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Theorem evlselv 42573
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 = ((𝐼𝐽) mPoly 𝑅)
evlselv.t 𝑇 = (𝐽 mPoly 𝑈)
evlselv.l 𝐿 = (algSc‘𝑈)
evlselv.i (𝜑𝐼𝑉)
evlselv.r (𝜑𝑅 ∈ CRing)
evlselv.j (𝜑𝐽𝐼)
evlselv.f (𝜑𝐹𝐵)
evlselv.a (𝜑𝐴 ∈ (𝐾m 𝐼))
Assertion
Ref Expression
evlselv (𝜑 → ((((𝐼𝐽) eval 𝑅)‘(((𝐽 eval 𝑈)‘(((𝐼 selectVars 𝑅)‘𝐽)‘𝐹))‘(𝐿 ∘ (𝐴𝐽))))‘(𝐴 ↾ (𝐼𝐽))) = (((𝐼 eval 𝑅)‘𝐹)‘𝐴))

Proof of Theorem evlselv
Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑒 𝑖 𝑗 𝑘 𝑢 𝑣 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2734 . . . . . . . . . . . . 13 (Base‘𝑈) = (Base‘𝑈)
2 eqid 2734 . . . . . . . . . . . . 13 (.r𝑈) = (.r𝑈)
3 evlselv.u . . . . . . . . . . . . . . . 16 𝑈 = ((𝐼𝐽) mPoly 𝑅)
4 evlselv.i . . . . . . . . . . . . . . . . 17 (𝜑𝐼𝑉)
5 difssd 4146 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐼𝐽) ⊆ 𝐼)
64, 5ssexd 5329 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐼𝐽) ∈ V)
7 evlselv.r . . . . . . . . . . . . . . . 16 (𝜑𝑅 ∈ CRing)
83, 6, 7mplcrngd 42533 . . . . . . . . . . . . . . 15 (𝜑𝑈 ∈ CRing)
98crngringd 20263 . . . . . . . . . . . . . 14 (𝜑𝑈 ∈ Ring)
109ad2antrr 726 . . . . . . . . . . . . 13 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → 𝑈 ∈ Ring)
11 evlselv.t . . . . . . . . . . . . . . . 16 𝑇 = (𝐽 mPoly 𝑈)
12 eqid 2734 . . . . . . . . . . . . . . . 16 (Base‘𝑇) = (Base‘𝑇)
13 eqid 2734 . . . . . . . . . . . . . . . 16 {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} = {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}
14 evlselv.p . . . . . . . . . . . . . . . . 17 𝑃 = (𝐼 mPoly 𝑅)
15 evlselv.b . . . . . . . . . . . . . . . . 17 𝐵 = (Base‘𝑃)
16 evlselv.j . . . . . . . . . . . . . . . . 17 (𝜑𝐽𝐼)
17 evlselv.f . . . . . . . . . . . . . . . . 17 (𝜑𝐹𝐵)
1814, 15, 3, 11, 12, 7, 16, 17selvcl 42569 . . . . . . . . . . . . . . . 16 (𝜑 → (((𝐼 selectVars 𝑅)‘𝐽)‘𝐹) ∈ (Base‘𝑇))
1911, 1, 12, 13, 18mplelf 22035 . . . . . . . . . . . . . . 15 (𝜑 → (((𝐼 selectVars 𝑅)‘𝐽)‘𝐹):{𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}⟶(Base‘𝑈))
2019adantr 480 . . . . . . . . . . . . . 14 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (((𝐼 selectVars 𝑅)‘𝐽)‘𝐹):{𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}⟶(Base‘𝑈))
2120ffvelcdmda 7103 . . . . . . . . . . . . 13 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒) ∈ (Base‘𝑈))
22 eqid 2734 . . . . . . . . . . . . . 14 (mulGrp‘𝑈) = (mulGrp‘𝑈)
23 eqid 2734 . . . . . . . . . . . . . 14 (.g‘(mulGrp‘𝑈)) = (.g‘(mulGrp‘𝑈))
244, 16ssexd 5329 . . . . . . . . . . . . . . 15 (𝜑𝐽 ∈ V)
2524ad2antrr 726 . . . . . . . . . . . . . 14 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → 𝐽 ∈ V)
268ad2antrr 726 . . . . . . . . . . . . . 14 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → 𝑈 ∈ CRing)
27 fvexd 6921 . . . . . . . . . . . . . . . . 17 (𝜑 → (Base‘𝑈) ∈ V)
28 evlselv.k . . . . . . . . . . . . . . . . . . 19 𝐾 = (Base‘𝑅)
2928fvexi 6920 . . . . . . . . . . . . . . . . . 18 𝐾 ∈ V
3029a1i 11 . . . . . . . . . . . . . . . . 17 (𝜑𝐾 ∈ V)
31 evlselv.l . . . . . . . . . . . . . . . . . 18 𝐿 = (algSc‘𝑈)
327crngringd 20263 . . . . . . . . . . . . . . . . . 18 (𝜑𝑅 ∈ Ring)
333, 1, 28, 31, 6, 32mplasclf 22106 . . . . . . . . . . . . . . . . 17 (𝜑𝐿:𝐾⟶(Base‘𝑈))
3427, 30, 33elmapdd 8879 . . . . . . . . . . . . . . . 16 (𝜑𝐿 ∈ ((Base‘𝑈) ↑m 𝐾))
35 evlselv.a . . . . . . . . . . . . . . . . 17 (𝜑𝐴 ∈ (𝐾m 𝐼))
3635, 16elmapssresd 42260 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐴𝐽) ∈ (𝐾m 𝐽))
3734, 36mapcod 42262 . . . . . . . . . . . . . . 15 (𝜑 → (𝐿 ∘ (𝐴𝐽)) ∈ ((Base‘𝑈) ↑m 𝐽))
3837ad2antrr 726 . . . . . . . . . . . . . 14 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → (𝐿 ∘ (𝐴𝐽)) ∈ ((Base‘𝑈) ↑m 𝐽))
39 simpr 484 . . . . . . . . . . . . . 14 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})
4013, 1, 22, 23, 25, 26, 38, 39evlsvvvallem 42547 . . . . . . . . . . . . 13 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → ((mulGrp‘𝑈) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴𝐽))‘𝑗)))) ∈ (Base‘𝑈))
411, 2, 10, 21, 40ringcld 20276 . . . . . . . . . . . 12 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r𝑈)((mulGrp‘𝑈) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴𝐽))‘𝑗))))) ∈ (Base‘𝑈))
42 eqidd 2735 . . . . . . . . . . . 12 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r𝑈)((mulGrp‘𝑈) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴𝐽))‘𝑗)))))) = (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r𝑈)((mulGrp‘𝑈) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴𝐽))‘𝑗)))))))
43 eqidd 2735 . . . . . . . . . . . 12 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑢 ∈ (Base‘𝑈) ↦ (𝑢𝑐)) = (𝑢 ∈ (Base‘𝑈) ↦ (𝑢𝑐)))
44 fveq1 6905 . . . . . . . . . . . 12 (𝑢 = (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r𝑈)((mulGrp‘𝑈) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴𝐽))‘𝑗))))) → (𝑢𝑐) = ((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r𝑈)((mulGrp‘𝑈) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴𝐽))‘𝑗)))))‘𝑐))
4541, 42, 43, 44fmptco 7148 . . . . . . . . . . 11 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → ((𝑢 ∈ (Base‘𝑈) ↦ (𝑢𝑐)) ∘ (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r𝑈)((mulGrp‘𝑈) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴𝐽))‘𝑗))))))) = (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ ((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r𝑈)((mulGrp‘𝑈) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴𝐽))‘𝑗)))))‘𝑐)))
4633ad2antrr 726 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → 𝐿:𝐾⟶(Base‘𝑈))
47 eqid 2734 . . . . . . . . . . . . . . . . . . . . . 22 (mulGrp‘𝑅) = (mulGrp‘𝑅)
4847, 28mgpbas 20157 . . . . . . . . . . . . . . . . . . . . 21 𝐾 = (Base‘(mulGrp‘𝑅))
49 eqid 2734 . . . . . . . . . . . . . . . . . . . . 21 (.g‘(mulGrp‘𝑅)) = (.g‘(mulGrp‘𝑅))
5047ringmgp 20256 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑅 ∈ Ring → (mulGrp‘𝑅) ∈ Mnd)
5132, 50syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (mulGrp‘𝑅) ∈ Mnd)
5251ad3antrrr 730 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) ∧ 𝑗𝐽) → (mulGrp‘𝑅) ∈ Mnd)
5313psrbagf 21955 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} → 𝑒:𝐽⟶ℕ0)
5453adantl 481 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → 𝑒:𝐽⟶ℕ0)
5554ffvelcdmda 7103 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) ∧ 𝑗𝐽) → (𝑒𝑗) ∈ ℕ0)
56 elmapi 8887 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐴 ∈ (𝐾m 𝐼) → 𝐴:𝐼𝐾)
5735, 56syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑𝐴:𝐼𝐾)
5857, 16fssresd 6775 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (𝐴𝐽):𝐽𝐾)
5958ad2antrr 726 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → (𝐴𝐽):𝐽𝐾)
6059ffvelcdmda 7103 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) ∧ 𝑗𝐽) → ((𝐴𝐽)‘𝑗) ∈ 𝐾)
6148, 49, 52, 55, 60mulgnn0cld 19125 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) ∧ 𝑗𝐽) → ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)) ∈ 𝐾)
6246, 61cofmpt 7151 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → (𝐿 ∘ (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))) = (𝑗𝐽 ↦ (𝐿‘((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))
633mplassa 22059 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝐼𝐽) ∈ V ∧ 𝑅 ∈ CRing) → 𝑈 ∈ AssAlg)
646, 7, 63syl2anc 584 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑𝑈 ∈ AssAlg)
65 eqid 2734 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (Scalar‘𝑈) = (Scalar‘𝑈)
6631, 65asclrhm 21927 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑈 ∈ AssAlg → 𝐿 ∈ ((Scalar‘𝑈) RingHom 𝑈))
6764, 66syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑𝐿 ∈ ((Scalar‘𝑈) RingHom 𝑈))
683, 6, 7mplsca 22050 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑𝑅 = (Scalar‘𝑈))
6968eqcomd 2740 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → (Scalar‘𝑈) = 𝑅)
7069oveq1d 7445 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → ((Scalar‘𝑈) RingHom 𝑈) = (𝑅 RingHom 𝑈))
7167, 70eleqtrd 2840 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑𝐿 ∈ (𝑅 RingHom 𝑈))
7247, 22rhmmhm 20495 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐿 ∈ (𝑅 RingHom 𝑈) → 𝐿 ∈ ((mulGrp‘𝑅) MndHom (mulGrp‘𝑈)))
7371, 72syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝐿 ∈ ((mulGrp‘𝑅) MndHom (mulGrp‘𝑈)))
7473ad3antrrr 730 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) ∧ 𝑗𝐽) → 𝐿 ∈ ((mulGrp‘𝑅) MndHom (mulGrp‘𝑈)))
7548, 49, 23mhmmulg 19145 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐿 ∈ ((mulGrp‘𝑅) MndHom (mulGrp‘𝑈)) ∧ (𝑒𝑗) ∈ ℕ0 ∧ ((𝐴𝐽)‘𝑗) ∈ 𝐾) → (𝐿‘((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))) = ((𝑒𝑗)(.g‘(mulGrp‘𝑈))(𝐿‘((𝐴𝐽)‘𝑗))))
7674, 55, 60, 75syl3anc 1370 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) ∧ 𝑗𝐽) → (𝐿‘((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))) = ((𝑒𝑗)(.g‘(mulGrp‘𝑈))(𝐿‘((𝐴𝐽)‘𝑗))))
7758ad3antrrr 730 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) ∧ 𝑗𝐽) → (𝐴𝐽):𝐽𝐾)
78 simpr 484 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) ∧ 𝑗𝐽) → 𝑗𝐽)
7977, 78fvco3d 7008 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) ∧ 𝑗𝐽) → ((𝐿 ∘ (𝐴𝐽))‘𝑗) = (𝐿‘((𝐴𝐽)‘𝑗)))
8079oveq2d 7446 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) ∧ 𝑗𝐽) → ((𝑒𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴𝐽))‘𝑗)) = ((𝑒𝑗)(.g‘(mulGrp‘𝑈))(𝐿‘((𝐴𝐽)‘𝑗))))
8176, 80eqtr4d 2777 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) ∧ 𝑗𝐽) → (𝐿‘((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))) = ((𝑒𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴𝐽))‘𝑗)))
8281mpteq2dva 5247 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → (𝑗𝐽 ↦ (𝐿‘((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))) = (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴𝐽))‘𝑗))))
8362, 82eqtrd 2774 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → (𝐿 ∘ (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))) = (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴𝐽))‘𝑗))))
8483oveq2d 7446 . . . . . . . . . . . . . . . . 17 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → ((mulGrp‘𝑈) Σg (𝐿 ∘ (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))) = ((mulGrp‘𝑈) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴𝐽))‘𝑗)))))
85 eqid 2734 . . . . . . . . . . . . . . . . . 18 (Base‘(mulGrp‘(Scalar‘𝑈))) = (Base‘(mulGrp‘(Scalar‘𝑈)))
86 eqid 2734 . . . . . . . . . . . . . . . . . 18 (0g‘(mulGrp‘(Scalar‘𝑈))) = (0g‘(mulGrp‘(Scalar‘𝑈)))
8768, 7eqeltrrd 2839 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (Scalar‘𝑈) ∈ CRing)
88 eqid 2734 . . . . . . . . . . . . . . . . . . . . 21 (mulGrp‘(Scalar‘𝑈)) = (mulGrp‘(Scalar‘𝑈))
8988crngmgp 20258 . . . . . . . . . . . . . . . . . . . 20 ((Scalar‘𝑈) ∈ CRing → (mulGrp‘(Scalar‘𝑈)) ∈ CMnd)
9087, 89syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (mulGrp‘(Scalar‘𝑈)) ∈ CMnd)
9190ad2antrr 726 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → (mulGrp‘(Scalar‘𝑈)) ∈ CMnd)
9222ringmgp 20256 . . . . . . . . . . . . . . . . . . . 20 (𝑈 ∈ Ring → (mulGrp‘𝑈) ∈ Mnd)
939, 92syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (mulGrp‘𝑈) ∈ Mnd)
