Theorem evlselv 43597

Description: Evaluating a selection of variable assignments, then evaluating the rest of the variables, is the same as evaluating with all assignments. (Contributed by SN, 10-Mar-2025.)

Hypotheses

Ref	Expression
evlselv.p	⊢ 𝑃 = (𝐼 mPoly 𝑅)
evlselv.k	⊢ 𝐾 = (Base‘𝑅)
evlselv.b	⊢ 𝐵 = (Base‘𝑃)
evlselv.u	⊢ 𝑈 = ((𝐼 ∖ 𝐽) mPoly 𝑅)
evlselv.t	⊢ 𝑇 = (𝐽 mPoly 𝑈)
evlselv.l	⊢ 𝐿 = (algSc‘𝑈)
evlselv.i	⊢ (𝜑 → 𝐼 ∈ 𝑉)
evlselv.r	⊢ (𝜑 → 𝑅 ∈ CRing)
evlselv.j	⊢ (𝜑 → 𝐽 ⊆ 𝐼)
evlselv.f	⊢ (𝜑 → 𝐹 ∈ 𝐵)
evlselv.a	⊢ (𝜑 → 𝐴 ∈ (𝐾 ↑m 𝐼))

Assertion

Ref	Expression
evlselv	⊢ (𝜑 → ((((𝐼 ∖ 𝐽) eval 𝑅)‘(((𝐽 eval 𝑈)‘(((𝐼 selectVars 𝑅)‘𝐽)‘𝐹))‘(𝐿 ∘ (𝐴 ↾ 𝐽))))‘(𝐴 ↾ (𝐼 ∖ 𝐽))) = (((𝐼 eval 𝑅)‘𝐹)‘𝐴))

Proof of Theorem evlselv

Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑒 𝑖 𝑗 𝑘 𝑢 𝑣 𝑓 ℎ 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2761	. . . . . . . . . . . . 13 ⊢ (Base‘𝑈) = (Base‘𝑈)
2		eqid 2761	. . . . . . . . . . . . 13 ⊢ (.r‘𝑈) = (.r‘𝑈)
3		evlselv.u	. . . . . . . . . . . . . . . 16 ⊢ 𝑈 = ((𝐼 ∖ 𝐽) mPoly 𝑅)
4		evlselv.i	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
5		difssd 4084	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐼 ∖ 𝐽) ⊆ 𝐼)
6	4, 5	ssexd 5286	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐼 ∖ 𝐽) ∈ V)
7		evlselv.r	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑅 ∈ CRing)
8	3, 6, 7	mplcrngd 22324	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑈 ∈ CRing)
9	8	crngringd 20466	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑈 ∈ Ring)
10	9	ad2antrr 739	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin}) → 𝑈 ∈ Ring)
11		evlselv.t	. . . . . . . . . . . . . . . 16 ⊢ 𝑇 = (𝐽 mPoly 𝑈)
12		eqid 2761	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝑇) = (Base‘𝑇)
13		eqid 2761	. . . . . . . . . . . . . . . 16 ⊢ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} = {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin}
14		evlselv.p	. . . . . . . . . . . . . . . . 17 ⊢ 𝑃 = (𝐼 mPoly 𝑅)
15		evlselv.b	. . . . . . . . . . . . . . . . 17 ⊢ 𝐵 = (Base‘𝑃)
16		evlselv.j	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐽 ⊆ 𝐼)
17		evlselv.f	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐹 ∈ 𝐵)
18	14, 15, 3, 11, 12, 7, 16, 17	selvcl 22442	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (((𝐼 selectVars 𝑅)‘𝐽)‘𝐹) ∈ (Base‘𝑇))
19	11, 1, 12, 13, 18	mplelf 22298	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (((𝐼 selectVars 𝑅)‘𝐽)‘𝐹):{𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin}⟶(Base‘𝑈))
20	19	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (((𝐼 selectVars 𝑅)‘𝐽)‘𝐹):{𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin}⟶(Base‘𝑈))
21	20	ffvelcdmda 7082	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin}) → ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒) ∈ (Base‘𝑈))
22		eqid 2761	. . . . . . . . . . . . . 14 ⊢ (mulGrp‘𝑈) = (mulGrp‘𝑈)
23		eqid 2761	. . . . . . . . . . . . . 14 ⊢ (.g‘(mulGrp‘𝑈)) = (.g‘(mulGrp‘𝑈))
24	4, 16	ssexd 5286	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐽 ∈ V)
25	24	ad2antrr 739	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin}) → 𝐽 ∈ V)
26	8	ad2antrr 739	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin}) → 𝑈 ∈ CRing)
27		fvexd 6898	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (Base‘𝑈) ∈ V)
28		evlselv.k	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐾 = (Base‘𝑅)
29	28	fvexi 6897	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐾 ∈ V
30	29	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐾 ∈ V)
31		evlselv.l	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐿 = (algSc‘𝑈)
32	7	crngringd 20466	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑅 ∈ Ring)
33	3, 1, 28, 31, 6, 32	mplasclf 22367	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐿:𝐾⟶(Base‘𝑈))
34	27, 30, 33	elmapdd 8854	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐿 ∈ ((Base‘𝑈) ↑m 𝐾))
35		evlselv.a	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐴 ∈ (𝐾 ↑m 𝐼))
36	35, 16	elmapssresd 8888	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐴 ↾ 𝐽) ∈ (𝐾 ↑m 𝐽))
37	34, 36	mapcod 43274	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐿 ∘ (𝐴 ↾ 𝐽)) ∈ ((Base‘𝑈) ↑m 𝐽))
38	37	ad2antrr 739	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin}) → (𝐿 ∘ (𝐴 ↾ 𝐽)) ∈ ((Base‘𝑈) ↑m 𝐽))
39		simpr 490	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin}) → 𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin})
40	13, 1, 22, 23, 25, 26, 38, 39	evlsvvvallem 22393	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin}) → ((mulGrp‘𝑈) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴 ↾ 𝐽))‘𝑗)))) ∈ (Base‘𝑈))
41	1, 2, 10, 21, 40	ringcld 20477	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin}) → (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r‘𝑈)((mulGrp‘𝑈) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴 ↾ 𝐽))‘𝑗))))) ∈ (Base‘𝑈))
42		eqidd 2762	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r‘𝑈)((mulGrp‘𝑈) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴 ↾ 𝐽))‘𝑗)))))) = (𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r‘𝑈)((mulGrp‘𝑈) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴 ↾ 𝐽))‘𝑗)))))))
43		eqidd 2762	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝑢 ∈ (Base‘𝑈) ↦ (𝑢‘𝑐)) = (𝑢 ∈ (Base‘𝑈) ↦ (𝑢‘𝑐)))
44		fveq1 6882	. . . . . . . . . . . 12 ⊢ (𝑢 = (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r‘𝑈)((mulGrp‘𝑈) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴 ↾ 𝐽))‘𝑗))))) → (𝑢‘𝑐) = ((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r‘𝑈)((mulGrp‘𝑈) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴 ↾ 𝐽))‘𝑗)))))‘𝑐))
45	41, 42, 43, 44	fmptco 7128	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → ((𝑢 ∈ (Base‘𝑈) ↦ (𝑢‘𝑐)) ∘ (𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r‘𝑈)((mulGrp‘𝑈) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴 ↾ 𝐽))‘𝑗))))))) = (𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ ((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r‘𝑈)((mulGrp‘𝑈) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴 ↾ 𝐽))‘𝑗)))))‘𝑐)))
46	33	ad2antrr 739	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin}) → 𝐿:𝐾⟶(Base‘𝑈))
47		eqid 2761	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
48	47, 28	mgpbas 20358	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝐾 = (Base‘(mulGrp‘𝑅))
49		eqid 2761	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (.g‘(mulGrp‘𝑅)) = (.g‘(mulGrp‘𝑅))
50	47	ringmgp 20458	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑅 ∈ Ring → (mulGrp‘𝑅) ∈ Mnd)
51	32, 50	syl 18	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (mulGrp‘𝑅) ∈ Mnd)
52	51	ad3antrrr 743	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin}) ∧ 𝑗 ∈ 𝐽) → (mulGrp‘𝑅) ∈ Mnd)
53	13	psrbagf 22219	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} → 𝑒:𝐽⟶ℕ0)
54	53	adantl 487	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin}) → 𝑒:𝐽⟶ℕ0)
55	54	ffvelcdmda 7082	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin}) ∧ 𝑗 ∈ 𝐽) → (𝑒‘𝑗) ∈ ℕ0)
56		elmapi 8862	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐴 ∈ (𝐾 ↑m 𝐼) → 𝐴:𝐼⟶𝐾)
57	35, 56	syl 18	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝐴:𝐼⟶𝐾)
58	57, 16	fssresd 6747	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (𝐴 ↾ 𝐽):𝐽⟶𝐾)
59	58	ad2antrr 739	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin}) → (𝐴 ↾ 𝐽):𝐽⟶𝐾)
60	59	ffvelcdmda 7082	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin}) ∧ 𝑗 ∈ 𝐽) → ((𝐴 ↾ 𝐽)‘𝑗) ∈ 𝐾)
61	48, 49, 52, 55, 60	mulgnn0cld 19298	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin}) ∧ 𝑗 ∈ 𝐽) → ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)) ∈ 𝐾)
62	46, 61	cofmpt 7131	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin}) → (𝐿 ∘ (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))) = (𝑗 ∈ 𝐽 ↦ (𝐿‘((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))
63	3	mplassa 22322	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝐼 ∖ 𝐽) ∈ V ∧ 𝑅 ∈ CRing) → 𝑈 ∈ AssAlg)
64	6, 7, 63	syl2anc 596	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → 𝑈 ∈ AssAlg)
65		eqid 2761	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (Scalar‘𝑈) = (Scalar‘𝑈)
66	31, 65	asclrhm 22191	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑈 ∈ AssAlg → 𝐿 ∈ ((Scalar‘𝑈) RingHom 𝑈))
67	64, 66	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → 𝐿 ∈ ((Scalar‘𝑈) RingHom 𝑈))
68	3, 6, 7	mplsca 22313	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → 𝑅 = (Scalar‘𝑈))
69	68	eqcomd 2767	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → (Scalar‘𝑈) = 𝑅)
70	69	oveq1d 7433	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → ((Scalar‘𝑈) RingHom 𝑈) = (𝑅 RingHom 𝑈))
71	67, 70	eleqtrd 2863	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝐿 ∈ (𝑅 RingHom 𝑈))
72	47, 22	rhmmhm 20703	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐿 ∈ (𝑅 RingHom 𝑈) → 𝐿 ∈ ((mulGrp‘𝑅) MndHom (mulGrp‘𝑈)))
73	71, 72	syl 18	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝐿 ∈ ((mulGrp‘𝑅) MndHom (mulGrp‘𝑈)))
74	73	ad3antrrr 743	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin}) ∧ 𝑗 ∈ 𝐽) → 𝐿 ∈ ((mulGrp‘𝑅) MndHom (mulGrp‘𝑈)))
75	48, 49, 23	mhmmulg 19318	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐿 ∈ ((mulGrp‘𝑅) MndHom (mulGrp‘𝑈)) ∧ (𝑒‘𝑗) ∈ ℕ0 ∧ ((𝐴 ↾ 𝐽)‘𝑗) ∈ 𝐾) → (𝐿‘((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))) = ((𝑒‘𝑗)(.g‘(mulGrp‘𝑈))(𝐿‘((𝐴 ↾ 𝐽)‘𝑗))))
76	74, 55, 60, 75	syl3anc 1398	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin}) ∧ 𝑗 ∈ 𝐽) → (𝐿‘((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))) = ((𝑒‘𝑗)(.g‘(mulGrp‘𝑈))(𝐿‘((𝐴 ↾ 𝐽)‘𝑗))))
77	58	ad3antrrr 743	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin}) ∧ 𝑗 ∈ 𝐽) → (𝐴 ↾ 𝐽):𝐽⟶𝐾)
78		simpr 490	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin}) ∧ 𝑗 ∈ 𝐽) → 𝑗 ∈ 𝐽)
79	77, 78	fvco3d 6984	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin}) ∧ 𝑗 ∈ 𝐽) → ((𝐿 ∘ (𝐴 ↾ 𝐽))‘𝑗) = (𝐿‘((𝐴 ↾ 𝐽)‘𝑗)))
80	79	oveq2d 7434	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin}) ∧ 𝑗 ∈ 𝐽) → ((𝑒‘𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴 ↾ 𝐽))‘𝑗)) = ((𝑒‘𝑗)(.g‘(mulGrp‘𝑈))(𝐿‘((𝐴 ↾ 𝐽)‘𝑗))))
81	76, 80	eqtr4d 2799	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin}) ∧ 𝑗 ∈ 𝐽) → (𝐿‘((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))) = ((𝑒‘𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴 ↾ 𝐽))‘𝑗)))
82	81	mpteq2dva 5198	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin}) → (𝑗 ∈ 𝐽 ↦ (𝐿‘((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))) = (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴 ↾ 𝐽))‘𝑗))))
83	62, 82	eqtrd 2796	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin}) → (𝐿 ∘ (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))) = (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴 ↾ 𝐽))‘𝑗))))
84	83	oveq2d 7434	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin}) → ((mulGrp‘𝑈) Σg (𝐿 ∘ (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))))) = ((mulGrp‘𝑈) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴 ↾ 𝐽))‘𝑗)))))
85		eqid 2761	. . . . . . . . . . . . . . . . . 18 ⊢ (Base‘(mulGrp‘(Scalar‘𝑈))) = (Base‘(mulGrp‘(Scalar‘𝑈)))
86		eqid 2761	. . . . . . . . . . . . . . . . . 18 ⊢ (0g‘(mulGrp‘(Scalar‘𝑈))) = (0g‘(mulGrp‘(Scalar‘𝑈)))
87	68, 7	eqeltrrd 2862	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (Scalar‘𝑈) ∈ CRing)
