selvvvval - Mathbox for Steven Nguyen

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Theorem selvvvval 42706

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 2733	. . . . . 6 ⊢ ((𝐼 ∖ 𝐽) mPoly 𝑅) = ((𝐼 ∖ 𝐽) mPoly 𝑅)
4		eqid 2733	. . . . . 6 ⊢ (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)) = (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))
5		eqid 2733	. . . . . 6 ⊢ (algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) = (algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))
6		eqid 2733	. . . . . 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 42705	. . . . 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 2733	. . . . . 6 ⊢ (𝐼 eval (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) = (𝐼 eval (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))
12		eqid 2733	. . . . . 6 ⊢ (𝐼 mPoly (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) = (𝐼 mPoly (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))
13		eqid 2733	. . . . . 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 2733	. . . . . 6 ⊢ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) = (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))
16		eqid 2733	. . . . . 6 ⊢ (mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) = (mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))
17		eqid 2733	. . . . . 6 ⊢ (._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) = (._g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
18		eqid 2733	. . . . . 6 ⊢ (._r‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) = (._r‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))
19	1, 2	mplrcl 21934	. . . . . . 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 42668	. . . . . . 7 ⊢ (𝜑 → ((𝐼 ∖ 𝐽) mPoly 𝑅) ∈ CRing)
24	4, 21, 23	mplcrngd 42668	. . . . . 6 ⊢ (𝜑 → (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)) ∈ CRing)
25	4	mplassa 21962	. . . . . . . . . . 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 2733	. . . . . . . . . . 11 ⊢ (Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) = (Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))
28	5, 27	asclrhm 21831	. . . . . . . . . 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 21962	. . . . . . . . . . . 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 2733	. . . . . . . . . . . 12 ⊢ (algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅)) = (algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))
33		eqid 2733	. . . . . . . . . . . 12 ⊢ (Scalar‘((𝐼 ∖ 𝐽) mPoly 𝑅)) = (Scalar‘((𝐼 ∖ 𝐽) mPoly 𝑅))
34	32, 33	asclrhm 21831	. . . . . . . . . . 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 21953	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑅 = (Scalar‘((𝐼 ∖ 𝐽) mPoly 𝑅)))
37	36	eqcomd 2739	. . . . . . . . . . 11 ⊢ (𝜑 → (Scalar‘((𝐼 ∖ 𝐽) mPoly 𝑅)) = 𝑅)
38	4, 21, 23	mplsca 21953	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐼 ∖ 𝐽) mPoly 𝑅) = (Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
39	37, 38	oveq12d 7372	. . . . . . . . . 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 2835	. . . . . . . . 9 ⊢ (𝜑 → (algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅)) ∈ (𝑅 RingHom (Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))
41		rhmco 20420	. . . . . . . . 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 20405	. . . . . . . 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 19142	. . . . . . . 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 22298	. . . . . 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 6845	. . . . . . 7 ⊢ (𝜑 → (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∈ V)
48		eqid 2733	. . . . . . . . . . . 12 ⊢ (𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅)) = (𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))
49	23	crngringd 20168	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐼 ∖ 𝐽) mPoly 𝑅) ∈ Ring)
50	4, 48, 15, 21, 49	mvrf2 21933	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅)):𝐽⟶(Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
51	50	ffvelcdmda 7025	. . . . . . . . . 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 3908	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (𝐼 ∖ 𝐽) ↔ (𝑧 ∈ 𝐼 ∧ ¬ 𝑧 ∈ 𝐽))
54		eqid 2733	. . . . . . . . . . . . . 14 ⊢ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)) = (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅))
55	4, 15, 54, 5, 21, 49	mplasclf 22003	. . . . . . . . . . . . 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 2733	. . . . . . . . . . . . . 14 ⊢ ((𝐼 ∖ 𝐽) mVar 𝑅) = ((𝐼 ∖ 𝐽) mVar 𝑅)
58	7	crngringd 20168	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑅 ∈ Ring)
59	3, 57, 54, 22, 58	mvrf2 21933	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐼 ∖ 𝐽) mVar 𝑅):(𝐼 ∖ 𝐽)⟶(Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)))
60	59	ffvelcdmda 7025	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐼 ∖ 𝐽)) → (((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧) ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)))
61	56, 60	ffvelcdmd 7026	. . . . . . . . . . 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 4512	. . . . . . . 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 7056	. . . . . . 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 8773	. . . . . 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 42694	. . . . 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 2733	. . . . . . . . . . . . 13 ⊢ (Base‘𝑅) = (Base‘𝑅)
69	1, 68, 2, 14, 9	mplelf 21938	. . . . . . . . . . . 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 6930	. . . . . . . . . 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 22003	. . . . . . . . . . . 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 7025	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (𝐹‘𝑔) ∈ (Base‘𝑅))
76	74, 75	fvco3d 6930	. . . . . . . . . 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 2768	. . . . . . . . 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 20067	. . . . . . . . . . 11 ⊢ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) = (Base‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
79		eqid 2733	. . . . . . . . . . 11 ⊢ (0_g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) = (0_g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
80	16, 18	mgpplusg 20066	. . . . . . . . . . 11 ⊢ (._r‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) = (+_g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
81	16	crngmgp 20163	. . . . . . . . . . . . 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 19720	. . . . . . . . . . . . . 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 21859	. . . . . . . . . . . . . . 15 ⊢ (𝑔 ∈ 𝐷 → 𝑔:𝐼⟶ℕ₀)
88	87	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → 𝑔:𝐼⟶ℕ₀)
89	88	ffvelcdmda 7025	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ 𝐼) → (𝑔‘𝑘) ∈ ℕ₀)
90		eqid 2733	. . . . . . . . . . . . . . 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 2816	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑘 → (𝑧 ∈ 𝐽 ↔ 𝑘 ∈ 𝐽))
92		fveq2 6830	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑘 → ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧) = ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))
93		fveq2 6830	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = 𝑘 → (((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧) = (((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))
94	93	fveq2d 6834	. . . . . . . . . . . . . . . 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 4505	. . . . . . . . . . . . . . 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 7025	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ 𝐼) ∧ 𝑘 ∈ 𝐽) → ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘) ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
99		eldif 3908	. . . . . . . . . . . . . . . . . . 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 7025	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → (((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘) ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)))
102	100, 101	ffvelcdmd 7026	. . . . . . . . . . . . . . . . . . 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 4512	. . . . . . . . . . . . . . 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 6960	. . . . . . . . . . . . . 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 2833	. . . . . . . . . . . . 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 19012	. . . . . . . . . . . 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 7056	. . . . . . . . . . 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 6898	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → 𝑔 = (𝑘 ∈ 𝐼 ↦ (𝑔‘𝑘)))
112	14	psrbagfsupp 21860	. . . . . . . . . . . . . 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 18991	. . . . . . . . . . . . 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 6845	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (0_g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) ∈ V)
118	114, 116, 89, 108, 117	fsuppssov1 9277	. . . . . . . . . . 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 4422	. . . . . . . . . . . 12 ⊢ ((𝐼 ∖ 𝐽) ∩ 𝐽) = ∅
120	119	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((𝐼 ∖ 𝐽) ∩ 𝐽) = ∅)