9493ad2antrr 726 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → (mulGrp‘𝑈) ∈ Mnd)
9588, 22rhmmhm 20495 . . . . . . . . . . . . . . . . . . . 20 (𝐿 ∈ ((Scalar‘𝑈) RingHom 𝑈) → 𝐿 ∈ ((mulGrp‘(Scalar‘𝑈)) MndHom (mulGrp‘𝑈)))
9667, 95syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐿 ∈ ((mulGrp‘(Scalar‘𝑈)) MndHom (mulGrp‘𝑈)))
9796ad2antrr 726 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → 𝐿 ∈ ((mulGrp‘(Scalar‘𝑈)) MndHom (mulGrp‘𝑈)))
9868fveq2d 6910 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (Base‘𝑅) = (Base‘(Scalar‘𝑈)))
9928, 98eqtrid 2786 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑𝐾 = (Base‘(Scalar‘𝑈)))
10099ad3antrrr 730 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) ∧ 𝑗𝐽) → 𝐾 = (Base‘(Scalar‘𝑈)))
10161, 100eleqtrd 2840 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) ∧ 𝑗𝐽) → ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)) ∈ (Base‘(Scalar‘𝑈)))
102 eqid 2734 . . . . . . . . . . . . . . . . . . . . 21 (Base‘(Scalar‘𝑈)) = (Base‘(Scalar‘𝑈))
10388, 102mgpbas 20157 . . . . . . . . . . . . . . . . . . . 20 (Base‘(Scalar‘𝑈)) = (Base‘(mulGrp‘(Scalar‘𝑈)))
104101, 103eleqtrdi 2848 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) ∧ 𝑗𝐽) → ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)) ∈ (Base‘(mulGrp‘(Scalar‘𝑈))))
105104fmpttd 7134 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))):𝐽⟶(Base‘(mulGrp‘(Scalar‘𝑈))))
10654feqmptd 6976 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → 𝑒 = (𝑗𝐽 ↦ (𝑒𝑗)))
10713psrbagfsupp 21956 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} → 𝑒 finSupp 0)
108107adantl 481 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → 𝑒 finSupp 0)
109106, 108eqbrtrrd 5171 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → (𝑗𝐽 ↦ (𝑒𝑗)) finSupp 0)
110 eqid 2734 . . . . . . . . . . . . . . . . . . . . . . 23 (0g‘(mulGrp‘𝑅)) = (0g‘(mulGrp‘𝑅))
11148, 110, 49mulg0 19104 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘𝐾 → (0(.g‘(mulGrp‘𝑅))𝑘) = (0g‘(mulGrp‘𝑅)))
112111adantl 481 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) ∧ 𝑘𝐾) → (0(.g‘(mulGrp‘𝑅))𝑘) = (0g‘(mulGrp‘𝑅)))
113 fvexd 6921 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → (0g‘(mulGrp‘𝑅)) ∈ V)
114109, 112, 55, 60, 113fsuppssov1 9421 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))) finSupp (0g‘(mulGrp‘𝑅)))
115 eqid 2734 . . . . . . . . . . . . . . . . . . . . 21 (1r𝑅) = (1r𝑅)
11647, 115ringidval 20200 . . . . . . . . . . . . . . . . . . . 20 (1r𝑅) = (0g‘(mulGrp‘𝑅))
117114, 116breqtrrdi 5189 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))) finSupp (1r𝑅))
11868fveq2d 6910 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (1r𝑅) = (1r‘(Scalar‘𝑈)))
119 eqid 2734 . . . . . . . . . . . . . . . . . . . . . 22 (1r‘(Scalar‘𝑈)) = (1r‘(Scalar‘𝑈))
12088, 119ringidval 20200 . . . . . . . . . . . . . . . . . . . . 21 (1r‘(Scalar‘𝑈)) = (0g‘(mulGrp‘(Scalar‘𝑈)))
121118, 120eqtrdi 2790 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (1r𝑅) = (0g‘(mulGrp‘(Scalar‘𝑈))))
122121ad2antrr 726 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → (1r𝑅) = (0g‘(mulGrp‘(Scalar‘𝑈))))
123117, 122breqtrd 5173 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))) finSupp (0g‘(mulGrp‘(Scalar‘𝑈))))
12485, 86, 91, 94, 25, 97, 105, 123gsummhm 19970 . . . . . . . . . . . . . . . . 17 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → ((mulGrp‘𝑈) Σg (𝐿 ∘ (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))) = (𝐿‘((mulGrp‘(Scalar‘𝑈)) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))))
12584, 124eqtr3d 2776 . . . . . . . . . . . . . . . 16 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → ((mulGrp‘𝑈) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴𝐽))‘𝑗)))) = (𝐿‘((mulGrp‘(Scalar‘𝑈)) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))))
126125oveq2d 7446 . . . . . . . . . . . . . . 15 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r𝑈)((mulGrp‘𝑈) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴𝐽))‘𝑗))))) = (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r𝑈)(𝐿‘((mulGrp‘(Scalar‘𝑈)) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))))
12764ad2antrr 726 . . . . . . . . . . . . . . . 16 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → 𝑈 ∈ AssAlg)
128101fmpttd 7134 . . . . . . . . . . . . . . . . 17 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))):𝐽⟶(Base‘(Scalar‘𝑈)))
129123, 120breqtrrdi 5189 . . . . . . . . . . . . . . . . 17 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))) finSupp (1r‘(Scalar‘𝑈)))
130103, 120, 91, 25, 128, 129gsumcl 19947 . . . . . . . . . . . . . . . 16 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → ((mulGrp‘(Scalar‘𝑈)) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))) ∈ (Base‘(Scalar‘𝑈)))
131 eqid 2734 . . . . . . . . . . . . . . . . 17 ( ·𝑠𝑈) = ( ·𝑠𝑈)
13231, 65, 102, 1, 2, 131asclmul2 21924 . . . . . . . . . . . . . . . 16 ((𝑈 ∈ AssAlg ∧ ((mulGrp‘(Scalar‘𝑈)) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))) ∈ (Base‘(Scalar‘𝑈)) ∧ ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒) ∈ (Base‘𝑈)) → (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r𝑈)(𝐿‘((mulGrp‘(Scalar‘𝑈)) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))) = (((mulGrp‘(Scalar‘𝑈)) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))( ·𝑠𝑈)((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)))
133127, 130, 21, 132syl3anc 1370 . . . . . . . . . . . . . . 15 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r𝑈)(𝐿‘((mulGrp‘(Scalar‘𝑈)) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))) = (((mulGrp‘(Scalar‘𝑈)) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))( ·𝑠𝑈)((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)))
134126, 133eqtrd 2774 . . . . . . . . . . . . . 14 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r𝑈)((mulGrp‘𝑈) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴𝐽))‘𝑗))))) = (((mulGrp‘(Scalar‘𝑈)) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))( ·𝑠𝑈)((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)))
135134fveq1d 6908 . . . . . . . . . . . . 13 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → ((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r𝑈)((mulGrp‘𝑈) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴𝐽))‘𝑗)))))‘𝑐) = ((((mulGrp‘(Scalar‘𝑈)) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))( ·𝑠𝑈)((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒))‘𝑐))
136 eqid 2734 . . . . . . . . . . . . . 14 (.r𝑅) = (.r𝑅)
137 eqid 2734 . . . . . . . . . . . . . 14 {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}
13899ad2antrr 726 . . . . . . . . . . . . . . 15 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → 𝐾 = (Base‘(Scalar‘𝑈)))
139130, 138eleqtrrd 2841 . . . . . . . . . . . . . 14 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → ((mulGrp‘(Scalar‘𝑈)) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))) ∈ 𝐾)
140 simplr 769 . . . . . . . . . . . . . 14 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → 𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin})
1413, 131, 28, 1, 136, 137, 139, 21, 140mplvscaval 22053 . . . . . . . . . . . . 13 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → ((((mulGrp‘(Scalar‘𝑈)) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))( ·𝑠𝑈)((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒))‘𝑐) = (((mulGrp‘(Scalar‘𝑈)) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))(.r𝑅)(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)))
142135, 141eqtrd 2774 . . . . . . . . . . . 12 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → ((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r𝑈)((mulGrp‘𝑈) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴𝐽))‘𝑗)))))‘𝑐) = (((mulGrp‘(Scalar‘𝑈)) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))(.r𝑅)(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)))
143142mpteq2dva 5247 . . . . . . . . . . 11 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ ((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r𝑈)((mulGrp‘𝑈) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴𝐽))‘𝑗)))))‘𝑐)) = (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((mulGrp‘(Scalar‘𝑈)) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))(.r𝑅)(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐))))
14445, 143eqtrd 2774 . . . . . . . . . 10 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → ((𝑢 ∈ (Base‘𝑈) ↦ (𝑢𝑐)) ∘ (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r𝑈)((mulGrp‘𝑈) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴𝐽))‘𝑗))))))) = (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((mulGrp‘(Scalar‘𝑈)) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))(.r𝑅)(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐))))
145144oveq2d 7446 . . . . . . . . 9 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑅 Σg ((𝑢 ∈ (Base‘𝑈) ↦ (𝑢𝑐)) ∘ (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r𝑈)((mulGrp‘𝑈) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴𝐽))‘𝑗)))))))) = (𝑅 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((mulGrp‘(Scalar‘𝑈)) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))(.r𝑅)(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)))))
14669fveq2d 6910 . . . . . . . . . . . . . . 15 (𝜑 → (mulGrp‘(Scalar‘𝑈)) = (mulGrp‘𝑅))
147146ad2antrr 726 . . . . . . . . . . . . . 14 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → (mulGrp‘(Scalar‘𝑈)) = (mulGrp‘𝑅))
148147oveq1d 7445 . . . . . . . . . . . . 13 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → ((mulGrp‘(Scalar‘𝑈)) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))) = ((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))
149148oveq1d 7445 . . . . . . . . . . . 12 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → (((mulGrp‘(Scalar‘𝑈)) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))(.r𝑅)(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)) = (((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))(.r𝑅)(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)))
1507ad2antrr 726 . . . . . . . . . . . . 13 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → 𝑅 ∈ CRing)
151148, 139eqeltrrd 2839 . . . . . . . . . . . . 13 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → ((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))) ∈ 𝐾)
15219ffvelcdmda 7103 . . . . . . . . . . . . . . . 16 ((𝜑𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒) ∈ (Base‘𝑈))
1533, 28, 1, 137, 152mplelf 22035 . . . . . . . . . . . . . . 15 ((𝜑𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒):{𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}⟶𝐾)
154153ffvelcdmda 7103 . . . . . . . . . . . . . 14 (((𝜑𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐) ∈ 𝐾)
155154an32s 652 . . . . . . . . . . . . 13 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐) ∈ 𝐾)
15628, 136, 150, 151, 155crngcomd 20272 . . . . . . . . . . . 12 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → (((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))(.r𝑅)(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)) = ((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))))
