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Theorem evlselv 41625
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 2731 . . . . . . . . . . . . 13 (Base‘𝑈) = (Base‘𝑈)
2 eqid 2731 . . . . . . . . . . . . 13 (.r𝑈) = (.r𝑈)
3 evlselv.u . . . . . . . . . . . . . . . 16 𝑈 = ((𝐼𝐽) mPoly 𝑅)
4 evlselv.i . . . . . . . . . . . . . . . . 17 (𝜑𝐼𝑉)
5 difssd 4132 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐼𝐽) ⊆ 𝐼)
64, 5ssexd 5324 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐼𝐽) ∈ V)
7 evlselv.r . . . . . . . . . . . . . . . 16 (𝜑𝑅 ∈ CRing)
83, 6, 7mplcrngd 41584 . . . . . . . . . . . . . . 15 (𝜑𝑈 ∈ CRing)
98crngringd 20147 . . . . . . . . . . . . . 14 (𝜑𝑈 ∈ Ring)
109ad2antrr 723 . . . . . . . . . . . . 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 2731 . . . . . . . . . . . . . . . 16 (Base‘𝑇) = (Base‘𝑇)
13 eqid 2731 . . . . . . . . . . . . . . . 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, 4, 7, 16, 17selvcl 41621 . . . . . . . . . . . . . . . 16 (𝜑 → (((𝐼 selectVars 𝑅)‘𝐽)‘𝐹) ∈ (Base‘𝑇))
1911, 1, 12, 13, 18mplelf 21869 . . . . . . . . . . . . . . 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 7086 . . . . . . . . . . . . 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 2731 . . . . . . . . . . . . . 14 (mulGrp‘𝑈) = (mulGrp‘𝑈)
23 eqid 2731 . . . . . . . . . . . . . 14 (.g‘(mulGrp‘𝑈)) = (.g‘(mulGrp‘𝑈))
244, 16ssexd 5324 . . . . . . . . . . . . . . 15 (𝜑𝐽 ∈ V)
2524ad2antrr 723 . . . . . . . . . . . . . 14 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → 𝐽 ∈ V)
268ad2antrr 723 . . . . . . . . . . . . . 14 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → 𝑈 ∈ CRing)
27 fvexd 6906 . . . . . . . . . . . . . . . . 17 (𝜑 → (Base‘𝑈) ∈ V)
28 evlselv.k . . . . . . . . . . . . . . . . . . 19 𝐾 = (Base‘𝑅)
2928fvexi 6905 . . . . . . . . . . . . . . . . . 18 𝐾 ∈ V
3029a1i 11 . . . . . . . . . . . . . . . . 17 (𝜑𝐾 ∈ V)
31 evlselv.l . . . . . . . . . . . . . . . . . 18 𝐿 = (algSc‘𝑈)
327crngringd 20147 . . . . . . . . . . . . . . . . . 18 (𝜑𝑅 ∈ Ring)
333, 1, 28, 31, 6, 32mplasclf 21938 . . . . . . . . . . . . . . . . 17 (𝜑𝐿:𝐾⟶(Base‘𝑈))
3427, 30, 33elmapdd 8841 . . . . . . . . . . . . . . . 16 (𝜑𝐿 ∈ ((Base‘𝑈) ↑m 𝐾))
35 evlselv.a . . . . . . . . . . . . . . . . 17 (𝜑𝐴 ∈ (𝐾m 𝐼))
3635, 16elmapssresd 41536 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐴𝐽) ∈ (𝐾m 𝐽))
3734, 36mapcod 41537 . . . . . . . . . . . . . . 15 (𝜑 → (𝐿 ∘ (𝐴𝐽)) ∈ ((Base‘𝑈) ↑m 𝐽))
3837ad2antrr 723 . . . . . . . . . . . . . 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 41599 . . . . . . . . . . . . 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 20158 . . . . . . . . . . . 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 2732 . . . . . . . . . . . 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 2732 . . . . . . . . . . . 12 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑢 ∈ (Base‘𝑈) ↦ (𝑢𝑐)) = (𝑢 ∈ (Base‘𝑈) ↦ (𝑢𝑐)))
44 fveq1 6890 . . . . . . . . . . . 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 7129 . . . . . . . . . . 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 723 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → 𝐿:𝐾⟶(Base‘𝑈))
47 eqid 2731 . . . . . . . . . . . . . . . . . . . . . 22 (mulGrp‘𝑅) = (mulGrp‘𝑅)
4847, 28mgpbas 20041 . . . . . . . . . . . . . . . . . . . . 21 𝐾 = (Base‘(mulGrp‘𝑅))
49 eqid 2731 . . . . . . . . . . . . . . . . . . . . 21 (.g‘(mulGrp‘𝑅)) = (.g‘(mulGrp‘𝑅))
5047ringmgp 20140 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑅 ∈ Ring → (mulGrp‘𝑅) ∈ Mnd)
5132, 50syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (mulGrp‘𝑅) ∈ Mnd)
5251ad3antrrr 727 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) ∧ 𝑗𝐽) → (mulGrp‘𝑅) ∈ Mnd)
5313psrbagf 21782 . . . . . . . . . . . . . . . . . . . . . . 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 7086 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) ∧ 𝑗𝐽) → (𝑒𝑗) ∈ ℕ0)
56 elmapi 8849 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐴 ∈ (𝐾m 𝐼) → 𝐴:𝐼𝐾)
5735, 56syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑𝐴:𝐼𝐾)
5857, 16fssresd 6758 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (𝐴𝐽):𝐽𝐾)
5958ad2antrr 723 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → (𝐴𝐽):𝐽𝐾)
6059ffvelcdmda 7086 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) ∧ 𝑗𝐽) → ((𝐴𝐽)‘𝑗) ∈ 𝐾)
6148, 49, 52, 55, 60mulgnn0cld 19018 . . . . . . . . . . . . . . . . . . . 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 7132 . . . . . . . . . . . . . . . . . . 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 21893 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝐼𝐽) ∈ V ∧ 𝑅 ∈ CRing) → 𝑈 ∈ AssAlg)
646, 7, 63syl2anc 583 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑𝑈 ∈ AssAlg)
65 eqid 2731 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (Scalar‘𝑈) = (Scalar‘𝑈)
6631, 65asclrhm 21755 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑈 ∈ AssAlg → 𝐿 ∈ ((Scalar‘𝑈) RingHom 𝑈))
6764, 66syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑𝐿 ∈ ((Scalar‘𝑈) RingHom 𝑈))
683, 6, 7mplsca 21884 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑𝑅 = (Scalar‘𝑈))
6968eqcomd 2737 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → (Scalar‘𝑈) = 𝑅)
7069oveq1d 7427 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → ((Scalar‘𝑈) RingHom 𝑈) = (𝑅 RingHom 𝑈))
7167, 70eleqtrd 2834 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑𝐿 ∈ (𝑅 RingHom 𝑈))
7247, 22rhmmhm 20377 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐿 ∈ (𝑅 RingHom 𝑈) → 𝐿 ∈ ((mulGrp‘𝑅) MndHom (mulGrp‘𝑈)))
7371, 72syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝐿 ∈ ((mulGrp‘𝑅) MndHom (mulGrp‘𝑈)))
