selvvvval - Mathbox for Steven Nguyen

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Theorem selvvvval 42561

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 2729	. . . . . 6 ⊢ ((𝐼 ∖ 𝐽) mPoly 𝑅) = ((𝐼 ∖ 𝐽) mPoly 𝑅)
4		eqid 2729	. . . . . 6 ⊢ (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)) = (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))
5		eqid 2729	. . . . . 6 ⊢ (algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) = (algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))
6		eqid 2729	. . . . . 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 42560	. . . . 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 2729	. . . . . 6 ⊢ (𝐼 eval (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) = (𝐼 eval (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))
12		eqid 2729	. . . . . 6 ⊢ (𝐼 mPoly (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) = (𝐼 mPoly (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))
13		eqid 2729	. . . . . 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 2729	. . . . . 6 ⊢ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) = (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))
16		eqid 2729	. . . . . 6 ⊢ (mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) = (mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))
17		eqid 2729	. . . . . 6 ⊢ (._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) = (._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
18		eqid 2729	. . . . . 6 ⊢ (._r‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) = (._r‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))
19	1, 2	mplrcl 21919	. . . . . . 7 ⊢ (𝐹 ∈ 𝐵 → 𝐼 ∈ V)
20	9, 19	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ V)
21	20, 8	ssexd 5266	. . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ V)
22	20	difexd 5273	. . . . . . . 8 ⊢ (𝜑 → (𝐼 ∖ 𝐽) ∈ V)
23	3, 22, 7	mplcrngd 42523	. . . . . . 7 ⊢ (𝜑 → ((𝐼 ∖ 𝐽) mPoly 𝑅) ∈ CRing)
24	4, 21, 23	mplcrngd 42523	. . . . . 6 ⊢ (𝜑 → (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)) ∈ CRing)
25	4	mplassa 21947	. . . . . . . . . . 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 2729	. . . . . . . . . . 11 ⊢ (Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) = (Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))
28	5, 27	asclrhm 21815	. . . . . . . . . 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 21947	. . . . . . . . . . . 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 2729	. . . . . . . . . . . 12 ⊢ (algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅)) = (algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))
33		eqid 2729	. . . . . . . . . . . 12 ⊢ (Scalar‘((𝐼 ∖ 𝐽) mPoly 𝑅)) = (Scalar‘((𝐼 ∖ 𝐽) mPoly 𝑅))
34	32, 33	asclrhm 21815	. . . . . . . . . . 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 21938	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑅 = (Scalar‘((𝐼 ∖ 𝐽) mPoly 𝑅)))
37	36	eqcomd 2735	. . . . . . . . . . 11 ⊢ (𝜑 → (Scalar‘((𝐼 ∖ 𝐽) mPoly 𝑅)) = 𝑅)
38	4, 21, 23	mplsca 21938	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐼 ∖ 𝐽) mPoly 𝑅) = (Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
39	37, 38	oveq12d 7371	. . . . . . . . . 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 2830	. . . . . . . . 9 ⊢ (𝜑 → (algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅)) ∈ (𝑅 RingHom (Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))
41		rhmco 20404	. . . . . . . . 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 20387	. . . . . . . 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 19123	. . . . . . . 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 22283	. . . . . 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 6841	. . . . . . 7 ⊢ (𝜑 → (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∈ V)
48		eqid 2729	. . . . . . . . . . . 12 ⊢ (𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅)) = (𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))
49	23	crngringd 20149	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐼 ∖ 𝐽) mPoly 𝑅) ∈ Ring)
50	4, 48, 15, 21, 49	mvrf2 21918	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅)):𝐽⟶(Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
51	50	ffvelcdmda 7022	. . . . . . . . . 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 3915	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (𝐼 ∖ 𝐽) ↔ (𝑧 ∈ 𝐼 ∧ ¬ 𝑧 ∈ 𝐽))
54		eqid 2729	. . . . . . . . . . . . . 14 ⊢ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)) = (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅))
55	4, 15, 54, 5, 21, 49	mplasclf 21988	. . . . . . . . . . . . 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 2729	. . . . . . . . . . . . . 14 ⊢ ((𝐼 ∖ 𝐽) mVar 𝑅) = ((𝐼 ∖ 𝐽) mVar 𝑅)