157149, 156eqtrd 2774 . . . . . . . . . . 11 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → (((mulGrp‘(Scalar‘𝑈)) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))(.r𝑅)(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)) = ((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))))
158157mpteq2dva 5247 . . . . . . . . . 10 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((mulGrp‘(Scalar‘𝑈)) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))(.r𝑅)(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐))) = (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ ((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))))
159158oveq2d 7446 . . . . . . . . 9 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑅 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((mulGrp‘(Scalar‘𝑈)) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))(.r𝑅)(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)))) = (𝑅 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ ((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))))))
160145, 159eqtrd 2774 . . . . . . . 8 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑅 Σg ((𝑢 ∈ (Base‘𝑈) ↦ (𝑢𝑐)) ∘ (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r𝑈)((mulGrp‘𝑈) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴𝐽))‘𝑗)))))))) = (𝑅 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ ((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))))))
161160oveq1d 7445 . . . . . . 7 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → ((𝑅 Σg ((𝑢 ∈ (Base‘𝑈) ↦ (𝑢𝑐)) ∘ (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r𝑈)((mulGrp‘𝑈) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴𝐽))‘𝑗))))))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑐𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))) = ((𝑅 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ ((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑐𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))))
162 eqid 2734 . . . . . . . . . 10 (𝑢 ∈ (Base‘𝑈) ↦ (𝑢𝑐)) = (𝑢 ∈ (Base‘𝑈) ↦ (𝑢𝑐))
163 fveq1 6905 . . . . . . . . . 10 (𝑢 = (𝑈 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r𝑈)((mulGrp‘𝑈) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴𝐽))‘𝑗))))))) → (𝑢𝑐) = ((𝑈 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r𝑈)((mulGrp‘𝑈) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴𝐽))‘𝑗)))))))‘𝑐))
164 eqid 2734 . . . . . . . . . . . . 13 (𝐽 eval 𝑈) = (𝐽 eval 𝑈)
165164, 11, 12, 13, 1, 22, 23, 2, 24, 8, 18, 37evlvvval 42559 . . . . . . . . . . . 12 (𝜑 → (((𝐽 eval 𝑈)‘(((𝐼 selectVars 𝑅)‘𝐽)‘𝐹))‘(𝐿 ∘ (𝐴𝐽))) = (𝑈 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r𝑈)((mulGrp‘𝑈) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴𝐽))‘𝑗))))))))
166164, 11, 12, 1, 24, 8, 18, 37evlcl 42558 . . . . . . . . . . . 12 (𝜑 → (((𝐽 eval 𝑈)‘(((𝐼 selectVars 𝑅)‘𝐽)‘𝐹))‘(𝐿 ∘ (𝐴𝐽))) ∈ (Base‘𝑈))
167165, 166eqeltrrd 2839 . . . . . . . . . . 11 (𝜑 → (𝑈 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r𝑈)((mulGrp‘𝑈) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴𝐽))‘𝑗))))))) ∈ (Base‘𝑈))
168167adantr 480 . . . . . . . . . 10 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑈 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r𝑈)((mulGrp‘𝑈) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴𝐽))‘𝑗))))))) ∈ (Base‘𝑈))
169 fvexd 6921 . . . . . . . . . 10 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → ((𝑈 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r𝑈)((mulGrp‘𝑈) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴𝐽))‘𝑗)))))))‘𝑐) ∈ V)
170162, 163, 168, 169fvmptd3 7038 . . . . . . . . 9 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → ((𝑢 ∈ (Base‘𝑈) ↦ (𝑢𝑐))‘(𝑈 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r𝑈)((mulGrp‘𝑈) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴𝐽))‘𝑗)))))))) = ((𝑈 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r𝑈)((mulGrp‘𝑈) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴𝐽))‘𝑗)))))))‘𝑐))
171 eqid 2734 . . . . . . . . . 10 (0g𝑈) = (0g𝑈)
1729ringcmnd 20297 . . . . . . . . . . 11 (𝜑𝑈 ∈ CMnd)
173172adantr 480 . . . . . . . . . 10 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → 𝑈 ∈ CMnd)
1747crnggrpd 20264 . . . . . . . . . . . 12 (𝜑𝑅 ∈ Grp)
175174grpmndd 18976 . . . . . . . . . . 11 (𝜑𝑅 ∈ Mnd)
176175adantr 480 . . . . . . . . . 10 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → 𝑅 ∈ Mnd)
177 ovex 7463 . . . . . . . . . . . 12 (ℕ0m 𝐽) ∈ V
178177rabex 5344 . . . . . . . . . . 11 {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ∈ V
179178a1i 11 . . . . . . . . . 10 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ∈ V)
1806adantr 480 . . . . . . . . . . . 12 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝐼𝐽) ∈ V)
181174adantr 480 . . . . . . . . . . . 12 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → 𝑅 ∈ Grp)
182 simpr 484 . . . . . . . . . . . 12 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → 𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin})
1833, 1, 137, 162, 180, 181, 182mplmapghm 42542 . . . . . . . . . . 11 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑢 ∈ (Base‘𝑈) ↦ (𝑢𝑐)) ∈ (𝑈 GrpHom 𝑅))
184 ghmmhm 19256 . . . . . . . . . . 11 ((𝑢 ∈ (Base‘𝑈) ↦ (𝑢𝑐)) ∈ (𝑈 GrpHom 𝑅) → (𝑢 ∈ (Base‘𝑈) ↦ (𝑢𝑐)) ∈ (𝑈 MndHom 𝑅))
185183, 184syl 17 . . . . . . . . . 10 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑢 ∈ (Base‘𝑈) ↦ (𝑢𝑐)) ∈ (𝑈 MndHom 𝑅))
18641fmpttd 7134 . . . . . . . . . 10 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r𝑈)((mulGrp‘𝑈) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴𝐽))‘𝑗)))))):{𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}⟶(Base‘𝑈))
18724adantr 480 . . . . . . . . . . 11 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → 𝐽 ∈ V)
1888adantr 480 . . . . . . . . . . 11 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → 𝑈 ∈ CRing)
18918adantr 480 . . . . . . . . . . 11 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (((𝐼 selectVars 𝑅)‘𝐽)‘𝐹) ∈ (Base‘𝑇))
19037adantr 480 . . . . . . . . . . 11 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝐿 ∘ (𝐴𝐽)) ∈ ((Base‘𝑈) ↑m 𝐽))
19113, 11, 12, 1, 22, 23, 2, 187, 188, 189, 190evlvvvallem 42560 . . . . . . . . . 10 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r𝑈)((mulGrp‘𝑈) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴𝐽))‘𝑗)))))) finSupp (0g𝑈))
1921, 171, 173, 176, 179, 185, 186, 191gsummhm 19970 . . . . . . . . 9 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑅 Σg ((𝑢 ∈ (Base‘𝑈) ↦ (𝑢𝑐)) ∘ (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r𝑈)((mulGrp‘𝑈) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴𝐽))‘𝑗)))))))) = ((𝑢 ∈ (Base‘𝑈) ↦ (𝑢𝑐))‘(𝑈 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r𝑈)((mulGrp‘𝑈) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴𝐽))‘𝑗)))))))))
193165adantr 480 . . . . . . . . . 10 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (((𝐽 eval 𝑈)‘(((𝐼 selectVars 𝑅)‘𝐽)‘𝐹))‘(𝐿 ∘ (𝐴𝐽))) = (𝑈 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r𝑈)((mulGrp‘𝑈) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴𝐽))‘𝑗))))))))
194193fveq1d 6908 . . . . . . . . 9 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → ((((𝐽 eval 𝑈)‘(((𝐼 selectVars 𝑅)‘𝐽)‘𝐹))‘(𝐿 ∘ (𝐴𝐽)))‘𝑐) = ((𝑈 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r𝑈)((mulGrp‘𝑈) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴𝐽))‘𝑗)))))))‘𝑐))
195170, 192, 1943eqtr4rd 2785 . . . . . . . 8 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → ((((𝐽 eval 𝑈)‘(((𝐼 selectVars 𝑅)‘𝐽)‘𝐹))‘(𝐿 ∘ (𝐴𝐽)))‘𝑐) = (𝑅 Σg ((𝑢 ∈ (Base‘𝑈) ↦ (𝑢𝑐)) ∘ (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r𝑈)((mulGrp‘𝑈) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴𝐽))‘𝑗)))))))))
196195oveq1d 7445 . . . . . . 7 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (((((𝐽 eval 𝑈)‘(((𝐼 selectVars 𝑅)‘𝐽)‘𝐹))‘(𝐿 ∘ (𝐴𝐽)))‘𝑐)(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑐𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))) = ((𝑅 Σg ((𝑢 ∈ (Base‘𝑈) ↦ (𝑢𝑐)) ∘ (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r𝑈)((mulGrp‘𝑈) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴𝐽))‘𝑗))))))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑐𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))))
197 eqid 2734 . . . . . . . 8 (0g𝑅) = (0g𝑅)
19832adantr 480 . . . . . . . 8 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → 𝑅 ∈ Ring)
19947crngmgp 20258 . . . . . . . . . . 11 (𝑅 ∈ CRing → (mulGrp‘𝑅) ∈ CMnd)
2007, 199syl 17 . . . . . . . . . 10 (𝜑 → (mulGrp‘𝑅) ∈ CMnd)
201200adantr 480 . . . . . . . . 9 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (mulGrp‘𝑅) ∈ CMnd)
20251ad2antrr 726 . . . . . . . . . . 11 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ (𝐼𝐽)) → (mulGrp‘𝑅) ∈ Mnd)
203137psrbagf 21955 . . . . . . . . . . . . 13 (𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} → 𝑐:(𝐼𝐽)⟶ℕ0)
204203adantl 481 . . . . . . . . . . . 12 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → 𝑐:(𝐼𝐽)⟶ℕ0)
205204ffvelcdmda 7103 . . . . . . . . . . 11 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ (𝐼𝐽)) → (𝑐𝑘) ∈ ℕ0)
20657, 5fssresd 6775 . . . . . . . . . . . . 13 (𝜑 → (𝐴 ↾ (𝐼𝐽)):(𝐼𝐽)⟶𝐾)
207206adantr 480 . . . . . . . . . . . 12 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝐴 ↾ (𝐼𝐽)):(𝐼𝐽)⟶𝐾)
208207ffvelcdmda 7103 . . . . . . . . . . 11 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ (𝐼𝐽)) → ((𝐴 ↾ (𝐼𝐽))‘𝑘) ∈ 𝐾)
20948, 49, 202, 205, 208mulgnn0cld 19125 . . . . . . . . . 10 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ (𝐼𝐽)) → ((𝑐𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)) ∈ 𝐾)
210209fmpttd 7134 . . . . . . . . 9 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑘 ∈ (𝐼𝐽) ↦ ((𝑐𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))):(𝐼𝐽)⟶𝐾)
211204feqmptd 6976 . . . . . . . . . . 11 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → 𝑐 = (𝑘 ∈ (𝐼𝐽) ↦ (𝑐𝑘)))
212137psrbagfsupp 21956 . . . . . . . . . . . 12 (𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} → 𝑐 finSupp 0)
213212adantl 481 . . . . . . . . . . 11 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → 𝑐 finSupp 0)
214211, 213eqbrtrrd 5171 . . . . . . . . . 10 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑘 ∈ (𝐼𝐽) ↦ (𝑐𝑘)) finSupp 0)
21548, 110, 49mulg0 19104 . . . . . . . . . . 11 (𝑣𝐾 → (0(.g‘(mulGrp‘𝑅))𝑣) = (0g‘(mulGrp‘𝑅)))
216215adantl 481 . . . . . . . . . 10 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑣𝐾) → (0(.g‘(mulGrp‘𝑅))𝑣) = (0g‘(mulGrp‘𝑅)))
217 fvexd 6921 . . . . . . . . . 10 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ (𝐼𝐽)) → (𝑐𝑘) ∈ V)
218 fvexd 6921 . . . . . . . . . 10 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (0g‘(mulGrp‘𝑅)) ∈ V)
219214, 216, 217, 208, 218fsuppssov1 9421 . . . . . . . . 9 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑘 ∈ (𝐼𝐽) ↦ ((𝑐𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))) finSupp (0g‘(mulGrp‘𝑅)))
22048, 110, 201, 180, 210, 219gsumcl 19947 . . . . . . . 8 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → ((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑐𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)))) ∈ 𝐾)
