selvvvval - Mathbox for Steven Nguyen

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Theorem selvvvval 42828

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 2736	. . . . . 6 ⊢ ((𝐼 ∖ 𝐽) mPoly 𝑅) = ((𝐼 ∖ 𝐽) mPoly 𝑅)
4		eqid 2736	. . . . . 6 ⊢ (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)) = (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))
5		eqid 2736	. . . . . 6 ⊢ (algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) = (algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))
6		eqid 2736	. . . . . 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 42827	. . . . 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 2736	. . . . . 6 ⊢ (𝐼 eval (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) = (𝐼 eval (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))
12		eqid 2736	. . . . . 6 ⊢ (𝐼 mPoly (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) = (𝐼 mPoly (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))
13		eqid 2736	. . . . . 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 2736	. . . . . 6 ⊢ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) = (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))
16		eqid 2736	. . . . . 6 ⊢ (mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) = (mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))
17		eqid 2736	. . . . . 6 ⊢ (._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) = (._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
18		eqid 2736	. . . . . 6 ⊢ (._r‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) = (._r‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))
19	1, 2	mplrcl 21949	. . . . . . 7 ⊢ (𝐹 ∈ 𝐵 → 𝐼 ∈ V)
20	9, 19	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ V)
21	20, 8	ssexd 5269	. . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ V)
22	20	difexd 5276	. . . . . . . 8 ⊢ (𝜑 → (𝐼 ∖ 𝐽) ∈ V)
23	3, 22, 7	mplcrngd 42800	. . . . . . 7 ⊢ (𝜑 → ((𝐼 ∖ 𝐽) mPoly 𝑅) ∈ CRing)
24	4, 21, 23	mplcrngd 42800	. . . . . 6 ⊢ (𝜑 → (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)) ∈ CRing)
25	4	mplassa 21977	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ V ∧ ((𝐼 ∖ 𝐽) mPoly 𝑅) ∈ CRing) → (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)) ∈ AssAlg)
26	21, 23, 25	syl2anc 584	. . . . . . . . . 10 ⊢ (𝜑 → (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)) ∈ AssAlg)
27		eqid 2736	. . . . . . . . . . 11 ⊢ (Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) = (Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))
28	5, 27	asclrhm 21846	. . . . . . . . . 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 21977	. . . . . . . . . . . 12 ⊢ (((𝐼 ∖ 𝐽) ∈ V ∧ 𝑅 ∈ CRing) → ((𝐼 ∖ 𝐽) mPoly 𝑅) ∈ AssAlg)
31	22, 7, 30	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐼 ∖ 𝐽) mPoly 𝑅) ∈ AssAlg)
32		eqid 2736	. . . . . . . . . . . 12 ⊢ (algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅)) = (algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))
33		eqid 2736	. . . . . . . . . . . 12 ⊢ (Scalar‘((𝐼 ∖ 𝐽) mPoly 𝑅)) = (Scalar‘((𝐼 ∖ 𝐽) mPoly 𝑅))
34	32, 33	asclrhm 21846	. . . . . . . . . . 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 21968	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑅 = (Scalar‘((𝐼 ∖ 𝐽) mPoly 𝑅)))
37	36	eqcomd 2742	. . . . . . . . . . 11 ⊢ (𝜑 → (Scalar‘((𝐼 ∖ 𝐽) mPoly 𝑅)) = 𝑅)
38	4, 21, 23	mplsca 21968	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐼 ∖ 𝐽) mPoly 𝑅) = (Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
39	37, 38	oveq12d 7376	. . . . . . . . . 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 2838	. . . . . . . . 9 ⊢ (𝜑 → (algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅)) ∈ (𝑅 RingHom (Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))
41		rhmco 20434	. . . . . . . . 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 584	. . . . . . . 8 ⊢ (𝜑 → ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∘ (algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))) ∈ (𝑅 RingHom (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
43		rhmghm 20419	. . . . . . . 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 19155	. . . . . . . 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 22324	. . . . . 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 2736	. . . . . . . . . . . 12 ⊢ (𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅)) = (𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))