88		eqid 2761	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (mulGrp‘(Scalar‘𝑈)) = (mulGrp‘(Scalar‘𝑈))
89	88	crngmgp 20460	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((Scalar‘𝑈) ∈ CRing → (mulGrp‘(Scalar‘𝑈)) ∈ CMnd)
90	87, 89	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (mulGrp‘(Scalar‘𝑈)) ∈ CMnd)
91	90	ad2antrr 739	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin}) → (mulGrp‘(Scalar‘𝑈)) ∈ CMnd)
92	22	ringmgp 20458	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑈 ∈ Ring → (mulGrp‘𝑈) ∈ Mnd)
93	9, 92	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (mulGrp‘𝑈) ∈ Mnd)
94	93	ad2antrr 739	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin}) → (mulGrp‘𝑈) ∈ Mnd)
95	88, 22	rhmmhm 20703	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐿 ∈ ((Scalar‘𝑈) RingHom 𝑈) → 𝐿 ∈ ((mulGrp‘(Scalar‘𝑈)) MndHom (mulGrp‘𝑈)))
96	67, 95	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐿 ∈ ((mulGrp‘(Scalar‘𝑈)) MndHom (mulGrp‘𝑈)))
97	96	ad2antrr 739	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin}) → 𝐿 ∈ ((mulGrp‘(Scalar‘𝑈)) MndHom (mulGrp‘𝑈)))
98	68	fveq2d 6887	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (Base‘𝑅) = (Base‘(Scalar‘𝑈)))
99	28, 98	eqtrid 2808	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝐾 = (Base‘(Scalar‘𝑈)))
100	99	ad3antrrr 743	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin}) ∧ 𝑗 ∈ 𝐽) → 𝐾 = (Base‘(Scalar‘𝑈)))
101	61, 100	eleqtrd 2863	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin}) ∧ 𝑗 ∈ 𝐽) → ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)) ∈ (Base‘(Scalar‘𝑈)))
102		eqid 2761	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (Base‘(Scalar‘𝑈)) = (Base‘(Scalar‘𝑈))
103	88, 102	mgpbas 20358	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Base‘(Scalar‘𝑈)) = (Base‘(mulGrp‘(Scalar‘𝑈)))
104	101, 103	eleqtrdi 2871	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin}) ∧ 𝑗 ∈ 𝐽) → ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)) ∈ (Base‘(mulGrp‘(Scalar‘𝑈))))
105	104	fmpttd 7113	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin}) → (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))):𝐽⟶(Base‘(mulGrp‘(Scalar‘𝑈))))
106	54	feqmptd 6951	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin}) → 𝑒 = (𝑗 ∈ 𝐽 ↦ (𝑒‘𝑗)))
107	13	psrbagfsupp 22220	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} → 𝑒 finSupp 0)
108	107	adantl 487	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin}) → 𝑒 finSupp 0)
109	106, 108	eqbrtrrd 5129	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin}) → (𝑗 ∈ 𝐽 ↦ (𝑒‘𝑗)) finSupp 0)
110		eqid 2761	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (0g‘(mulGrp‘𝑅)) = (0g‘(mulGrp‘𝑅))
111	48, 110, 49	mulg0 19277	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 ∈ 𝐾 → (0(.g‘(mulGrp‘𝑅))𝑘) = (0g‘(mulGrp‘𝑅)))
112	111	adantl 487	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ 𝐾) → (0(.g‘(mulGrp‘𝑅))𝑘) = (0g‘(mulGrp‘𝑅)))
113		fvexd 6898	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin}) → (0g‘(mulGrp‘𝑅)) ∈ V)
114	109, 112, 55, 60, 113	fsuppssov1 9369	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin}) → (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))) finSupp (0g‘(mulGrp‘𝑅)))
115		eqid 2761	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (1r‘𝑅) = (1r‘𝑅)
116	47, 115	ringidval 20402	. . . . . . . . . . . . . . . . . . . 20 ⊢ (1r‘𝑅) = (0g‘(mulGrp‘𝑅))
117	114, 116	breqtrrdi 5147	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin}) → (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))) finSupp (1r‘𝑅))
118	68	fveq2d 6887	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (1r‘𝑅) = (1r‘(Scalar‘𝑈)))
119		eqid 2761	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (1r‘(Scalar‘𝑈)) = (1r‘(Scalar‘𝑈))
120	88, 119	ringidval 20402	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (1r‘(Scalar‘𝑈)) = (0g‘(mulGrp‘(Scalar‘𝑈)))
121	118, 120	eqtrdi 2812	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (1r‘𝑅) = (0g‘(mulGrp‘(Scalar‘𝑈))))
122	121	ad2antrr 739	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin}) → (1r‘𝑅) = (0g‘(mulGrp‘(Scalar‘𝑈))))
123	117, 122	breqtrd 5131	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin}) → (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))) finSupp (0g‘(mulGrp‘(Scalar‘𝑈))))
124	85, 86, 91, 94, 25, 97, 105, 123	gsummhm 20145	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin}) → ((mulGrp‘𝑈) Σg (𝐿 ∘ (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))))) = (𝐿‘((mulGrp‘(Scalar‘𝑈)) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))))))
125	84, 124	eqtr3d 2798	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin}) → ((mulGrp‘𝑈) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴 ↾ 𝐽))‘𝑗)))) = (𝐿‘((mulGrp‘(Scalar‘𝑈)) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))))))
126	125	oveq2d 7434	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin}) → (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r‘𝑈)((mulGrp‘𝑈) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴 ↾ 𝐽))‘𝑗))))) = (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r‘𝑈)(𝐿‘((mulGrp‘(Scalar‘𝑈)) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))))
127	64	ad2antrr 739	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin}) → 𝑈 ∈ AssAlg)
128	101	fmpttd 7113	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin}) → (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))):𝐽⟶(Base‘(Scalar‘𝑈)))
129	123, 120	breqtrrdi 5147	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin}) → (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))) finSupp (1r‘(Scalar‘𝑈)))
130	103, 120, 91, 25, 128, 129	gsumcl 20122	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin}) → ((mulGrp‘(Scalar‘𝑈)) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))) ∈ (Base‘(Scalar‘𝑈)))
131		eqid 2761	. . . . . . . . . . . . . . . . 17 ⊢ ( ·𝑠 ‘𝑈) = ( ·𝑠 ‘𝑈)
132	31, 65, 102, 1, 2, 131	asclmul2 22188	. . . . . . . . . . . . . . . 16 ⊢ ((𝑈 ∈ AssAlg ∧ ((mulGrp‘(Scalar‘𝑈)) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))) ∈ (Base‘(Scalar‘𝑈)) ∧ ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒) ∈ (Base‘𝑈)) → (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r‘𝑈)(𝐿‘((mulGrp‘(Scalar‘𝑈)) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))) = (((mulGrp‘(Scalar‘𝑈)) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))))( ·𝑠 ‘𝑈)((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)))
133	127, 130, 21, 132	syl3anc 1398	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin}) → (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r‘𝑈)(𝐿‘((mulGrp‘(Scalar‘𝑈)) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))) = (((mulGrp‘(Scalar‘𝑈)) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))))( ·𝑠 ‘𝑈)((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)))
134	126, 133	eqtrd 2796	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin}) → (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r‘𝑈)((mulGrp‘𝑈) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴 ↾ 𝐽))‘𝑗))))) = (((mulGrp‘(Scalar‘𝑈)) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))))( ·𝑠 ‘𝑈)((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)))
135	134	fveq1d 6885	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin}) → ((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r‘𝑈)((mulGrp‘𝑈) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴 ↾ 𝐽))‘𝑗)))))‘𝑐) = ((((mulGrp‘(Scalar‘𝑈)) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))))( ·𝑠 ‘𝑈)((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒))‘𝑐))
136		eqid 2761	. . . . . . . . . . . . . 14 ⊢ (.r‘𝑅) = (.r‘𝑅)
137		eqid 2761	. . . . . . . . . . . . . 14 ⊢ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}
138	99	ad2antrr 739	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin}) → 𝐾 = (Base‘(Scalar‘𝑈)))
139	130, 138	eleqtrrd 2864	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin}) → ((mulGrp‘(Scalar‘𝑈)) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))) ∈ 𝐾)
140		simplr 781	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin}) → 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin})
141	3, 131, 28, 1, 136, 137, 139, 21, 140	mplvscaval 22316	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin}) → ((((mulGrp‘(Scalar‘𝑈)) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))))( ·𝑠 ‘𝑈)((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒))‘𝑐) = (((mulGrp‘(Scalar‘𝑈)) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))))(.r‘𝑅)(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)))
142	135, 141	eqtrd 2796	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin}) → ((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r‘𝑈)((mulGrp‘𝑈) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴 ↾ 𝐽))‘𝑗)))))‘𝑐) = (((mulGrp‘(Scalar‘𝑈)) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))))(.r‘𝑅)(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)))
143	142	mpteq2dva 5198	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ ((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r‘𝑈)((mulGrp‘𝑈) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴 ↾ 𝐽))‘𝑗)))))‘𝑐)) = (𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (((mulGrp‘(Scalar‘𝑈)) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))))(.r‘𝑅)(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐))))
144	45, 143	eqtrd 2796	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → ((𝑢 ∈ (Base‘𝑈) ↦ (𝑢‘𝑐)) ∘ (𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r‘𝑈)((mulGrp‘𝑈) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴 ↾ 𝐽))‘𝑗))))))) = (𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (((mulGrp‘(Scalar‘𝑈)) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))))(.r‘𝑅)(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐))))
145	144	oveq2d 7434	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝑅 Σg ((𝑢 ∈ (Base‘𝑈) ↦ (𝑢‘𝑐)) ∘ (𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r‘𝑈)((mulGrp‘𝑈) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴 ↾ 𝐽))‘𝑗)))))))) = (𝑅 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (((mulGrp‘(Scalar‘𝑈)) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))))(.r‘𝑅)(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)))))
146	69	fveq2d 6887	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (mulGrp‘(Scalar‘𝑈)) = (mulGrp‘𝑅))
147	146	ad2antrr 739	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin}) → (mulGrp‘(Scalar‘𝑈)) = (mulGrp‘𝑅))
148	147	oveq1d 7433	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin}) → ((mulGrp‘(Scalar‘𝑈)) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))) = ((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))
149	148	oveq1d 7433	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin}) → (((mulGrp‘(Scalar‘𝑈)) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))))(.r‘𝑅)(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)) = (((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))))(.r‘𝑅)(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)))
150	7	ad2antrr 739	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin}) → 𝑅 ∈ CRing)
151	148, 139	eqeltrrd 2862	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin}) → ((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))) ∈ 𝐾)