22132ad2antrr 726 . . . . . . . . 9 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → 𝑅 ∈ Ring)
22228, 136, 221, 155, 151ringcld 20276 . . . . . . . 8 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → ((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))) ∈ 𝐾)
223178mptex 7242 . . . . . . . . . . 11 (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)) ∈ V
224223a1i 11 . . . . . . . . . 10 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)) ∈ V)
225 fvexd 6921 . . . . . . . . . 10 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (0g𝑅) ∈ V)
226 funmpt 6605 . . . . . . . . . . 11 Fun (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐))
227226a1i 11 . . . . . . . . . 10 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → Fun (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)))
22811, 12, 171, 18mplelsfi 22032 . . . . . . . . . . 11 (𝜑 → (((𝐼 selectVars 𝑅)‘𝐽)‘𝐹) finSupp (0g𝑈))
229228adantr 480 . . . . . . . . . 10 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (((𝐼 selectVars 𝑅)‘𝐽)‘𝐹) finSupp (0g𝑈))
230 ssidd 4018 . . . . . . . . . . . . . 14 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹) supp (0g𝑈)) ⊆ ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹) supp (0g𝑈)))
231 fvexd 6921 . . . . . . . . . . . . . 14 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (0g𝑈) ∈ V)
23220, 230, 179, 231suppssr 8218 . . . . . . . . . . . . 13 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ ({𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ∖ ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹) supp (0g𝑈)))) → ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒) = (0g𝑈))
233232fveq1d 6908 . . . . . . . . . . . 12 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ ({𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ∖ ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹) supp (0g𝑈)))) → (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐) = ((0g𝑈)‘𝑐))
2343, 137, 197, 171, 6, 174mpl0 22043 . . . . . . . . . . . . . . . 16 (𝜑 → (0g𝑈) = ({𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} × {(0g𝑅)}))
235234adantr 480 . . . . . . . . . . . . . . 15 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (0g𝑈) = ({𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} × {(0g𝑅)}))
236235fveq1d 6908 . . . . . . . . . . . . . 14 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → ((0g𝑈)‘𝑐) = (({𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} × {(0g𝑅)})‘𝑐))
237 fvex 6919 . . . . . . . . . . . . . . . 16 (0g𝑅) ∈ V
238237fvconst2 7223 . . . . . . . . . . . . . . 15 (𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} → (({𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} × {(0g𝑅)})‘𝑐) = (0g𝑅))
239238adantl 481 . . . . . . . . . . . . . 14 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (({𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} × {(0g𝑅)})‘𝑐) = (0g𝑅))
240236, 239eqtrd 2774 . . . . . . . . . . . . 13 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → ((0g𝑈)‘𝑐) = (0g𝑅))
241240adantr 480 . . . . . . . . . . . 12 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ ({𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ∖ ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹) supp (0g𝑈)))) → ((0g𝑈)‘𝑐) = (0g𝑅))
242233, 241eqtrd 2774 . . . . . . . . . . 11 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ ({𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ∖ ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹) supp (0g𝑈)))) → (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐) = (0g𝑅))
243242, 179suppss2 8223 . . . . . . . . . 10 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → ((𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)) supp (0g𝑅)) ⊆ ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹) supp (0g𝑈)))
244224, 225, 227, 229, 243fsuppsssuppgd 9419 . . . . . . . . 9 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)) finSupp (0g𝑅))
24532ad2antrr 726 . . . . . . . . . 10 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑣𝐾) → 𝑅 ∈ Ring)
246 simpr 484 . . . . . . . . . 10 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑣𝐾) → 𝑣𝐾)
24728, 136, 197, 245, 246ringlzd 20308 . . . . . . . . 9 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑣𝐾) → ((0g𝑅)(.r𝑅)𝑣) = (0g𝑅))
248244, 247, 155, 151, 225fsuppssov1 9421 . . . . . . . 8 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ ((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))) finSupp (0g𝑅))
24928, 197, 136, 198, 179, 220, 222, 248gsummulc1 20329 . . . . . . 7 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑅 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑐𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))))) = ((𝑅 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ ((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑐𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))))
250161, 196, 2493eqtr4d 2784 . . . . . 6 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (((((𝐽 eval 𝑈)‘(((𝐼 selectVars 𝑅)‘𝐽)‘𝐹))‘(𝐿 ∘ (𝐴𝐽)))‘𝑐)(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑐𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))) = (𝑅 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑐𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))))))
251 fveq2 6906 . . . . . . . . . . . . . 14 (𝑎 = 𝑒 → ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎) = ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒))
252251adantl 481 . . . . . . . . . . . . 13 ((𝑏 = 𝑐𝑎 = 𝑒) → ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎) = ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒))
253 simpl 482 . . . . . . . . . . . . 13 ((𝑏 = 𝑐𝑎 = 𝑒) → 𝑏 = 𝑐)
254252, 253fveq12d 6913 . . . . . . . . . . . 12 ((𝑏 = 𝑐𝑎 = 𝑒) → (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏) = (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐))
255 fveq1 6905 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑒 → (𝑎𝑗) = (𝑒𝑗))
256255adantl 481 . . . . . . . . . . . . . . 15 ((𝑏 = 𝑐𝑎 = 𝑒) → (𝑎𝑗) = (𝑒𝑗))
257256oveq1d 7445 . . . . . . . . . . . . . 14 ((𝑏 = 𝑐𝑎 = 𝑒) → ((𝑎𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)) = ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))
258257mpteq2dv 5249 . . . . . . . . . . . . 13 ((𝑏 = 𝑐𝑎 = 𝑒) → (𝑗𝐽 ↦ ((𝑎𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))) = (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))
259258oveq2d 7446 . . . . . . . . . . . 12 ((𝑏 = 𝑐𝑎 = 𝑒) → ((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑎𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))) = ((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))
260254, 259oveq12d 7448 . . . . . . . . . . 11 ((𝑏 = 𝑐𝑎 = 𝑒) → ((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑎𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))) = ((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))))
261 fveq1 6905 . . . . . . . . . . . . . . 15 (𝑏 = 𝑐 → (𝑏𝑘) = (𝑐𝑘))
262261adantr 480 . . . . . . . . . . . . . 14 ((𝑏 = 𝑐𝑎 = 𝑒) → (𝑏𝑘) = (𝑐𝑘))
263262oveq1d 7445 . . . . . . . . . . . . 13 ((𝑏 = 𝑐𝑎 = 𝑒) → ((𝑏𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)) = ((𝑐𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)))
264263mpteq2dv 5249 . . . . . . . . . . . 12 ((𝑏 = 𝑐𝑎 = 𝑒) → (𝑘 ∈ (𝐼𝐽) ↦ ((𝑏𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))) = (𝑘 ∈ (𝐼𝐽) ↦ ((𝑐𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))
265264oveq2d 7446 . . . . . . . . . . 11 ((𝑏 = 𝑐𝑎 = 𝑒) → ((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑏𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)))) = ((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑐𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)))))
266260, 265oveq12d 7448 . . . . . . . . . 10 ((𝑏 = 𝑐𝑎 = 𝑒) → (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑎𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑏𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))) = (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑐𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))))
267 eqid 2734 . . . . . . . . . 10 (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑎𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑏𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)))))) = (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑎𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑏𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))))
268 ovex 7463 . . . . . . . . . 10 (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑐𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))) ∈ V
269266, 267, 268ovmpoa 7587 . . . . . . . . 9 ((𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → (𝑐(𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑎𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑏𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))))𝑒) = (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑐𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))))
270269adantll 714 . . . . . . . 8 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → (𝑐(𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑎𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑏𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))))𝑒) = (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑐𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))))
271270mpteq2dva 5247 . . . . . . 7 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (𝑐(𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑎𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑏𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))))𝑒)) = (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑐𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)))))))
272271oveq2d 7446 . . . . . 6 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑅 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (𝑐(𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑎𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑏𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))))𝑒))) = (𝑅 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑐𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))))))
273250, 272eqtr4d 2777 . . . . 5 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (((((𝐽 eval 𝑈)‘(((𝐼 selectVars 𝑅)‘𝐽)‘𝐹))‘(𝐿 ∘ (𝐴𝐽)))‘𝑐)(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑐𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))) = (𝑅 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (𝑐(𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑎𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑏𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))))𝑒))))
274273mpteq2dva 5247 . . . 4 (𝜑 → (𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ↦ (((((𝐽 eval 𝑈)‘(((𝐼 selectVars 𝑅)‘𝐽)‘𝐹))‘(𝐿 ∘ (𝐴𝐽)))‘𝑐)(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑐𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)))))) = (𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ↦ (𝑅 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (𝑐(𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑎𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑏𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))))𝑒)))))