49	23	crngringd 20181	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐼 ∖ 𝐽) mPoly 𝑅) ∈ Ring)
50	4, 48, 15, 21, 49	mvrf2 21948	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅)):𝐽⟶(Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
51	50	ffvelcdmda 7029	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐽) → ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧) ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
52	51	adantlr 715	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐼) ∧ 𝑧 ∈ 𝐽) → ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧) ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
53		eldif 3911	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (𝐼 ∖ 𝐽) ↔ (𝑧 ∈ 𝐼 ∧ ¬ 𝑧 ∈ 𝐽))
54		eqid 2736	. . . . . . . . . . . . . 14 ⊢ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)) = (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅))
55	4, 15, 54, 5, 21, 49	mplasclf 22020	. . . . . . . . . . . . 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 2736	. . . . . . . . . . . . . 14 ⊢ ((𝐼 ∖ 𝐽) mVar 𝑅) = ((𝐼 ∖ 𝐽) mVar 𝑅)
58	7	crngringd 20181	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑅 ∈ Ring)
59	3, 57, 54, 22, 58	mvrf2 21948	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐼 ∖ 𝐽) mVar 𝑅):(𝐼 ∖ 𝐽)⟶(Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)))
60	59	ffvelcdmda 7029	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐼 ∖ 𝐽)) → (((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧) ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)))
61	56, 60	ffvelcdmd 7030	. . . . . . . . . . 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 595	. . . . . . . . . 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 4515	. . . . . . . 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 7060	. . . . . . 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 8778	. . . . . 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 42818	. . . . 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 2736	. . . . . . . . . . . . 13 ⊢ (Base‘𝑅) = (Base‘𝑅)
69	1, 68, 2, 14, 9	mplelf 21953	. . . . . . . . . . . 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 22020	. . . . . . . . . . . 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 7029	. . . . . . . . . . 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 2771	. . . . . . . . 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 20080	. . . . . . . . . . 11 ⊢ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) = (Base‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
79		eqid 2736	. . . . . . . . . . 11 ⊢ (0_g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) = (0_g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
80	16, 18	mgpplusg 20079	. . . . . . . . . . 11 ⊢ (._r‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) = (+_g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
81	16	crngmgp 20176	. . . . . . . . . . . . 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 19733	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∈ Mnd)
86	85	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ 𝐼) → (mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∈ Mnd)
87	14	psrbagf 21874	. . . . . . . . . . . . . . 15 ⊢ (𝑔 ∈ 𝐷 → 𝑔:𝐼⟶ℕ₀)
88	87	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → 𝑔:𝐼⟶ℕ₀)
89	88	ffvelcdmda 7029	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ 𝐼) → (𝑔‘𝑘) ∈ ℕ₀)
90		eqid 2736	. . . . . . . . . . . . . . 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 2819	. . . . . . . . . . . . . . . 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 4508	. . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ 𝐼) → (𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅)):𝐽⟶(Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
98	97	ffvelcdmda 7029	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ 𝐼) ∧ 𝑘 ∈ 𝐽) → ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘) ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
99		eldif 3911	. . . . . . . . . . . . . . . . . . 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 7029	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → (((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘) ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)))
