𝐼 ∖ 𝐽)) = (𝑔 ↾ (𝐼 ∖ 𝐽))), (1_r‘𝑅), (0_g‘𝑅))) = if(𝑌 = 𝑔, ((𝐹‘𝑔)(._r‘𝑅)(1_r‘𝑅)), ((𝐹‘𝑔)(._r‘𝑅)(0_g‘𝑅))))
406	387, 405	eqtr3id 2786	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((𝐹‘𝑔)(._r‘𝑅)if((𝑌 ↾ 𝐽) = (𝑔 ↾ 𝐽), if((𝑌 ↾ (𝐼 ∖ 𝐽)) = (𝑔 ↾ (𝐼 ∖ 𝐽)), (1_r‘𝑅), (0_g‘𝑅)), (0_g‘𝑅))) = if(𝑌 = 𝑔, ((𝐹‘𝑔)(._r‘𝑅)(1_r‘𝑅)), ((𝐹‘𝑔)(._r‘𝑅)(0_g‘𝑅))))
407	385, 406	eqtrd 2772	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((𝐹‘𝑔)(._r‘𝑅)(((𝑖 ∈ {𝑦 ∈ (ℕ₀ ↑_m (𝐼 ∖ 𝐽)) ∣ (^◡𝑦 “ ℕ) ∈ Fin} ↦ if(𝑖 = (𝑔 ↾ (𝐼 ∖ 𝐽)), (1_r‘𝑅), (0_g‘𝑅)))(._r‘((𝐼 ∖ 𝐽) mPoly 𝑅))if((𝑌 ↾ 𝐽) = (𝑔 ↾ 𝐽), (1_r‘((𝐼 ∖ 𝐽) mPoly 𝑅)), (0_g‘((𝐼 ∖ 𝐽) mPoly 𝑅))))‘(𝑌 ↾ (𝐼 ∖ 𝐽)))) = if(𝑌 = 𝑔, ((𝐹‘𝑔)(._r‘𝑅)(1_r‘𝑅)), ((𝐹‘𝑔)(._r‘𝑅)(0_g‘𝑅))))
408	7	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → 𝑅 ∈ CRing)
409	3, 257, 286, 365, 176, 158, 160, 57, 408, 367	mplcoe2 22029	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (𝑖 ∈ {𝑦 ∈ (ℕ₀ ↑_m (𝐼 ∖ 𝐽)) ∣ (^◡𝑦 “ ℕ) ∈ Fin} ↦ if(𝑖 = (𝑔 ↾ (𝐼 ∖ 𝐽)), (1_r‘𝑅), (0_g‘𝑅))) = ((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (((𝑔 ↾ (𝐼 ∖ 𝐽))‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘)))))
410		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → 𝑘 ∈ (𝐼 ∖ 𝐽))
411	410	fvresd 6854	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → ((𝑔 ↾ (𝐼 ∖ 𝐽))‘𝑘) = (𝑔‘𝑘))
412	411	oveq1d 7375	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → (((𝑔 ↾ (𝐼 ∖ 𝐽))‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘)) = ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘)))
413	412	mpteq2dva 5179	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (((𝑔 ↾ (𝐼 ∖ 𝐽))‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))) = (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))
414	413	oveq2d 7376	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (((𝑔 ↾ (𝐼 ∖ 𝐽))‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘)))) = ((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘)))))
415	409, 414	eqtrd 2772	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (𝑖 ∈ {𝑦 ∈ (ℕ₀ ↑_m (𝐼 ∖ 𝐽)) ∣ (^◡𝑦 “ ℕ) ∈ Fin} ↦ if(𝑖 = (𝑔 ↾ (𝐼 ∖ 𝐽)), (1_r‘𝑅), (0_g‘𝑅))) = ((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘)))))
416		eqid 2737	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ {𝑥 ∈ (ℕ₀ ↑_m 𝐽) ∣ (^◡𝑥 “ ℕ) ∈ Fin} ↦ if(𝑗 = (𝑔 ↾ 𝐽), (1_r‘((𝐼 ∖ 𝐽) mPoly 𝑅)), (0_g‘((𝐼 ∖ 𝐽) mPoly 𝑅)))) = (𝑗 ∈ {𝑥 ∈ (ℕ₀ ↑_m 𝐽) ∣ (^◡𝑥 “ ℕ) ∈ Fin} ↦ if(𝑗 = (𝑔 ↾ 𝐽), (1_r‘((𝐼 ∖ 𝐽) mPoly 𝑅)), (0_g‘((𝐼 ∖ 𝐽) mPoly 𝑅))))