275274oveq2d 7446 . . 3 (𝜑 → (𝑅 Σg (𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ↦ (((((𝐽 eval 𝑈)‘(((𝐼 selectVars 𝑅)‘𝐽)‘𝐹))‘(𝐿 ∘ (𝐴𝐽)))‘𝑐)(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑐𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))))) = (𝑅 Σg (𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ↦ (𝑅 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (𝑐(𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑎𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑏𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))))𝑒))))))
27632ringcmnd 20297 . . . . 5 (𝜑𝑅 ∈ CMnd)
277 ovex 7463 . . . . . . 7 (ℕ0m 𝐼) ∈ V
278277rabex 5344 . . . . . 6 { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∈ V
279278a1i 11 . . . . 5 (𝜑 → { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∈ V)
28032adantr 480 . . . . . . 7 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → 𝑅 ∈ Ring)
28119adantr 480 . . . . . . . . . . 11 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (((𝐼 selectVars 𝑅)‘𝐽)‘𝐹):{𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}⟶(Base‘𝑈))
282 eqid 2734 . . . . . . . . . . . 12 { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} = { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}
2834adantr 480 . . . . . . . . . . . 12 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → 𝐼𝑉)
28416adantr 480 . . . . . . . . . . . 12 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → 𝐽𝐼)
285 simpr 484 . . . . . . . . . . . 12 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → 𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin})
286282, 13, 283, 284, 285psrbagres 42532 . . . . . . . . . . 11 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝑑𝐽) ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})
287281, 286ffvelcdmd 7104 . . . . . . . . . 10 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑𝐽)) ∈ (Base‘𝑈))
2883, 28, 1, 137, 287mplelf 22035 . . . . . . . . 9 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑𝐽)):{𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}⟶𝐾)
289 difssd 4146 . . . . . . . . . 10 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝐼𝐽) ⊆ 𝐼)
290282, 137, 283, 289, 285psrbagres 42532 . . . . . . . . 9 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝑑 ↾ (𝐼𝐽)) ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin})
291288, 290ffvelcdmd 7104 . . . . . . . 8 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑𝐽))‘(𝑑 ↾ (𝐼𝐽))) ∈ 𝐾)
292200adantr 480 . . . . . . . . 9 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (mulGrp‘𝑅) ∈ CMnd)
29324adantr 480 . . . . . . . . 9 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → 𝐽 ∈ V)
29451ad2antrr 726 . . . . . . . . . . 11 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑗𝐽) → (mulGrp‘𝑅) ∈ Mnd)
295282psrbagf 21955 . . . . . . . . . . . . . 14 (𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} → 𝑑:𝐼⟶ℕ0)
296295adantl 481 . . . . . . . . . . . . 13 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → 𝑑:𝐼⟶ℕ0)
297296, 284fssresd 6775 . . . . . . . . . . . 12 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝑑𝐽):𝐽⟶ℕ0)
298297ffvelcdmda 7103 . . . . . . . . . . 11 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑗𝐽) → ((𝑑𝐽)‘𝑗) ∈ ℕ0)
29958ffvelcdmda 7103 . . . . . . . . . . . 12 ((𝜑𝑗𝐽) → ((𝐴𝐽)‘𝑗) ∈ 𝐾)
300299adantlr 715 . . . . . . . . . . 11 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑗𝐽) → ((𝐴𝐽)‘𝑗) ∈ 𝐾)
30148, 49, 294, 298, 300mulgnn0cld 19125 . . . . . . . . . 10 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑗𝐽) → (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)) ∈ 𝐾)
302301fmpttd 7134 . . . . . . . . 9 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))):𝐽𝐾)
30324mptexd 7243 . . . . . . . . . . 11 (𝜑 → (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))) ∈ V)
304303adantr 480 . . . . . . . . . 10 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))) ∈ V)
305 fvexd 6921 . . . . . . . . . 10 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (0g‘(mulGrp‘𝑅)) ∈ V)
306 funmpt 6605 . . . . . . . . . . 11 Fun (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))
307306a1i 11 . . . . . . . . . 10 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → Fun (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))
308282psrbagfsupp 21956 . . . . . . . . . . . 12 (𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} → 𝑑 finSupp 0)
309308adantl 481 . . . . . . . . . . 11 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → 𝑑 finSupp 0)
310 0zd 12622 . . . . . . . . . . 11 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → 0 ∈ ℤ)
311309, 310fsuppres 9430 . . . . . . . . . 10 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝑑𝐽) finSupp 0)
312 ssidd 4018 . . . . . . . . . . . . . 14 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → ((𝑑𝐽) supp 0) ⊆ ((𝑑𝐽) supp 0))
313297, 312, 293, 310suppssr 8218 . . . . . . . . . . . . 13 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑗 ∈ (𝐽 ∖ ((𝑑𝐽) supp 0))) → ((𝑑𝐽)‘𝑗) = 0)
314313oveq1d 7445 . . . . . . . . . . . 12 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑗 ∈ (𝐽 ∖ ((𝑑𝐽) supp 0))) → (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)) = (0(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))
315 eldifi 4140 . . . . . . . . . . . . 13 (𝑗 ∈ (𝐽 ∖ ((𝑑𝐽) supp 0)) → 𝑗𝐽)
31648, 110, 49mulg0 19104 . . . . . . . . . . . . . 14 (((𝐴𝐽)‘𝑗) ∈ 𝐾 → (0(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)) = (0g‘(mulGrp‘𝑅)))
317300, 316syl 17 . . . . . . . . . . . . 13 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑗𝐽) → (0(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)) = (0g‘(mulGrp‘𝑅)))
318315, 317sylan2 593 . . . . . . . . . . . 12 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑗 ∈ (𝐽 ∖ ((𝑑𝐽) supp 0))) → (0(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)) = (0g‘(mulGrp‘𝑅)))
319314, 318eqtrd 2774 . . . . . . . . . . 11 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑗 ∈ (𝐽 ∖ ((𝑑𝐽) supp 0))) → (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)) = (0g‘(mulGrp‘𝑅)))
320319, 293suppss2 8223 . . . . . . . . . 10 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → ((𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))) supp (0g‘(mulGrp‘𝑅))) ⊆ ((𝑑𝐽) supp 0))
321304, 305, 307, 311, 320fsuppsssuppgd 9419 . . . . . . . . 9 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))) finSupp (0g‘(mulGrp‘𝑅)))
32248, 110, 292, 293, 302, 321gsumcl 19947 . . . . . . . 8 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → ((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))) ∈ 𝐾)
32328, 136, 280, 291, 322ringcld 20276 . . . . . . 7 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → ((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑𝐽))‘(𝑑 ↾ (𝐼𝐽)))(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))) ∈ 𝐾)
3246adantr 480 . . . . . . . 8 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝐼𝐽) ∈ V)
32551ad2antrr 726 . . . . . . . . . 10 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ (𝐼𝐽)) → (mulGrp‘𝑅) ∈ Mnd)
326296, 289fssresd 6775 . . . . . . . . . . 11 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝑑 ↾ (𝐼𝐽)):(𝐼𝐽)⟶ℕ0)
327326ffvelcdmda 7103 . . . . . . . . . 10 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ (𝐼𝐽)) → ((𝑑 ↾ (𝐼𝐽))‘𝑘) ∈ ℕ0)
328206ffvelcdmda 7103 . . . . . . . . . . 11 ((𝜑𝑘 ∈ (𝐼𝐽)) → ((𝐴 ↾ (𝐼𝐽))‘𝑘) ∈ 𝐾)
329328adantlr 715 . . . . . . . . . 10 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ (𝐼𝐽)) → ((𝐴 ↾ (𝐼𝐽))‘𝑘) ∈ 𝐾)
33048, 49, 325, 327, 329mulgnn0cld 19125 . . . . . . . . 9 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ (𝐼𝐽)) → (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)) ∈ 𝐾)
331330fmpttd 7134 . . . . . . . 8 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝑘 ∈ (𝐼𝐽) ↦ (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))):(𝐼𝐽)⟶𝐾)
332324mptexd 7243 . . . . . . . . 9 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝑘 ∈ (𝐼𝐽) ↦ (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))) ∈ V)
333 funmpt 6605 . . . . . . . . . 10 Fun (𝑘 ∈ (𝐼𝐽) ↦ (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)))
334333a1i 11 . . . . . . . . 9 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → Fun (𝑘 ∈ (𝐼𝐽) ↦ (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))
335309, 310fsuppres 9430 . . . . . . . . 9 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝑑 ↾ (𝐼𝐽)) finSupp 0)
336 ssidd 4018 . . . . . . . . . . . . 13 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → ((𝑑 ↾ (𝐼𝐽)) supp 0) ⊆ ((𝑑 ↾ (𝐼𝐽)) supp 0))
337326, 336, 324, 310suppssr 8218 . . . . . . . . . . . 12 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ ((𝐼𝐽) ∖ ((𝑑 ↾ (𝐼𝐽)) supp 0))) → ((𝑑 ↾ (𝐼𝐽))‘𝑘) = 0)
338337oveq1d 7445 . . . . . . . . . . 11 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ ((𝐼𝐽) ∖ ((𝑑 ↾ (𝐼𝐽)) supp 0))) → (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)) = (0(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)))
339 eldifi 4140 . . . . . . . . . . . . 13 (𝑘 ∈ ((𝐼𝐽) ∖ ((𝑑 ↾ (𝐼𝐽)) supp 0)) → 𝑘 ∈ (𝐼𝐽))
340339, 329sylan2 593 . . . . . . . . . . . 12 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ ((𝐼𝐽) ∖ ((𝑑 ↾ (𝐼𝐽)) supp 0))) → ((𝐴 ↾ (𝐼𝐽))‘𝑘) ∈ 𝐾)
34148, 110, 49mulg0 19104 . . . . . . . . . . . 12 (((𝐴 ↾ (𝐼𝐽))‘𝑘) ∈ 𝐾 → (0(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)) = (0g‘(mulGrp‘𝑅)))
342340, 341syl 17 . . . . . . . . . . 11 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ ((𝐼𝐽) ∖ ((𝑑 ↾ (𝐼𝐽)) supp 0))) → (0(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)) = (0g‘(mulGrp‘𝑅)))
343338, 342eqtrd 2774 . . . . . . . . . 10 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ ((𝐼𝐽) ∖ ((𝑑 ↾ (𝐼𝐽)) supp 0))) → (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)) = (0g‘(mulGrp‘𝑅)))
344343, 324suppss2 8223 . . . . . . . . 9 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → ((𝑘 ∈ (𝐼𝐽) ↦ (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))) supp (0g‘(mulGrp‘𝑅))) ⊆ ((𝑑 ↾ (𝐼𝐽)) supp 0))
345332, 305, 334, 335, 344fsuppsssuppgd 9419 . . . . . . . 8 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝑘 ∈ (𝐼𝐽) ↦ (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))) finSupp (0g‘(mulGrp‘𝑅)))
34648, 110, 292, 324, 331, 345gsumcl 19947 . . . . . . 7 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → ((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)))) ∈ 𝐾)
34728, 136, 280, 323, 346ringcld 20276 . . . . . 6 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑𝐽))‘(𝑑 ↾ (𝐼𝐽)))(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))) ∈ 𝐾)
348347fmpttd 7134 . . . . 5 (𝜑 → (𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑𝐽))‘(𝑑 ↾ (𝐼𝐽)))(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)))))):{ ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}⟶𝐾)