102	100, 101	ffvelcdmd 7030	. . . . . . . . . . . . . . . . . . 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 595	. . . . . . . . . . . . . . . . . 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 719	. . . . . . . . . . . . . . . 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 4515	. . . . . . . . . . . . . . 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 6964	. . . . . . . . . . . . . 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 2836	. . . . . . . . . . . . 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 19025	. . . . . . . . . . . 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 7060	. . . . . . . . . . 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 21875	. . . . . . . . . . . . . 14 ⊢ (𝑔 ∈ 𝐷 → 𝑔 finSupp 0)
113	112	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → 𝑔 finSupp 0)
114	111, 113	eqbrtrrd 5122	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (𝑘 ∈ 𝐼 ↦ (𝑔‘𝑘)) finSupp 0)
115	78, 79, 17	mulg0 19004	. . . . . . . . . . . . 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 9287	. . . . . . . . . . 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 4425	. . . . . . . . . . . 12 ⊢ ((𝐼 ∖ 𝐽) ∩ 𝐽) = ∅
120	119	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((𝐼 ∖ 𝐽) ∩ 𝐽) = ∅)
121		undifr 4435	. . . . . . . . . . . . . 14 ⊢ (𝐽 ⊆ 𝐼 ↔ ((𝐼 ∖ 𝐽) ∪ 𝐽) = 𝐼)
122	8, 121	sylib 218	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐼 ∖ 𝐽) ∪ 𝐽) = 𝐼)
123	122	eqcomd 2742	. . . . . . . . . . . 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 19857	. . . . . . . . . 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 4083	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ (𝐼 ∖ 𝐽) → 𝑘 ∈ 𝐼)
127	126	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → 𝑘 ∈ 𝐼)
128	126, 106	sylan2 593	. . . . . . . . . . . . . . . . . . 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 6964	. . . . . . . . . . . . . . . . . 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 4084	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ (𝐼 ∖ 𝐽) → ¬ 𝑘 ∈ 𝐽)
131	130	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → ¬ 𝑘 ∈ 𝐽)
132	131	iffalsed 4490	. . . . . . . . . . . . . . . . . 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 2771	. . . . . . . . . . . . . . . . 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 7374	. . . . . . . . . . . . . . . 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 2736	. . . . . . . . . . . . . . . . . . . 20 ⊢ (mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) = (mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
136	135, 16	rhmmhm 20415	. . . . . . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . . . 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 593	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → (𝑔‘𝑘) ∈ ℕ₀)
140	101	adantlr 715	. . . . . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)) = (Base‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))
143	140, 142	eleqtrd 2838	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → (((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘) ∈ (Base‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))
144		eqid 2736	. . . . . . . . . . . . . . . . . . 19 ⊢ (Base‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) = (Base‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
145	135, 144	mgpbas 20080	. . . . . . . . . . . . . . . . . 18 ⊢ (Base‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) = (Base‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))
146		eqid 2736	. . . . . . . . . . . . . . . . . 18 ⊢ (._g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))) = (._g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))
147	145, 146, 17	mhmmulg 19045	. . . . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . . . 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 2774	. . . . . . . . . . . . . . 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 5191	. . . . . . . . . . . . . 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 4089	. . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . . . 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 7375	. . . . . . . . . . . . . . . 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 2736	. . . . . . . . . . . . . . . . . 18 ⊢ (mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) = (mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅))
159	158, 54	mgpbas 20080	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)) = (Base‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))