121		undifr 4432	. . . . . . . . . . . . . 14 ⊢ (𝐽 ⊆ 𝐼 ↔ ((𝐼 ∖ 𝐽) ∪ 𝐽) = 𝐼)
122	8, 121	sylib 218	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐼 ∖ 𝐽) ∪ 𝐽) = 𝐼)
123	122	eqcomd 2739	. . . . . . . . . . . 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 19844	. . . . . . . . . 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 4080	. . . . . . . . . . . . . . . . . . . 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 6960	. . . . . . . . . . . . . . . . . 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 4081	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ (𝐼 ∖ 𝐽) → ¬ 𝑘 ∈ 𝐽)
131	130	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → ¬ 𝑘 ∈ 𝐽)
132	131	iffalsed 4487	. . . . . . . . . . . . . . . . . 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 2768	. . . . . . . . . . . . . . . . 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 7370	. . . . . . . . . . . . . . . 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 2733	. . . . . . . . . . . . . . . . . . . 20 ⊢ (mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) = (mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
136	135, 16	rhmmhm 20401	. . . . . . . . . . . . . . . . . . 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 6834	. . . . . . . . . . . . . . . . . . 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 2835	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → (((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘) ∈ (Base‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))
144		eqid 2733	. . . . . . . . . . . . . . . . . . 19 ⊢ (Base‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) = (Base‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
145	135, 144	mgpbas 20067	. . . . . . . . . . . . . . . . . 18 ⊢ (Base‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) = (Base‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))
146		eqid 2733	. . . . . . . . . . . . . . . . . 18 ⊢ (._g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))) = (._g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))
147	145, 146, 17	mhmmulg 19032	. . . . . . . . . . . . . . . . 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 2771	. . . . . . . . . . . . . . 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 4086	. . . . . . . . . . . . . . 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 6834	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) = (mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))
155	154	fveq2d 6834	. . . . . . . . . . . . . . . . . 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 7371	. . . . . . . . . . . . . . . 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 2733	. . . . . . . . . . . . . . . . . 18 ⊢ (mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) = (mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅))
159	158, 54	mgpbas 20067	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)) = (Base‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))
160		eqid 2733	. . . . . . . . . . . . . . . . 17 ⊢ (._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅))) = (._g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))
161	158	crngmgp 20163	. . . . . . . . . . . . . . . . . . . 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 19720	. . . . . . . . . . . . . . . . . 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 19012	. . . . . . . . . . . . . . . 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 2834	. . . . . . . . . . . . . . 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 7073	. . . . . . . . . . . . . 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 2778	. . . . . . . . . . . . 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 7370	. . . . . . . . . . . 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 2733	. . . . . . . . . . . . 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 2834	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∈ CRing)
172	135	crngmgp 20163	. . . . . . . . . . . . . . 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 2835	. . . . . . . . . . . . . 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 7056	. . . . . . . . . . . . 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 12489	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → 0 ∈ ℤ)
181	114, 151, 180	fmptssfisupp 9287	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (𝑔‘𝑘)) finSupp 0)
182	141	eqimssd 3987	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)) ⊆ (Base‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))