3497adantr 480 . . . . . . . . . 10 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → 𝑅 ∈ CRing)
35017adantr 480 . . . . . . . . . 10 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → 𝐹𝐵)
351282, 14, 15, 349, 284, 350, 285selvvvval 42571 . . . . . . . . 9 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑𝐽))‘(𝑑 ↾ (𝐼𝐽))) = (𝐹𝑑))
352351mpteq2dva 5247 . . . . . . . 8 (𝜑 → (𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑𝐽))‘(𝑑 ↾ (𝐼𝐽)))) = (𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (𝐹𝑑)))
353 eqid 2734 . . . . . . . . . . 11 (Base‘𝑅) = (Base‘𝑅)
35414, 353, 15, 282, 17mplelf 22035 . . . . . . . . . 10 (𝜑𝐹:{ ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}⟶(Base‘𝑅))
355354feqmptd 6976 . . . . . . . . 9 (𝜑𝐹 = (𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (𝐹𝑑)))
35614, 15, 197, 17mplelsfi 22032 . . . . . . . . 9 (𝜑𝐹 finSupp (0g𝑅))
357355, 356eqbrtrrd 5171 . . . . . . . 8 (𝜑 → (𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (𝐹𝑑)) finSupp (0g𝑅))
358352, 357eqbrtrd 5169 . . . . . . 7 (𝜑 → (𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑𝐽))‘(𝑑 ↾ (𝐼𝐽)))) finSupp (0g𝑅))
35932adantr 480 . . . . . . . 8 ((𝜑𝑣𝐾) → 𝑅 ∈ Ring)
360 simpr 484 . . . . . . . 8 ((𝜑𝑣𝐾) → 𝑣𝐾)
36128, 136, 197, 359, 360ringlzd 20308 . . . . . . 7 ((𝜑𝑣𝐾) → ((0g𝑅)(.r𝑅)𝑣) = (0g𝑅))
362 fvexd 6921 . . . . . . 7 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑𝐽))‘(𝑑 ↾ (𝐼𝐽))) ∈ V)
363 fvexd 6921 . . . . . . 7 (𝜑 → (0g𝑅) ∈ V)
364358, 361, 362, 322, 363fsuppssov1 9421 . . . . . 6 (𝜑 → (𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ ((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑𝐽))‘(𝑑 ↾ (𝐼𝐽)))(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))) finSupp (0g𝑅))
365 ovexd 7465 . . . . . 6 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → ((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑𝐽))‘(𝑑 ↾ (𝐼𝐽)))(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))) ∈ V)
366364, 361, 365, 346, 363fsuppssov1 9421 . . . . 5 (𝜑 → (𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑𝐽))‘(𝑑 ↾ (𝐼𝐽)))(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)))))) finSupp (0g𝑅))
367 eqid 2734 . . . . . 6 (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (𝑏𝑎)) = (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (𝑏𝑎))
368282, 13, 137, 367, 4, 16evlselvlem 42572 . . . . 5 (𝜑 → (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (𝑏𝑎)):({𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} × {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})–1-1-onto→{ ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin})
36928, 197, 276, 279, 348, 366, 368gsumf1o 19948 . . . 4 (𝜑 → (𝑅 Σg (𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑𝐽))‘(𝑑 ↾ (𝐼𝐽)))(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))))) = (𝑅 Σg ((𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑𝐽))‘(𝑑 ↾ (𝐼𝐽)))(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)))))) ∘ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (𝑏𝑎)))))
370137psrbagf 21955 . . . . . . . . . 10 (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} → 𝑏:(𝐼𝐽)⟶ℕ0)
371370ad2antrl 728 . . . . . . . . 9 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → 𝑏:(𝐼𝐽)⟶ℕ0)
37213psrbagf 21955 . . . . . . . . . 10 (𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} → 𝑎:𝐽⟶ℕ0)
373372ad2antll 729 . . . . . . . . 9 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → 𝑎:𝐽⟶ℕ0)
374 disjdifr 4478 . . . . . . . . . 10 ((𝐼𝐽) ∩ 𝐽) = ∅
375374a1i 11 . . . . . . . . 9 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → ((𝐼𝐽) ∩ 𝐽) = ∅)
376371, 373, 375fun2d 6772 . . . . . . . 8 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → (𝑏𝑎):((𝐼𝐽) ∪ 𝐽)⟶ℕ0)
377 undifr 4488 . . . . . . . . . . 11 (𝐽𝐼 ↔ ((𝐼𝐽) ∪ 𝐽) = 𝐼)
37816, 377sylib 218 . . . . . . . . . 10 (𝜑 → ((𝐼𝐽) ∪ 𝐽) = 𝐼)
379378adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → ((𝐼𝐽) ∪ 𝐽) = 𝐼)
380379feq2d 6722 . . . . . . . 8 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → ((𝑏𝑎):((𝐼𝐽) ∪ 𝐽)⟶ℕ0 ↔ (𝑏𝑎):𝐼⟶ℕ0))
381376, 380mpbid 232 . . . . . . 7 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → (𝑏𝑎):𝐼⟶ℕ0)
382 vex 3481 . . . . . . . . . . 11 𝑏 ∈ V
383 vex 3481 . . . . . . . . . . 11 𝑎 ∈ V
384382, 383unex 7762 . . . . . . . . . 10 (𝑏𝑎) ∈ V
385384a1i 11 . . . . . . . . 9 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → (𝑏𝑎) ∈ V)
386 0zd 12622 . . . . . . . . 9 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → 0 ∈ ℤ)
387381ffund 6740 . . . . . . . . 9 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → Fun (𝑏𝑎))
388137psrbagfsupp 21956 . . . . . . . . . . 11 (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} → 𝑏 finSupp 0)
389388ad2antrl 728 . . . . . . . . . 10 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → 𝑏 finSupp 0)
39013psrbagfsupp 21956 . . . . . . . . . . 11 (𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} → 𝑎 finSupp 0)
391390ad2antll 729 . . . . . . . . . 10 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → 𝑎 finSupp 0)
392389, 391fsuppun 9424 . . . . . . . . 9 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → ((𝑏𝑎) supp 0) ∈ Fin)
393385, 386, 387, 392isfsuppd 9403 . . . . . . . 8 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → (𝑏𝑎) finSupp 0)
394 fcdmnn0fsuppg 12583 . . . . . . . . 9 (((𝑏𝑎) ∈ V ∧ (𝑏𝑎):𝐼⟶ℕ0) → ((𝑏𝑎) finSupp 0 ↔ ((𝑏𝑎) “ ℕ) ∈ Fin))
395385, 381, 394syl2anc 584 . . . . . . . 8 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → ((𝑏𝑎) finSupp 0 ↔ ((𝑏𝑎) “ ℕ) ∈ Fin))
396393, 395mpbid 232 . . . . . . 7 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → ((𝑏𝑎) “ ℕ) ∈ Fin)
3974adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → 𝐼𝑉)
398282psrbag 21954 . . . . . . . 8 (𝐼𝑉 → ((𝑏𝑎) ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↔ ((𝑏𝑎):𝐼⟶ℕ0 ∧ ((𝑏𝑎) “ ℕ) ∈ Fin)))
399397, 398syl 17 . . . . . . 7 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → ((𝑏𝑎) ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↔ ((𝑏𝑎):𝐼⟶ℕ0 ∧ ((𝑏𝑎) “ ℕ) ∈ Fin)))
400381, 396, 399mpbir2and 713 . . . . . 6 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → (𝑏𝑎) ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin})
401 eqidd 2735 . . . . . 6 (𝜑 → (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (𝑏𝑎)) = (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (𝑏𝑎)))
402 eqidd 2735 . . . . . 6 (𝜑 → (𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑𝐽))‘(𝑑 ↾ (𝐼𝐽)))(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)))))) = (𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑𝐽))‘(𝑑 ↾ (𝐼𝐽)))(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)))))))
403 reseq1 5993 . . . . . . . . . . . 12 (𝑑 = (𝑏𝑎) → (𝑑𝐽) = ((𝑏𝑎) ↾ 𝐽))
404403fveq2d 6910 . . . . . . . . . . 11 (𝑑 = (𝑏𝑎) → ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑𝐽)) = ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘((𝑏𝑎) ↾ 𝐽)))
405 reseq1 5993 . . . . . . . . . . 11 (𝑑 = (𝑏𝑎) → (𝑑 ↾ (𝐼𝐽)) = ((𝑏𝑎) ↾ (𝐼𝐽)))
406404, 405fveq12d 6913 . . . . . . . . . 10 (𝑑 = (𝑏𝑎) → (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑𝐽))‘(𝑑 ↾ (𝐼𝐽))) = (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘((𝑏𝑎) ↾ 𝐽))‘((𝑏𝑎) ↾ (𝐼𝐽))))
407403fveq1d 6908 . . . . . . . . . . . . 13 (𝑑 = (𝑏𝑎) → ((𝑑𝐽)‘𝑗) = (((𝑏𝑎) ↾ 𝐽)‘𝑗))
408407oveq1d 7445 . . . . . . . . . . . 12 (𝑑 = (𝑏𝑎) → (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)) = ((((𝑏𝑎) ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))
409408mpteq2dv 5249 . . . . . . . . . . 11 (𝑑 = (𝑏𝑎) → (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))) = (𝑗𝐽 ↦ ((((𝑏𝑎) ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))
410409oveq2d 7446 . . . . . . . . . 10 (𝑑 = (𝑏𝑎) → ((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))) = ((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((((𝑏𝑎) ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))
411406, 410oveq12d 7448 . . . . . . . . 9 (𝑑 = (𝑏𝑎) → ((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑𝐽))‘(𝑑 ↾ (𝐼𝐽)))(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))) = ((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘((𝑏𝑎) ↾ 𝐽))‘((𝑏𝑎) ↾ (𝐼𝐽)))(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((((𝑏𝑎) ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))))
412405fveq1d 6908 . . . . . . . . . . . 12 (𝑑 = (𝑏𝑎) → ((𝑑 ↾ (𝐼𝐽))‘𝑘) = (((𝑏𝑎) ↾ (𝐼𝐽))‘𝑘))
413412oveq1d 7445 . . . . . . . . . . 11 (𝑑 = (𝑏𝑎) → (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)) = ((((𝑏𝑎) ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)))
414413mpteq2dv 5249 . . . . . . . . . 10 (𝑑 = (𝑏𝑎) → (𝑘 ∈ (𝐼𝐽) ↦ (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))) = (𝑘 ∈ (𝐼𝐽) ↦ ((((𝑏𝑎) ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))
415414oveq2d 7446 . . . . . . . . 9 (𝑑 = (𝑏𝑎) → ((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)))) = ((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((((𝑏𝑎) ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)))))
416411, 415oveq12d 7448 . . . . . . . 8 (𝑑 = (𝑏𝑎) → (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑𝐽))‘(𝑑 ↾ (𝐼𝐽)))(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))) = (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘((𝑏𝑎) ↾ 𝐽))‘((𝑏𝑎) ↾ (𝐼𝐽)))(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((((𝑏𝑎) ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((((𝑏𝑎) ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))))
417384, 416csbie 3943 . . . . . . 7 (𝑏𝑎) / 𝑑(((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑𝐽))‘(𝑑 ↾ (𝐼𝐽)))(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))) = (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘((𝑏𝑎) ↾ 𝐽))‘((𝑏𝑎) ↾ (𝐼𝐽)))(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((((𝑏𝑎) ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((((𝑏𝑎) ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)))))
418370ffnd 6737 . . . . . . . . . . . . 13 (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} → 𝑏 Fn (𝐼𝐽))
419418ad2antrl 728 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → 𝑏 Fn (𝐼𝐽))
420373ffnd 6737 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → 𝑎 Fn 𝐽)
421 fnunres2 6681 . . . . . . . . . . . 12 ((𝑏 Fn (𝐼𝐽) ∧ 𝑎 Fn 𝐽 ∧ ((𝐼𝐽) ∩ 𝐽) = ∅) → ((𝑏𝑎) ↾ 𝐽) = 𝑎)
422419, 420, 375, 421syl3anc 1370 . . . . . . . . . . 11 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → ((𝑏𝑎) ↾ 𝐽) = 𝑎)
423422fveq2d 6910 . . . . . . . . . 10 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘((𝑏𝑎) ↾ 𝐽)) = ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎))
424 fnunres1 6680 . . . . . . . . . . 11 ((𝑏 Fn (𝐼𝐽) ∧ 𝑎 Fn 𝐽 ∧ ((𝐼𝐽) ∩ 𝐽) = ∅) → ((𝑏𝑎) ↾ (𝐼𝐽)) = 𝑏)
425419, 420, 375, 424syl3anc 1370 . . . . . . . . . 10 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → ((𝑏𝑎) ↾ (𝐼𝐽)) = 𝑏)
426423, 425fveq12d 6913 . . . . . . . . 9 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘((𝑏𝑎) ↾ 𝐽))‘((𝑏𝑎) ↾ (𝐼𝐽))) = (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏))
427422fveq1d 6908 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → (((𝑏𝑎) ↾ 𝐽)‘𝑗) = (𝑎𝑗))