152	19	ffvelcdmda 7082	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin}) → ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒) ∈ (Base‘𝑈))
153	3, 28, 1, 137, 152	mplelf 22298	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin}) → ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒):{𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶𝐾)
154	153	ffvelcdmda 7082	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin}) ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐) ∈ 𝐾)
155	154	an32s 665	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin}) → (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐) ∈ 𝐾)
156	28, 136, 150, 151, 155	crngcomd 20475	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin}) → (((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))))(.r‘𝑅)(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)) = ((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))))))
157	149, 156	eqtrd 2796	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin}) → (((mulGrp‘(Scalar‘𝑈)) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))))(.r‘𝑅)(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)) = ((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))))))
158	157	mpteq2dva 5198	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (((mulGrp‘(Scalar‘𝑈)) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))))(.r‘𝑅)(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐))) = (𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ ((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))))
159	158	oveq2d 7434	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝑅 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (((mulGrp‘(Scalar‘𝑈)) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))))(.r‘𝑅)(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)))) = (𝑅 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ ((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))))))))
160	145, 159	eqtrd 2796	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝑅 Σg ((𝑢 ∈ (Base‘𝑈) ↦ (𝑢‘𝑐)) ∘ (𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r‘𝑈)((mulGrp‘𝑈) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴 ↾ 𝐽))‘𝑗)))))))) = (𝑅 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ ((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))))))))
161	160	oveq1d 7433	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → ((𝑅 Σg ((𝑢 ∈ (Base‘𝑈) ↦ (𝑢‘𝑐)) ∘ (𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r‘𝑈)((mulGrp‘𝑈) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴 ↾ 𝐽))‘𝑗))))))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑐‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))) = ((𝑅 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ ((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑐‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))))
162		eqid 2761	. . . . . . . . . 10 ⊢ (𝑢 ∈ (Base‘𝑈) ↦ (𝑢‘𝑐)) = (𝑢 ∈ (Base‘𝑈) ↦ (𝑢‘𝑐))
163		fveq1 6882	. . . . . . . . . 10 ⊢ (𝑢 = (𝑈 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r‘𝑈)((mulGrp‘𝑈) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴 ↾ 𝐽))‘𝑗))))))) → (𝑢‘𝑐) = ((𝑈 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r‘𝑈)((mulGrp‘𝑈) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴 ↾ 𝐽))‘𝑗)))))))‘𝑐))
164		eqid 2761	. . . . . . . . . . . . 13 ⊢ (𝐽 eval 𝑈) = (𝐽 eval 𝑈)
165	164, 11, 12, 13, 1, 22, 23, 2, 24, 8, 18, 37	evlvvval 22435	. . . . . . . . . . . 12 ⊢ (𝜑 → (((𝐽 eval 𝑈)‘(((𝐼 selectVars 𝑅)‘𝐽)‘𝐹))‘(𝐿 ∘ (𝐴 ↾ 𝐽))) = (𝑈 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r‘𝑈)((mulGrp‘𝑈) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴 ↾ 𝐽))‘𝑗))))))))
166	164, 11, 12, 1, 24, 8, 18, 37	evlcl 22404	. . . . . . . . . . . 12 ⊢ (𝜑 → (((𝐽 eval 𝑈)‘(((𝐼 selectVars 𝑅)‘𝐽)‘𝐹))‘(𝐿 ∘ (𝐴 ↾ 𝐽))) ∈ (Base‘𝑈))
167	165, 166	eqeltrrd 2862	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑈 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r‘𝑈)((mulGrp‘𝑈) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴 ↾ 𝐽))‘𝑗))))))) ∈ (Base‘𝑈))
168	167	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝑈 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r‘𝑈)((mulGrp‘𝑈) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴 ↾ 𝐽))‘𝑗))))))) ∈ (Base‘𝑈))
169		fvexd 6898	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → ((𝑈 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r‘𝑈)((mulGrp‘𝑈) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴 ↾ 𝐽))‘𝑗)))))))‘𝑐) ∈ V)
170	162, 163, 168, 169	fvmptd3 7015	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → ((𝑢 ∈ (Base‘𝑈) ↦ (𝑢‘𝑐))‘(𝑈 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r‘𝑈)((mulGrp‘𝑈) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴 ↾ 𝐽))‘𝑗)))))))) = ((𝑈 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r‘𝑈)((mulGrp‘𝑈) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴 ↾ 𝐽))‘𝑗)))))))‘𝑐))
171		eqid 2761	. . . . . . . . . 10 ⊢ (0g‘𝑈) = (0g‘𝑈)
172	9	ringcmnd 20506	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑈 ∈ CMnd)
173	172	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → 𝑈 ∈ CMnd)
174	7	crnggrpd 20467	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑅 ∈ Grp)
175	174	grpmndd 19150	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ Mnd)
176	175	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → 𝑅 ∈ Mnd)
177		ovex 7451	. . . . . . . . . . . 12 ⊢ (ℕ0 ↑m 𝐽) ∈ V
178	177	rabex 5300	. . . . . . . . . . 11 ⊢ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} ∈ V
179	178	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} ∈ V)
180	6	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝐼 ∖ 𝐽) ∈ V)
181	174	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → 𝑅 ∈ Grp)
182		simpr 490	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin})
183	3, 1, 137, 162, 180, 181, 182	mplmapghm 22424	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝑢 ∈ (Base‘𝑈) ↦ (𝑢‘𝑐)) ∈ (𝑈 GrpHom 𝑅))
184		ghmmhm 19433	. . . . . . . . . . 11 ⊢ ((𝑢 ∈ (Base‘𝑈) ↦ (𝑢‘𝑐)) ∈ (𝑈 GrpHom 𝑅) → (𝑢 ∈ (Base‘𝑈) ↦ (𝑢‘𝑐)) ∈ (𝑈 MndHom 𝑅))
185	183, 184	syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝑢 ∈ (Base‘𝑈) ↦ (𝑢‘𝑐)) ∈ (𝑈 MndHom 𝑅))
186	41	fmpttd 7113	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r‘𝑈)((mulGrp‘𝑈) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴 ↾ 𝐽))‘𝑗)))))):{𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin}⟶(Base‘𝑈))
187	24	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → 𝐽 ∈ V)
188	8	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → 𝑈 ∈ CRing)
189	18	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (((𝐼 selectVars 𝑅)‘𝐽)‘𝐹) ∈ (Base‘𝑇))
190	37	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝐿 ∘ (𝐴 ↾ 𝐽)) ∈ ((Base‘𝑈) ↑m 𝐽))
191	13, 11, 12, 1, 22, 23, 2, 187, 188, 189, 190	evlvvvallem 43595	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r‘𝑈)((mulGrp‘𝑈) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴 ↾ 𝐽))‘𝑗)))))) finSupp (0g‘𝑈))
192	1, 171, 173, 176, 179, 185, 186, 191	gsummhm 20145	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝑅 Σg ((𝑢 ∈ (Base‘𝑈) ↦ (𝑢‘𝑐)) ∘ (𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r‘𝑈)((mulGrp‘𝑈) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴 ↾ 𝐽))‘𝑗)))))))) = ((𝑢 ∈ (Base‘𝑈) ↦ (𝑢‘𝑐))‘(𝑈 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r‘𝑈)((mulGrp‘𝑈) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴 ↾ 𝐽))‘𝑗)))))))))
193	165	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (((𝐽 eval 𝑈)‘(((𝐼 selectVars 𝑅)‘𝐽)‘𝐹))‘(𝐿 ∘ (𝐴 ↾ 𝐽))) = (𝑈 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r‘𝑈)((mulGrp‘𝑈) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴 ↾ 𝐽))‘𝑗))))))))
194	193	fveq1d 6885	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → ((((𝐽 eval 𝑈)‘(((𝐼 selectVars 𝑅)‘𝐽)‘𝐹))‘(𝐿 ∘ (𝐴 ↾ 𝐽)))‘𝑐) = ((𝑈 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r‘𝑈)((mulGrp‘𝑈) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴 ↾ 𝐽))‘𝑗)))))))‘𝑐))
195	170, 192, 194	3eqtr4rd 2807	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → ((((𝐽 eval 𝑈)‘(((𝐼 selectVars 𝑅)‘𝐽)‘𝐹))‘(𝐿 ∘ (𝐴 ↾ 𝐽)))‘𝑐) = (𝑅 Σg ((𝑢 ∈ (Base‘𝑈) ↦ (𝑢‘𝑐)) ∘ (𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r‘𝑈)((mulGrp‘𝑈) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴 ↾ 𝐽))‘𝑗)))))))))
196	195	oveq1d 7433	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (((((𝐽 eval 𝑈)‘(((𝐼 selectVars 𝑅)‘𝐽)‘𝐹))‘(𝐿 ∘ (𝐴 ↾ 𝐽)))‘𝑐)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑐‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))) = ((𝑅 Σg ((𝑢 ∈ (Base‘𝑈) ↦ (𝑢‘𝑐)) ∘ (𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r‘𝑈)((mulGrp‘𝑈) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴 ↾ 𝐽))‘𝑗))))))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑐‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))))
197		eqid 2761	. . . . . . . 8 ⊢ (0g‘𝑅) = (0g‘𝑅)
198	32	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → 𝑅 ∈ Ring)
199	47	crngmgp 20460	. . . . . . . . . . 11 ⊢ (𝑅 ∈ CRing → (mulGrp‘𝑅) ∈ CMnd)
200	7, 199	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (mulGrp‘𝑅) ∈ CMnd)
201	200	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (mulGrp‘𝑅) ∈ CMnd)
202	51	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → (mulGrp‘𝑅) ∈ Mnd)
203	137	psrbagf 22219	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} → 𝑐:(𝐼 ∖ 𝐽)⟶ℕ0)
204	203	adantl 487	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → 𝑐:(𝐼 ∖ 𝐽)⟶ℕ0)
205	204	ffvelcdmda 7082	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → (𝑐‘𝑘) ∈ ℕ0)
206	57, 5	fssresd 6747	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴 ↾ (𝐼 ∖ 𝐽)):(𝐼 ∖ 𝐽)⟶𝐾)
207	206	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝐴 ↾ (𝐼 ∖ 𝐽)):(𝐼 ∖ 𝐽)⟶𝐾)
208	207	ffvelcdmda 7082	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → ((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘) ∈ 𝐾)
209	48, 49, 202, 205, 208	mulgnn0cld 19298	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → ((𝑐‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)) ∈ 𝐾)
210	209	fmpttd 7113	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑐‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))):(𝐼 ∖ 𝐽)⟶𝐾)
211	204	feqmptd 6951	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → 𝑐 = (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (𝑐‘𝑘)))
212	137	psrbagfsupp 22220	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} → 𝑐 finSupp 0)
213	212	adantl 487	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → 𝑐 finSupp 0)
214	211, 213	eqbrtrrd 5129	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (𝑐‘𝑘)) finSupp 0)
215	48, 110, 49	mulg0 19277	. . . . . . . . . . 11 ⊢ (𝑣 ∈ 𝐾 → (0(.g‘(mulGrp‘𝑅))𝑣) = (0g‘(mulGrp‘𝑅)))
216	215	adantl 487	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑣 ∈ 𝐾) → (0(.g‘(mulGrp‘𝑅))𝑣) = (0g‘(mulGrp‘𝑅)))
217		fvexd 6898	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → (𝑐‘𝑘) ∈ V)
218		fvexd 6898	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (0g‘(mulGrp‘𝑅)) ∈ V)
219	214, 216, 217, 208, 218	fsuppssov1 9369	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑐‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))) finSupp (0g‘(mulGrp‘𝑅)))
220	48, 110, 201, 180, 210, 219	gsumcl 20122	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → ((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑐‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)))) ∈ 𝐾)