417		eqeq1 2741	. . . . . . . . . . . . . 14 ⊢ (𝑗 = (𝑌 ↾ 𝐽) → (𝑗 = (𝑔 ↾ 𝐽) ↔ (𝑌 ↾ 𝐽) = (𝑔 ↾ 𝐽)))
418	417	ifbid 4491	. . . . . . . . . . . . 13 ⊢ (𝑗 = (𝑌 ↾ 𝐽) → if(𝑗 = (𝑔 ↾ 𝐽), (1_r‘((𝐼 ∖ 𝐽) mPoly 𝑅)), (0_g‘((𝐼 ∖ 𝐽) mPoly 𝑅))) = if((𝑌 ↾ 𝐽) = (𝑔 ↾ 𝐽), (1_r‘((𝐼 ∖ 𝐽) mPoly 𝑅)), (0_g‘((𝐼 ∖ 𝐽) mPoly 𝑅))))
419		fvexd 6849	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (1_r‘((𝐼 ∖ 𝐽) mPoly 𝑅)) ∈ V)
420		fvexd 6849	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (0_g‘((𝐼 ∖ 𝐽) mPoly 𝑅)) ∈ V)
421	419, 420	ifcld 4514	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → if((𝑌 ↾ 𝐽) = (𝑔 ↾ 𝐽), (1_r‘((𝐼 ∖ 𝐽) mPoly 𝑅)), (0_g‘((𝐼 ∖ 𝐽) mPoly 𝑅))) ∈ V)
422	416, 418, 328, 421	fvmptd3 6965	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((𝑗 ∈ {𝑥 ∈ (ℕ₀ ↑_m 𝐽) ∣ (^◡𝑥 “ ℕ) ∈ Fin} ↦ if(𝑗 = (𝑔 ↾ 𝐽), (1_r‘((𝐼 ∖ 𝐽) mPoly 𝑅)), (0_g‘((𝐼 ∖ 𝐽) mPoly 𝑅))))‘(𝑌 ↾ 𝐽)) = if((𝑌 ↾ 𝐽) = (𝑔 ↾ 𝐽), (1_r‘((𝐼 ∖ 𝐽) mPoly 𝑅)), (0_g‘((𝐼 ∖ 𝐽) mPoly 𝑅))))
423	23	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((𝐼 ∖ 𝐽) mPoly 𝑅) ∈ CRing)
424	14, 265, 84, 191, 71	psrbagres 43003	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (𝑔 ↾ 𝐽) ∈ {𝑥 ∈ (ℕ₀ ↑_m 𝐽) ∣ (^◡𝑥 “ ℕ) ∈ Fin})
425	4, 265, 250, 364, 206, 16, 17, 48, 423, 424	mplcoe2 22029	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (𝑗 ∈ {𝑥 ∈ (ℕ₀ ↑_m 𝐽) ∣ (^◡𝑥 “ ℕ) ∈ Fin} ↦ if(𝑗 = (𝑔 ↾ 𝐽), (1_r‘((𝐼 ∖ 𝐽) mPoly 𝑅)), (0_g‘((𝐼 ∖ 𝐽) mPoly 𝑅)))) = ((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ (((𝑔 ↾ 𝐽)‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘)))))
426		simpr 484	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ 𝐽) → 𝑘 ∈ 𝐽)
427	426	fvresd 6854	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ 𝐽) → ((𝑔 ↾ 𝐽)‘𝑘) = (𝑔‘𝑘))
428	427	oveq1d 7375	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ 𝐽) → (((𝑔 ↾ 𝐽)‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘)) = ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘)))
429	428	mpteq2dva 5179	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (𝑘 ∈ 𝐽 ↦ (((𝑔 ↾ 𝐽)‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))) = (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))
430	429	oveq2d 7376	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ (((𝑔 ↾ 𝐽)‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘)))) = ((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘)))))