160		eqid 2736	. . . . . . . . . . . . . . . . 17 ⊢ (._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅))) = (._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))
161	158	crngmgp 20176	. . . . . . . . . . . . . . . . . . . 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 19733	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) ∈ Mnd)
164	163	ad2antrr 726	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → (mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) ∈ Mnd)
165	159, 160, 164, 139, 140	mulgnn0cld 19025	. . . . . . . . . . . . . . . 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 2837	. . . . . . . . . . . . . . 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 7077	. . . . . . . . . . . . . 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 2781	. . . . . . . . . . . . 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 7374	. . . . . . . . . . . 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 2736	. . . . . . . . . . . . 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 2837	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∈ CRing)
172	135	crngmgp 20176	. . . . . . . . . . . . . . 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 2838	. . . . . . . . . . . . . 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 7060	. . . . . . . . . . . . 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 12500	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → 0 ∈ ℤ)
181	114, 151, 180	fmptssfisupp 9297	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (𝑔‘𝑘)) finSupp 0)
182	141	eqimssd 3990	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)) ⊆ (Base‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))
183	182	sselda 3933	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑢 ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅))) → 𝑢 ∈ (Base‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))
184	183	adantlr 715	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑢 ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅))) → 𝑢 ∈ (Base‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))
185	145, 170, 146	mulg0 19004	. . . . . . . . . . . . . . 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 9287	. . . . . . . . . . . . 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 19867	. . . . . . . . . . . 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 2771	. . . . . . . . . . 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 3933	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ 𝐽) → 𝑘 ∈ 𝐼)
194	193, 106	syldan 591	. . . . . . . . . . . . . . . . 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 6964	. . . . . . . . . . . . . . . 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 4485	. . . . . . . . . . . . . . . . 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 2771	. . . . . . . . . . . . . . 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 7374	. . . . . . . . . . . . . 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 5191	. . . . . . . . . . . . 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 2771	. . . . . . . . . . . 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 7374	. . . . . . . . . . 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 7376	. . . . . . . . . 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 19844	. . . . . . . . . . . 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 726	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ 𝐽) → (mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∈ Mnd)
208	193, 89	syldan 591	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ 𝐽) → (𝑔‘𝑘) ∈ ℕ₀)
209	50	ffvelcdmda 7029	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐽) → ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘) ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
210	209	adantlr 715	. . . . . . . . . . . . . . 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 19025	. . . . . . . . . . . . . 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 7060	. . . . . . . . . . . . 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 9297	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (𝑘 ∈ 𝐽 ↦ (𝑔‘𝑘)) finSupp 0)
214	213, 116, 208, 210, 117	fsuppssov1 9287	. . . . . . . . . . . . 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 19844	. . . . . . . . . . . 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 2736	. . . . . . . . . . . . 13 ⊢ ( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) = ( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))