221	32	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin}) → 𝑅 ∈ Ring)
222	28, 136, 221, 155, 151	ringcld 20477	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin}) → ((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))))) ∈ 𝐾)
223	178	mptex 7227	. . . . . . . . . . 11 ⊢ (𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)) ∈ V
224	223	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)) ∈ V)
225		fvexd 6898	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (0g‘𝑅) ∈ V)
226		funmpt 6576	. . . . . . . . . . 11 ⊢ Fun (𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐))
227	226	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → Fun (𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)))
228	11, 12, 171, 18	mplelsfi 22295	. . . . . . . . . . 11 ⊢ (𝜑 → (((𝐼 selectVars 𝑅)‘𝐽)‘𝐹) finSupp (0g‘𝑈))
229	228	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (((𝐼 selectVars 𝑅)‘𝐽)‘𝐹) finSupp (0g‘𝑈))
230		ssidd 3954	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹) supp (0g‘𝑈)) ⊆ ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹) supp (0g‘𝑈)))
231		fvexd 6898	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (0g‘𝑈) ∈ V)
232	20, 230, 179, 231	suppssr 8205	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ ({𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} ∖ ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹) supp (0g‘𝑈)))) → ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒) = (0g‘𝑈))
233	232	fveq1d 6885	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ ({𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} ∖ ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹) supp (0g‘𝑈)))) → (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐) = ((0g‘𝑈)‘𝑐))
234	3, 137, 197, 171, 6, 174	mpl0 22306	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (0g‘𝑈) = ({𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {(0g‘𝑅)}))
235	234	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (0g‘𝑈) = ({𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {(0g‘𝑅)}))
236	235	fveq1d 6885	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → ((0g‘𝑈)‘𝑐) = (({𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {(0g‘𝑅)})‘𝑐))
237		fvex 6896	. . . . . . . . . . . . . . . 16 ⊢ (0g‘𝑅) ∈ V
238	237	fvconst2 7208	. . . . . . . . . . . . . . 15 ⊢ (𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} → (({𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {(0g‘𝑅)})‘𝑐) = (0g‘𝑅))
239	238	adantl 487	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (({𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {(0g‘𝑅)})‘𝑐) = (0g‘𝑅))
240	236, 239	eqtrd 2796	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → ((0g‘𝑈)‘𝑐) = (0g‘𝑅))
241	240	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ ({𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} ∖ ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹) supp (0g‘𝑈)))) → ((0g‘𝑈)‘𝑐) = (0g‘𝑅))
242	233, 241	eqtrd 2796	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ ({𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} ∖ ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹) supp (0g‘𝑈)))) → (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐) = (0g‘𝑅))
243	242, 179	suppss2 8210	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → ((𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)) supp (0g‘𝑅)) ⊆ ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹) supp (0g‘𝑈)))
244	224, 225, 227, 229, 243	fsuppsssuppgd 9367	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)) finSupp (0g‘𝑅))
245	32	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑣 ∈ 𝐾) → 𝑅 ∈ Ring)
246		simpr 490	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑣 ∈ 𝐾) → 𝑣 ∈ 𝐾)
247	28, 136, 197, 245, 246	ringlzd 20519	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑣 ∈ 𝐾) → ((0g‘𝑅)(.r‘𝑅)𝑣) = (0g‘𝑅))
248	244, 247, 155, 151, 225	fsuppssov1 9369	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ ((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))) finSupp (0g‘𝑅))
249	28, 197, 136, 198, 179, 220, 222, 248	gsummulc1 20538	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝑅 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑐‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))))) = ((𝑅 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ ((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑐‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))))
250	161, 196, 249	3eqtr4d 2806	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (((((𝐽 eval 𝑈)‘(((𝐼 selectVars 𝑅)‘𝐽)‘𝐹))‘(𝐿 ∘ (𝐴 ↾ 𝐽)))‘𝑐)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑐‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))) = (𝑅 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑐‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))))))
251		fveq2 6883	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑒 → ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎) = ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒))
252	251	adantl 487	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 𝑐 ∧ 𝑎 = 𝑒) → ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎) = ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒))
253		simpl 488	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 𝑐 ∧ 𝑎 = 𝑒) → 𝑏 = 𝑐)
254	252, 253	fveq12d 6890	. . . . . . . . . . . 12 ⊢ ((𝑏 = 𝑐 ∧ 𝑎 = 𝑒) → (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏) = (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐))
255		fveq1 6882	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝑒 → (𝑎‘𝑗) = (𝑒‘𝑗))
256	255	adantl 487	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 𝑐 ∧ 𝑎 = 𝑒) → (𝑎‘𝑗) = (𝑒‘𝑗))
257	256	oveq1d 7433	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 𝑐 ∧ 𝑎 = 𝑒) → ((𝑎‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)) = ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))
258	257	mpteq2dv 5199	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 𝑐 ∧ 𝑎 = 𝑒) → (𝑗 ∈ 𝐽 ↦ ((𝑎‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))) = (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))))
259	258	oveq2d 7434	. . . . . . . . . . . 12 ⊢ ((𝑏 = 𝑐 ∧ 𝑎 = 𝑒) → ((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑎‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))) = ((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))
260	254, 259	oveq12d 7436	. . . . . . . . . . 11 ⊢ ((𝑏 = 𝑐 ∧ 𝑎 = 𝑒) → ((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑎‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))))) = ((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))))))
261		fveq1 6882	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑐 → (𝑏‘𝑘) = (𝑐‘𝑘))
262	261	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 𝑐 ∧ 𝑎 = 𝑒) → (𝑏‘𝑘) = (𝑐‘𝑘))
263	262	oveq1d 7433	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 𝑐 ∧ 𝑎 = 𝑒) → ((𝑏‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)) = ((𝑐‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)))
264	263	mpteq2dv 5199	. . . . . . . . . . . 12 ⊢ ((𝑏 = 𝑐 ∧ 𝑎 = 𝑒) → (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑏‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))) = (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑐‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))
265	264	oveq2d 7434	. . . . . . . . . . 11 ⊢ ((𝑏 = 𝑐 ∧ 𝑎 = 𝑒) → ((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑏‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)))) = ((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑐‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)))))
266	260, 265	oveq12d 7436	. . . . . . . . . 10 ⊢ ((𝑏 = 𝑐 ∧ 𝑎 = 𝑒) → (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑎‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑏‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))) = (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑐‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))))
267		eqid 2761	. . . . . . . . . 10 ⊢ (𝑏 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑎‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑏‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)))))) = (𝑏 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑎‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑏‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))))
268		ovex 7451	. . . . . . . . . 10 ⊢ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑐‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))) ∈ V
269	266, 267, 268	ovmpoa 7573	. . . . . . . . 9 ⊢ ((𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin}) → (𝑐(𝑏 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑎‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑏‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))))𝑒) = (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑐‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))))
270	269	adantll 727	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin}) → (𝑐(𝑏 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑎‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑏‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))))𝑒) = (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑐‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))))
271	270	mpteq2dva 5198	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (𝑐(𝑏 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑎‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑏‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))))𝑒)) = (𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑐‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)))))))
272	271	oveq2d 7434	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝑅 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (𝑐(𝑏 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑎‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑏‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))))𝑒))) = (𝑅 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑐‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))))))
273	250, 272	eqtr4d 2799	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (((((𝐽 eval 𝑈)‘(((𝐼 selectVars 𝑅)‘𝐽)‘𝐹))‘(𝐿 ∘ (𝐴 ↾ 𝐽)))‘𝑐)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑐‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))) = (𝑅 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (𝑐(𝑏 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑎‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑏‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))))𝑒))))
274	273	mpteq2dva 5198	. . . 4 ⊢ (𝜑 → (𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ (((((𝐽 eval 𝑈)‘(((𝐼 selectVars 𝑅)‘𝐽)‘𝐹))‘(𝐿 ∘ (𝐴 ↾ 𝐽)))‘𝑐)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑐‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)))))) = (𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ (𝑅 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (𝑐(𝑏 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑎‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑏‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))))𝑒)))))