431	425, 430	eqtrd 2772	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (𝑗 ∈ {𝑥 ∈ (ℕ₀ ↑_m 𝐽) ∣ (^◡𝑥 “ ℕ) ∈ Fin} ↦ if(𝑗 = (𝑔 ↾ 𝐽), (1_r‘((𝐼 ∖ 𝐽) mPoly 𝑅)), (0_g‘((𝐼 ∖ 𝐽) mPoly 𝑅)))) = ((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘)))))
432	431	fveq1d 6836	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((𝑗 ∈ {𝑥 ∈ (ℕ₀ ↑_m 𝐽) ∣ (^◡𝑥 “ ℕ) ∈ Fin} ↦ if(𝑗 = (𝑔 ↾ 𝐽), (1_r‘((𝐼 ∖ 𝐽) mPoly 𝑅)), (0_g‘((𝐼 ∖ 𝐽) mPoly 𝑅))))‘(𝑌 ↾ 𝐽)) = (((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))‘(𝑌 ↾ 𝐽)))
433	422, 432	eqtr3d 2774	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → if((𝑌 ↾ 𝐽) = (𝑔 ↾ 𝐽), (1_r‘((𝐼 ∖ 𝐽) mPoly 𝑅)), (0_g‘((𝐼 ∖ 𝐽) mPoly 𝑅))) = (((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))‘(𝑌 ↾ 𝐽)))
434	415, 433	oveq12d 7378	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((𝑖 ∈ {𝑦 ∈ (ℕ₀ ↑_m (𝐼 ∖ 𝐽)) ∣ (^◡𝑦 “ ℕ) ∈ Fin} ↦ if(𝑖 = (𝑔 ↾ (𝐼 ∖ 𝐽)), (1_r‘𝑅), (0_g‘𝑅)))(._r‘((𝐼 ∖ 𝐽) mPoly 𝑅))if((𝑌 ↾ 𝐽) = (𝑔 ↾ 𝐽), (1_r‘((𝐼 ∖ 𝐽) mPoly 𝑅)), (0_g‘((𝐼 ∖ 𝐽) mPoly 𝑅)))) = (((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))(._r‘((𝐼 ∖ 𝐽) mPoly 𝑅))(((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))‘(𝑌 ↾ 𝐽))))
435	434	fveq1d 6836	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (((𝑖 ∈ {𝑦 ∈ (ℕ₀ ↑_m (𝐼 ∖ 𝐽)) ∣ (^◡𝑦 “ ℕ) ∈ Fin} ↦ if(𝑖 = (𝑔 ↾ (𝐼 ∖ 𝐽)), (1_r‘𝑅), (0_g‘𝑅)))(._r‘((𝐼 ∖ 𝐽) mPoly 𝑅))if((𝑌 ↾ 𝐽) = (𝑔 ↾ 𝐽), (1_r‘((𝐼 ∖ 𝐽) mPoly 𝑅)), (0_g‘((𝐼 ∖ 𝐽) mPoly 𝑅))))‘(𝑌 ↾ (𝐼 ∖ 𝐽))) = ((((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))(._r‘((𝐼 ∖ 𝐽) mPoly 𝑅))(((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))‘(𝑌 ↾ 𝐽)))‘(𝑌 ↾ (𝐼 ∖ 𝐽))))
436	435	oveq2d 7376	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((𝐹‘𝑔)(._r‘𝑅)(((𝑖 ∈ {𝑦 ∈ (ℕ₀ ↑_m (𝐼 ∖ 𝐽)) ∣ (^◡𝑦 “ ℕ) ∈ Fin} ↦ if(𝑖 = (𝑔 ↾ (𝐼 ∖ 𝐽)), (1_r‘𝑅), (0_g‘𝑅)))(._r‘((𝐼 ∖ 𝐽) mPoly 𝑅))if((𝑌 ↾ 𝐽) = (𝑔 ↾ 𝐽), (1_r‘((𝐼 ∖ 𝐽) mPoly 𝑅)), (0_g‘((𝐼 ∖ 𝐽) mPoly 𝑅))))‘(𝑌 ↾ (𝐼 ∖ 𝐽)))) = ((𝐹‘𝑔)(._r‘𝑅)((((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))(._r‘((𝐼 ∖ 𝐽) mPoly 𝑅))(((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))‘(𝑌 ↾ 𝐽)))‘(𝑌 ↾ (𝐼 ∖ 𝐽)))))
437	68, 355, 365, 366, 75	ringridmd 20245	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((𝐹‘𝑔)(._r‘𝑅)(1_r‘𝑅)) = (𝐹‘𝑔))