7473ad3antrrr 727 . . . . . . . . . . . . . . . . . . . . . 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 19038 . . . . . . . . . . . . . . . . . . . . . 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 727 . . . . . . . . . . . . . . . . . . . . . . 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 6991 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) ∧ 𝑗𝐽) → ((𝐿 ∘ (𝐴𝐽))‘𝑗) = (𝐿‘((𝐴𝐽)‘𝑗)))
8079oveq2d 7428 . . . . . . . . . . . . . . . . . . . . 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 2774 . . . . . . . . . . . . . . . . . . . 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 5248 . . . . . . . . . . . . . . . . . . 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 2771 . . . . . . . . . . . . . . . . . 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 7428 . . . . . . . . . . . . . . . . 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 2731 . . . . . . . . . . . . . . . . . 18 (Base‘(mulGrp‘(Scalar‘𝑈))) = (Base‘(mulGrp‘(Scalar‘𝑈)))
86 eqid 2731 . . . . . . . . . . . . . . . . . 18 (0g‘(mulGrp‘(Scalar‘𝑈))) = (0g‘(mulGrp‘(Scalar‘𝑈)))
8768, 7eqeltrrd 2833 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (Scalar‘𝑈) ∈ CRing)
88 eqid 2731 . . . . . . . . . . . . . . . . . . . . 21 (mulGrp‘(Scalar‘𝑈)) = (mulGrp‘(Scalar‘𝑈))
8988crngmgp 20142 . . . . . . . . . . . . . . . . . . . 20 ((Scalar‘𝑈) ∈ CRing → (mulGrp‘(Scalar‘𝑈)) ∈ CMnd)
9087, 89syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (mulGrp‘(Scalar‘𝑈)) ∈ CMnd)
9190ad2antrr 723 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → (mulGrp‘(Scalar‘𝑈)) ∈ CMnd)
9222ringmgp 20140 . . . . . . . . . . . . . . . . . . . 20 (𝑈 ∈ Ring → (mulGrp‘𝑈) ∈ Mnd)
939, 92syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (mulGrp‘𝑈) ∈ Mnd)
9493ad2antrr 723 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → (mulGrp‘𝑈) ∈ Mnd)
9588, 22rhmmhm 20377 . . . . . . . . . . . . . . . . . . . 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 723 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → 𝐿 ∈ ((mulGrp‘(Scalar‘𝑈)) MndHom (mulGrp‘𝑈)))
9868fveq2d 6895 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (Base‘𝑅) = (Base‘(Scalar‘𝑈)))
9928, 98eqtrid 2783 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑𝐾 = (Base‘(Scalar‘𝑈)))
10099ad3antrrr 727 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) ∧ 𝑗𝐽) → 𝐾 = (Base‘(Scalar‘𝑈)))
10161, 100eleqtrd 2834 . . . . . . . . . . . . . . . . . . . 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 2731 . . . . . . . . . . . . . . . . . . . . 21 (Base‘(Scalar‘𝑈)) = (Base‘(Scalar‘𝑈))
10388, 102mgpbas 20041 . . . . . . . . . . . . . . . . . . . 20 (Base‘(Scalar‘𝑈)) = (Base‘(mulGrp‘(Scalar‘𝑈)))
104101, 103eleqtrdi 2842 . . . . . . . . . . . . . . . . . . 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 7116 . . . . . . . . . . . . . . . . . 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))))
10624mptexd 7228 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))) ∈ V)
107106ad2antrr 723 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))) ∈ V)
108 fvexd 6906 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → (0g‘(mulGrp‘𝑅)) ∈ V)
109 funmpt 6586 . . . . . . . . . . . . . . . . . . . . . 22 Fun (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))
110109a1i 11 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → Fun (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))
11154feqmptd 6960 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → 𝑒 = (𝑗𝐽 ↦ (𝑒𝑗)))
11213psrbagfsupp 21784 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} → 𝑒 finSupp 0)
113112adantl 481 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → 𝑒 finSupp 0)
114111, 113eqbrtrrd 5172 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → (𝑗𝐽 ↦ (𝑒𝑗)) finSupp 0)
115 ssidd 4005 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → ((𝑗𝐽 ↦ (𝑒𝑗)) supp 0) ⊆ ((𝑗𝐽 ↦ (𝑒𝑗)) supp 0))
116 eqid 2731 . . . . . . . . . . . . . . . . . . . . . . . 24 (0g‘(mulGrp‘𝑅)) = (0g‘(mulGrp‘𝑅))
11748, 116, 49mulg0 19000 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘𝐾 → (0(.g‘(mulGrp‘𝑅))𝑘) = (0g‘(mulGrp‘𝑅)))
118117adantl 481 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) ∧ 𝑘𝐾) → (0(.g‘(mulGrp‘𝑅))𝑘) = (0g‘(mulGrp‘𝑅)))
119 0zd 12577 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → 0 ∈ ℤ)
120115, 118, 55, 60, 119suppssov1 8188 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → ((𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))) supp (0g‘(mulGrp‘𝑅))) ⊆ ((𝑗𝐽 ↦ (𝑒𝑗)) supp 0))
121107, 108, 110, 114, 120fsuppsssuppgd 41534 . . . . . . . . . . . . . . . . . . . 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)))
122 eqid 2731 . . . . . . . . . . . . . . . . . . . . 21 (1r𝑅) = (1r𝑅)
12347, 122ringidval 20084 . . . . . . . . . . . . . . . . . . . 20 (1r𝑅) = (0g‘(mulGrp‘𝑅))
124121, 123breqtrrdi 5190 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))) finSupp (1r𝑅))
12568fveq2d 6895 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (1r𝑅) = (1r‘(Scalar‘𝑈)))
126 eqid 2731 . . . . . . . . . . . . . . . . . . . . . 22 (1r‘(Scalar‘𝑈)) = (1r‘(Scalar‘𝑈))
12788, 126ringidval 20084 . . . . . . . . . . . . . . . . . . . . 21 (1r‘(Scalar‘𝑈)) = (0g‘(mulGrp‘(Scalar‘𝑈)))
128125, 127eqtrdi 2787 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (1r𝑅) = (0g‘(mulGrp‘(Scalar‘𝑈))))
129128ad2antrr 723 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → (1r𝑅) = (0g‘(mulGrp‘(Scalar‘𝑈))))
130124, 129breqtrd 5174 . . . . . . . . . . . . . . . . . 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))))