428427oveq1d 7445 . . . . . . . . . . 11 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → ((((𝑏𝑎) ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)) = ((𝑎𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))
429428mpteq2dv 5249 . . . . . . . . . 10 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → (𝑗𝐽 ↦ ((((𝑏𝑎) ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))) = (𝑗𝐽 ↦ ((𝑎𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))
430429oveq2d 7446 . . . . . . . . 9 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → ((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((((𝑏𝑎) ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))) = ((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑎𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))
431426, 430oveq12d 7448 . . . . . . . 8 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → ((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘((𝑏𝑎) ↾ 𝐽))‘((𝑏𝑎) ↾ (𝐼𝐽)))(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((((𝑏𝑎) ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))) = ((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑎𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))))
432425fveq1d 6908 . . . . . . . . . . 11 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → (((𝑏𝑎) ↾ (𝐼𝐽))‘𝑘) = (𝑏𝑘))
433432oveq1d 7445 . . . . . . . . . 10 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → ((((𝑏𝑎) ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)) = ((𝑏𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)))
434433mpteq2dv 5249 . . . . . . . . 9 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → (𝑘 ∈ (𝐼𝐽) ↦ ((((𝑏𝑎) ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))) = (𝑘 ∈ (𝐼𝐽) ↦ ((𝑏𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))
435434oveq2d 7446 . . . . . . . 8 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → ((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((((𝑏𝑎) ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)))) = ((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑏𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)))))
436431, 435oveq12d 7448 . . . . . . 7 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘((𝑏𝑎) ↾ 𝐽))‘((𝑏𝑎) ↾ (𝐼𝐽)))(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((((𝑏𝑎) ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((((𝑏𝑎) ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))) = (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑎𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑏𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))))
437417, 436eqtrid 2786 . . . . . 6 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → (𝑏𝑎) / 𝑑(((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑𝐽))‘(𝑑 ↾ (𝐼𝐽)))(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))) = (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑎𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑏𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))))
438400, 401, 402, 437fmpocos 42253 . . . . 5 (𝜑 → ((𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑𝐽))‘(𝑑 ↾ (𝐼𝐽)))(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)))))) ∘ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (𝑏𝑎))) = (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑎𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑏𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)))))))
439438oveq2d 7446 . . . 4 (𝜑 → (𝑅 Σg ((𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑𝐽))‘(𝑑 ↾ (𝐼𝐽)))(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)))))) ∘ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (𝑏𝑎)))) = (𝑅 Σg (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑎𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑏𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))))))
440 ovex 7463 . . . . . . 7 (ℕ0m (𝐼𝐽)) ∈ V
441440rabex 5344 . . . . . 6 {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∈ V
442441a1i 11 . . . . 5 (𝜑 → {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∈ V)
443178a1i 11 . . . . 5 (𝜑 → {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ∈ V)
44432adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → 𝑅 ∈ Ring)
44519ffvelcdmda 7103 . . . . . . . . . . . . 13 ((𝜑𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎) ∈ (Base‘𝑈))
4463, 28, 1, 137, 445mplelf 22035 . . . . . . . . . . . 12 ((𝜑𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎):{𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}⟶𝐾)
447446ffvelcdmda 7103 . . . . . . . . . . 11 (((𝜑𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) ∧ 𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏) ∈ 𝐾)
448447an32s 652 . . . . . . . . . 10 (((𝜑𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏) ∈ 𝐾)
449448anasss 466 . . . . . . . . 9 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏) ∈ 𝐾)
45024adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → 𝐽 ∈ V)
4517adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → 𝑅 ∈ CRing)
45236adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → (𝐴𝐽) ∈ (𝐾m 𝐽))
453 simprr 773 . . . . . . . . . 10 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})
45413, 28, 47, 49, 450, 451, 452, 453evlsvvvallem 42547 . . . . . . . . 9 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → ((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑎𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))) ∈ 𝐾)
45528, 136, 444, 449, 454ringcld 20276 . . . . . . . 8 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → ((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑎𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))) ∈ 𝐾)
4566adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → (𝐼𝐽) ∈ V)
45735, 5elmapssresd 42260 . . . . . . . . . 10 (𝜑 → (𝐴 ↾ (𝐼𝐽)) ∈ (𝐾m (𝐼𝐽)))
458457adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → (𝐴 ↾ (𝐼𝐽)) ∈ (𝐾m (𝐼𝐽)))
459 simprl 771 . . . . . . . . 9 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → 𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin})
460137, 28, 47, 49, 456, 451, 458, 459evlsvvvallem 42547 . . . . . . . 8 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → ((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑏𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)))) ∈ 𝐾)
46128, 136, 444, 455, 460ringcld 20276 . . . . . . 7 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑎𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑏𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))) ∈ 𝐾)
462461ralrimivva 3199 . . . . . 6 (𝜑 → ∀𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}∀𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑎𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑏𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))) ∈ 𝐾)
463267fmpo 8091 . . . . . 6 (∀𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}∀𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑎𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑏𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))) ∈ 𝐾 ↔ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑎𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑏𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)))))):({𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} × {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})⟶𝐾)
464462, 463sylib 218 . . . . 5 (𝜑 → (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑎𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑏𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)))))):({𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} × {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})⟶𝐾)
465 f1of1 6847 . . . . . . . 8 ((𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (𝑏𝑎)):({𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} × {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})–1-1-onto→{ ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} → (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (𝑏𝑎)):({𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} × {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})–1-1→{ ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin})
466368, 465syl 17 . . . . . . 7 (𝜑 → (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (𝑏𝑎)):({𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} × {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})–1-1→{ ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin})
467278mptex 7242 . . . . . . . 8 (𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑𝐽))‘(𝑑 ↾ (𝐼𝐽)))(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)))))) ∈ V
468467a1i 11 . . . . . . 7 (𝜑 → (𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑𝐽))‘(𝑑 ↾ (𝐼𝐽)))(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)))))) ∈ V)
469366, 466, 363, 468fsuppco 9439 . . . . . 6 (𝜑 → ((𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑𝐽))‘(𝑑 ↾ (𝐼𝐽)))(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)))))) ∘ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (𝑏𝑎))) finSupp (0g𝑅))
470438, 469eqbrtrrd 5171 . . . . 5 (𝜑 → (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑎𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑏𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)))))) finSupp (0g𝑅))
47128, 197, 276, 442, 443, 464, 470gsumxp 20008 . . . 4 (𝜑 → (𝑅 Σg (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑎𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑏𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))))) = (𝑅 Σg (𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ↦ (𝑅 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (𝑐(𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑎𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑏𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))))𝑒))))))
472369, 439, 4713eqtrd 2778 . . 3 (𝜑 → (𝑅 Σg (𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑𝐽))‘(𝑑 ↾ (𝐼𝐽)))(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))))) = (𝑅 Σg (𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ↦ (𝑅 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (𝑐(𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑎𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑏𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))))𝑒))))))
47328, 136, 280, 291, 322, 346ringassd 20274 . . . . . 6 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑𝐽))‘(𝑑 ↾ (𝐼𝐽)))(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))) = ((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑𝐽))‘(𝑑 ↾ (𝐼𝐽)))(.r𝑅)(((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)))))))
47447, 136mgpplusg 20155 . . . . . . . . 9 (.r𝑅) = (+g‘(mulGrp‘𝑅))
47551ad2antrr 726 . . . . . . . . . . 11 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑖𝐼) → (mulGrp‘𝑅) ∈ Mnd)
476296ffvelcdmda 7103 . . . . . . . . . . 11 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑖𝐼) → (𝑑𝑖) ∈ ℕ0)
47757adantr 480 . . . . . . . . . . . 12 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → 𝐴:𝐼𝐾)
478477ffvelcdmda 7103 . . . . . . . . . . 11 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑖𝐼) → (𝐴𝑖) ∈ 𝐾)