275	274	oveq2d 7434	. . 3 ⊢ (𝜑 → (𝑅 Σg (𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ (((((𝐽 eval 𝑈)‘(((𝐼 selectVars 𝑅)‘𝐽)‘𝐹))‘(𝐿 ∘ (𝐴 ↾ 𝐽)))‘𝑐)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑐‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))))) = (𝑅 Σg (𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ (𝑅 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (𝑐(𝑏 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑎‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑏‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))))𝑒))))))
276	32	ringcmnd 20506	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ CMnd)
277		ovex 7451	. . . . . . 7 ⊢ (ℕ0 ↑m 𝐼) ∈ V
278	277	rabex 5300	. . . . . 6 ⊢ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∈ V
279	278	a1i 11	. . . . 5 ⊢ (𝜑 → {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∈ V)
280	32	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑅 ∈ Ring)
281	19	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (((𝐼 selectVars 𝑅)‘𝐽)‘𝐹):{𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin}⟶(Base‘𝑈))
282		eqid 2761	. . . . . . . . . . . 12 ⊢ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}
283	4	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝐼 ∈ 𝑉)
284	16	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝐽 ⊆ 𝐼)
285		simpr 490	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
286	282, 13, 283, 284, 285	psrbagres 22231	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑑 ↾ 𝐽) ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin})
287	281, 286	ffvelcdmd 7083	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽)) ∈ (Base‘𝑈))
288	3, 28, 1, 137, 287	mplelf 22298	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽)):{𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶𝐾)
289		difssd 4084	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝐼 ∖ 𝐽) ⊆ 𝐼)
290	282, 137, 283, 289, 285	psrbagres 22231	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑑 ↾ (𝐼 ∖ 𝐽)) ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin})
291	288, 290	ffvelcdmd 7083	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽))‘(𝑑 ↾ (𝐼 ∖ 𝐽))) ∈ 𝐾)
292	200	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (mulGrp‘𝑅) ∈ CMnd)
293	24	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝐽 ∈ V)
294	51	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑗 ∈ 𝐽) → (mulGrp‘𝑅) ∈ Mnd)
295	282	psrbagf 22219	. . . . . . . . . . . . . 14 ⊢ (𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} → 𝑑:𝐼⟶ℕ0)
296	295	adantl 487	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑑:𝐼⟶ℕ0)
297	296, 284	fssresd 6747	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑑 ↾ 𝐽):𝐽⟶ℕ0)
298	297	ffvelcdmda 7082	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑗 ∈ 𝐽) → ((𝑑 ↾ 𝐽)‘𝑗) ∈ ℕ0)
299	58	ffvelcdmda 7082	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐽) → ((𝐴 ↾ 𝐽)‘𝑗) ∈ 𝐾)
300	299	adantlr 728	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑗 ∈ 𝐽) → ((𝐴 ↾ 𝐽)‘𝑗) ∈ 𝐾)
301	48, 49, 294, 298, 300	mulgnn0cld 19298	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑗 ∈ 𝐽) → (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)) ∈ 𝐾)
302	301	fmpttd 7113	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))):𝐽⟶𝐾)
303	24	mptexd 7228	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))) ∈ V)
304	303	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))) ∈ V)
305		fvexd 6898	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (0g‘(mulGrp‘𝑅)) ∈ V)
306		funmpt 6576	. . . . . . . . . . 11 ⊢ Fun (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))
307	306	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → Fun (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))))
308	282	psrbagfsupp 22220	. . . . . . . . . . . 12 ⊢ (𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} → 𝑑 finSupp 0)
309	308	adantl 487	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑑 finSupp 0)
310		0zd 12698	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 0 ∈ ℤ)
311	309, 310	fsuppres 9378	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑑 ↾ 𝐽) finSupp 0)
312		ssidd 3954	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → ((𝑑 ↾ 𝐽) supp 0) ⊆ ((𝑑 ↾ 𝐽) supp 0))
313	297, 312, 293, 310	suppssr 8205	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑗 ∈ (𝐽 ∖ ((𝑑 ↾ 𝐽) supp 0))) → ((𝑑 ↾ 𝐽)‘𝑗) = 0)
314	313	oveq1d 7433	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑗 ∈ (𝐽 ∖ ((𝑑 ↾ 𝐽) supp 0))) → (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)) = (0(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))
315		eldifi 4078	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (𝐽 ∖ ((𝑑 ↾ 𝐽) supp 0)) → 𝑗 ∈ 𝐽)
316	48, 110, 49	mulg0 19277	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ↾ 𝐽)‘𝑗) ∈ 𝐾 → (0(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)) = (0g‘(mulGrp‘𝑅)))
317	300, 316	syl 18	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑗 ∈ 𝐽) → (0(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)) = (0g‘(mulGrp‘𝑅)))
318	315, 317	sylan2 605	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑗 ∈ (𝐽 ∖ ((𝑑 ↾ 𝐽) supp 0))) → (0(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)) = (0g‘(mulGrp‘𝑅)))
319	314, 318	eqtrd 2796	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑗 ∈ (𝐽 ∖ ((𝑑 ↾ 𝐽) supp 0))) → (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)) = (0g‘(mulGrp‘𝑅)))
320	319, 293	suppss2 8210	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → ((𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))) supp (0g‘(mulGrp‘𝑅))) ⊆ ((𝑑 ↾ 𝐽) supp 0))
321	304, 305, 307, 311, 320	fsuppsssuppgd 9367	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))) finSupp (0g‘(mulGrp‘𝑅)))
322	48, 110, 292, 293, 302, 321	gsumcl 20122	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → ((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))) ∈ 𝐾)
323	28, 136, 280, 291, 322	ringcld 20477	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → ((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽))‘(𝑑 ↾ (𝐼 ∖ 𝐽)))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))))) ∈ 𝐾)
324	6	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝐼 ∖ 𝐽) ∈ V)
325	51	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → (mulGrp‘𝑅) ∈ Mnd)
326	296, 289	fssresd 6747	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑑 ↾ (𝐼 ∖ 𝐽)):(𝐼 ∖ 𝐽)⟶ℕ0)
327	326	ffvelcdmda 7082	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → ((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘) ∈ ℕ0)
328	206	ffvelcdmda 7082	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → ((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘) ∈ 𝐾)
329	328	adantlr 728	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → ((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘) ∈ 𝐾)
330	48, 49, 325, 327, 329	mulgnn0cld 19298	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)) ∈ 𝐾)
331	330	fmpttd 7113	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))):(𝐼 ∖ 𝐽)⟶𝐾)
332	324	mptexd 7228	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))) ∈ V)
333		funmpt 6576	. . . . . . . . . 10 ⊢ Fun (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)))
334	333	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → Fun (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))
335	309, 310	fsuppres 9378	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑑 ↾ (𝐼 ∖ 𝐽)) finSupp 0)
336		ssidd 3954	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → ((𝑑 ↾ (𝐼 ∖ 𝐽)) supp 0) ⊆ ((𝑑 ↾ (𝐼 ∖ 𝐽)) supp 0))
337	326, 336, 324, 310	suppssr 8205	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ ((𝐼 ∖ 𝐽) ∖ ((𝑑 ↾ (𝐼 ∖ 𝐽)) supp 0))) → ((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘) = 0)
338	337	oveq1d 7433	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ ((𝐼 ∖ 𝐽) ∖ ((𝑑 ↾ (𝐼 ∖ 𝐽)) supp 0))) → (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)) = (0(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)))
339		eldifi 4078	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ((𝐼 ∖ 𝐽) ∖ ((𝑑 ↾ (𝐼 ∖ 𝐽)) supp 0)) → 𝑘 ∈ (𝐼 ∖ 𝐽))
340	339, 329	sylan2 605	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ ((𝐼 ∖ 𝐽) ∖ ((𝑑 ↾ (𝐼 ∖ 𝐽)) supp 0))) → ((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘) ∈ 𝐾)
341	48, 110, 49	mulg0 19277	. . . . . . . . . . . 12 ⊢ (((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘) ∈ 𝐾 → (0(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)) = (0g‘(mulGrp‘𝑅)))
342	340, 341	syl 18	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ ((𝐼 ∖ 𝐽) ∖ ((𝑑 ↾ (𝐼 ∖ 𝐽)) supp 0))) → (0(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)) = (0g‘(mulGrp‘𝑅)))
343	338, 342	eqtrd 2796	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ ((𝐼 ∖ 𝐽) ∖ ((𝑑 ↾ (𝐼 ∖ 𝐽)) supp 0))) → (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)) = (0g‘(mulGrp‘𝑅)))
344	343, 324	suppss2 8210	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → ((𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))) supp (0g‘(mulGrp‘𝑅))) ⊆ ((𝑑 ↾ (𝐼 ∖ 𝐽)) supp 0))
345	332, 305, 334, 335, 344	fsuppsssuppgd 9367	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))) finSupp (0g‘(mulGrp‘𝑅)))
346	48, 110, 292, 324, 331, 345	gsumcl 20122	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → ((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)))) ∈ 𝐾)
347	28, 136, 280, 323, 346	ringcld 20477	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽))‘(𝑑 ↾ (𝐼 ∖ 𝐽)))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))) ∈ 𝐾)
348	347	fmpttd 7113	. . . . 5 ⊢ (𝜑 → (𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽))‘(𝑑 ↾ (𝐼 ∖ 𝐽)))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)))))):{ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}⟶𝐾)
349	7	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑅 ∈ CRing)
350	17	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝐹 ∈ 𝐵)
351	282, 14, 15, 349, 284, 350, 285	selvvvval 22444	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽))‘(𝑑 ↾ (𝐼 ∖ 𝐽))) = (𝐹‘𝑑))
352	351	mpteq2dva 5198	. . . . . . . 8 ⊢ (𝜑 → (𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽))‘(𝑑 ↾ (𝐼 ∖ 𝐽)))) = (𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝐹‘𝑑)))
353		eqid 2761	. . . . . . . . . . 11 ⊢ (Base‘𝑅) = (Base‘𝑅)
354	14, 353, 15, 282, 17	mplelf 22298	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:{ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}⟶(Base‘𝑅))
355	354	feqmptd 6951	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 = (𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝐹‘𝑑)))
356	14, 15, 197, 17	mplelsfi 22295	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 finSupp (0g‘𝑅))
357	355, 356	eqbrtrrd 5129	. . . . . . . 8 ⊢ (𝜑 → (𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝐹‘𝑑)) finSupp (0g‘𝑅))
358	352, 357	eqbrtrd 5127	. . . . . . 7 ⊢ (𝜑 → (𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽))‘(𝑑 ↾ (𝐼 ∖ 𝐽)))) finSupp (0g‘𝑅))
359	32	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐾) → 𝑅 ∈ Ring)
360		simpr 490	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐾) → 𝑣 ∈ 𝐾)
361	28, 136, 197, 359, 360	ringlzd 20519	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐾) → ((0g‘𝑅)(.r‘𝑅)𝑣) = (0g‘𝑅))
362		fvexd 6898	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽))‘(𝑑 ↾ (𝐼 ∖ 𝐽))) ∈ V)
363		fvexd 6898	. . . . . . 7 ⊢ (𝜑 → (0g‘𝑅) ∈ V)