58	7	crngringd 20149	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑅 ∈ Ring)
59	3, 57, 54, 22, 58	mvrf2 21918	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐼 ∖ 𝐽) mVar 𝑅):(𝐼 ∖ 𝐽)⟶(Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)))
60	59	ffvelcdmda 7022	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐼 ∖ 𝐽)) → (((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧) ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)))
61	56, 60	ffvelcdmd 7023	. . . . . . . . . . 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 4514	. . . . . . . 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 7053	. . . . . . 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 8775	. . . . . 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 42549	. . . . 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 2729	. . . . . . . . . . . . 13 ⊢ (Base‘𝑅) = (Base‘𝑅)
69	1, 68, 2, 14, 9	mplelf 21923	. . . . . . . . . . . 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 6927	. . . . . . . . . 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 21988	. . . . . . . . . . . 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 7022	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (𝐹‘𝑔) ∈ (Base‘𝑅))
76	74, 75	fvco3d 6927	. . . . . . . . . 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 2764	. . . . . . . . 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 20048	. . . . . . . . . . 11 ⊢ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) = (Base‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
79		eqid 2729	. . . . . . . . . . 11 ⊢ (0_g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) = (0_g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
80	16, 18	mgpplusg 20047	. . . . . . . . . . 11 ⊢ (._r‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) = (+_g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
81	16	crngmgp 20144	. . . . . . . . . . . . 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 19701	. . . . . . . . . . . . . 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 21843	. . . . . . . . . . . . . . 15 ⊢ (𝑔 ∈ 𝐷 → 𝑔:𝐼⟶ℕ₀)
88	87	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → 𝑔:𝐼⟶ℕ₀)
89	88	ffvelcdmda 7022	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ 𝐼) → (𝑔‘𝑘) ∈ ℕ₀)
90		eqid 2729	. . . . . . . . . . . . . . 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 2811	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑘 → (𝑧 ∈ 𝐽 ↔ 𝑘 ∈ 𝐽))
92		fveq2 6826	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑘 → ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧) = ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))
93		fveq2 6826	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = 𝑘 → (((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧) = (((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))
94	93	fveq2d 6830	. . . . . . . . . . . . . . . 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 4507	. . . . . . . . . . . . . . 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 7022	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ 𝐼) ∧ 𝑘 ∈ 𝐽) → ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘) ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
99		eldif 3915	. . . . . . . . . . . . . . . . . . 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 7022	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → (((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘) ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)))
102	100, 101	ffvelcdmd 7023	. . . . . . . . . . . . . . . . . . 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 4514	. . . . . . . . . . . . . . 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 6957	. . . . . . . . . . . . . 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 2828	. . . . . . . . . . . . 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 18992	. . . . . . . . . . . 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 7053	. . . . . . . . . . 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 6895	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → 𝑔 = (𝑘 ∈ 𝐼 ↦ (𝑔‘𝑘)))
112	14	psrbagfsupp 21844	. . . . . . . . . . . . . 14 ⊢ (𝑔 ∈ 𝐷 → 𝑔 finSupp 0)
113	112	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → 𝑔 finSupp 0)
114	111, 113	eqbrtrrd 5119	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (𝑘 ∈ 𝐼 ↦ (𝑔‘𝑘)) finSupp 0)
115	78, 79, 17	mulg0 18971	. . . . . . . . . . . . 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 6841	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (0_g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) ∈ V)