183	182	sselda 3930	. . . . . . . . . . . . . . . 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 18991	. . . . . . . . . . . . . . 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 6845	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (0_g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))) ∈ V)
188	181, 186, 139, 140, 187	fsuppssov1 9277	. . . . . . . . . . . . 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 19854	. . . . . . . . . . . 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 2768	. . . . . . . . . . 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 3930	. . . . . . . . . . . . . . . . 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 6960	. . . . . . . . . . . . . . . 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 4482	. . . . . . . . . . . . . . . . 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 2768	. . . . . . . . . . . . . . 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 7370	. . . . . . . . . . . . . 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 2768	. . . . . . . . . . . 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 7370	. . . . . . . . . . 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 7372	. . . . . . . . . 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 19831	. . . . . . . . . . . 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 7025	. . . . . . . . . . . . . . . 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 19012	. . . . . . . . . . . . . 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 7056	. . . . . . . . . . . . 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 9287	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (𝑘 ∈ 𝐽 ↦ (𝑔‘𝑘)) finSupp 0)
214	213, 116, 208, 210, 117	fsuppssov1 9277	. . . . . . . . . . . . 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 19831	. . . . . . . . . . . 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 2733	. . . . . . . . . . . . 13 ⊢ ( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) = ( ·_𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))
217	5, 27, 144, 15, 18, 216	asclmul1 21827	. . . . . . . . . . . 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 7371	. . . . . . . . . . . . . . 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 7372	. . . . . . . . . . . . 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 7369	. . . . . . . . . . 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 2771	. . . . . . . . . 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 2772	. . . . . . . . 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 7372	. . . . . . . 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 7026	. . . . . . . . . 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 2835	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔)) ∈ (Base‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))
230	4, 21, 49	mpllmodd 21964	. . . . . . . . . . 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 2733	. . . . . . . . . . . 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 7056	. . . . . . . . . . . 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 18991	. . . . . . . . . . . . . 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 6845	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (0_g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅))) ∈ V)
238	181, 236, 139, 140, 237	fsuppssov1 9277	. . . . . . . . . . . 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 19831	. . . . . . . . . . 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 2835	. . . . . . . . . 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 20816	. . . . . . . . 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 21827	. . . . . . . . 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 2768	. . . . . . 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 7370	. . . . 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 2772	. . . 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 6832	. . 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 6832	. 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 2733	. . . 4 ⊢ (0_g‘((𝐼 ∖ 𝐽) mPoly 𝑅)) = (0_g‘((𝐼 ∖ 𝐽) mPoly 𝑅))
251	49	ringcmnd 20206	. . . 4 ⊢ (𝜑 → ((𝐼 ∖ 𝐽) mPoly 𝑅) ∈ CMnd)
252	7	crnggrpd 20169	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ Grp)
253	252	grpmndd 18863	. . . 4 ⊢ (𝜑 → 𝑅 ∈ Mnd)
254		ovex 7387	. . . . . 6 ⊢ (ℕ₀ ↑_m 𝐼) ∈ V
255	14, 254	rabex2 5283	. . . . 5 ⊢ 𝐷 ∈ V
256	255	a1i 11	. . . 4 ⊢ (𝜑 → 𝐷 ∈ V)
257		eqid 2733	. . . . . 6 ⊢ {𝑦 ∈ (ℕ₀ ↑_m (𝐼 ∖ 𝐽)) ∣ (^◡𝑦 “ ℕ) ∈ Fin} = {𝑦 ∈ (ℕ₀ ↑_m (𝐼 ∖ 𝐽)) ∣ (^◡𝑦 “ ℕ) ∈ Fin}
258		eqid 2733	. . . . . 6 ⊢ (𝑣 ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)) ↦ (𝑣‘(𝑌 ↾ (𝐼 ∖ 𝐽)))) = (𝑣 ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)) ↦ (𝑣‘(𝑌 ↾ (𝐼 ∖ 𝐽))))