438	68, 355, 286, 366, 75	ringrzd 20268	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((𝐹‘𝑔)(._r‘𝑅)(0_g‘𝑅)) = (0_g‘𝑅))
439	437, 438	ifeq12d 4489	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → if(𝑌 = 𝑔, ((𝐹‘𝑔)(._r‘𝑅)(1_r‘𝑅)), ((𝐹‘𝑔)(._r‘𝑅)(0_g‘𝑅))) = if(𝑌 = 𝑔, (𝐹‘𝑔), (0_g‘𝑅)))
440	407, 436, 439	3eqtr3d 2780	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((𝐹‘𝑔)(._r‘𝑅)((((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))(._r‘((𝐼 ∖ 𝐽) mPoly 𝑅))(((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))‘(𝑌 ↾ 𝐽)))‘(𝑌 ↾ (𝐼 ∖ 𝐽)))) = if(𝑌 = 𝑔, (𝐹‘𝑔), (0_g‘𝑅)))
441	354, 357, 440	3eqtrd 2776	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (((((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))))‘(𝑌 ↾ 𝐽))‘(𝑌 ↾ (𝐼 ∖ 𝐽))) = if(𝑌 = 𝑔, (𝐹‘𝑔), (0_g‘𝑅)))
442	441	mpteq2dva 5179	. . . . 5 ⊢ (𝜑 → (𝑔 ∈ 𝐷 ↦ (((((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))))‘(𝑌 ↾ 𝐽))‘(𝑌 ↾ (𝐼 ∖ 𝐽)))) = (𝑔 ∈ 𝐷 ↦ if(𝑌 = 𝑔, (𝐹‘𝑔), (0_g‘𝑅))))
443	336, 442	eqtrd 2772	. . . 4 ⊢ (𝜑 → ((𝑣 ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)) ↦ (𝑣‘(𝑌 ↾ (𝐼 ∖ 𝐽)))) ∘ ((𝑤 ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ↦ (𝑤‘(𝑌 ↾ 𝐽))) ∘ (𝑔 ∈ 𝐷 ↦ (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))))))) = (𝑔 ∈ 𝐷 ↦ if(𝑌 = 𝑔, (𝐹‘𝑔), (0_g‘𝑅))))
444	443	oveq2d 7376	. . 3 ⊢ (𝜑 → (𝑅 Σ_g ((𝑣 ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)) ↦ (𝑣‘(𝑌 ↾ (𝐼 ∖ 𝐽)))) ∘ ((𝑤 ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ↦ (𝑤‘(𝑌 ↾ 𝐽))) ∘ (𝑔 ∈ 𝐷 ↦ (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘)))))))))) = (𝑅 Σ_g (𝑔 ∈ 𝐷 ↦ if(𝑌 = 𝑔, (𝐹‘𝑔), (0_g‘𝑅)))))
445	58	ringcmnd 20256	. . . 4 ⊢ (𝜑 → 𝑅 ∈ CMnd)
446	68, 286	ring0cl 20239	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → (0_g‘𝑅) ∈ (Base‘𝑅))
447	58, 446	syl 17	. . . . . . 7 ⊢ (𝜑 → (0_g‘𝑅) ∈ (Base‘𝑅))
448	447	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (0_g‘𝑅) ∈ (Base‘𝑅))
449	75, 448	ifcld 4514	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → if(𝑌 = 𝑔, (𝐹‘𝑔), (0_g‘𝑅)) ∈ (Base‘𝑅))
450	449	fmpttd 7061	. . . 4 ⊢ (𝜑 → (𝑔 ∈ 𝐷 ↦ if(𝑌 = 𝑔, (𝐹‘𝑔), (0_g‘𝑅))):𝐷⟶(Base‘𝑅))
451		eldifsnneq 4735	. . . . . . . 8 ⊢ (𝑔 ∈ (𝐷 ∖ {𝑌}) → ¬ 𝑔 = 𝑌)
452	451	neqcomd 2747	. . . . . . 7 ⊢ (𝑔 ∈ (𝐷 ∖ {𝑌}) → ¬ 𝑌 = 𝑔)
453	452	iffalsed 4478	. . . . . 6 ⊢ (𝑔 ∈ (𝐷 ∖ {𝑌}) → if(𝑌 = 𝑔, (𝐹‘𝑔), (0_g‘𝑅)) = (0_g‘𝑅))
454	453	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐷 ∖ {𝑌})) → if(𝑌 = 𝑔, (𝐹‘𝑔), (0_g‘𝑅)) = (0_g‘𝑅))