217	5, 27, 144, 15, 18, 216	asclmul1 21842	. . . . . . . . . . . 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 1373	. . . . . . . . . . 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 7375	. . . . . . . . . . . . . . 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 5192	. . . . . . . . . . . . . 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 7376	. . . . . . . . . . . . 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 7373	. . . . . . . . . . 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 2774	. . . . . . . . . 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 2775	. . . . . . . . 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 7376	. . . . . . . 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 7030	. . . . . . . . . 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 2838	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔)) ∈ (Base‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))
230	4, 21, 49	mpllmodd 21979	. . . . . . . . . . 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 2736	. . . . . . . . . . . 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 7060	. . . . . . . . . . . 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 19004	. . . . . . . . . . . . . 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 9287	. . . . . . . . . . . 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 19844	. . . . . . . . . . 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 2838	. . . . . . . . . 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 20830	. . . . . . . . 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 21842	. . . . . . . . 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 1373	. . . . . . . 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 2771	. . . . . . 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 5191	. . . . . 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 7374	. . . . 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 2775	. . . 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 2736	. . . 4 ⊢ (0_g‘((𝐼 ∖ 𝐽) mPoly 𝑅)) = (0_g‘((𝐼 ∖ 𝐽) mPoly 𝑅))
251	49	ringcmnd 20219	. . . 4 ⊢ (𝜑 → ((𝐼 ∖ 𝐽) mPoly 𝑅) ∈ CMnd)
252	7	crnggrpd 20182	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ Grp)
253	252	grpmndd 18876	. . . 4 ⊢ (𝜑 → 𝑅 ∈ Mnd)
254		ovex 7391	. . . . . 6 ⊢ (ℕ₀ ↑_m 𝐼) ∈ V
255	14, 254	rabex2 5286	. . . . 5 ⊢ 𝐷 ∈ V
256	255	a1i 11	. . . 4 ⊢ (𝜑 → 𝐷 ∈ V)
257		eqid 2736	. . . . . 6 ⊢ {𝑦 ∈ (ℕ₀ ↑_m (𝐼 ∖ 𝐽)) ∣ (^◡𝑦 “ ℕ) ∈ Fin} = {𝑦 ∈ (ℕ₀ ↑_m (𝐼 ∖ 𝐽)) ∣ (^◡𝑦 “ ℕ) ∈ Fin}
258		eqid 2736	. . . . . 6 ⊢ (𝑣 ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)) ↦ (𝑣‘(𝑌 ↾ (𝐼 ∖ 𝐽)))) = (𝑣 ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)) ↦ (𝑣‘(𝑌 ↾ (𝐼 ∖ 𝐽))))
259		difssd 4089	. . . . . . 7 ⊢ (𝜑 → (𝐼 ∖ 𝐽) ⊆ 𝐼)
260		selvvvval.y	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ 𝐷)
261	14, 257, 20, 259, 260	psrbagres 42799	. . . . . 6 ⊢ (𝜑 → (𝑌 ↾ (𝐼 ∖ 𝐽)) ∈ {𝑦 ∈ (ℕ₀ ↑_m (𝐼 ∖ 𝐽)) ∣ (^◡𝑦 “ ℕ) ∈ Fin})
262	3, 54, 257, 258, 22, 252, 261	mplmapghm 42807	. . . . 5 ⊢ (𝜑 → (𝑣 ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)) ↦ (𝑣‘(𝑌 ↾ (𝐼 ∖ 𝐽)))) ∈ (((𝐼 ∖ 𝐽) mPoly 𝑅) GrpHom 𝑅))
263		ghmmhm 19155	. . . . 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 2736	. . . . . . . 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 21953	. . . . . . 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 42799	. . . . . . . 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 7030	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) → (𝑤‘(𝑌 ↾ 𝐽)) ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)))
271	270	fmpttd 7060	. . . . 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 20830	. . . . . 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 7060	. . . . 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 20181	. . . . . 6 ⊢ (𝜑 → (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)) ∈ Ring)
277		eqid 2736	. . . . . . 7 ⊢ (0_g‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) = (0_g‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))
278	15, 277	ring0cl 20202	. . . . . 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 3957	. . . . 5 ⊢ (𝜑 → (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ⊆ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
281	255	mptex 7169	. . . . . . . 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 2736	. . . . . . . 8 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