259		difssd 4086	. . . . . . 7 ⊢ (𝜑 → (𝐼 ∖ 𝐽) ⊆ 𝐼)
260		selvvvval.y	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ 𝐷)
261	14, 257, 20, 259, 260	psrbagres 42667	. . . . . 6 ⊢ (𝜑 → (𝑌 ↾ (𝐼 ∖ 𝐽)) ∈ {𝑦 ∈ (ℕ₀ ↑_m (𝐼 ∖ 𝐽)) ∣ (^◡𝑦 “ ℕ) ∈ Fin})
262	3, 54, 257, 258, 22, 252, 261	mplmapghm 42677	. . . . 5 ⊢ (𝜑 → (𝑣 ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)) ↦ (𝑣‘(𝑌 ↾ (𝐼 ∖ 𝐽)))) ∈ (((𝐼 ∖ 𝐽) mPoly 𝑅) GrpHom 𝑅))
263		ghmmhm 19142	. . . . 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 2733	. . . . . . . 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 21938	. . . . . . 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 42667	. . . . . . . 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 7026	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) → (𝑤‘(𝑌 ↾ 𝐽)) ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)))
271	270	fmpttd 7056	. . . . 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 20816	. . . . . 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 7056	. . . . 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 6683	. . . 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 6845	. . . . 5 ⊢ (𝜑 → (0_g‘((𝐼 ∖ 𝐽) mPoly 𝑅)) ∈ V)
276	24	crngringd 20168	. . . . . 6 ⊢ (𝜑 → (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)) ∈ Ring)
277		eqid 2733	. . . . . . 7 ⊢ (0_g‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) = (0_g‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))
278	15, 277	ring0cl 20189	. . . . . 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 3954	. . . . 5 ⊢ (𝜑 → (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ⊆ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))
281	255	mptex 7165	. . . . . . . 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 6845	. . . . . . 7 ⊢ (𝜑 → (0_g‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) ∈ V)
284		funmpt 6526	. . . . . . . 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 2733	. . . . . . . 8 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
287	1, 2, 286, 9	mplelsfi 21935	. . . . . . 7 ⊢ (𝜑 → 𝐹 finSupp (0_g‘𝑅))
288		ssidd 3954	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹 supp (0_g‘𝑅)) ⊆ (𝐹 supp (0_g‘𝑅)))
289		fvexd 6845	. . . . . . . . . . 11 ⊢ (𝜑 → (0_g‘𝑅) ∈ V)
290	69, 288, 9, 289	suppssrg 8134	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐷 ∖ (𝐹 supp (0_g‘𝑅)))) → (𝐹‘𝑔) = (0_g‘𝑅))
291	290	fveq2d 6834	. . . . . . . . 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 42675	. . . . . . . . . . 11 ⊢ (𝜑 → ((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(0_g‘𝑅)) = (0_g‘((𝐼 ∖ 𝐽) mPoly 𝑅)))
293	38	fveq2d 6834	. . . . . . . . . . 11 ⊢ (𝜑 → (0_g‘((𝐼 ∖ 𝐽) mPoly 𝑅)) = (0_g‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))
294	292, 293	eqtrd 2768	. . . . . . . . . 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 2768	. . . . . . . 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 8138	. . . . . . 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 9275	. . . . . 6 ⊢ (𝜑 → (𝑔 ∈ 𝐷 ↦ ((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))) finSupp (0_g‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))
299		eqid 2733	. . . . . . . 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 20832	. . . . . . 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 6845	. . . . . 6 ⊢ (𝜑 → (0_g‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∈ V)
303	298, 301, 227, 241, 302	fsuppssov1 9277	. . . . 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 2733	. . . . . . . 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 20169	. . . . . . . 8 ⊢ (𝜑 → ((𝐼 ∖ 𝐽) mPoly 𝑅) ∈ Grp)
306	4, 15, 265, 304, 21, 305, 268	mplmapghm 42677	. . . . . . 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 19142	. . . . . . 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 18706	. . . . . 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 9297	. . . 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 19854	. . 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 6829	. . . 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 19831	. . . 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 6845	. . . 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 6960	. . 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 20206	. . . . . 6 ⊢ (𝜑 → (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)) ∈ CMnd)
318	305	grpmndd 18863	. . . . . 6 ⊢ (𝜑 → ((𝐼 ∖ 𝐽) mPoly 𝑅) ∈ Mnd)