455	454, 256	suppss2 8143	. . . 4 ⊢ (𝜑 → ((𝑔 ∈ 𝐷 ↦ if(𝑌 = 𝑔, (𝐹‘𝑔), (0_g‘𝑅))) supp (0_g‘𝑅)) ⊆ {𝑌})
456	256	mptexd 7172	. . . . 5 ⊢ (𝜑 → (𝑔 ∈ 𝐷 ↦ if(𝑌 = 𝑔, (𝐹‘𝑔), (0_g‘𝑅))) ∈ V)
457		funmpt 6530	. . . . . 6 ⊢ Fun (𝑔 ∈ 𝐷 ↦ if(𝑌 = 𝑔, (𝐹‘𝑔), (0_g‘𝑅)))
458	457	a1i 11	. . . . 5 ⊢ (𝜑 → Fun (𝑔 ∈ 𝐷 ↦ if(𝑌 = 𝑔, (𝐹‘𝑔), (0_g‘𝑅))))
459		snfi 8983	. . . . . . 7 ⊢ {𝑌} ∈ Fin
460	459	a1i 11	. . . . . 6 ⊢ (𝜑 → {𝑌} ∈ Fin)
461	460, 455	ssfid 9172	. . . . 5 ⊢ (𝜑 → ((𝑔 ∈ 𝐷 ↦ if(𝑌 = 𝑔, (𝐹‘𝑔), (0_g‘𝑅))) supp (0_g‘𝑅)) ∈ Fin)
462	456, 447, 458, 461	isfsuppd 9272	. . . 4 ⊢ (𝜑 → (𝑔 ∈ 𝐷 ↦ if(𝑌 = 𝑔, (𝐹‘𝑔), (0_g‘𝑅))) finSupp (0_g‘𝑅))
463	68, 286, 445, 256, 450, 455, 462	gsumres 19879	. . 3 ⊢ (𝜑 → (𝑅 Σ_g ((𝑔 ∈ 𝐷 ↦ if(𝑌 = 𝑔, (𝐹‘𝑔), (0_g‘𝑅))) ↾ {𝑌})) = (𝑅 Σ_g (𝑔 ∈ 𝐷 ↦ if(𝑌 = 𝑔, (𝐹‘𝑔), (0_g‘𝑅)))))
464	260	snssd 4753	. . . . . 6 ⊢ (𝜑 → {𝑌} ⊆ 𝐷)
465	464	resmptd 5999	. . . . 5 ⊢ (𝜑 → ((𝑔 ∈ 𝐷 ↦ if(𝑌 = 𝑔, (𝐹‘𝑔), (0_g‘𝑅))) ↾ {𝑌}) = (𝑔 ∈ {𝑌} ↦ if(𝑌 = 𝑔, (𝐹‘𝑔), (0_g‘𝑅))))
466	465	oveq2d 7376	. . . 4 ⊢ (𝜑 → (𝑅 Σ_g ((𝑔 ∈ 𝐷 ↦ if(𝑌 = 𝑔, (𝐹‘𝑔), (0_g‘𝑅))) ↾ {𝑌})) = (𝑅 Σ_g (𝑔 ∈ {𝑌} ↦ if(𝑌 = 𝑔, (𝐹‘𝑔), (0_g‘𝑅)))))
467	69, 260	ffvelcdmd 7031	. . . . 5 ⊢ (𝜑 → (𝐹‘𝑌) ∈ (Base‘𝑅))
468		iftrue 4473	. . . . . . . 8 ⊢ (𝑌 = 𝑔 → if(𝑌 = 𝑔, (𝐹‘𝑔), (0_g‘𝑅)) = (𝐹‘𝑔))
469	468	eqcoms 2745	. . . . . . 7 ⊢ (𝑔 = 𝑌 → if(𝑌 = 𝑔, (𝐹‘𝑔), (0_g‘𝑅)) = (𝐹‘𝑔))
470		fveq2 6834	. . . . . . 7 ⊢ (𝑔 = 𝑌 → (𝐹‘𝑔) = (𝐹‘𝑌))
471	469, 470	eqtrd 2772	. . . . . 6 ⊢ (𝑔 = 𝑌 → if(𝑌 = 𝑔, (𝐹‘𝑔), (0_g‘𝑅)) = (𝐹‘𝑌))
472	68, 471	gsumsn 19920	. . . . 5 ⊢ ((𝑅 ∈ Mnd ∧ 𝑌 ∈ 𝐷 ∧ (𝐹‘𝑌) ∈ (Base‘𝑅)) → (𝑅 Σ_g (𝑔 ∈ {𝑌} ↦ if(𝑌 = 𝑔, (𝐹‘𝑔), (0_g‘𝑅)))) = (𝐹‘𝑌))
473	253, 260, 467, 472	syl3anc 1374	. . . 4 ⊢ (𝜑 → (𝑅 Σ_g (𝑔 ∈ {𝑌} ↦ if(𝑌 = 𝑔, (𝐹‘𝑔), (0_g‘𝑅)))) = (𝐹‘𝑌))
474	466, 473	eqtrd 2772	. . 3 ⊢ (𝜑 → (𝑅 Σ_g ((𝑔 ∈ 𝐷 ↦ if(𝑌 = 𝑔, (𝐹‘𝑔), (0_g‘𝑅))) ↾ {𝑌})) = (𝐹‘𝑌))
475	444, 463, 474	3eqtr2d 2778	. 2 ⊢ (𝜑 → (𝑅 Σ_g ((𝑣 ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)) ↦ (𝑣‘(𝑌 ↾ (𝐼 ∖ 𝐽)))) ∘ ((𝑤 ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ↦ (𝑤‘(𝑌 ↾ 𝐽))) ∘ (𝑔 ∈ 𝐷 ↦ (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σ_g (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σ_g (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘)))))))))) = (𝐹‘𝑌))