364	358, 361, 362, 322, 363	fsuppssov1 9369	. . . . . 6 ⊢ (𝜑 → (𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ ((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽))‘(𝑑 ↾ (𝐼 ∖ 𝐽)))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))) finSupp (0g‘𝑅))
365		ovexd 7453	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → ((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽))‘(𝑑 ↾ (𝐼 ∖ 𝐽)))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))))) ∈ V)
366	364, 361, 365, 346, 363	fsuppssov1 9369	. . . . 5 ⊢ (𝜑 → (𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽))‘(𝑑 ↾ (𝐼 ∖ 𝐽)))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)))))) finSupp (0g‘𝑅))
367		eqid 2761	. . . . . 6 ⊢ (𝑏 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (𝑏 ∪ 𝑎)) = (𝑏 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (𝑏 ∪ 𝑎))
368	282, 13, 137, 367, 4, 16	evlselvlem 43596	. . . . 5 ⊢ (𝜑 → (𝑏 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (𝑏 ∪ 𝑎)):({𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin})–1-1-onto→{ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
369	28, 197, 276, 279, 348, 366, 368	gsumf1o 20123	. . . 4 ⊢ (𝜑 → (𝑅 Σg (𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽))‘(𝑑 ↾ (𝐼 ∖ 𝐽)))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))))) = (𝑅 Σg ((𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽))‘(𝑑 ↾ (𝐼 ∖ 𝐽)))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)))))) ∘ (𝑏 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (𝑏 ∪ 𝑎)))))
370	137	psrbagf 22219	. . . . . . . . . 10 ⊢ (𝑏 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} → 𝑏:(𝐼 ∖ 𝐽)⟶ℕ0)
371	370	ad2antrl 741	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin})) → 𝑏:(𝐼 ∖ 𝐽)⟶ℕ0)
372	13	psrbagf 22219	. . . . . . . . . 10 ⊢ (𝑎 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} → 𝑎:𝐽⟶ℕ0)
373	372	ad2antll 742	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin})) → 𝑎:𝐽⟶ℕ0)
374		disjdifr 4427	. . . . . . . . . 10 ⊢ ((𝐼 ∖ 𝐽) ∩ 𝐽) = ∅
375	374	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin})) → ((𝐼 ∖ 𝐽) ∩ 𝐽) = ∅)
376	371, 373, 375	fun2d 6744	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin})) → (𝑏 ∪ 𝑎):((𝐼 ∖ 𝐽) ∪ 𝐽)⟶ℕ0)
377		undifr 4439	. . . . . . . . . . 11 ⊢ (𝐽 ⊆ 𝐼 ↔ ((𝐼 ∖ 𝐽) ∪ 𝐽) = 𝐼)
378	16, 377	sylib 221	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐼 ∖ 𝐽) ∪ 𝐽) = 𝐼)
379	378	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin})) → ((𝐼 ∖ 𝐽) ∪ 𝐽) = 𝐼)
380	379	feq2d 6691	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin})) → ((𝑏 ∪ 𝑎):((𝐼 ∖ 𝐽) ∪ 𝐽)⟶ℕ0 ↔ (𝑏 ∪ 𝑎):𝐼⟶ℕ0))
381	376, 380	mpbid 235	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin})) → (𝑏 ∪ 𝑎):𝐼⟶ℕ0)
382		vex 3455	. . . . . . . . . . 11 ⊢ 𝑏 ∈ V
383		vex 3455	. . . . . . . . . . 11 ⊢ 𝑎 ∈ V
384	382, 383	unex 7759	. . . . . . . . . 10 ⊢ (𝑏 ∪ 𝑎) ∈ V
385	384	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin})) → (𝑏 ∪ 𝑎) ∈ V)
386		0zd 12698	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin})) → 0 ∈ ℤ)
387	381	ffund 6712	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin})) → Fun (𝑏 ∪ 𝑎))
388	137	psrbagfsupp 22220	. . . . . . . . . . 11 ⊢ (𝑏 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} → 𝑏 finSupp 0)
389	388	ad2antrl 741	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin})) → 𝑏 finSupp 0)
390	13	psrbagfsupp 22220	. . . . . . . . . . 11 ⊢ (𝑎 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} → 𝑎 finSupp 0)
391	390	ad2antll 742	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin})) → 𝑎 finSupp 0)
392	389, 391	fsuppun 9372	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin})) → ((𝑏 ∪ 𝑎) supp 0) ∈ Fin)
393	385, 386, 387, 392	isfsuppd 9351	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin})) → (𝑏 ∪ 𝑎) finSupp 0)
394		fcdmnn0fsuppg 12659	. . . . . . . . 9 ⊢ (((𝑏 ∪ 𝑎) ∈ V ∧ (𝑏 ∪ 𝑎):𝐼⟶ℕ0) → ((𝑏 ∪ 𝑎) finSupp 0 ↔ (◡(𝑏 ∪ 𝑎) “ ℕ) ∈ Fin))
395	385, 381, 394	syl2anc 596	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin})) → ((𝑏 ∪ 𝑎) finSupp 0 ↔ (◡(𝑏 ∪ 𝑎) “ ℕ) ∈ Fin))
396	393, 395	mpbid 235	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin})) → (◡(𝑏 ∪ 𝑎) “ ℕ) ∈ Fin)
397	4	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin})) → 𝐼 ∈ 𝑉)
398	282	psrbag 22218	. . . . . . . 8 ⊢ (𝐼 ∈ 𝑉 → ((𝑏 ∪ 𝑎) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↔ ((𝑏 ∪ 𝑎):𝐼⟶ℕ0 ∧ (◡(𝑏 ∪ 𝑎) “ ℕ) ∈ Fin)))
399	397, 398	syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin})) → ((𝑏 ∪ 𝑎) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↔ ((𝑏 ∪ 𝑎):𝐼⟶ℕ0 ∧ (◡(𝑏 ∪ 𝑎) “ ℕ) ∈ Fin)))
400	381, 396, 399	mpbir2and 726	. . . . . 6 ⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin})) → (𝑏 ∪ 𝑎) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
401		eqidd 2762	. . . . . 6 ⊢ (𝜑 → (𝑏 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (𝑏 ∪ 𝑎)) = (𝑏 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (𝑏 ∪ 𝑎)))
402		eqidd 2762	. . . . . 6 ⊢ (𝜑 → (𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽))‘(𝑑 ↾ (𝐼 ∖ 𝐽)))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)))))) = (𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽))‘(𝑑 ↾ (𝐼 ∖ 𝐽)))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)))))))
403		reseq1 5964	. . . . . . . . . . . 12 ⊢ (𝑑 = (𝑏 ∪ 𝑎) → (𝑑 ↾ 𝐽) = ((𝑏 ∪ 𝑎) ↾ 𝐽))
404	403	fveq2d 6887	. . . . . . . . . . 11 ⊢ (𝑑 = (𝑏 ∪ 𝑎) → ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽)) = ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘((𝑏 ∪ 𝑎) ↾ 𝐽)))
405		reseq1 5964	. . . . . . . . . . 11 ⊢ (𝑑 = (𝑏 ∪ 𝑎) → (𝑑 ↾ (𝐼 ∖ 𝐽)) = ((𝑏 ∪ 𝑎) ↾ (𝐼 ∖ 𝐽)))
406	404, 405	fveq12d 6890	. . . . . . . . . 10 ⊢ (𝑑 = (𝑏 ∪ 𝑎) → (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽))‘(𝑑 ↾ (𝐼 ∖ 𝐽))) = (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘((𝑏 ∪ 𝑎) ↾ 𝐽))‘((𝑏 ∪ 𝑎) ↾ (𝐼 ∖ 𝐽))))
407	403	fveq1d 6885	. . . . . . . . . . . . 13 ⊢ (𝑑 = (𝑏 ∪ 𝑎) → ((𝑑 ↾ 𝐽)‘𝑗) = (((𝑏 ∪ 𝑎) ↾ 𝐽)‘𝑗))
408	407	oveq1d 7433	. . . . . . . . . . . 12 ⊢ (𝑑 = (𝑏 ∪ 𝑎) → (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)) = ((((𝑏 ∪ 𝑎) ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))
409	408	mpteq2dv 5199	. . . . . . . . . . 11 ⊢ (𝑑 = (𝑏 ∪ 𝑎) → (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))) = (𝑗 ∈ 𝐽 ↦ ((((𝑏 ∪ 𝑎) ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))))
410	409	oveq2d 7434	. . . . . . . . . 10 ⊢ (𝑑 = (𝑏 ∪ 𝑎) → ((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))) = ((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((((𝑏 ∪ 𝑎) ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))
411	406, 410	oveq12d 7436	. . . . . . . . 9 ⊢ (𝑑 = (𝑏 ∪ 𝑎) → ((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽))‘(𝑑 ↾ (𝐼 ∖ 𝐽)))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))))) = ((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘((𝑏 ∪ 𝑎) ↾ 𝐽))‘((𝑏 ∪ 𝑎) ↾ (𝐼 ∖ 𝐽)))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((((𝑏 ∪ 𝑎) ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))))))
412	405	fveq1d 6885	. . . . . . . . . . . 12 ⊢ (𝑑 = (𝑏 ∪ 𝑎) → ((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘) = (((𝑏 ∪ 𝑎) ↾ (𝐼 ∖ 𝐽))‘𝑘))
413	412	oveq1d 7433	. . . . . . . . . . 11 ⊢ (𝑑 = (𝑏 ∪ 𝑎) → (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)) = ((((𝑏 ∪ 𝑎) ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)))
414	413	mpteq2dv 5199	. . . . . . . . . 10 ⊢ (𝑑 = (𝑏 ∪ 𝑎) → (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))) = (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((((𝑏 ∪ 𝑎) ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))
415	414	oveq2d 7434	. . . . . . . . 9 ⊢ (𝑑 = (𝑏 ∪ 𝑎) → ((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)))) = ((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((((𝑏 ∪ 𝑎) ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)))))
416	411, 415	oveq12d 7436	. . . . . . . 8 ⊢ (𝑑 = (𝑏 ∪ 𝑎) → (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽))‘(𝑑 ↾ (𝐼 ∖ 𝐽)))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))) = (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘((𝑏 ∪ 𝑎) ↾ 𝐽))‘((𝑏 ∪ 𝑎) ↾ (𝐼 ∖ 𝐽)))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((((𝑏 ∪ 𝑎) ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((((𝑏 ∪ 𝑎) ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))))
417	384, 416	csbie 3882	. . . . . . 7 ⊢ ⦋(𝑏 ∪ 𝑎) / 𝑑⦌(((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽))‘(𝑑 ↾ (𝐼 ∖ 𝐽)))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))) = (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘((𝑏 ∪ 𝑎) ↾ 𝐽))‘((𝑏 ∪ 𝑎) ↾ (𝐼 ∖ 𝐽)))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((((𝑏 ∪ 𝑎) ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((((𝑏 ∪ 𝑎) ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)))))
418	370	ffnd 6708	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} → 𝑏 Fn (𝐼 ∖ 𝐽))
419	418	ad2antrl 741	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin})) → 𝑏 Fn (𝐼 ∖ 𝐽))
420	373	ffnd 6708	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin})) → 𝑎 Fn 𝐽)
421		fnunres2 6650	. . . . . . . . . . . 12 ⊢ ((𝑏 Fn (𝐼 ∖ 𝐽) ∧ 𝑎 Fn 𝐽 ∧ ((𝐼 ∖ 𝐽) ∩ 𝐽) = ∅) → ((𝑏 ∪ 𝑎) ↾ 𝐽) = 𝑎)
422	419, 420, 375, 421	syl3anc 1398	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin})) → ((𝑏 ∪ 𝑎) ↾ 𝐽) = 𝑎)
423	422	fveq2d 6887	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin})) → ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘((𝑏 ∪ 𝑎) ↾ 𝐽)) = ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎))
424		fnunres1 6649	. . . . . . . . . . 11 ⊢ ((𝑏 Fn (𝐼 ∖ 𝐽) ∧ 𝑎 Fn 𝐽 ∧ ((𝐼 ∖ 𝐽) ∩ 𝐽) = ∅) → ((𝑏 ∪ 𝑎) ↾ (𝐼 ∖ 𝐽)) = 𝑏)
425	419, 420, 375, 424	syl3anc 1398	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin})) → ((𝑏 ∪ 𝑎) ↾ (𝐼 ∖ 𝐽)) = 𝑏)
426	423, 425	fveq12d 6890	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin})) → (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘((𝑏 ∪ 𝑎) ↾ 𝐽))‘((𝑏 ∪ 𝑎) ↾ (𝐼 ∖ 𝐽))) = (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏))
427	422	fveq1d 6885	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin})) → (((𝑏 ∪ 𝑎) ↾ 𝐽)‘𝑗) = (𝑎‘𝑗))
428	427	oveq1d 7433	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin})) → ((((𝑏 ∪ 𝑎) ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)) = ((𝑎‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))
429	428	mpteq2dv 5199	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin})) → (𝑗 ∈ 𝐽 ↦ ((((𝑏 ∪ 𝑎) ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))) = (𝑗 ∈ 𝐽 ↦ ((𝑎‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))))
430	429	oveq2d 7434	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin})) → ((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((((𝑏 ∪ 𝑎) ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))) = ((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑎‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))
431	426, 430	oveq12d 7436	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin})) → ((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘((𝑏 ∪ 𝑎) ↾ 𝐽))‘((𝑏 ∪ 𝑎) ↾ (𝐼 ∖ 𝐽)))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((((𝑏 ∪ 𝑎) ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))))) = ((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑎‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))))))
432	425	fveq1d 6885	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin})) → (((𝑏 ∪ 𝑎) ↾ (𝐼 ∖ 𝐽))‘𝑘) = (𝑏‘𝑘))
433	432	oveq1d 7433	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin})) → ((((𝑏 ∪ 𝑎) ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)) = ((𝑏‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)))