118	114, 116, 89, 108, 117	fsuppssov1 9293	. . . . . . . . . . 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 4426	. . . . . . . . . . . 12 ⊢ ((𝐼 ∖ 𝐽) ∩ 𝐽) = ∅
120	119	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((𝐼 ∖ 𝐽) ∩ 𝐽) = ∅)
121		undifr 4436	. . . . . . . . . . . . . 14 ⊢ (𝐽 ⊆ 𝐼 ↔ ((𝐼 ∖ 𝐽) ∪ 𝐽) = 𝐼)
122	8, 121	sylib 218	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐼 ∖ 𝐽) ∪ 𝐽) = 𝐼)
123	122	eqcomd 2735	. . . . . . . . . . . 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 19825	. . . . . . . . . 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 4084	. . . . . . . . . . . . . . . . . . . 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 6957	. . . . . . . . . . . . . . . . . 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 4085	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ (𝐼 ∖ 𝐽) → ¬ 𝑘 ∈ 𝐽)
131	130	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → ¬ 𝑘 ∈ 𝐽)
132	131	iffalsed 4489	. . . . . . . . . . . . . . . . . 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 2764	. . . . . . . . . . . . . . . . 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 7369	. . . . . . . . . . . . . . . 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 2729	. . . . . . . . . . . . . . . . . . . 20 ⊢ (mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) = (mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
136	135, 16	rhmmhm 20382	. . . . . . . . . . . . . . . . . . 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 6830	. . . . . . . . . . . . . . . . . . 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 2830	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → (((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘) ∈ (Base‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))
144		eqid 2729	. . . . . . . . . . . . . . . . . . 19 ⊢ (Base‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) = (Base‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
145	135, 144	mgpbas 20048	. . . . . . . . . . . . . . . . . 18 ⊢ (Base‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) = (Base‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))
146		eqid 2729	. . . . . . . . . . . . . . . . . 18 ⊢ (._g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))) = (._g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))
147	145, 146, 17	mhmmulg 19012	. . . . . . . . . . . . . . . . 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 2767	. . . . . . . . . . . . . . 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 5188	. . . . . . . . . . . . . 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 4090	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (𝐼 ∖ 𝐽) ⊆ 𝐼)
152	151	resmptd 5995	. . . . . . . . . . . . . 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 6830	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) = (mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))
155	154	fveq2d 6830	. . . . . . . . . . . . . . . . . 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 7370	. . . . . . . . . . . . . . . 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 2729	. . . . . . . . . . . . . . . . . 18 ⊢ (mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) = (mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅))
159	158, 54	mgpbas 20048	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)) = (Base‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))
160		eqid 2729	. . . . . . . . . . . . . . . . 17 ⊢ (._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅))) = (._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))
161	158	crngmgp 20144	. . . . . . . . . . . . . . . . . . . 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 19701	. . . . . . . . . . . . . . . . . 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 18992	. . . . . . . . . . . . . . . 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 2829	. . . . . . . . . . . . . . 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 7070	. . . . . . . . . . . . . 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 2774	. . . . . . . . . . . . 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 7369	. . . . . . . . . . . 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 2729	. . . . . . . . . . . . 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 2829	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∈ CRing)
172	135	crngmgp 20144	. . . . . . . . . . . . . . 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 2830	. . . . . . . . . . . . . 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 7053	. . . . . . . . . . . . 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 12501	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → 0 ∈ ℤ)