47948, 49, 475, 476, 478mulgnn0cld 19125 . . . . . . . . . 10 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑖𝐼) → ((𝑑𝑖)(.g‘(mulGrp‘𝑅))(𝐴𝑖)) ∈ 𝐾)
480479fmpttd 7134 . . . . . . . . 9 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝑖𝐼 ↦ ((𝑑𝑖)(.g‘(mulGrp‘𝑅))(𝐴𝑖))):𝐼𝐾)
481296feqmptd 6976 . . . . . . . . . . 11 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → 𝑑 = (𝑖𝐼 ↦ (𝑑𝑖)))
482481, 309eqbrtrrd 5171 . . . . . . . . . 10 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝑖𝐼 ↦ (𝑑𝑖)) finSupp 0)
483111adantl 481 . . . . . . . . . 10 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑘𝐾) → (0(.g‘(mulGrp‘𝑅))𝑘) = (0g‘(mulGrp‘𝑅)))
484482, 483, 476, 478, 305fsuppssov1 9421 . . . . . . . . 9 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝑖𝐼 ↦ ((𝑑𝑖)(.g‘(mulGrp‘𝑅))(𝐴𝑖))) finSupp (0g‘(mulGrp‘𝑅)))
485 disjdif 4477 . . . . . . . . . 10 (𝐽 ∩ (𝐼𝐽)) = ∅
486485a1i 11 . . . . . . . . 9 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝐽 ∩ (𝐼𝐽)) = ∅)
487 undif 4487 . . . . . . . . . . . 12 (𝐽𝐼 ↔ (𝐽 ∪ (𝐼𝐽)) = 𝐼)
48816, 487sylib 218 . . . . . . . . . . 11 (𝜑 → (𝐽 ∪ (𝐼𝐽)) = 𝐼)
489488eqcomd 2740 . . . . . . . . . 10 (𝜑𝐼 = (𝐽 ∪ (𝐼𝐽)))
490489adantr 480 . . . . . . . . 9 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → 𝐼 = (𝐽 ∪ (𝐼𝐽)))
49148, 110, 474, 292, 283, 480, 484, 486, 490gsumsplit 19960 . . . . . . . 8 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → ((mulGrp‘𝑅) Σg (𝑖𝐼 ↦ ((𝑑𝑖)(.g‘(mulGrp‘𝑅))(𝐴𝑖)))) = (((mulGrp‘𝑅) Σg ((𝑖𝐼 ↦ ((𝑑𝑖)(.g‘(mulGrp‘𝑅))(𝐴𝑖))) ↾ 𝐽))(.r𝑅)((mulGrp‘𝑅) Σg ((𝑖𝐼 ↦ ((𝑑𝑖)(.g‘(mulGrp‘𝑅))(𝐴𝑖))) ↾ (𝐼𝐽)))))
492284resmptd 6059 . . . . . . . . . . 11 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → ((𝑖𝐼 ↦ ((𝑑𝑖)(.g‘(mulGrp‘𝑅))(𝐴𝑖))) ↾ 𝐽) = (𝑖𝐽 ↦ ((𝑑𝑖)(.g‘(mulGrp‘𝑅))(𝐴𝑖))))
493 fveq2 6906 . . . . . . . . . . . . . 14 (𝑖 = 𝑗 → (𝑑𝑖) = (𝑑𝑗))
494 fveq2 6906 . . . . . . . . . . . . . 14 (𝑖 = 𝑗 → (𝐴𝑖) = (𝐴𝑗))
495493, 494oveq12d 7448 . . . . . . . . . . . . 13 (𝑖 = 𝑗 → ((𝑑𝑖)(.g‘(mulGrp‘𝑅))(𝐴𝑖)) = ((𝑑𝑗)(.g‘(mulGrp‘𝑅))(𝐴𝑗)))
496495cbvmptv 5260 . . . . . . . . . . . 12 (𝑖𝐽 ↦ ((𝑑𝑖)(.g‘(mulGrp‘𝑅))(𝐴𝑖))) = (𝑗𝐽 ↦ ((𝑑𝑗)(.g‘(mulGrp‘𝑅))(𝐴𝑗)))
497 simpr 484 . . . . . . . . . . . . . . . 16 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑗𝐽) → 𝑗𝐽)
498497fvresd 6926 . . . . . . . . . . . . . . 15 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑗𝐽) → ((𝑑𝐽)‘𝑗) = (𝑑𝑗))
499497fvresd 6926 . . . . . . . . . . . . . . 15 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑗𝐽) → ((𝐴𝐽)‘𝑗) = (𝐴𝑗))
500498, 499oveq12d 7448 . . . . . . . . . . . . . 14 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑗𝐽) → (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)) = ((𝑑𝑗)(.g‘(mulGrp‘𝑅))(𝐴𝑗)))
501500eqcomd 2740 . . . . . . . . . . . . 13 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑗𝐽) → ((𝑑𝑗)(.g‘(mulGrp‘𝑅))(𝐴𝑗)) = (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))
502501mpteq2dva 5247 . . . . . . . . . . . 12 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝑗𝐽 ↦ ((𝑑𝑗)(.g‘(mulGrp‘𝑅))(𝐴𝑗))) = (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))
503496, 502eqtrid 2786 . . . . . . . . . . 11 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝑖𝐽 ↦ ((𝑑𝑖)(.g‘(mulGrp‘𝑅))(𝐴𝑖))) = (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))
504492, 503eqtrd 2774 . . . . . . . . . 10 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → ((𝑖𝐼 ↦ ((𝑑𝑖)(.g‘(mulGrp‘𝑅))(𝐴𝑖))) ↾ 𝐽) = (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))
505504oveq2d 7446 . . . . . . . . 9 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → ((mulGrp‘𝑅) Σg ((𝑖𝐼 ↦ ((𝑑𝑖)(.g‘(mulGrp‘𝑅))(𝐴𝑖))) ↾ 𝐽)) = ((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))
506289resmptd 6059 . . . . . . . . . . 11 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → ((𝑖𝐼 ↦ ((𝑑𝑖)(.g‘(mulGrp‘𝑅))(𝐴𝑖))) ↾ (𝐼𝐽)) = (𝑖 ∈ (𝐼𝐽) ↦ ((𝑑𝑖)(.g‘(mulGrp‘𝑅))(𝐴𝑖))))
507 fveq2 6906 . . . . . . . . . . . . . 14 (𝑖 = 𝑘 → (𝑑𝑖) = (𝑑𝑘))
508 fveq2 6906 . . . . . . . . . . . . . 14 (𝑖 = 𝑘 → (𝐴𝑖) = (𝐴𝑘))
509507, 508oveq12d 7448 . . . . . . . . . . . . 13 (𝑖 = 𝑘 → ((𝑑𝑖)(.g‘(mulGrp‘𝑅))(𝐴𝑖)) = ((𝑑𝑘)(.g‘(mulGrp‘𝑅))(𝐴𝑘)))
510509cbvmptv 5260 . . . . . . . . . . . 12 (𝑖 ∈ (𝐼𝐽) ↦ ((𝑑𝑖)(.g‘(mulGrp‘𝑅))(𝐴𝑖))) = (𝑘 ∈ (𝐼𝐽) ↦ ((𝑑𝑘)(.g‘(mulGrp‘𝑅))(𝐴𝑘)))
511 simpr 484 . . . . . . . . . . . . . . . 16 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ (𝐼𝐽)) → 𝑘 ∈ (𝐼𝐽))
512511fvresd 6926 . . . . . . . . . . . . . . 15 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ (𝐼𝐽)) → ((𝑑 ↾ (𝐼𝐽))‘𝑘) = (𝑑𝑘))
513511fvresd 6926 . . . . . . . . . . . . . . 15 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ (𝐼𝐽)) → ((𝐴 ↾ (𝐼𝐽))‘𝑘) = (𝐴𝑘))
514512, 513oveq12d 7448 . . . . . . . . . . . . . 14 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ (𝐼𝐽)) → (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)) = ((𝑑𝑘)(.g‘(mulGrp‘𝑅))(𝐴𝑘)))
515514eqcomd 2740 . . . . . . . . . . . . 13 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ (𝐼𝐽)) → ((𝑑𝑘)(.g‘(mulGrp‘𝑅))(𝐴𝑘)) = (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)))
516515mpteq2dva 5247 . . . . . . . . . . . 12 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝑘 ∈ (𝐼𝐽) ↦ ((𝑑𝑘)(.g‘(mulGrp‘𝑅))(𝐴𝑘))) = (𝑘 ∈ (𝐼𝐽) ↦ (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))
517510, 516eqtrid 2786 . . . . . . . . . . 11 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝑖 ∈ (𝐼𝐽) ↦ ((𝑑𝑖)(.g‘(mulGrp‘𝑅))(𝐴𝑖))) = (𝑘 ∈ (𝐼𝐽) ↦ (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))
518506, 517eqtrd 2774 . . . . . . . . . 10 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → ((𝑖𝐼 ↦ ((𝑑𝑖)(.g‘(mulGrp‘𝑅))(𝐴𝑖))) ↾ (𝐼𝐽)) = (𝑘 ∈ (𝐼𝐽) ↦ (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))
519518oveq2d 7446 . . . . . . . . 9 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → ((mulGrp‘𝑅) Σg ((𝑖𝐼 ↦ ((𝑑𝑖)(.g‘(mulGrp‘𝑅))(𝐴𝑖))) ↾ (𝐼𝐽))) = ((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)))))
520505, 519oveq12d 7448 . . . . . . . 8 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (((mulGrp‘𝑅) Σg ((𝑖𝐼 ↦ ((𝑑𝑖)(.g‘(mulGrp‘𝑅))(𝐴𝑖))) ↾ 𝐽))(.r𝑅)((mulGrp‘𝑅) Σg ((𝑖𝐼 ↦ ((𝑑𝑖)(.g‘(mulGrp‘𝑅))(𝐴𝑖))) ↾ (𝐼𝐽)))) = (((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))))
521491, 520eqtr2d 2775 . . . . . . 7 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))) = ((mulGrp‘𝑅) Σg (𝑖𝐼 ↦ ((𝑑𝑖)(.g‘(mulGrp‘𝑅))(𝐴𝑖)))))
522351, 521oveq12d 7448 . . . . . 6 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → ((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑𝐽))‘(𝑑 ↾ (𝐼𝐽)))(.r𝑅)(((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)))))) = ((𝐹𝑑)(.r𝑅)((mulGrp‘𝑅) Σg (𝑖𝐼 ↦ ((𝑑𝑖)(.g‘(mulGrp‘𝑅))(𝐴𝑖))))))
523473, 522eqtrd 2774 . . . . 5 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑𝐽))‘(𝑑 ↾ (𝐼𝐽)))(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))) = ((𝐹𝑑)(.r𝑅)((mulGrp‘𝑅) Σg (𝑖𝐼 ↦ ((𝑑𝑖)(.g‘(mulGrp‘𝑅))(𝐴𝑖))))))
524523mpteq2dva 5247 . . . 4 (𝜑 → (𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑𝐽))‘(𝑑 ↾ (𝐼𝐽)))(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)))))) = (𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ ((𝐹𝑑)(.r𝑅)((mulGrp‘𝑅) Σg (𝑖𝐼 ↦ ((𝑑𝑖)(.g‘(mulGrp‘𝑅))(𝐴𝑖)))))))
525524oveq2d 7446 . . 3 (𝜑 → (𝑅 Σg (𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑𝐽))‘(𝑑 ↾ (𝐼𝐽)))(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))))) = (𝑅 Σg (𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ ((𝐹𝑑)(.r𝑅)((mulGrp‘𝑅) Σg (𝑖𝐼 ↦ ((𝑑𝑖)(.g‘(mulGrp‘𝑅))(𝐴𝑖))))))))
526275, 472, 5253eqtr2d 2780 . 2 (𝜑 → (𝑅 Σg (𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ↦ (((((𝐽 eval 𝑈)‘(((𝐼 selectVars 𝑅)‘𝐽)‘𝐹))‘(𝐿 ∘ (𝐴𝐽)))‘𝑐)(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑐𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))))) = (𝑅 Σg (𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ ((𝐹𝑑)(.r𝑅)((mulGrp‘𝑅) Σg (𝑖𝐼 ↦ ((𝑑𝑖)(.g‘(mulGrp‘𝑅))(𝐴𝑖))))))))
527 eqid 2734 . . 3 ((𝐼𝐽) eval 𝑅) = ((𝐼𝐽) eval 𝑅)
528527, 3, 1, 137, 28, 47, 49, 136, 6, 7, 166, 457evlvvval 42559 . 2 (𝜑 → ((((𝐼𝐽) eval 𝑅)‘(((𝐽 eval 𝑈)‘(((𝐼 selectVars 𝑅)‘𝐽)‘𝐹))‘(𝐿 ∘ (𝐴𝐽))))‘(𝐴 ↾ (𝐼𝐽))) = (𝑅 Σg (𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ↦ (((((𝐽 eval 𝑈)‘(((𝐼 selectVars 𝑅)‘𝐽)‘𝐹))‘(𝐿 ∘ (𝐴𝐽)))‘𝑐)(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑐𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))))))
529 eqid 2734 . . 3 (𝐼 eval 𝑅) = (𝐼 eval 𝑅)
530529, 14, 15, 282, 28, 47, 49, 136, 4, 7, 17, 35evlvvval 42559 . 2 (𝜑 → (((𝐼 eval 𝑅)‘𝐹)‘𝐴) = (𝑅 Σg (𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ ((𝐹𝑑)(.r𝑅)((mulGrp‘𝑅) Σg (𝑖𝐼 ↦ ((𝑑𝑖)(.g‘(mulGrp‘𝑅))(𝐴𝑖))))))))
531526, 528, 5303eqtr4d 2784 1 (𝜑 → ((((𝐼𝐽) eval 𝑅)‘(((𝐽 eval 𝑈)‘(((𝐼 selectVars 𝑅)‘𝐽)‘𝐹))‘(𝐿 ∘ (𝐴𝐽))))‘(𝐴 ↾ (𝐼𝐽))) = (((𝐼 eval 𝑅)‘𝐹)‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1536  wcel 2105  wral 3058  {crab 3432  Vcvv 3477  csb 3907  cdif 3959  cun 3960  cin 3961  wss 3962  c0 4338  {csn 4630   class class class wbr 5147  cmpt 5230   × cxp 5686  ccnv 5687  cres 5690  cima 5691  ccom 5692  Fun wfun 6556   Fn wfn 6557  wf 6558  1-1wf1 6559  1-1-ontowf1o 6561  cfv 6562  (class class class)co 7430  cmpo 7432   supp csupp 8183  m cmap 8864  Fincfn 8983   finSupp cfsupp 9398  0cc0 11152  cn 12263  0cn0 12523  cz 12610  Basecbs 17244  .rcmulr 17298  Scalarcsca 17300   ·𝑠 cvsca 17301  0gc0g 17485   Σg cgsu 17486  Mndcmnd 18759   MndHom cmhm 18806  Grpcgrp 18963  .gcmg 19097   GrpHom cghm 19242  CMndccmn 19812  mulGrpcmgp 20151  1rcur 20198  Ringcrg 20250  CRingccrg 20251   RingHom crh 20485  AssAlgcasa 21887  algSccascl 21889   mPoly cmpl 21943   eval cevl 22114   selectVars cslv 22149
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-iin 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-of 7696  df-ofr 7697  df-om 7887  df-1st 8012  df-2nd 8013  df-supp 8184  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-2o 8505  df-er 8743  df-map 8866  df-pm 8867  df-ixp 8936  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-fsupp 9399  df-sup 9479  df-oi 9547  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-2 12326  df-3 12327  df-4 12328  df-5 12329  df-6 12330  df-7 12331  df-8 12332  df-9 12333  df-n0 12524  df-z 12611  df-dec 12731  df-uz 12876  df-fz 13544  df-fzo 13691  df-seq 14039  df-hash 14366  df-struct 17180  df-sets 17197  df-slot 17215  df-ndx 17227  df-base 17245  df-ress 17274  df-plusg 17310  df-mulr 17311  df-sca 17313  df-vsca 17314  df-ip 17315  df-tset 17316  df-ple 17317  df-ds 17319  df-hom 17321  df-cco 17322  df-0g 17487  df-gsum 17488  df-prds 17493  df-pws 17495  df-mre 17630  df-mrc 17631  df-acs 17633  df-mgm 18665  df-sgrp 18744  df-mnd 18760  df-mhm 18808  df-submnd 18809  df-grp 18966  df-minusg 18967  df-sbg 18968  df-mulg 19098  df-subg 19153  df-ghm 19243  df-cntz 19347  df-cmn 19814  df-abl 19815  df-mgp 20152  df-rng 20170  df-ur 20199  df-srg 20204  df-ring 20252  df-cring 20253  df-rhm 20488  df-subrng 20562  df-subrg 20586  df-lmod 20876  df-lss 20947  df-lsp 20987  df-assa 21890  df-asp 21891  df-ascl 21892  df-psr 21946  df-mvr 21947  df-mpl 21948  df-evls 22115  df-evl 22116  df-selv 22153
This theorem is referenced by: (None)
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