13185, 86, 91, 94, 25, 97, 105, 130gsummhm 19854 . . . . . . . . . . . . . . . . 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))))))
13284, 131eqtr3d 2773 . . . . . . . . . . . . . . . 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))))))
133132oveq2d 7428 . . . . . . . . . . . . . . 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)))))))
13464ad2antrr 723 . . . . . . . . . . . . . . . 16 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → 𝑈 ∈ AssAlg)
135101fmpttd 7116 . . . . . . . . . . . . . . . . 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)))
136130, 127breqtrrdi 5190 . . . . . . . . . . . . . . . . 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)))
137103, 127, 91, 25, 135, 136gsumcl 19831 . . . . . . . . . . . . . . . 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)))
138 eqid 2731 . . . . . . . . . . . . . . . . 17 ( ·𝑠𝑈) = ( ·𝑠𝑈)
13931, 65, 102, 1, 2, 138asclmul2 21752 . . . . . . . . . . . . . . . 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)))
140134, 137, 21, 139syl3anc 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)))
141133, 140eqtrd 2771 . . . . . . . . . . . . . 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)))
142141fveq1d 6893 . . . . . . . . . . . . 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))
143 eqid 2731 . . . . . . . . . . . . . 14 (.r𝑅) = (.r𝑅)
144 eqid 2731 . . . . . . . . . . . . . 14 {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}
14599ad2antrr 723 . . . . . . . . . . . . . . 15 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → 𝐾 = (Base‘(Scalar‘𝑈)))
146137, 145eleqtrrd 2835 . . . . . . . . . . . . . 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)
147 simplr 766 . . . . . . . . . . . . . 14 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → 𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin})
1483, 138, 28, 1, 143, 144, 146, 21, 147mplvscaval 21887 . . . . . . . . . . . . 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)))
149142, 148eqtrd 2771 . . . . . . . . . . . 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)))
150149mpteq2dva 5248 . . . . . . . . . . 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))))
15145, 150eqtrd 2771 . . . . . . . . . 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))))
152151oveq2d 7428 . . . . . . . . 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)))))
15369fveq2d 6895 . . . . . . . . . . . . . . 15 (𝜑 → (mulGrp‘(Scalar‘𝑈)) = (mulGrp‘𝑅))
154153ad2antrr 723 . . . . . . . . . . . . . 14 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → (mulGrp‘(Scalar‘𝑈)) = (mulGrp‘𝑅))
155154oveq1d 7427 . . . . . . . . . . . . 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)))))
156155oveq1d 7427 . . . . . . . . . . . 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)))
1577ad2antrr 723 . . . . . . . . . . . . 13 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → 𝑅 ∈ CRing)
158155, 146eqeltrrd 2833 . . . . . . . . . . . . 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)
15919ffvelcdmda 7086 . . . . . . . . . . . . . . . 16 ((𝜑𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒) ∈ (Base‘𝑈))
1603, 28, 1, 144, 159mplelf 21869 . . . . . . . . . . . . . . 15 ((𝜑𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒):{𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}⟶𝐾)
161160ffvelcdmda 7086 . . . . . . . . . . . . . 14 (((𝜑𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐) ∈ 𝐾)
162161an32s 649 . . . . . . . . . . . . 13 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐) ∈ 𝐾)
16328, 143, 157, 158, 162crngcomd 41555 . . . . . . . . . . . 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))))))
164156, 163eqtrd 2771 . . . . . . . . . . 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))))))
165164mpteq2dva 5248 . . . . . . . . . 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)))))))
166165oveq2d 7428 . . . . . . . . 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))))))))
167152, 166eqtrd 2771 . . . . . . . 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))))))))
168167oveq1d 7427 . . . . . . 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))))))
169 eqid 2731 . . . . . . . . . 10 (𝑢 ∈ (Base‘𝑈) ↦ (𝑢𝑐)) = (𝑢 ∈ (Base‘𝑈) ↦ (𝑢𝑐))
170 fveq1 6890 . . . . . . . . . 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))
171 eqid 2731 . . . . . . . . . . . . 13 (𝐽 eval 𝑈) = (𝐽 eval 𝑈)
172171, 11, 12, 13, 1, 22, 23, 2, 24, 8, 18, 37evlvvval 41611 . . . . . . . . . . . 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))))))))
173171, 11, 12, 1, 24, 8, 18, 37evlcl 41610 . . . . . . . . . . . 12 (𝜑 → (((𝐽 eval 𝑈)‘(((𝐼 selectVars 𝑅)‘𝐽)‘𝐹))‘(𝐿 ∘ (𝐴𝐽))) ∈ (Base‘𝑈))
174172, 173eqeltrrd 2833 . . . . . . . . . . 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))
175174adantr 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))
176 fvexd 6906 . . . . . . . . . 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)
177169, 170, 175, 176fvmptd3 7021 . . . . . . . . 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))
178 eqid 2731 . . . . . . . . . 10 (0g𝑈) = (0g𝑈)
1799ringcmnd 20179 . . . . . . . . . . 11 (𝜑𝑈 ∈ CMnd)
180179adantr 480 . . . . . . . . . 10 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → 𝑈 ∈ CMnd)
1817crnggrpd 20148 . . . . . . . . . . . 12 (𝜑𝑅 ∈ Grp)
182181grpmndd 18874 . . . . . . . . . . 11 (𝜑𝑅 ∈ Mnd)
183182adantr 480 . . . . . . . . . 10 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → 𝑅 ∈ Mnd)
184 ovex 7445 . . . . . . . . . . . 12 (ℕ0m 𝐽) ∈ V
185184rabex 5332 . . . . . . . . . . 11 {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ∈ V
186185a1i 11 . . . . . . . . . 10 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ∈ V)