287	1, 2, 286, 9	mplelsfi 21950	. . . . . . 7 ⊢ (𝜑 → 𝐹 finSupp (0_g‘𝑅))
288		ssidd 3957	. . . . . . . . . . 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 8138	. . . . . . . . . 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 21980	. . . . . . . . . . 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 2771	. . . . . . . . . 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 2771	. . . . . . . 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 8142	. . . . . . 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 9285	. . . . . 6 ⊢ (𝜑 → (𝑔 ∈ 𝐷 ↦ ((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))) finSupp (0_g‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))
299		eqid 2736	. . . . . . . 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 20846	. . . . . . 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 580	. . . . . 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 9287	. . . . 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 2736	. . . . . . . 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 20182	. . . . . . . 8 ⊢ (𝜑 → ((𝐼 ∖ 𝐽) mPoly 𝑅) ∈ Grp)
306	4, 15, 265, 304, 21, 305, 268	mplmapghm 42807	. . . . . . 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 19155	. . . . . . 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 18719	. . . . . 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 9307	. . . 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 19867	. . 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 19844	. . . 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 6964	. . 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 20219	. . . . . 6 ⊢ (𝜑 → (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)) ∈ CMnd)
318	305	grpmndd 18876	. . . . . 6 ⊢ (𝜑 → ((𝐼 ∖ 𝐽) mPoly 𝑅) ∈ Mnd)
319	15, 277, 317, 318, 256, 308, 273, 303	gsummhm 19867	. . . . 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 19844	. . . . . 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 6964	. . . . 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 2771	. . . 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 2776	. 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 21953	. . . . . . 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 7030	. . . . . 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 2737	. . . . . . 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 2737	. . . . . . 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 7074	. . . . . 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 2737	. . . . . 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 7074	. . . . 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 2736	. . . . . . . . . 10 ⊢ (._r‘((𝐼 ∖ 𝐽) mPoly 𝑅)) = (._r‘((𝐼 ∖ 𝐽) mPoly 𝑅))
338	4, 216, 54, 15, 337, 265, 227, 241, 328	mplvscaval 21971	. . . . . . . . 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 21971	. . . . . . . . . 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 7374	. . . . . . . . 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 2838	. . . . . . . . . 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 21953	. . . . . . . . . . . 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 7030	. . . . . . . . . . 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 20195	. . . . . . . . . 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 2736	. . . . . . . . . . 11 ⊢ (Base‘(Scalar‘((𝐼 ∖ 𝐽) mPoly 𝑅))) = (Base‘(Scalar‘((𝐼 ∖ 𝐽) mPoly 𝑅)))
350		eqid 2736	. . . . . . . . . . 11 ⊢ ( ·_𝑠 ‘((𝐼 ∖ 𝐽) mPoly 𝑅)) = ( ·_𝑠 ‘((𝐼 ∖ 𝐽) mPoly 𝑅))
351	32, 33, 349, 54, 337, 350	asclmul1 21842	. . . . . . . . . 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 1373	. . . . . . . . 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 2775	. . . . . . . 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 2736	. . . . . . . 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 21971	. . . . . . 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 7457	. . . . . . . . . . . . 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 2759	. . . . . . . . . . 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 2740	. . . . . . . . . . . . . 14 ⊢ (𝑖 = (𝑌 ↾ (𝐼 ∖ 𝐽)) → (𝑖 = (𝑔 ↾ (𝐼 ∖ 𝐽)) ↔ (𝑌 ↾ (𝐼 ∖ 𝐽)) = (𝑔 ↾ (𝐼 ∖ 𝐽))))
363	362	ifbid 4503	. . . . . . . . . . . . 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 2736	. . . . . . . . . . . . . 14 ⊢ (1_r‘((𝐼 ∖ 𝐽) mPoly 𝑅)) = (1_r‘((𝐼 ∖ 𝐽) mPoly 𝑅))