319	15, 277, 317, 318, 256, 308, 273, 303	gsummhm 19854	. . . . 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 6829	. . . . . 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 19831	. . . . . 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 6845	. . . . . 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 6960	. . . . 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 2768	. . . 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 6832	. . 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 2773	. 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 21938	. . . . . . 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 7026	. . . . . 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 2734	. . . . . . 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 2734	. . . . . . 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 6829	. . . . . . 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 7070	. . . . . 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 2734	. . . . . 6 ⊢ (𝜑 → (𝑣 ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)) ↦ (𝑣‘(𝑌 ↾ (𝐼 ∖ 𝐽)))) = (𝑣 ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)) ↦ (𝑣‘(𝑌 ↾ (𝐼 ∖ 𝐽)))))
335		fveq1 6829	. . . . . 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 7070	. . . . 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 2733	. . . . . . . . . 10 ⊢ (._r‘((𝐼 ∖ 𝐽) mPoly 𝑅)) = (._r‘((𝐼 ∖ 𝐽) mPoly 𝑅))
338	4, 216, 54, 15, 337, 265, 227, 241, 328	mplvscaval 21956	. . . . . . . . 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 21956	. . . . . . . . . 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 7370	. . . . . . . . 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 6834	. . . . . . . . . . . 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 2835	. . . . . . . . . 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 21938	. . . . . . . . . . . 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 7026	. . . . . . . . . . 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 20182	. . . . . . . . . 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 2733	. . . . . . . . . . 11 ⊢ (Base‘(Scalar‘((𝐼 ∖ 𝐽) mPoly 𝑅))) = (Base‘(Scalar‘((𝐼 ∖ 𝐽) mPoly 𝑅)))
350		eqid 2733	. . . . . . . . . . 11 ⊢ ( ·_𝑠 ‘((𝐼 ∖ 𝐽) mPoly 𝑅)) = ( ·_𝑠 ‘((𝐼 ∖ 𝐽) mPoly 𝑅))
351	32, 33, 349, 54, 337, 350	asclmul1 21827	. . . . . . . . . 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 2772	. . . . . . . 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 6832	. . . . . . 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 2733	. . . . . . . 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 21956	. . . . . . 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 7453	. . . . . . . . . . . . 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 6831	. . . . . . . . . . . 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 6847	. . . . . . . . . . . 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 2756	. . . . . . . . . . 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 2737	. . . . . . . . . . . . . 14 ⊢ (𝑖 = (𝑌 ↾ (𝐼 ∖ 𝐽)) → (𝑖 = (𝑔 ↾ (𝐼 ∖ 𝐽)) ↔ (𝑌 ↾ (𝐼 ∖ 𝐽)) = (𝑔 ↾ (𝐼 ∖ 𝐽))))
363	362	ifbid 4500	. . . . . . . . . . . . 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 2733	. . . . . . . . . . . . . 14 ⊢ (1_r‘((𝐼 ∖ 𝐽) mPoly 𝑅)) = (1_r‘((𝐼 ∖ 𝐽) mPoly 𝑅))
365		eqid 2733	. . . . . . . . . . . . . . 15 ⊢ (1_r‘𝑅) = (1_r‘𝑅)
366	58	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → 𝑅 ∈ Ring)
367	14, 257, 84, 151, 71	psrbagres 42667	. . . . . . . . . . . . . . 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 21973	. . . . . . . . . . . . . 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 20195	. . . . . . . . . . . . 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 6845	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (1_r‘𝑅) ∈ V)
371		fvexd 6845	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (0_g‘𝑅) ∈ V)
372	370, 371	ifcld 4523	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → if((𝑌 ↾ (𝐼 ∖ 𝐽)) = (𝑔 ↾ (𝐼 ∖ 𝐽)), (1_r‘𝑅), (0_g‘𝑅)) ∈ V)
373	363, 369, 356, 372	fvmptd4 6961	. . . . . . . . . . . 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 20218	. . . . . . . . . . . . . . 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 21946	. . . . . . . . . . . . . . . 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 2768	. . . . . . . . . . . . . 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 6832	. . . . . . . . . . . . 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 6843	. . . . . . . . . . . . . . 15 ⊢ (0_g‘𝑅) ∈ V