181	114, 151, 180	fmptssfisupp 9303	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (𝑔‘𝑘)) finSupp 0)
182	141	eqimssd 3994	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)) ⊆ (Base‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))
183	182	sselda 3937	. . . . . . . . . . . . . . . 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 18971	. . . . . . . . . . . . . . 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 6841	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (0_g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))) ∈ V)
188	181, 186, 139, 140, 187	fsuppssov1 9293	. . . . . . . . . . . . 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 19835	. . . . . . . . . . . 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 2764	. . . . . . . . . . 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 5995	. . . . . . . . . . . . 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 3937	. . . . . . . . . . . . . . . . 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 6957	. . . . . . . . . . . . . . . 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 4484	. . . . . . . . . . . . . . . . 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 2764	. . . . . . . . . . . . . . 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 7369	. . . . . . . . . . . . . 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 5188	. . . . . . . . . . . . 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 2764	. . . . . . . . . . . 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 7369	. . . . . . . . . . 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 7371	. . . . . . . . . 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 19812	. . . . . . . . . . . 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 7022	. . . . . . . . . . . . . . . 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 18992	. . . . . . . . . . . . . 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 7053	. . . . . . . . . . . . 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 9303	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (𝑘 ∈ 𝐽 ↦ (𝑔‘𝑘)) finSupp 0)
214	213, 116, 208, 210, 117	fsuppssov1 9293	. . . . . . . . . . . . 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 19812	. . . . . . . . . . . 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 2729	. . . . . . . . . . . . 13 ⊢ ( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) = ( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))
217	5, 27, 144, 15, 18, 216	asclmul1 21811	. . . . . . . . . . . 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 7370	. . . . . . . . . . . . . . 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 5189	. . . . . . . . . . . . . 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 7371	. . . . . . . . . . . . 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 7368	. . . . . . . . . . 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 2767	. . . . . . . . . 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 2768	. . . . . . . . 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 7371	. . . . . . . 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 7023	. . . . . . . . . 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 2830	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔)) ∈ (Base‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))
230	4, 21, 49	mpllmodd 21949	. . . . . . . . . . 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 2729	. . . . . . . . . . . 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 7053	. . . . . . . . . . . 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 18971	. . . . . . . . . . . . . 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 6841	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (0_g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅))) ∈ V)
238	181, 236, 139, 140, 237	fsuppssov1 9293	. . . . . . . . . . . 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 19812	. . . . . . . . . . 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 2830	. . . . . . . . . 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 20800	. . . . . . . . 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 21811	. . . . . . . . 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 2764	. . . . . . 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 5188	. . . . . 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 7369	. . . . 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 2768	. . . 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 6828	. . 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 6828	. 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 2729	. . . 4 ⊢ (0_g‘((𝐼 ∖ 𝐽) mPoly 𝑅)) = (0_g‘((𝐼 ∖ 𝐽) mPoly 𝑅))