1876adantr 480 . . . . . . . . . . . 12 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝐼𝐽) ∈ V)
188181adantr 480 . . . . . . . . . . . 12 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → 𝑅 ∈ Grp)
189 simpr 484 . . . . . . . . . . . 12 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → 𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin})
1903, 1, 144, 169, 187, 188, 189mplmapghm 41594 . . . . . . . . . . 11 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑢 ∈ (Base‘𝑈) ↦ (𝑢𝑐)) ∈ (𝑈 GrpHom 𝑅))
191 ghmmhm 19147 . . . . . . . . . . 11 ((𝑢 ∈ (Base‘𝑈) ↦ (𝑢𝑐)) ∈ (𝑈 GrpHom 𝑅) → (𝑢 ∈ (Base‘𝑈) ↦ (𝑢𝑐)) ∈ (𝑈 MndHom 𝑅))
192190, 191syl 17 . . . . . . . . . 10 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑢 ∈ (Base‘𝑈) ↦ (𝑢𝑐)) ∈ (𝑈 MndHom 𝑅))
19341fmpttd 7116 . . . . . . . . . 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))
19424adantr 480 . . . . . . . . . . 11 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → 𝐽 ∈ V)
1958adantr 480 . . . . . . . . . . 11 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → 𝑈 ∈ CRing)
19618adantr 480 . . . . . . . . . . 11 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (((𝐼 selectVars 𝑅)‘𝐽)‘𝐹) ∈ (Base‘𝑇))
19737adantr 480 . . . . . . . . . . 11 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝐿 ∘ (𝐴𝐽)) ∈ ((Base‘𝑈) ↑m 𝐽))
19813, 11, 12, 1, 22, 23, 2, 194, 195, 196, 197evlvvvallem 41612 . . . . . . . . . 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))
1991, 178, 180, 183, 186, 192, 193, 198gsummhm 19854 . . . . . . . . 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)))))))))
200172adantr 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))))))))
201200fveq1d 6893 . . . . . . . . 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))
202177, 199, 2013eqtr4rd 2782 . . . . . . . 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)))))))))
203202oveq1d 7427 . . . . . . 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))))))
204 eqid 2731 . . . . . . . 8 (0g𝑅) = (0g𝑅)
20532adantr 480 . . . . . . . 8 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → 𝑅 ∈ Ring)
20647crngmgp 20142 . . . . . . . . . . 11 (𝑅 ∈ CRing → (mulGrp‘𝑅) ∈ CMnd)
2077, 206syl 17 . . . . . . . . . 10 (𝜑 → (mulGrp‘𝑅) ∈ CMnd)
208207adantr 480 . . . . . . . . 9 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (mulGrp‘𝑅) ∈ CMnd)
20951ad2antrr 723 . . . . . . . . . . 11 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ (𝐼𝐽)) → (mulGrp‘𝑅) ∈ Mnd)
210144psrbagf 21782 . . . . . . . . . . . . 13 (𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} → 𝑐:(𝐼𝐽)⟶ℕ0)
211210adantl 481 . . . . . . . . . . . 12 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → 𝑐:(𝐼𝐽)⟶ℕ0)
212211ffvelcdmda 7086 . . . . . . . . . . 11 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ (𝐼𝐽)) → (𝑐𝑘) ∈ ℕ0)
21357, 5fssresd 6758 . . . . . . . . . . . . 13 (𝜑 → (𝐴 ↾ (𝐼𝐽)):(𝐼𝐽)⟶𝐾)
214213adantr 480 . . . . . . . . . . . 12 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝐴 ↾ (𝐼𝐽)):(𝐼𝐽)⟶𝐾)
215214ffvelcdmda 7086 . . . . . . . . . . 11 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ (𝐼𝐽)) → ((𝐴 ↾ (𝐼𝐽))‘𝑘) ∈ 𝐾)
21648, 49, 209, 212, 215mulgnn0cld 19018 . . . . . . . . . 10 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ (𝐼𝐽)) → ((𝑐𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)) ∈ 𝐾)
217216fmpttd 7116 . . . . . . . . 9 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑘 ∈ (𝐼𝐽) ↦ ((𝑐𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))):(𝐼𝐽)⟶𝐾)
2186mptexd 7228 . . . . . . . . . . 11 (𝜑 → (𝑘 ∈ (𝐼𝐽) ↦ ((𝑐𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))) ∈ V)
219218adantr 480 . . . . . . . . . 10 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑘 ∈ (𝐼𝐽) ↦ ((𝑐𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))) ∈ V)
220 fvexd 6906 . . . . . . . . . 10 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (0g‘(mulGrp‘𝑅)) ∈ V)
221 funmpt 6586 . . . . . . . . . . 11 Fun (𝑘 ∈ (𝐼𝐽) ↦ ((𝑐𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)))
222221a1i 11 . . . . . . . . . 10 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → Fun (𝑘 ∈ (𝐼𝐽) ↦ ((𝑐𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))
223211feqmptd 6960 . . . . . . . . . . 11 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → 𝑐 = (𝑘 ∈ (𝐼𝐽) ↦ (𝑐𝑘)))
224144psrbagfsupp 21784 . . . . . . . . . . . 12 (𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} → 𝑐 finSupp 0)
225224adantl 481 . . . . . . . . . . 11 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → 𝑐 finSupp 0)
226223, 225eqbrtrrd 5172 . . . . . . . . . 10 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑘 ∈ (𝐼𝐽) ↦ (𝑐𝑘)) finSupp 0)
227 ssidd 4005 . . . . . . . . . . 11 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → ((𝑘 ∈ (𝐼𝐽) ↦ (𝑐𝑘)) supp 0) ⊆ ((𝑘 ∈ (𝐼𝐽) ↦ (𝑐𝑘)) supp 0))
22848, 116, 49mulg0 19000 . . . . . . . . . . . 12 (𝑣𝐾 → (0(.g‘(mulGrp‘𝑅))𝑣) = (0g‘(mulGrp‘𝑅)))
229228adantl 481 . . . . . . . . . . 11 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑣𝐾) → (0(.g‘(mulGrp‘𝑅))𝑣) = (0g‘(mulGrp‘𝑅)))
230 fvexd 6906 . . . . . . . . . . 11 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ (𝐼𝐽)) → (𝑐𝑘) ∈ V)
231 0zd 12577 . . . . . . . . . . 11 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → 0 ∈ ℤ)
232227, 229, 230, 215, 231suppssov1 8188 . . . . . . . . . 10 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → ((𝑘 ∈ (𝐼𝐽) ↦ ((𝑐𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))) supp (0g‘(mulGrp‘𝑅))) ⊆ ((𝑘 ∈ (𝐼𝐽) ↦ (𝑐𝑘)) supp 0))
233219, 220, 222, 226, 232fsuppsssuppgd 41534 . . . . . . . . 9 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑘 ∈ (𝐼𝐽) ↦ ((𝑐𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))) finSupp (0g‘(mulGrp‘𝑅)))