476	249, 326, 475	3eqtrd 2776	1 ⊢ (𝜑 → (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑌 ↾ 𝐽))‘(𝑌 ↾ (𝐼 ∖ 𝐽))) = (𝐹‘𝑌))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1542 ∈ wcel 2114 {crab 3390 Vcvv 3430 ∖ cdif 3887 ∪ cun 3888 ∩ cin 3889 ⊆ wss 3890 ∅c0 4274 ifcif 4467 {csn 4568 class class class wbr 5086 ↦ cmpt 5167 × cxp 5622 ^◡ccnv 5623 ↾ cres 5626 “ cima 5627 ∘ ccom 5628 Fun wfun 6486 Fn wfn 6487 ⟶wf 6488 ‘cfv 6492 (class class class)co 7360 supp csupp 8103 ↑_m cmap 8766 Fincfn 8886 finSupp cfsupp 9267 0cc0 11029 ℕcn 12165 ℕ₀cn0 12428 ℤcz 12515 Basecbs 17170 ._rcmulr 17212 Scalarcsca 17214 ·_𝑠 cvsca 17215 0_gc0g 17393 Σ_g cgsu 17394 Mndcmnd 18693 MndHom cmhm 18740 ._gcmg 19034 GrpHom cghm 19178 CMndccmn 19746 mulGrpcmgp 20112 1_rcur 20153 Ringcrg 20205 CRingccrg 20206 RingHom crh 20440 LModclmod 20846 AssAlgcasa 21840 algSccascl 21842 mVar cmvr 21895 mPoly cmpl 21896 eval cevl 22061 selectVars cslv 22104

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 5212 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106

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 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-of 7624 df-ofr 7625 df-om 7811 df-1st 7935 df-2nd 7936 df-supp 8104 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8342 df-1o 8398 df-2o 8399 df-er 8636 df-map 8768 df-pm 8769 df-ixp 8839 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-fsupp 9268 df-sup 9348 df-oi 9418 df-card 9854 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-7 12240 df-8 12241 df-9 12242 df-n0 12429 df-z 12516 df-dec 12636 df-uz 12780 df-fz 13453 df-fzo 13600 df-seq 13955 df-hash 14284 df-struct 17108 df-sets 17125 df-slot 17143 df-ndx 17155 df-base 17171 df-ress 17192 df-plusg 17224 df-mulr 17225 df-sca 17227 df-vsca 17228 df-ip 17229 df-tset 17230 df-ple 17231 df-ds 17233 df-hom 17235 df-cco 17236 df-0g 17395 df-gsum 17396 df-prds 17401 df-pws 17403 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-mhm 18742 df-submnd 18743 df-grp 18903 df-minusg 18904 df-sbg 18905 df-mulg 19035 df-subg 19090 df-ghm 19179 df-cntz 19283 df-cmn 19748 df-abl 19749 df-mgp 20113 df-rng 20125 df-ur 20154 df-srg 20159 df-ring 20207 df-cring 20208 df-rhm 20443 df-subrng 20514 df-subrg 20538 df-lmod 20848 df-lss 20918 df-lsp 20958 df-assa 21843 df-asp 21844 df-ascl 21845 df-psr 21899 df-mvr 21900 df-mpl 21901 df-evls 22062 df-evl 22063 df-selv 22108

This theorem is referenced by: evlselv 43034

W3C validator