251	49	ringcmnd 20187	. . . 4 ⊢ (𝜑 → ((𝐼 ∖ 𝐽) mPoly 𝑅) ∈ CMnd)
252	7	crnggrpd 20150	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ Grp)
253	252	grpmndd 18843	. . . 4 ⊢ (𝜑 → 𝑅 ∈ Mnd)
254		ovex 7386	. . . . . 6 ⊢ (ℕ₀ ↑_m 𝐼) ∈ V
255	14, 254	rabex2 5283	. . . . 5 ⊢ 𝐷 ∈ V
256	255	a1i 11	. . . 4 ⊢ (𝜑 → 𝐷 ∈ V)
257		eqid 2729	. . . . . 6 ⊢ {𝑦 ∈ (ℕ₀ ↑_m (𝐼 ∖ 𝐽)) ∣ (^◡𝑦 “ ℕ) ∈ Fin} = {𝑦 ∈ (ℕ₀ ↑_m (𝐼 ∖ 𝐽)) ∣ (^◡𝑦 “ ℕ) ∈ Fin}
258		eqid 2729	. . . . . 6 ⊢ (𝑣 ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)) ↦ (𝑣‘(𝑌 ↾ (𝐼 ∖ 𝐽)))) = (𝑣 ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)) ↦ (𝑣‘(𝑌 ↾ (𝐼 ∖ 𝐽))))
259		difssd 4090	. . . . . . 7 ⊢ (𝜑 → (𝐼 ∖ 𝐽) ⊆ 𝐼)
260		selvvvval.y	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ 𝐷)
261	14, 257, 20, 259, 260	psrbagres 42522	. . . . . 6 ⊢ (𝜑 → (𝑌 ↾ (𝐼 ∖ 𝐽)) ∈ {𝑦 ∈ (ℕ₀ ↑_m (𝐼 ∖ 𝐽)) ∣ (^◡𝑦 “ ℕ) ∈ Fin})
262	3, 54, 257, 258, 22, 252, 261	mplmapghm 42532	. . . . 5 ⊢ (𝜑 → (𝑣 ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)) ↦ (𝑣‘(𝑌 ↾ (𝐼 ∖ 𝐽)))) ∈ (((𝐼 ∖ 𝐽) mPoly 𝑅) GrpHom 𝑅))
263		ghmmhm 19123	. . . . 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 2729	. . . . . . . 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 21923	. . . . . . 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 42522	. . . . . . . 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 7023	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) → (𝑤‘(𝑌 ↾ 𝐽)) ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)))
271	270	fmpttd 7053	. . . . 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 20800	. . . . . 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 7053	. . . . 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 6681	. . . 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 6841	. . . . 5 ⊢ (𝜑 → (0_g‘((𝐼 ∖ 𝐽) mPoly 𝑅)) ∈ V)
276	24	crngringd 20149	. . . . . 6 ⊢ (𝜑 → (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)) ∈ Ring)
277		eqid 2729	. . . . . . 7 ⊢ (0_g‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) = (0_g‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))
278	15, 277	ring0cl 20170	. . . . . 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 3961	. . . . 5 ⊢ (𝜑 → (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ⊆ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
281	255	mptex 7163	. . . . . . . 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 6841	. . . . . . 7 ⊢ (𝜑 → (0_g‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) ∈ V)
284		funmpt 6524	. . . . . . . 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 2729	. . . . . . . 8 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
287	1, 2, 286, 9	mplelsfi 21920	. . . . . . 7 ⊢ (𝜑 → 𝐹 finSupp (0_g‘𝑅))
288		ssidd 3961	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹 supp (0_g‘𝑅)) ⊆ (𝐹 supp (0_g‘𝑅)))
289		fvexd 6841	. . . . . . . . . . 11 ⊢ (𝜑 → (0_g‘𝑅) ∈ V)
290	69, 288, 9, 289	suppssrg 8136	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐷 ∖ (𝐹 supp (0_g‘𝑅)))) → (𝐹‘𝑔) = (0_g‘𝑅))
291	290	fveq2d 6830	. . . . . . . . 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 42530	. . . . . . . . . . 11 ⊢ (𝜑 → ((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(0_g‘𝑅)) = (0_g‘((𝐼 ∖ 𝐽) mPoly 𝑅)))
293	38	fveq2d 6830	. . . . . . . . . . 11 ⊢ (𝜑 → (0_g‘((𝐼 ∖ 𝐽) mPoly 𝑅)) = (0_g‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))
294	292, 293	eqtrd 2764	. . . . . . . . . 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 2764	. . . . . . . 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 8140	. . . . . . 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 9291	. . . . . 6 ⊢ (𝜑 → (𝑔 ∈ 𝐷 ↦ ((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))) finSupp (0_g‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))
299		eqid 2729	. . . . . . . 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 20816	. . . . . . 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 6841	. . . . . 6 ⊢ (𝜑 → (0_g‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∈ V)
303	298, 301, 227, 241, 302	fsuppssov1 9293	. . . . 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 2729	. . . . . . . 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 20150	. . . . . . . 8 ⊢ (𝜑 → ((𝐼 ∖ 𝐽) mPoly 𝑅) ∈ Grp)