365		eqid 2736	. . . . . . . . . . . . . . 15 ⊢ (1_r‘𝑅) = (1_r‘𝑅)
366	58	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → 𝑅 ∈ Ring)
367	14, 257, 84, 151, 71	psrbagres 42799	. . . . . . . . . . . . . . 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 21990	. . . . . . . . . . . . . 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 20208	. . . . . . . . . . . . 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 4526	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → if((𝑌 ↾ (𝐼 ∖ 𝐽)) = (𝑔 ↾ (𝐼 ∖ 𝐽)), (1_r‘𝑅), (0_g‘𝑅)) ∈ V)
373	363, 369, 356, 372	fvmptd4 6965	. . . . . . . . . . . 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 20231	. . . . . . . . . . . . . . 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 21961	. . . . . . . . . . . . . . . 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 2771	. . . . . . . . . . . . . 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 7150	. . . . . . . . . . . . . 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 2771	. . . . . . . . . . . 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 4501	. . . . . . . . . . 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 2783	. . . . . . . . . 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 7374	. . . . . . . . 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 4533	. . . . . . . . . . 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 7369	. . . . . . . . . 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 21874	. . . . . . . . . . . . . . . . . 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 4434	. . . . . . . . . . . . . . . . . 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 6982	. . . . . . . . . . . . . 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 584	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (𝑌 = 𝑔 ↔ ((𝑌 ↾ 𝐽) = (𝑔 ↾ 𝐽) ∧ (𝑌 ↾ (𝐼 ∖ 𝐽)) = (𝑔 ↾ (𝐼 ∖ 𝐽)))))
402	401	ifbid 4503	. . . . . . . . . . . 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 7374	. . . . . . . . . . 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 7457	. . . . . . . . . . 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 2786	. . . . . . . . . 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 2785	. . . . . . . . 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 2771	. . . . . . . 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 21996	. . . . . . . . . . . 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 7373	. . . . . . . . . . . . . 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 5191	. . . . . . . . . . . . 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 7374	. . . . . . . . . . . 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 2771	. . . . . . . . . . 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 2736	. . . . . . . . . . . . 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 2740	. . . . . . . . . . . . . 14 ⊢ (𝑗 = (𝑌 ↾ 𝐽) → (𝑗 = (𝑔 ↾ 𝐽) ↔ (𝑌 ↾ 𝐽) = (𝑔 ↾ 𝐽)))
418	417	ifbid 4503	. . . . . . . . . . . . 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 4526	. . . . . . . . . . . . 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 6964	. . . . . . . . . . . 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 42799	. . . . . . . . . . . . . . 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 21996	. . . . . . . . . . . . . 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 7373	. . . . . . . . . . . . . . . 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 5191	. . . . . . . . . . . . . . 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 7374	. . . . . . . . . . . . . 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 2771	. . . . . . . . . . . . 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 2773	. . . . . . . . . . 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 7376	. . . . . . . . . 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 7374	. . . . . . . 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 20208	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((𝐹‘𝑔)(._r‘𝑅)(1_r‘𝑅)) = (𝐹‘𝑔))
438	68, 355, 286, 366, 75	ringrzd 20231	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((𝐹‘𝑔)(._r‘𝑅)(0_g‘𝑅)) = (0_g‘𝑅))
439	437, 438	ifeq12d 4501	. . . . . . . 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 2779	. . . . . . 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 2775	. . . . . 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 5191	. . . . 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 2771	. . . 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 7374	. . 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 20219	. . . 4 ⊢ (𝜑 → 𝑅 ∈ CMnd)