380	379	fvconst2 7146	. . . . . . . . . . . . . 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 2768	. . . . . . . . . . . 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 4498	. . . . . . . . . . 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 2780	. . . . . . . . . 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 7370	. . . . . . . . 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 4530	. . . . . . . . . . 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 7365	. . . . . . . . . 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 21859	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑌 ∈ 𝐷 → 𝑌:𝐼⟶ℕ₀)
389	260, 388	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑌:𝐼⟶ℕ₀)
390	389	ffnd 6659	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑌 Fn 𝐼)
391	390	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → 𝑌 Fn 𝐼)
392		undif 4431	. . . . . . . . . . . . . . . . . 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 6582	. . . . . . . . . . . . . . 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 6659	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → 𝑔 Fn 𝐼)
398	394	fneq2d 6582	. . . . . . . . . . . . . . 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 6978	. . . . . . . . . . . . . 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 4500	. . . . . . . . . . . 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 7370	. . . . . . . . . . 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 7453	. . . . . . . . . . 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 2783	. . . . . . . . . 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 2782	. . . . . . . . 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 2768	. . . . . . . 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 21979	. . . . . . . . . . . 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 6850	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → ((𝑔 ↾ (𝐼 ∖ 𝐽))‘𝑘) = (𝑔‘𝑘))
412	411	oveq1d 7369	. . . . . . . . . . . . . 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 7370	. . . . . . . . . . . 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 2768	. . . . . . . . . . 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 2733	. . . . . . . . . . . . 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 2737	. . . . . . . . . . . . . 14 ⊢ (𝑗 = (𝑌 ↾ 𝐽) → (𝑗 = (𝑔 ↾ 𝐽) ↔ (𝑌 ↾ 𝐽) = (𝑔 ↾ 𝐽)))
418	417	ifbid 4500	. . . . . . . . . . . . 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 6845	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (1_r‘((𝐼 ∖ 𝐽) mPoly 𝑅)) ∈ V)
420		fvexd 6845	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (0_g‘((𝐼 ∖ 𝐽) mPoly 𝑅)) ∈ V)
421	419, 420	ifcld 4523	. . . . . . . . . . . . 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 6960	. . . . . . . . . . . 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 42667	. . . . . . . . . . . . . . 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 21979	. . . . . . . . . . . . . 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 6850	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ 𝐽) → ((𝑔 ↾ 𝐽)‘𝑘) = (𝑔‘𝑘))
428	427	oveq1d 7369	. . . . . . . . . . . . . . . 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 7370	. . . . . . . . . . . . . 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 2768	. . . . . . . . . . . . 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 6832	. . . . . . . . . . . 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 2770	. . . . . . . . . . 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 7372	. . . . . . . . . 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 6832	. . . . . . . . 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 7370	. . . . . . . 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 20195	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((𝐹‘𝑔)(._r‘𝑅)(1_r‘𝑅)) = (𝐹‘𝑔))
438	68, 355, 286, 366, 75	ringrzd 20218	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((𝐹‘𝑔)(._r‘𝑅)(0_g‘𝑅)) = (0_g‘𝑅))
439	437, 438	ifeq12d 4498	. . . . . . . 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 2776	. . . . . . 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 2772	. . . . . 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 2768	. . . 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 7370	. . 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 20206	. . . 4 ⊢ (𝜑 → 𝑅 ∈ CMnd)
446	68, 286	ring0cl 20189	. . . . . . . 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 4523	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → if(𝑌 = 𝑔, (𝐹‘𝑔), (0_g‘𝑅)) ∈ (Base‘𝑅))
450	449	fmpttd 7056	. . . 4 ⊢ (𝜑 → (𝑔 ∈ 𝐷 ↦ if(𝑌 = 𝑔, (𝐹‘𝑔), (0_g‘𝑅))):𝐷⟶(Base‘𝑅))
451		eldifsnneq 4744	. . . . . . . 8 ⊢ (𝑔 ∈ (𝐷 ∖ {𝑌}) → ¬ 𝑔 = 𝑌)
452	451	neqcomd 2743	. . . . . . 7 ⊢ (𝑔 ∈ (𝐷 ∖ {𝑌}) → ¬ 𝑌 = 𝑔)