306	4, 15, 265, 304, 21, 305, 268	mplmapghm 42532	. . . . . . 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 19123	. . . . . . 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 18686	. . . . . 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 9313	. . . 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 19835	. . 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 6825	. . . 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 19812	. . . 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 6841	. . . 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 6957	. . 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 20187	. . . . . 6 ⊢ (𝜑 → (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)) ∈ CMnd)
318	305	grpmndd 18843	. . . . . 6 ⊢ (𝜑 → ((𝐼 ∖ 𝐽) mPoly 𝑅) ∈ Mnd)
319	15, 277, 317, 318, 256, 308, 273, 303	gsummhm 19835	. . . . 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 6825	. . . . . 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 19812	. . . . . 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 6841	. . . . . 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 6957	. . . . 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 2764	. . . 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 6828	. . 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 2769	. 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 21923	. . . . . . 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 7023	. . . . . 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 2730	. . . . . . 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 2730	. . . . . . 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 6825	. . . . . . 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 7067	. . . . . 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 2730	. . . . . 6 ⊢ (𝜑 → (𝑣 ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)) ↦ (𝑣‘(𝑌 ↾ (𝐼 ∖ 𝐽)))) = (𝑣 ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)) ↦ (𝑣‘(𝑌 ↾ (𝐼 ∖ 𝐽)))))
335		fveq1 6825	. . . . . 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 7067	. . . . 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 2729	. . . . . . . . . 10 ⊢ (._r‘((𝐼 ∖ 𝐽) mPoly 𝑅)) = (._r‘((𝐼 ∖ 𝐽) mPoly 𝑅))
338	4, 216, 54, 15, 337, 265, 227, 241, 328	mplvscaval 21941	. . . . . . . . 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 21941	. . . . . . . . . 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 7369	. . . . . . . . 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 6830	. . . . . . . . . . . 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 2830	. . . . . . . . . 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 21923	. . . . . . . . . . . 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 7023	. . . . . . . . . . 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 20163	. . . . . . . . . 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 2729	. . . . . . . . . . 11 ⊢ (Base‘(Scalar‘((𝐼 ∖ 𝐽) mPoly 𝑅))) = (Base‘(Scalar‘((𝐼 ∖ 𝐽) mPoly 𝑅)))
350		eqid 2729	. . . . . . . . . . 11 ⊢ ( ·_𝑠 ‘((𝐼 ∖ 𝐽) mPoly 𝑅)) = ( ·_𝑠 ‘((𝐼 ∖ 𝐽) mPoly 𝑅))
351	32, 33, 349, 54, 337, 350	asclmul1 21811	. . . . . . . . . 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 2768	. . . . . . . 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 6828	. . . . . . 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 2729	. . . . . . . 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 21941	. . . . . . 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 7452	. . . . . . . . . . . . 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 6827	. . . . . . . . . . . 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 6843	. . . . . . . . . . . 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 2752	. . . . . . . . . . 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 2733	. . . . . . . . . . . . . 14 ⊢ (𝑖 = (𝑌 ↾ (𝐼 ∖ 𝐽)) → (𝑖 = (𝑔 ↾ (𝐼 ∖ 𝐽)) ↔ (𝑌 ↾ (𝐼 ∖ 𝐽)) = (𝑔 ↾ (𝐼 ∖ 𝐽))))
363	362	ifbid 4502	. . . . . . . . . . . . 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 2729	. . . . . . . . . . . . . 14 ⊢ (1_r‘((𝐼 ∖ 𝐽) mPoly 𝑅)) = (1_r‘((𝐼 ∖ 𝐽) mPoly 𝑅))
365		eqid 2729	. . . . . . . . . . . . . . 15 ⊢ (1_r‘𝑅) = (1_r‘𝑅)
366	58	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → 𝑅 ∈ Ring)
367	14, 257, 84, 151, 71	psrbagres 42522	. . . . . . . . . . . . . . 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 21958	. . . . . . . . . . . . . 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 20176	. . . . . . . . . . . . 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 6841	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (1_r‘𝑅) ∈ V)
371		fvexd 6841	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (0_g‘𝑅) ∈ V)
372	370, 371	ifcld 4525	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → if((𝑌 ↾ (𝐼 ∖ 𝐽)) = (𝑔 ↾ (𝐼 ∖ 𝐽)), (1_r‘𝑅), (0_g‘𝑅)) ∈ V)