434	433	mpteq2dv 5199	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin})) → (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((((𝑏 ∪ 𝑎) ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))) = (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑏‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))
435	434	oveq2d 7434	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin})) → ((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((((𝑏 ∪ 𝑎) ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)))) = ((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑏‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)))))
436	431, 435	oveq12d 7436	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin})) → (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘((𝑏 ∪ 𝑎) ↾ 𝐽))‘((𝑏 ∪ 𝑎) ↾ (𝐼 ∖ 𝐽)))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((((𝑏 ∪ 𝑎) ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((((𝑏 ∪ 𝑎) ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))) = (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑎‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑏‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))))
437	417, 436	eqtrid 2808	. . . . . 6 ⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin})) → ⦋(𝑏 ∪ 𝑎) / 𝑑⦌(((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽))‘(𝑑 ↾ (𝐼 ∖ 𝐽)))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))) = (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑎‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑏‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))))
438	400, 401, 402, 437	fmpocos 43267	. . . . 5 ⊢ (𝜑 → ((𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽))‘(𝑑 ↾ (𝐼 ∖ 𝐽)))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)))))) ∘ (𝑏 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (𝑏 ∪ 𝑎))) = (𝑏 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑎‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑏‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)))))))
439	438	oveq2d 7434	. . . 4 ⊢ (𝜑 → (𝑅 Σg ((𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽))‘(𝑑 ↾ (𝐼 ∖ 𝐽)))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)))))) ∘ (𝑏 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (𝑏 ∪ 𝑎)))) = (𝑅 Σg (𝑏 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑎‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑏‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))))))
440		ovex 7451	. . . . . . 7 ⊢ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∈ V
441	440	rabex 5300	. . . . . 6 ⊢ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∈ V
442	441	a1i 11	. . . . 5 ⊢ (𝜑 → {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∈ V)
443	178	a1i 11	. . . . 5 ⊢ (𝜑 → {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} ∈ V)
444	32	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin})) → 𝑅 ∈ Ring)
445	19	ffvelcdmda 7082	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin}) → ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎) ∈ (Base‘𝑈))
446	3, 28, 1, 137, 445	mplelf 22298	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin}) → ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎):{𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶𝐾)
447	446	ffvelcdmda 7082	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin}) ∧ 𝑏 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏) ∈ 𝐾)
448	447	an32s 665	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑏 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin}) → (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏) ∈ 𝐾)
449	448	anasss 472	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin})) → (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏) ∈ 𝐾)
450	24	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin})) → 𝐽 ∈ V)
451	7	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin})) → 𝑅 ∈ CRing)
452	36	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin})) → (𝐴 ↾ 𝐽) ∈ (𝐾 ↑m 𝐽))
453		simprr 785	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin})) → 𝑎 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin})
454	13, 28, 47, 49, 450, 451, 452, 453	evlsvvvallem 22393	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin})) → ((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑎‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))) ∈ 𝐾)
455	28, 136, 444, 449, 454	ringcld 20477	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin})) → ((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑎‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))))) ∈ 𝐾)
456	6	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin})) → (𝐼 ∖ 𝐽) ∈ V)
457	35, 5	elmapssresd 8888	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 ↾ (𝐼 ∖ 𝐽)) ∈ (𝐾 ↑m (𝐼 ∖ 𝐽)))
458	457	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin})) → (𝐴 ↾ (𝐼 ∖ 𝐽)) ∈ (𝐾 ↑m (𝐼 ∖ 𝐽)))
459		simprl 783	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin})) → 𝑏 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin})
460	137, 28, 47, 49, 456, 451, 458, 459	evlsvvvallem 22393	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin})) → ((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑏‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)))) ∈ 𝐾)
461	28, 136, 444, 455, 460	ringcld 20477	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin})) → (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑎‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑏‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))) ∈ 𝐾)
462	461	ralrimivva 3206	. . . . . 6 ⊢ (𝜑 → ∀𝑏 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}∀𝑎 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑎‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑏‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))) ∈ 𝐾)
463	267	fmpo 8077	. . . . . 6 ⊢ (∀𝑏 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}∀𝑎 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑎‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑏‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))) ∈ 𝐾 ↔ (𝑏 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑎‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑏‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)))))):({𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin})⟶𝐾)
464	462, 463	sylib 221	. . . . 5 ⊢ (𝜑 → (𝑏 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑎‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑏‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)))))):({𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin})⟶𝐾)
465		f1of1 6821	. . . . . . . 8 ⊢ ((𝑏 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (𝑏 ∪ 𝑎)):({𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin})–1-1-onto→{ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} → (𝑏 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (𝑏 ∪ 𝑎)):({𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin})–1-1→{ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
466	368, 465	syl 18	. . . . . . 7 ⊢ (𝜑 → (𝑏 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (𝑏 ∪ 𝑎)):({𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin})–1-1→{ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
467	278	mptex 7227	. . . . . . . 8 ⊢ (𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽))‘(𝑑 ↾ (𝐼 ∖ 𝐽)))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)))))) ∈ V
468	467	a1i 11	. . . . . . 7 ⊢ (𝜑 → (𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽))‘(𝑑 ↾ (𝐼 ∖ 𝐽)))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)))))) ∈ V)
469	366, 466, 363, 468	fsuppco 9387	. . . . . 6 ⊢ (𝜑 → ((𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽))‘(𝑑 ↾ (𝐼 ∖ 𝐽)))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)))))) ∘ (𝑏 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (𝑏 ∪ 𝑎))) finSupp (0g‘𝑅))
470	438, 469	eqbrtrrd 5129	. . . . 5 ⊢ (𝜑 → (𝑏 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑎‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑏‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)))))) finSupp (0g‘𝑅))
471	28, 197, 276, 442, 443, 464, 470	gsumxp 20183	. . . 4 ⊢ (𝜑 → (𝑅 Σg (𝑏 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑎‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑏‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))))) = (𝑅 Σg (𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ (𝑅 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (𝑐(𝑏 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑎‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑏‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))))𝑒))))))
472	369, 439, 471	3eqtrd 2800	. . 3 ⊢ (𝜑 → (𝑅 Σg (𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽))‘(𝑑 ↾ (𝐼 ∖ 𝐽)))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))))) = (𝑅 Σg (𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ (𝑅 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (𝑐(𝑏 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑎‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑏‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))))𝑒))))))
473	28, 136, 280, 291, 322, 346	ringassd 20478	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽))‘(𝑑 ↾ (𝐼 ∖ 𝐽)))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))) = ((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽))‘(𝑑 ↾ (𝐼 ∖ 𝐽)))(.r‘𝑅)(((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)))))))
474	47, 136	mgpplusg 20357	. . . . . . . . 9 ⊢ (.r‘𝑅) = (+g‘(mulGrp‘𝑅))
475	51	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑖 ∈ 𝐼) → (mulGrp‘𝑅) ∈ Mnd)
476	296	ffvelcdmda 7082	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑖 ∈ 𝐼) → (𝑑‘𝑖) ∈ ℕ0)
477	57	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝐴:𝐼⟶𝐾)
478	477	ffvelcdmda 7082	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑖 ∈ 𝐼) → (𝐴‘𝑖) ∈ 𝐾)
479	48, 49, 475, 476, 478	mulgnn0cld 19298	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑖 ∈ 𝐼) → ((𝑑‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖)) ∈ 𝐾)
480	479	fmpttd 7113	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑖 ∈ 𝐼 ↦ ((𝑑‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖))):𝐼⟶𝐾)
481	296	feqmptd 6951	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑑 = (𝑖 ∈ 𝐼 ↦ (𝑑‘𝑖)))
482	481, 309	eqbrtrrd 5129	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑖 ∈ 𝐼 ↦ (𝑑‘𝑖)) finSupp 0)
483	111	adantl 487	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ 𝐾) → (0(.g‘(mulGrp‘𝑅))𝑘) = (0g‘(mulGrp‘𝑅)))
484	482, 483, 476, 478, 305	fsuppssov1 9369	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑖 ∈ 𝐼 ↦ ((𝑑‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖))) finSupp (0g‘(mulGrp‘𝑅)))
485		disjdif 4426	. . . . . . . . . 10 ⊢ (𝐽 ∩ (𝐼 ∖ 𝐽)) = ∅
486	485	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝐽 ∩ (𝐼 ∖ 𝐽)) = ∅)
487		undif 4438	. . . . . . . . . . . 12 ⊢ (𝐽 ⊆ 𝐼 ↔ (𝐽 ∪ (𝐼 ∖ 𝐽)) = 𝐼)
488	16, 487	sylib 221	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐽 ∪ (𝐼 ∖ 𝐽)) = 𝐼)
489	488	eqcomd 2767	. . . . . . . . . 10 ⊢ (𝜑 → 𝐼 = (𝐽 ∪ (𝐼 ∖ 𝐽)))
490	489	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝐼 = (𝐽 ∪ (𝐼 ∖ 𝐽)))