23448, 116, 208, 187, 217, 233gsumcl 19831 . . . . . . . 8 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → ((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑐𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)))) ∈ 𝐾)
23532ad2antrr 723 . . . . . . . . 9 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → 𝑅 ∈ Ring)
23628, 143, 235, 162, 158ringcld 20158 . . . . . . . 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)
237185mptex 7227 . . . . . . . . . 10 (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)))))) ∈ V
238237a1i 11 . . . . . . . . 9 ((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)))))) ∈ V)
239 fvexd 6906 . . . . . . . . 9 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (0g𝑅) ∈ V)
240 funmpt 6586 . . . . . . . . . 10 Fun (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))))))
241240a1i 11 . . . . . . . . 9 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → Fun (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)))))))
242185mptex 7227 . . . . . . . . . . 11 (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)) ∈ V
243242a1i 11 . . . . . . . . . 10 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)) ∈ V)
244 funmpt 6586 . . . . . . . . . . 11 Fun (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐))
245244a1i 11 . . . . . . . . . 10 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → Fun (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)))
24611, 12, 178, 18, 8mplelsfi 21866 . . . . . . . . . . 11 (𝜑 → (((𝐼 selectVars 𝑅)‘𝐽)‘𝐹) finSupp (0g𝑈))
247246adantr 480 . . . . . . . . . 10 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (((𝐼 selectVars 𝑅)‘𝐽)‘𝐹) finSupp (0g𝑈))
248 ssidd 4005 . . . . . . . . . . . . . 14 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹) supp (0g𝑈)) ⊆ ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹) supp (0g𝑈)))
249 fvexd 6906 . . . . . . . . . . . . . 14 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (0g𝑈) ∈ V)
25020, 248, 186, 249suppssr 8186 . . . . . . . . . . . . 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))
251250fveq1d 6893 . . . . . . . . . . . 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))
2523, 144, 204, 178, 6, 181mpl0 21877 . . . . . . . . . . . . . . . 16 (𝜑 → (0g𝑈) = ({𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} × {(0g𝑅)}))
253252adantr 480 . . . . . . . . . . . . . . 15 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (0g𝑈) = ({𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} × {(0g𝑅)}))
254253fveq1d 6893 . . . . . . . . . . . . . 14 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → ((0g𝑈)‘𝑐) = (({𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} × {(0g𝑅)})‘𝑐))
255 fvex 6904 . . . . . . . . . . . . . . . 16 (0g𝑅) ∈ V
256255fvconst2 7207 . . . . . . . . . . . . . . 15 (𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} → (({𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} × {(0g𝑅)})‘𝑐) = (0g𝑅))
257256adantl 481 . . . . . . . . . . . . . 14 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (({𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} × {(0g𝑅)})‘𝑐) = (0g𝑅))
258254, 257eqtrd 2771 . . . . . . . . . . . . 13 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → ((0g𝑈)‘𝑐) = (0g𝑅))
259258adantr 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))
260251, 259eqtrd 2771 . . . . . . . . . . 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))
261260, 186suppss2 8191 . . . . . . . . . 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)))
262243, 239, 245, 247, 261fsuppsssuppgd 41534 . . . . . . . . 9 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)) finSupp (0g𝑅))
263 ssidd 4005 . . . . . . . . . 10 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → ((𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)) supp (0g𝑅)) ⊆ ((𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)) supp (0g𝑅)))
26432ad2antrr 723 . . . . . . . . . . 11 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑣𝐾) → 𝑅 ∈ Ring)
265 simpr 484 . . . . . . . . . . 11 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑣𝐾) → 𝑣𝐾)
26628, 143, 204, 264, 265ringlzd 20190 . . . . . . . . . 10 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑣𝐾) → ((0g𝑅)(.r𝑅)𝑣) = (0g𝑅))
267263, 266, 162, 158, 239suppssov1 8188 . . . . . . . . 9 ((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)))))) supp (0g𝑅)) ⊆ ((𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)) supp (0g𝑅)))
268238, 239, 241, 262, 267fsuppsssuppgd 41534 . . . . . . . 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))
26928, 204, 143, 205, 186, 234, 236, 268gsummulc1 20211 . . . . . . 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))))))
270168, 203, 2693eqtr4d 2781 . . . . . 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))))))))
271 fveq2 6891 . . . . . . . . . . . . . 14 (𝑎 = 𝑒 → ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎) = ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒))
272271adantl 481 . . . . . . . . . . . . 13 ((𝑏 = 𝑐𝑎 = 𝑒) → ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎) = ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒))
273 simpl 482 . . . . . . . . . . . . 13 ((𝑏 = 𝑐𝑎 = 𝑒) → 𝑏 = 𝑐)
274272, 273fveq12d 6898 . . . . . . . . . . . 12 ((𝑏 = 𝑐𝑎 = 𝑒) → (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏) = (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐))
275 fveq1 6890 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑒 → (𝑎𝑗) = (𝑒𝑗))