446	68, 286	ring0cl 20202	. . . . . . . 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 4526	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → if(𝑌 = 𝑔, (𝐹‘𝑔), (0_g‘𝑅)) ∈ (Base‘𝑅))
450	449	fmpttd 7060	. . . 4 ⊢ (𝜑 → (𝑔 ∈ 𝐷 ↦ if(𝑌 = 𝑔, (𝐹‘𝑔), (0_g‘𝑅))):𝐷⟶(Base‘𝑅))
451		eldifsnneq 4747	. . . . . . . 8 ⊢ (𝑔 ∈ (𝐷 ∖ {𝑌}) → ¬ 𝑔 = 𝑌)
452	451	neqcomd 2746	. . . . . . 7 ⊢ (𝑔 ∈ (𝐷 ∖ {𝑌}) → ¬ 𝑌 = 𝑔)
453	452	iffalsed 4490	. . . . . 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 8142	. . . 4 ⊢ (𝜑 → ((𝑔 ∈ 𝐷 ↦ if(𝑌 = 𝑔, (𝐹‘𝑔), (0_g‘𝑅))) supp (0_g‘𝑅)) ⊆ {𝑌})
456	256	mptexd 7170	. . . . 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 8980	. . . . . . 7 ⊢ {𝑌} ∈ Fin
460	459	a1i 11	. . . . . 6 ⊢ (𝜑 → {𝑌} ∈ Fin)
461	460, 455	ssfid 9169	. . . . 5 ⊢ (𝜑 → ((𝑔 ∈ 𝐷 ↦ if(𝑌 = 𝑔, (𝐹‘𝑔), (0_g‘𝑅))) supp (0_g‘𝑅)) ∈ Fin)
462	456, 447, 458, 461	isfsuppd 9269	. . . 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 19842	. . 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 4765	. . . . . 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 7374	. . . 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 7030	. . . . 5 ⊢ (𝜑 → (𝐹‘𝑌) ∈ (Base‘𝑅))
468		iftrue 4485	. . . . . . . 8 ⊢ (𝑌 = 𝑔 → if(𝑌 = 𝑔, (𝐹‘𝑔), (0_g‘𝑅)) = (𝐹‘𝑔))
469	468	eqcoms 2744	. . . . . . 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 2771	. . . . . 6 ⊢ (𝑔 = 𝑌 → if(𝑌 = 𝑔, (𝐹‘𝑔), (0_g‘𝑅)) = (𝐹‘𝑌))
472	68, 471	gsumsn 19883	. . . . 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 1373	. . . 4 ⊢ (𝜑 → (𝑅 Σ_g (𝑔 ∈ {𝑌} ↦ if(𝑌 = 𝑔, (𝐹‘𝑔), (0_g‘𝑅)))) = (𝐹‘𝑌))
474	466, 473	eqtrd 2771	. . 3 ⊢ (𝜑 → (𝑅 Σ_g ((𝑔 ∈ 𝐷 ↦ if(𝑌 = 𝑔, (𝐹‘𝑔), (0_g‘𝑅))) ↾ {𝑌})) = (𝐹‘𝑌))
475	444, 463, 474	3eqtr2d 2777	. 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 2775	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 1541 ∈ wcel 2113 {crab 3399 Vcvv 3440 ∖ cdif 3898 ∪ cun 3899 ∩ cin 3900 ⊆ wss 3901 ∅c0 4285 ifcif 4479 {csn 4580 class class class wbr 5098 ↦ cmpt 5179 × cxp 5622 ^◡ccnv 5623 ↾ cres 5626 “ cima 5627 ∘ ccom 5628 Fun wfun 6486 Fn wfn 6487 ⟶wf 6488 ‘cfv 6492 (class class class)co 7358 supp csupp 8102 ↑_m cmap 8763 Fincfn 8883 finSupp cfsupp 9264 0cc0 11026 ℕcn 12145 ℕ₀cn0 12401 ℤcz 12488 Basecbs 17136 ._rcmulr 17178 Scalarcsca 17180 ·_𝑠 cvsca 17181 0_gc0g 17359 Σ_g cgsu 17360 Mndcmnd 18659 MndHom cmhm 18706 ._gcmg 18997 GrpHom cghm 19141 CMndccmn 19709 mulGrpcmgp 20075 1_rcur 20116 Ringcrg 20168 CRingccrg 20169 RingHom crh 20405 LModclmod 20811 AssAlgcasa 21805 algSccascl 21807 mVar cmvr 21861 mPoly cmpl 21862 eval cevl 22028 selectVars cslv 22071

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-tp 4585 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-iin 4949 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 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 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-ofr 7623 df-om 7809 df-1st 7933 df-2nd 7934 df-supp 8103 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-2o 8398 df-er 8635 df-map 8765 df-pm 8766 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9265 df-sup 9345 df-oi 9415 df-card 9851 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-nn 12146 df-2 12208 df-3 12209 df-4 12210 df-5 12211 df-6 12212 df-7 12213 df-8 12214 df-9 12215 df-n0 12402 df-z 12489 df-dec 12608 df-uz 12752 df-fz 13424 df-fzo 13571 df-seq 13925 df-hash 14254 df-struct 17074 df-sets 17091 df-slot 17109 df-ndx 17121 df-base 17137 df-ress 17158 df-plusg 17190 df-mulr 17191 df-sca 17193 df-vsca 17194 df-ip 17195 df-tset 17196 df-ple 17197 df-ds 17199 df-hom 17201 df-cco 17202 df-0g 17361 df-gsum 17362 df-prds 17367 df-pws 17369 df-mre 17505 df-mrc 17506 df-acs 17508 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-mhm 18708 df-submnd 18709 df-grp 18866 df-minusg 18867 df-sbg 18868 df-mulg 18998 df-subg 19053 df-ghm 19142 df-cntz 19246 df-cmn 19711 df-abl 19712 df-mgp 20076 df-rng 20088 df-ur 20117 df-srg 20122 df-ring 20170 df-cring 20171 df-rhm 20408 df-subrng 20479 df-subrg 20503 df-lmod 20813 df-lss 20883 df-lsp 20923 df-assa 21808 df-asp 21809 df-ascl 21810 df-psr 21865 df-mvr 21866 df-mpl 21867 df-evls 22029 df-evl 22030 df-selv 22075

This theorem is referenced by: evlselv 42830

W3C validator