453	452	iffalsed 4487	. . . . . 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 8138	. . . 4 ⊢ (𝜑 → ((𝑔 ∈ 𝐷 ↦ if(𝑌 = 𝑔, (𝐹‘𝑔), (0_g‘𝑅))) supp (0_g‘𝑅)) ⊆ {𝑌})
456	256	mptexd 7166	. . . . 5 ⊢ (𝜑 → (𝑔 ∈ 𝐷 ↦ if(𝑌 = 𝑔, (𝐹‘𝑔), (0_g‘𝑅))) ∈ V)
457		funmpt 6526	. . . . . 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 8974	. . . . . . 7 ⊢ {𝑌} ∈ Fin
460	459	a1i 11	. . . . . 6 ⊢ (𝜑 → {𝑌} ∈ Fin)
461	460, 455	ssfid 9162	. . . . 5 ⊢ (𝜑 → ((𝑔 ∈ 𝐷 ↦ if(𝑌 = 𝑔, (𝐹‘𝑔), (0_g‘𝑅))) supp (0_g‘𝑅)) ∈ Fin)
462	456, 447, 458, 461	isfsuppd 9259	. . . 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 19829	. . 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 4762	. . . . . 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 7370	. . . 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 7026	. . . . 5 ⊢ (𝜑 → (𝐹‘𝑌) ∈ (Base‘𝑅))
468		iftrue 4482	. . . . . . . 8 ⊢ (𝑌 = 𝑔 → if(𝑌 = 𝑔, (𝐹‘𝑔), (0_g‘𝑅)) = (𝐹‘𝑔))
469	468	eqcoms 2741	. . . . . . 7 ⊢ (𝑔 = 𝑌 → if(𝑌 = 𝑔, (𝐹‘𝑔), (0_g‘𝑅)) = (𝐹‘𝑔))
470		fveq2 6830	. . . . . . 7 ⊢ (𝑔 = 𝑌 → (𝐹‘𝑔) = (𝐹‘𝑌))
471	469, 470	eqtrd 2768	. . . . . 6 ⊢ (𝑔 = 𝑌 → if(𝑌 = 𝑔, (𝐹‘𝑔), (0_g‘𝑅)) = (𝐹‘𝑌))
472	68, 471	gsumsn 19870	. . . . 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 2768	. . 3 ⊢ (𝜑 → (𝑅 Σ_g ((𝑔 ∈ 𝐷 ↦ if(𝑌 = 𝑔, (𝐹‘𝑔), (0_g‘𝑅))) ↾ {𝑌})) = (𝐹‘𝑌))
475	444, 463, 474	3eqtr2d 2774	. 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 2772	1 ⊢ (𝜑 → (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑌 ↾ 𝐽))‘(𝑌 ↾ (𝐼 ∖ 𝐽))) = (𝐹‘𝑌))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1541 ∈ wcel 2113 {crab 3396 Vcvv 3437 ∖ cdif 3895 ∪ cun 3896 ∩ cin 3897 ⊆ wss 3898 ∅c0 4282 ifcif 4476 {csn 4577 class class class wbr 5095 ↦ cmpt 5176 × cxp 5619 ^◡ccnv 5620 ↾ cres 5623 “ cima 5624 ∘ ccom 5625 Fun wfun 6482 Fn wfn 6483 ⟶wf 6484 ‘cfv 6488 (class class class)co 7354 supp csupp 8098 ↑_m cmap 8758 Fincfn 8877 finSupp cfsupp 9254 0cc0 11015 ℕcn 12134 ℕ₀cn0 12390 ℤcz 12477 Basecbs 17124 ._rcmulr 17166 Scalarcsca 17168 ·_𝑠 cvsca 17169 0_gc0g 17347 Σ_g cgsu 17348 Mndcmnd 18646 MndHom cmhm 18693 ._gcmg 18984 GrpHom cghm 19128 CMndccmn 19696 mulGrpcmgp 20062 1_rcur 20103 Ringcrg 20155 CRingccrg 20156 RingHom crh 20391 LModclmod 20797 AssAlgcasa 21791 algSccascl 21793 mVar cmvr 21846 mPoly cmpl 21847 eval cevl 22011 selectVars cslv 22046

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7676 ax-cnex 11071 ax-resscn 11072 ax-1cn 11073 ax-icn 11074 ax-addcl 11075 ax-addrcl 11076 ax-mulcl 11077 ax-mulrcl 11078 ax-mulcom 11079 ax-addass 11080 ax-mulass 11081 ax-distr 11082 ax-i2m1 11083 ax-1ne0 11084 ax-1rid 11085 ax-rnegex 11086 ax-rrecex 11087 ax-cnre 11088 ax-pre-lttri 11089 ax-pre-lttrn 11090 ax-pre-ltadd 11091 ax-pre-mulgt0 11092

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-rmo 3347 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-uni 4861 df-int 4900 df-iun 4945 df-iin 4946 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-se 5575 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6255 df-ord 6316 df-on 6317 df-lim 6318 df-suc 6319 df-iota 6444 df-fun 6490 df-fn 6491 df-f 6492 df-f1 6493 df-fo 6494 df-f1o 6495 df-fv 6496 df-isom 6497 df-riota 7311 df-ov 7357 df-oprab 7358 df-mpo 7359 df-of 7618 df-ofr 7619 df-om 7805 df-1st 7929 df-2nd 7930 df-supp 8099 df-frecs 8219 df-wrecs 8250 df-recs 8299 df-rdg 8337 df-1o 8393 df-2o 8394 df-er 8630 df-map 8760 df-pm 8761 df-ixp 8830 df-en 8878 df-dom 8879 df-sdom 8880 df-fin 8881 df-fsupp 9255 df-sup 9335 df-oi 9405 df-card 9841 df-pnf 11157 df-mnf 11158 df-xr 11159 df-ltxr 11160 df-le 11161 df-sub 11355 df-neg 11356 df-nn 12135 df-2 12197 df-3 12198 df-4 12199 df-5 12200 df-6 12201 df-7 12202 df-8 12203 df-9 12204 df-n0 12391 df-z 12478 df-dec 12597 df-uz 12741 df-fz 13412 df-fzo 13559 df-seq 13913 df-hash 14242 df-struct 17062 df-sets 17079 df-slot 17097 df-ndx 17109 df-base 17125 df-ress 17146 df-plusg 17178 df-mulr 17179 df-sca 17181 df-vsca 17182 df-ip 17183 df-tset 17184 df-ple 17185 df-ds 17187 df-hom 17189 df-cco 17190 df-0g 17349 df-gsum 17350 df-prds 17355 df-pws 17357 df-mre 17492 df-mrc 17493 df-acs 17495 df-mgm 18552 df-sgrp 18631 df-mnd 18647 df-mhm 18695 df-submnd 18696 df-grp 18853 df-minusg 18854 df-sbg 18855 df-mulg 18985 df-subg 19040 df-ghm 19129 df-cntz 19233 df-cmn 19698 df-abl 19699 df-mgp 20063 df-rng 20075 df-ur 20104 df-srg 20109 df-ring 20157 df-cring 20158 df-rhm 20394 df-subrng 20465 df-subrg 20489 df-lmod 20799 df-lss 20869 df-lsp 20909 df-assa 21794 df-asp 21795 df-ascl 21796 df-psr 21850 df-mvr 21851 df-mpl 21852 df-evls 22012 df-evl 22013 df-selv 22050

This theorem is referenced by: evlselv 42708

W3C validator