276275adantl 481 . . . . . . . . . . . . . . 15 ((𝑏 = 𝑐𝑎 = 𝑒) → (𝑎𝑗) = (𝑒𝑗))
277276oveq1d 7427 . . . . . . . . . . . . . 14 ((𝑏 = 𝑐𝑎 = 𝑒) → ((𝑎𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)) = ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))
278277mpteq2dv 5250 . . . . . . . . . . . . 13 ((𝑏 = 𝑐𝑎 = 𝑒) → (𝑗𝐽 ↦ ((𝑎𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))) = (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))
279278oveq2d 7428 . . . . . . . . . . . 12 ((𝑏 = 𝑐𝑎 = 𝑒) → ((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑎𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))) = ((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))
280274, 279oveq12d 7430 . . . . . . . . . . 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))))))
281 fveq1 6890 . . . . . . . . . . . . . . 15 (𝑏 = 𝑐 → (𝑏𝑘) = (𝑐𝑘))
282281adantr 480 . . . . . . . . . . . . . 14 ((𝑏 = 𝑐𝑎 = 𝑒) → (𝑏𝑘) = (𝑐𝑘))
283282oveq1d 7427 . . . . . . . . . . . . 13 ((𝑏 = 𝑐𝑎 = 𝑒) → ((𝑏𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)) = ((𝑐𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)))
284283mpteq2dv 5250 . . . . . . . . . . . 12 ((𝑏 = 𝑐𝑎 = 𝑒) → (𝑘 ∈ (𝐼𝐽) ↦ ((𝑏𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))) = (𝑘 ∈ (𝐼𝐽) ↦ ((𝑐𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))
285284oveq2d 7428 . . . . . . . . . . 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)))))
286280, 285oveq12d 7430 . . . . . . . . . 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))))))
287 eqid 2731 . . . . . . . . . 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))))))
288 ovex 7445 . . . . . . . . . 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
289286, 287, 288ovmpoa 7566 . . . . . . . . 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))))))
290289adantll 711 . . . . . . . 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))))))
291290mpteq2dva 5248 . . . . . . 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)))))))
292291oveq2d 7428 . . . . . 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))))))))
293270, 292eqtr4d 2774 . . . . 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))))
294293mpteq2dva 5248 . . . 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)))))
295294oveq2d 7428 . . 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))))))
29632ringcmnd 20179 . . . . 5 (𝜑𝑅 ∈ CMnd)
297 ovex 7445 . . . . . . 7 (ℕ0m 𝐼) ∈ V
298297rabex 5332 . . . . . 6 { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∈ V
299298a1i 11 . . . . 5 (𝜑 → { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∈ V)
30032adantr 480 . . . . . . 7 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → 𝑅 ∈ Ring)
30119adantr 480 . . . . . . . . . . 11 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (((𝐼 selectVars 𝑅)‘𝐽)‘𝐹):{𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}⟶(Base‘𝑈))
302 eqid 2731 . . . . . . . . . . . 12 { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} = { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}
3034adantr 480 . . . . . . . . . . . 12 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → 𝐼𝑉)
30416adantr 480 . . . . . . . . . . . 12 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → 𝐽𝐼)
305 simpr 484 . . . . . . . . . . . 12 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → 𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin})
306302, 13, 303, 304, 305psrbagres 41581 . . . . . . . . . . 11 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝑑𝐽) ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})
307301, 306ffvelcdmd 7087 . . . . . . . . . 10 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑𝐽)) ∈ (Base‘𝑈))
3083, 28, 1, 144, 307mplelf 21869 . . . . . . . . 9 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑𝐽)):{𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}⟶𝐾)
309 difssd 4132 . . . . . . . . . 10 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝐼𝐽) ⊆ 𝐼)
310302, 144, 303, 309, 305psrbagres 41581 . . . . . . . . 9 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝑑 ↾ (𝐼𝐽)) ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin})
311308, 310ffvelcdmd 7087 . . . . . . . 8 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑𝐽))‘(𝑑 ↾ (𝐼𝐽))) ∈ 𝐾)
312207adantr 480 . . . . . . . . 9 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (mulGrp‘𝑅) ∈ CMnd)
31324adantr 480 . . . . . . . . 9 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → 𝐽 ∈ V)
31451ad2antrr 723 . . . . . . . . . . 11 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑗𝐽) → (mulGrp‘𝑅) ∈ Mnd)
315302psrbagf 21782 . . . . . . . . . . . . . 14 (𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} → 𝑑:𝐼⟶ℕ0)
316315adantl 481 . . . . . . . . . . . . 13 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → 𝑑:𝐼⟶ℕ0)
317316, 304fssresd 6758 . . . . . . . . . . . 12 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝑑𝐽):𝐽⟶ℕ0)
318317ffvelcdmda 7086 . . . . . . . . . . 11 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑗𝐽) → ((𝑑𝐽)‘𝑗) ∈ ℕ0)
31958ffvelcdmda 7086 . . . . . . . . . . . 12 ((𝜑𝑗𝐽) → ((𝐴𝐽)‘𝑗) ∈ 𝐾)
320319adantlr 712 . . . . . . . . . . 11 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑗𝐽) → ((𝐴𝐽)‘𝑗) ∈ 𝐾)
32148, 49, 314, 318, 320mulgnn0cld 19018 . . . . . . . . . 10 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑗𝐽) → (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)) ∈ 𝐾)
322321fmpttd 7116 . . . . . . . . 9 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))):𝐽𝐾)
32324mptexd 7228 . . . . . . . . . . 11 (𝜑 → (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))) ∈ V)
324323adantr 480 . . . . . . . . . 10 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))) ∈ V)