373	363, 369, 356, 372	fvmptd4 6958	. . . . . . . . . . . 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 20199	. . . . . . . . . . . . . . 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 21931	. . . . . . . . . . . . . . . 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 2764	. . . . . . . . . . . . . 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 6828	. . . . . . . . . . . . 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 6839	. . . . . . . . . . . . . . 15 ⊢ (0_g‘𝑅) ∈ V
380	379	fvconst2 7144	. . . . . . . . . . . . . 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 2764	. . . . . . . . . . . 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 4500	. . . . . . . . . . 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 2776	. . . . . . . . . 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 7369	. . . . . . . . 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 4532	. . . . . . . . . . 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 7364	. . . . . . . . . 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 21843	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑌 ∈ 𝐷 → 𝑌:𝐼⟶ℕ₀)
389	260, 388	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑌:𝐼⟶ℕ₀)
390	389	ffnd 6657	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑌 Fn 𝐼)
391	390	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → 𝑌 Fn 𝐼)
392		undif 4435	. . . . . . . . . . . . . . . . . 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 6580	. . . . . . . . . . . . . . 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 6657	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → 𝑔 Fn 𝐼)
398	394	fneq2d 6580	. . . . . . . . . . . . . . 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 6975	. . . . . . . . . . . . . 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 4502	. . . . . . . . . . . 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 7369	. . . . . . . . . . 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 7452	. . . . . . . . . . 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 2779	. . . . . . . . . 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 2778	. . . . . . . . 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 2764	. . . . . . . 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 21964	. . . . . . . . . . . 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 6846	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → ((𝑔 ↾ (𝐼 ∖ 𝐽))‘𝑘) = (𝑔‘𝑘))
412	411	oveq1d 7368	. . . . . . . . . . . . . 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 5188	. . . . . . . . . . . . 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 7369	. . . . . . . . . . . 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 2764	. . . . . . . . . . 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 2729	. . . . . . . . . . . . 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 2733	. . . . . . . . . . . . . 14 ⊢ (𝑗 = (𝑌 ↾ 𝐽) → (𝑗 = (𝑔 ↾ 𝐽) ↔ (𝑌 ↾ 𝐽) = (𝑔 ↾ 𝐽)))
418	417	ifbid 4502	. . . . . . . . . . . . 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 6841	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (1_r‘((𝐼 ∖ 𝐽) mPoly 𝑅)) ∈ V)
420		fvexd 6841	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (0_g‘((𝐼 ∖ 𝐽) mPoly 𝑅)) ∈ V)
421	419, 420	ifcld 4525	. . . . . . . . . . . . 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 6957	. . . . . . . . . . . 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 42522	. . . . . . . . . . . . . . 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 21964	. . . . . . . . . . . . . 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 6846	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ 𝐽) → ((𝑔 ↾ 𝐽)‘𝑘) = (𝑔‘𝑘))
428	427	oveq1d 7368	. . . . . . . . . . . . . . . 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 5188	. . . . . . . . . . . . . . 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 7369	. . . . . . . . . . . . . 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 2764	. . . . . . . . . . . . 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 6828	. . . . . . . . . . . 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 2766	. . . . . . . . . . 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 7371	. . . . . . . . . 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 6828	. . . . . . . . 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 7369	. . . . . . . 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 20176	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((𝐹‘𝑔)(._r‘𝑅)(1_r‘𝑅)) = (𝐹‘𝑔))
438	68, 355, 286, 366, 75	ringrzd 20199	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((𝐹‘𝑔)(._r‘𝑅)(0_g‘𝑅)) = (0_g‘𝑅))