491	48, 110, 474, 292, 283, 480, 484, 486, 490	gsumsplit 20135	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → ((mulGrp‘𝑅) Σg (𝑖 ∈ 𝐼 ↦ ((𝑑‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖)))) = (((mulGrp‘𝑅) Σg ((𝑖 ∈ 𝐼 ↦ ((𝑑‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖))) ↾ 𝐽))(.r‘𝑅)((mulGrp‘𝑅) Σg ((𝑖 ∈ 𝐼 ↦ ((𝑑‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖))) ↾ (𝐼 ∖ 𝐽)))))
492	284	resmptd 6032	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → ((𝑖 ∈ 𝐼 ↦ ((𝑑‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖))) ↾ 𝐽) = (𝑖 ∈ 𝐽 ↦ ((𝑑‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖))))
493		fveq2 6883	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑗 → (𝑑‘𝑖) = (𝑑‘𝑗))
494		fveq2 6883	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑗 → (𝐴‘𝑖) = (𝐴‘𝑗))
495	493, 494	oveq12d 7436	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑗 → ((𝑑‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖)) = ((𝑑‘𝑗)(.g‘(mulGrp‘𝑅))(𝐴‘𝑗)))
496	495	cbvmptv 5209	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ 𝐽 ↦ ((𝑑‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖))) = (𝑗 ∈ 𝐽 ↦ ((𝑑‘𝑗)(.g‘(mulGrp‘𝑅))(𝐴‘𝑗)))
497		simpr 490	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑗 ∈ 𝐽) → 𝑗 ∈ 𝐽)
498	497	fvresd 6903	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑗 ∈ 𝐽) → ((𝑑 ↾ 𝐽)‘𝑗) = (𝑑‘𝑗))
499	497	fvresd 6903	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑗 ∈ 𝐽) → ((𝐴 ↾ 𝐽)‘𝑗) = (𝐴‘𝑗))
500	498, 499	oveq12d 7436	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑗 ∈ 𝐽) → (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)) = ((𝑑‘𝑗)(.g‘(mulGrp‘𝑅))(𝐴‘𝑗)))
501	500	eqcomd 2767	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑗 ∈ 𝐽) → ((𝑑‘𝑗)(.g‘(mulGrp‘𝑅))(𝐴‘𝑗)) = (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))
502	501	mpteq2dva 5198	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑗 ∈ 𝐽 ↦ ((𝑑‘𝑗)(.g‘(mulGrp‘𝑅))(𝐴‘𝑗))) = (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))))
503	496, 502	eqtrid 2808	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑖 ∈ 𝐽 ↦ ((𝑑‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖))) = (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))))
504	492, 503	eqtrd 2796	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → ((𝑖 ∈ 𝐼 ↦ ((𝑑‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖))) ↾ 𝐽) = (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))))
505	504	oveq2d 7434	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → ((mulGrp‘𝑅) Σg ((𝑖 ∈ 𝐼 ↦ ((𝑑‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖))) ↾ 𝐽)) = ((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))
506	289	resmptd 6032	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → ((𝑖 ∈ 𝐼 ↦ ((𝑑‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖))) ↾ (𝐼 ∖ 𝐽)) = (𝑖 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑑‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖))))
507		fveq2 6883	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑘 → (𝑑‘𝑖) = (𝑑‘𝑘))
508		fveq2 6883	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑘 → (𝐴‘𝑖) = (𝐴‘𝑘))
509	507, 508	oveq12d 7436	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑘 → ((𝑑‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖)) = ((𝑑‘𝑘)(.g‘(mulGrp‘𝑅))(𝐴‘𝑘)))
510	509	cbvmptv 5209	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑑‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖))) = (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑑‘𝑘)(.g‘(mulGrp‘𝑅))(𝐴‘𝑘)))
511		simpr 490	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → 𝑘 ∈ (𝐼 ∖ 𝐽))
512	511	fvresd 6903	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → ((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘) = (𝑑‘𝑘))
513	511	fvresd 6903	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → ((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘) = (𝐴‘𝑘))
514	512, 513	oveq12d 7436	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)) = ((𝑑‘𝑘)(.g‘(mulGrp‘𝑅))(𝐴‘𝑘)))
515	514	eqcomd 2767	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → ((𝑑‘𝑘)(.g‘(mulGrp‘𝑅))(𝐴‘𝑘)) = (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)))
516	515	mpteq2dva 5198	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑑‘𝑘)(.g‘(mulGrp‘𝑅))(𝐴‘𝑘))) = (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))
517	510, 516	eqtrid 2808	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑖 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑑‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖))) = (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))
518	506, 517	eqtrd 2796	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → ((𝑖 ∈ 𝐼 ↦ ((𝑑‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖))) ↾ (𝐼 ∖ 𝐽)) = (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))
519	518	oveq2d 7434	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → ((mulGrp‘𝑅) Σg ((𝑖 ∈ 𝐼 ↦ ((𝑑‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖))) ↾ (𝐼 ∖ 𝐽))) = ((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)))))
520	505, 519	oveq12d 7436	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (((mulGrp‘𝑅) Σg ((𝑖 ∈ 𝐼 ↦ ((𝑑‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖))) ↾ 𝐽))(.r‘𝑅)((mulGrp‘𝑅) Σg ((𝑖 ∈ 𝐼 ↦ ((𝑑‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖))) ↾ (𝐼 ∖ 𝐽)))) = (((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))))
521	491, 520	eqtr2d 2797	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))) = ((mulGrp‘𝑅) Σg (𝑖 ∈ 𝐼 ↦ ((𝑑‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖)))))
522	351, 521	oveq12d 7436	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → ((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽))‘(𝑑 ↾ (𝐼 ∖ 𝐽)))(.r‘𝑅)(((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)))))) = ((𝐹‘𝑑)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑖 ∈ 𝐼 ↦ ((𝑑‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖))))))
523	473, 522	eqtrd 2796	. . . . 5 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽))‘(𝑑 ↾ (𝐼 ∖ 𝐽)))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))) = ((𝐹‘𝑑)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑖 ∈ 𝐼 ↦ ((𝑑‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖))))))
524	523	mpteq2dva 5198	. . . 4 ⊢ (𝜑 → (𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽))‘(𝑑 ↾ (𝐼 ∖ 𝐽)))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)))))) = (𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ ((𝐹‘𝑑)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑖 ∈ 𝐼 ↦ ((𝑑‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖)))))))
525	524	oveq2d 7434	. . 3 ⊢ (𝜑 → (𝑅 Σg (𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽))‘(𝑑 ↾ (𝐼 ∖ 𝐽)))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))))) = (𝑅 Σg (𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ ((𝐹‘𝑑)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑖 ∈ 𝐼 ↦ ((𝑑‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖))))))))
526	275, 472, 525	3eqtr2d 2802	. 2 ⊢ (𝜑 → (𝑅 Σg (𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ (((((𝐽 eval 𝑈)‘(((𝐼 selectVars 𝑅)‘𝐽)‘𝐹))‘(𝐿 ∘ (𝐴 ↾ 𝐽)))‘𝑐)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑐‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))))) = (𝑅 Σg (𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ ((𝐹‘𝑑)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑖 ∈ 𝐼 ↦ ((𝑑‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖))))))))
527		eqid 2761	. . 3 ⊢ ((𝐼 ∖ 𝐽) eval 𝑅) = ((𝐼 ∖ 𝐽) eval 𝑅)
528	527, 3, 1, 137, 28, 47, 49, 136, 6, 7, 166, 457	evlvvval 22435	. 2 ⊢ (𝜑 → ((((𝐼 ∖ 𝐽) eval 𝑅)‘(((𝐽 eval 𝑈)‘(((𝐼 selectVars 𝑅)‘𝐽)‘𝐹))‘(𝐿 ∘ (𝐴 ↾ 𝐽))))‘(𝐴 ↾ (𝐼 ∖ 𝐽))) = (𝑅 Σg (𝑐 ∈ {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ (((((𝐽 eval 𝑈)‘(((𝐼 selectVars 𝑅)‘𝐽)‘𝐹))‘(𝐿 ∘ (𝐴 ↾ 𝐽)))‘𝑐)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑐‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))))))
529		eqid 2761	. . 3 ⊢ (𝐼 eval 𝑅) = (𝐼 eval 𝑅)
530	529, 14, 15, 282, 28, 47, 49, 136, 4, 7, 17, 35	evlvvval 22435	. 2 ⊢ (𝜑 → (((𝐼 eval 𝑅)‘𝐹)‘𝐴) = (𝑅 Σg (𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ ((𝐹‘𝑑)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑖 ∈ 𝐼 ↦ ((𝑑‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖))))))))
531	526, 528, 530	3eqtr4d 2806	1 ⊢ (𝜑 → ((((𝐼 ∖ 𝐽) eval 𝑅)‘(((𝐽 eval 𝑈)‘(((𝐼 selectVars 𝑅)‘𝐽)‘𝐹))‘(𝐿 ∘ (𝐴 ↾ 𝐽))))‘(𝐴 ↾ (𝐼 ∖ 𝐽))) = (((𝐼 eval 𝑅)‘𝐹)‘𝐴))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3077  {crab 3413  Vcvv 3451  ⦋csb 3847   ∖ cdif 3896   ∪ cun 3897   ∩ cin 3898   ⊆ wss 3899  ∅c0 4279  {csn 4584   class class class wbr 5103   ↦ cmpt 5186   × cxp 5649  ^◡ccnv 5650   ↾ cres 5653   “ cima 5654   ∘ ccom 5655  Fun wfun 6531   Fn wfn 6532  ⟶wf 6533  –1-1→wf1 6534  –1-1-onto→wf1o 6536  ‘cfv 6537  (class class class)co 7418   ∈ cmpo 7420   supp csupp 8170   ↑_m cmap 8840  Fincfn 8966   finSupp cfsupp 9346  0cc0 11193  ℕcn 12328  ℕ₀cn0 12599  ℤcz 12686  Basecbs 17380  ._rcmulr 17422  Scalarcsca 17424   ·_𝑠 cvsca 17425  0_gc0g 17603   Σ_g cgsu 17604  Mndcmnd 18916   MndHom cmhm 18969  Grpcgrp 19137  ._gcmg 19270   GrpHom cghm 19420  CMndccmn 19987  mulGrpcmgp 20353  1_rcur 20400  Ringcrg 20452  CRingccrg 20453   RingHom crh 20692  AssAlgcasa 22151  algSccascl 22153   mPoly cmpl 22207   eval cevl 22375   selectVars cslv 22418

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749  ax-cnex 11249  ax-resscn 11250  ax-1cn 11251  ax-icn 11252  ax-addcl 11253  ax-addrcl 11254  ax-mulcl 11255  ax-mulrcl 11256  ax-mulcom 11257  ax-addass 11258  ax-mulass 11259  ax-distr 11260  ax-i2m1 11261  ax-1ne0 11262  ax-1rid 11263  ax-rnegex 11264  ax-rrecex 11265  ax-cnre 11266  ax-pre-lttri 11267  ax-pre-lttrn 11268  ax-pre-ltadd 11269  ax-pre-mulgt0 11270

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-iin 4954  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7375  df-ov 7421  df-oprab 7422  df-mpo 7423  df-of 7691  df-ofr 7692  df-om 7876  df-1st 7999  df-2nd 8000  df-supp 8171  df-frecs 8292  df-wrecs 8323  df-recs 8372  df-rdg 8411  df-1o 8469  df-2o 8470  df-er 8710  df-map 8842  df-pm 8843  df-ixp 8919  df-en 8967  df-dom 8968  df-sdom 8969  df-fin 8970  df-fsupp 9347  df-sup 9427  df-oi 9497  df-card 10013  df-pnf 11338  df-mnf 11339  df-xr 11340  df-ltxr 11341  df-le 11342  df-sub 11536  df-neg 11537  df-nn 12329  df-2 12398  df-3 12399  df-4 12400  df-5 12401  df-6 12402  df-7 12403  df-8 12404  df-9 12405  df-n0 12600  df-z 12687  df-dec 12808  df-uz 12959  df-fz 13633  df-fzo 13782  df-seq 14138  df-hash 14468  df-struct 17318  df-sets 17335  df-slot 17353  df-ndx 17365  df-base 17381  df-ress 17402  df-plusg 17434  df-mulr 17435  df-sca 17437  df-vsca 17438  df-ip 17439  df-tset 17440  df-ple 17441  df-ds 17443  df-hom 17445  df-cco 17446  df-0g 17605  df-gsum 17606  df-prds 17611  df-pws 17613  df-mre 17749  df-mrc 17750  df-acs 17752  df-mgm 18809  df-sgrp 18901  df-mnd 18917  df-mhm 18971  df-submnd 18972  df-grp 19140  df-minusg 19141  df-sbg 19142  df-mulg 19271  df-subg 19326  df-ghm 19421  df-cntz 19524  df-cmn 19989  df-abl 19990  df-mgp 20354  df-rng 20368  df-ur 20401  df-srg 20406  df-ring 20454  df-cring 20455  df-rhm 20695  df-subrng 20791  df-subrg 20815  df-lmod 21130  df-lss 21200  df-lsp 21240  df-assa 22154  df-asp 22155  df-ascl 22156  df-psr 22210  df-mvr 22211  df-mpl 22212  df-evls 22376  df-evl 22377  df-selv 22419

This theorem is used by: (None)