325 fvexd 6906 . . . . . . . . . 10 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (0g‘(mulGrp‘𝑅)) ∈ V)
326 funmpt 6586 . . . . . . . . . . 11 Fun (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))
327326a1i 11 . . . . . . . . . 10 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → Fun (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))
328302psrbagfsupp 21784 . . . . . . . . . . . 12 (𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} → 𝑑 finSupp 0)
329328adantl 481 . . . . . . . . . . 11 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → 𝑑 finSupp 0)
330 0zd 12577 . . . . . . . . . . 11 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → 0 ∈ ℤ)
331329, 330fsuppres 9394 . . . . . . . . . 10 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝑑𝐽) finSupp 0)
332 ssidd 4005 . . . . . . . . . . . . . 14 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → ((𝑑𝐽) supp 0) ⊆ ((𝑑𝐽) supp 0))
333317, 332, 313, 330suppssr 8186 . . . . . . . . . . . . 13 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑗 ∈ (𝐽 ∖ ((𝑑𝐽) supp 0))) → ((𝑑𝐽)‘𝑗) = 0)
334333oveq1d 7427 . . . . . . . . . . . 12 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑗 ∈ (𝐽 ∖ ((𝑑𝐽) supp 0))) → (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)) = (0(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))
335 eldifi 4126 . . . . . . . . . . . . 13 (𝑗 ∈ (𝐽 ∖ ((𝑑𝐽) supp 0)) → 𝑗𝐽)
33648, 116, 49mulg0 19000 . . . . . . . . . . . . . 14 (((𝐴𝐽)‘𝑗) ∈ 𝐾 → (0(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)) = (0g‘(mulGrp‘𝑅)))
337320, 336syl 17 . . . . . . . . . . . . 13 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑗𝐽) → (0(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)) = (0g‘(mulGrp‘𝑅)))
338335, 337sylan2 592 . . . . . . . . . . . 12 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑗 ∈ (𝐽 ∖ ((𝑑𝐽) supp 0))) → (0(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)) = (0g‘(mulGrp‘𝑅)))
339334, 338eqtrd 2771 . . . . . . . . . . 11 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑗 ∈ (𝐽 ∖ ((𝑑𝐽) supp 0))) → (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)) = (0g‘(mulGrp‘𝑅)))
340339, 313suppss2 8191 . . . . . . . . . 10 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → ((𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))) supp (0g‘(mulGrp‘𝑅))) ⊆ ((𝑑𝐽) supp 0))
341324, 325, 327, 331, 340fsuppsssuppgd 41534 . . . . . . . . 9 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))) finSupp (0g‘(mulGrp‘𝑅)))
34248, 116, 312, 313, 322, 341gsumcl 19831 . . . . . . . 8 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → ((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))) ∈ 𝐾)
34328, 143, 300, 311, 342ringcld 20158 . . . . . . 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)
3446adantr 480 . . . . . . . 8 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝐼𝐽) ∈ V)
34551ad2antrr 723 . . . . . . . . . 10 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ (𝐼𝐽)) → (mulGrp‘𝑅) ∈ Mnd)
346316, 309fssresd 6758 . . . . . . . . . . 11 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝑑 ↾ (𝐼𝐽)):(𝐼𝐽)⟶ℕ0)
347346ffvelcdmda 7086 . . . . . . . . . 10 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ (𝐼𝐽)) → ((𝑑 ↾ (𝐼𝐽))‘𝑘) ∈ ℕ0)
348213ffvelcdmda 7086 . . . . . . . . . . 11 ((𝜑𝑘 ∈ (𝐼𝐽)) → ((𝐴 ↾ (𝐼𝐽))‘𝑘) ∈ 𝐾)
349348adantlr 712 . . . . . . . . . 10 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ (𝐼𝐽)) → ((𝐴 ↾ (𝐼𝐽))‘𝑘) ∈ 𝐾)
35048, 49, 345, 347, 349mulgnn0cld 19018 . . . . . . . . 9 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ (𝐼𝐽)) → (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)) ∈ 𝐾)
351350fmpttd 7116 . . . . . . . 8 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝑘 ∈ (𝐼𝐽) ↦ (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))):(𝐼𝐽)⟶𝐾)
352344mptexd 7228 . . . . . . . . 9 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝑘 ∈ (𝐼𝐽) ↦ (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))) ∈ V)
353 funmpt 6586 . . . . . . . . . 10 Fun (𝑘 ∈ (𝐼𝐽) ↦ (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)))
354353a1i 11 . . . . . . . . 9 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → Fun (𝑘 ∈ (𝐼𝐽) ↦ (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))
355329, 330fsuppres 9394 . . . . . . . . 9 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝑑 ↾ (𝐼𝐽)) finSupp 0)
356 ssidd 4005 . . . . . . . . . . . . 13 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → ((𝑑 ↾ (𝐼𝐽)) supp 0) ⊆ ((𝑑 ↾ (𝐼𝐽)) supp 0))
357346, 356, 344, 330suppssr 8186 . . . . . . . . . . . 12 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ ((𝐼𝐽) ∖ ((𝑑 ↾ (𝐼𝐽)) supp 0))) → ((𝑑 ↾ (𝐼𝐽))‘𝑘) = 0)
358357oveq1d 7427 . . . . . . . . . . 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)))
359 eldifi 4126 . . . . . . . . . . . . 13 (𝑘 ∈ ((𝐼𝐽) ∖ ((𝑑 ↾ (𝐼𝐽)) supp 0)) → 𝑘 ∈ (𝐼𝐽))
360359, 349sylan2 592 . . . . . . . . . . . 12 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ ((𝐼𝐽) ∖ ((𝑑 ↾ (𝐼𝐽)) supp 0))) → ((𝐴 ↾ (𝐼𝐽))‘𝑘) ∈ 𝐾)
36148, 116, 49mulg0 19000 . . . . . . . . . . . 12 (((𝐴 ↾ (𝐼𝐽))‘𝑘) ∈ 𝐾 → (0(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)) = (0g‘(mulGrp‘𝑅)))
362360, 361syl 17 . . . . . . . . . . 11 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ ((𝐼𝐽) ∖ ((𝑑 ↾ (𝐼𝐽)) supp 0))) → (0(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)) = (0g‘(mulGrp‘𝑅)))
363358, 362eqtrd 2771 . . . . . . . . . 10 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ ((𝐼𝐽) ∖ ((𝑑 ↾ (𝐼𝐽)) supp 0))) → (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)) = (0g‘(mulGrp‘𝑅)))