439	437, 438	ifeq12d 4500	. . . . . . . 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 2772	. . . . . . 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 2768	. . . . . 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 5188	. . . . 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 2764	. . . 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 7369	. . 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 20187	. . . 4 ⊢ (𝜑 → 𝑅 ∈ CMnd)
446	68, 286	ring0cl 20170	. . . . . . . 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 4525	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → if(𝑌 = 𝑔, (𝐹‘𝑔), (0_g‘𝑅)) ∈ (Base‘𝑅))
450	449	fmpttd 7053	. . . 4 ⊢ (𝜑 → (𝑔 ∈ 𝐷 ↦ if(𝑌 = 𝑔, (𝐹‘𝑔), (0_g‘𝑅))):𝐷⟶(Base‘𝑅))
451		eldifsnneq 4745	. . . . . . . 8 ⊢ (𝑔 ∈ (𝐷 ∖ {𝑌}) → ¬ 𝑔 = 𝑌)
452	451	neqcomd 2739	. . . . . . 7 ⊢ (𝑔 ∈ (𝐷 ∖ {𝑌}) → ¬ 𝑌 = 𝑔)
453	452	iffalsed 4489	. . . . . 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 8140	. . . 4 ⊢ (𝜑 → ((𝑔 ∈ 𝐷 ↦ if(𝑌 = 𝑔, (𝐹‘𝑔), (0_g‘𝑅))) supp (0_g‘𝑅)) ⊆ {𝑌})
456	256	mptexd 7164	. . . . 5 ⊢ (𝜑 → (𝑔 ∈ 𝐷 ↦ if(𝑌 = 𝑔, (𝐹‘𝑔), (0_g‘𝑅))) ∈ V)
457		funmpt 6524	. . . . . 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 8975	. . . . . . 7 ⊢ {𝑌} ∈ Fin
460	459	a1i 11	. . . . . 6 ⊢ (𝜑 → {𝑌} ∈ Fin)
461	460, 455	ssfid 9170	. . . . 5 ⊢ (𝜑 → ((𝑔 ∈ 𝐷 ↦ if(𝑌 = 𝑔, (𝐹‘𝑔), (0_g‘𝑅))) supp (0_g‘𝑅)) ∈ Fin)
462	456, 447, 458, 461	isfsuppd 9275	. . . 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 19810	. . 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 4763	. . . . . 6 ⊢ (𝜑 → {𝑌} ⊆ 𝐷)
465	464	resmptd 5995	. . . . 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 7369	. . . 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 7023	. . . . 5 ⊢ (𝜑 → (𝐹‘𝑌) ∈ (Base‘𝑅))
468		iftrue 4484	. . . . . . . 8 ⊢ (𝑌 = 𝑔 → if(𝑌 = 𝑔, (𝐹‘𝑔), (0_g‘𝑅)) = (𝐹‘𝑔))
469	468	eqcoms 2737	. . . . . . 7 ⊢ (𝑔 = 𝑌 → if(𝑌 = 𝑔, (𝐹‘𝑔), (0_g‘𝑅)) = (𝐹‘𝑔))
470		fveq2 6826	. . . . . . 7 ⊢ (𝑔 = 𝑌 → (𝐹‘𝑔) = (𝐹‘𝑌))
471	469, 470	eqtrd 2764	. . . . . 6 ⊢ (𝑔 = 𝑌 → if(𝑌 = 𝑔, (𝐹‘𝑔), (0_g‘𝑅)) = (𝐹‘𝑌))
472	68, 471	gsumsn 19851	. . . . 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 2764	. . 3 ⊢ (𝜑 → (𝑅 Σ_g ((𝑔 ∈ 𝐷 ↦ if(𝑌 = 𝑔, (𝐹‘𝑔), (0_g‘𝑅))) ↾ {𝑌})) = (𝐹‘𝑌))
475	444, 463, 474	3eqtr2d 2770	. 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 2768	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 1540 ∈ wcel 2109 {crab 3396 Vcvv 3438 ∖ cdif 3902 ∪ cun 3903 ∩ cin 3904 ⊆ wss 3905 ∅c0 4286 ifcif 4478 {csn 4579 class class class wbr 5095 ↦ cmpt 5176 × cxp 5621 ^◡ccnv 5622 ↾ cres 5625 “ cima 5626 ∘ ccom 5627 Fun wfun 6480 Fn wfn 6481 ⟶wf 6482 ‘cfv 6486 (class class class)co 7353 supp csupp 8100 ↑_m cmap 8760 Fincfn 8879 finSupp cfsupp 9270 0cc0 11028 ℕcn 12146 ℕ₀cn0 12402 ℤcz 12489 Basecbs 17138 ._rcmulr 17180 Scalarcsca 17182 ·_𝑠 cvsca 17183 0_gc0g 17361 Σ_g cgsu 17362 Mndcmnd 18626 MndHom cmhm 18673 ._gcmg 18964 GrpHom cghm 19109 CMndccmn 19677 mulGrpcmgp 20043 1_rcur 20084 Ringcrg 20136 CRingccrg 20137 RingHom crh 20372 LModclmod 20781 AssAlgcasa 21775 algSccascl 21777 mVar cmvr 21830 mPoly cmpl 21831 eval cevl 21996 selectVars cslv 22031

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3345 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-uni 4862 df-int 4900 df-iun 4946 df-iin 4947 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-se 5577 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-isom 6495 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-of 7617 df-ofr 7618 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8101 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-er 8632 df-map 8762 df-pm 8763 df-ixp 8832 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-fsupp 9271 df-sup 9351 df-oi 9421 df-card 9854 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11367 df-neg 11368 df-nn 12147 df-2 12209 df-3 12210 df-4 12211 df-5 12212 df-6 12213 df-7 12214 df-8 12215 df-9 12216 df-n0 12403 df-z 12490 df-dec 12610 df-uz 12754 df-fz 13429 df-fzo 13576 df-seq 13927 df-hash 14256 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17139 df-ress 17160 df-plusg 17192 df-mulr 17193 df-sca 17195 df-vsca 17196 df-ip 17197 df-tset 17198 df-ple 17199 df-ds 17201 df-hom 17203 df-cco 17204 df-0g 17363 df-gsum 17364 df-prds 17369 df-pws 17371 df-mre 17506 df-mrc 17507 df-acs 17509 df-mgm 18532 df-sgrp 18611 df-mnd 18627 df-mhm 18675 df-submnd 18676 df-grp 18833 df-minusg 18834 df-sbg 18835 df-mulg 18965 df-subg 19020 df-ghm 19110 df-cntz 19214 df-cmn 19679 df-abl 19680 df-mgp 20044 df-rng 20056 df-ur 20085 df-srg 20090 df-ring 20138 df-cring 20139 df-rhm 20375 df-subrng 20449 df-subrg 20473 df-lmod 20783 df-lss 20853 df-lsp 20893 df-assa 21778 df-asp 21779 df-ascl 21780 df-psr 21834 df-mvr 21835 df-mpl 21836 df-evls 21997 df-evl 21998 df-selv 22035

This theorem is referenced by: evlselv 42563

W3C validator