364363, 344suppss2 8191 . . . . . . . . 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))
365352, 325, 354, 355, 364fsuppsssuppgd 41534 . . . . . . . 8 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝑘 ∈ (𝐼𝐽) ↦ (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))) finSupp (0g‘(mulGrp‘𝑅)))
36648, 116, 312, 344, 351, 365gsumcl 19831 . . . . . . 7 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → ((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)))) ∈ 𝐾)
36728, 143, 300, 343, 366ringcld 20158 . . . . . 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)
368367fmpttd 7116 . . . . 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)
369298mptex 7227 . . . . . . 7 (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
370369a1i 11 . . . . . 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)))))) ∈ V)
371 fvexd 6906 . . . . . 6 (𝜑 → (0g𝑅) ∈ V)
372 funmpt 6586 . . . . . . 7 Fun (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))))))
373372a1i 11 . . . . . 6 (𝜑 → Fun (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)))))))
374298mptex 7227 . . . . . . . 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)))))) ∈ V
375374a1i 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)))))) ∈ V)
376 funmpt 6586 . . . . . . . 8 Fun (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))))))
377376a1i 11 . . . . . . 7 (𝜑 → Fun (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)))))))
3787adantr 480 . . . . . . . . . 10 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → 𝑅 ∈ CRing)
37917adantr 480 . . . . . . . . . 10 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → 𝐹𝐵)
380302, 14, 15, 303, 378, 304, 379, 305selvvvval 41623 . . . . . . . . 9 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑𝐽))‘(𝑑 ↾ (𝐼𝐽))) = (𝐹𝑑))
381380mpteq2dva 5248 . . . . . . . 8 (𝜑 → (𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑𝐽))‘(𝑑 ↾ (𝐼𝐽)))) = (𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (𝐹𝑑)))
382 eqid 2731 . . . . . . . . . . 11 (Base‘𝑅) = (Base‘𝑅)
38314, 382, 15, 302, 17mplelf 21869 . . . . . . . . . 10 (𝜑𝐹:{ ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}⟶(Base‘𝑅))
384383feqmptd 6960 . . . . . . . . 9 (𝜑𝐹 = (𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (𝐹𝑑)))
38514, 15, 204, 17, 7mplelsfi 21866 . . . . . . . . 9 (𝜑𝐹 finSupp (0g𝑅))
386384, 385eqbrtrrd 5172 . . . . . . . 8 (𝜑 → (𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (𝐹𝑑)) finSupp (0g𝑅))
387381, 386eqbrtrd 5170 . . . . . . 7 (𝜑 → (𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑𝐽))‘(𝑑 ↾ (𝐼𝐽)))) finSupp (0g𝑅))
388 ssidd 4005 . . . . . . . 8 (𝜑 → ((𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑𝐽))‘(𝑑 ↾ (𝐼𝐽)))) supp (0g𝑅)) ⊆ ((𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑𝐽))‘(𝑑 ↾ (𝐼𝐽)))) supp (0g𝑅)))
38932adantr 480 . . . . . . . . 9 ((𝜑𝑣𝐾) → 𝑅 ∈ Ring)
390 simpr 484 . . . . . . . . 9 ((𝜑𝑣𝐾) → 𝑣𝐾)
39128, 143, 204, 389, 390ringlzd 20190 . . . . . . . 8 ((𝜑𝑣𝐾) → ((0g𝑅)(.r𝑅)𝑣) = (0g𝑅))
392 fvexd 6906 . . . . . . . 8 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑𝐽))‘(𝑑 ↾ (𝐼𝐽))) ∈ V)
393388, 391, 392, 342, 371suppssov1 8188 . . . . . . 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)))))) supp (0g𝑅)) ⊆ ((𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑𝐽))‘(𝑑 ↾ (𝐼𝐽)))) supp (0g𝑅)))
394375, 371, 377, 387, 393fsuppsssuppgd 41534 . . . . . 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))
395 ssidd 4005 . . . . . . 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)))))) supp (0g𝑅)) ⊆ ((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)))))) supp (0g𝑅)))
396 ovexd 7447 . . . . . . 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))))) ∈ V)
397395, 391, 396, 366, 371suppssov1 8188 . . . . . 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)))))) supp (0g𝑅)) ⊆ ((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)))))) supp (0g𝑅)))
398370, 371, 373, 394, 397fsuppsssuppgd 41534 . . . . 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))
399 eqid 2731 . . . . . 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))
400302, 13, 144, 399, 4, 16evlselvlem 41624 . . . . 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})
40128, 204, 296, 299, 368, 398, 400gsumf1o 19832 . . . 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)))))
402144psrbagf 21782 . . . . . . . . . 10 (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} → 𝑏:(𝐼𝐽)⟶ℕ0)
403402ad2antrl 725 . . . . . . . . 9 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → 𝑏:(𝐼𝐽)⟶ℕ0)
40413psrbagf 21782 . . . . . . . . . 10 (𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} → 𝑎:𝐽⟶ℕ0)
405404ad2antll 726 . . . . . . . . 9 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → 𝑎:𝐽⟶ℕ0)
406 disjdifr 4472 . . . . . . . . . 10 ((𝐼𝐽) ∩ 𝐽) = ∅
407406a1i 11 . . . . . . . . 9 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → ((𝐼𝐽) ∩ 𝐽) = ∅)
408403, 405, 407fun2d 6755 . . . . . . . 8 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → (𝑏𝑎):((𝐼𝐽) ∪ 𝐽)⟶ℕ0)
409 undifr 4482 . . . . . . . . . . 11 (𝐽𝐼 ↔ ((𝐼𝐽) ∪ 𝐽) = 𝐼)
41016, 409sylib 217 . . . . . . . . . 10 (𝜑 → ((𝐼𝐽) ∪ 𝐽) = 𝐼)
411410adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → ((𝐼𝐽) ∪ 𝐽) = 𝐼)
412411feq2d 6703 . . . . . . . 8 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → ((𝑏𝑎):((𝐼𝐽) ∪ 𝐽)⟶ℕ0 ↔ (𝑏𝑎):𝐼⟶ℕ0))
413408, 412mpbid 231 . . . . . . 7 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → (𝑏𝑎):𝐼⟶ℕ0)
414 vex 3477 . . . . . . . . . . 11 𝑏 ∈ V
415 vex 3477 . . . . . . . . . . 11 𝑎 ∈ V
416414, 415unex 7737 . . . . . . . . . 10 (𝑏𝑎) ∈ V
417416a1i 11 . . . . . . . . 9 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → (𝑏𝑎) ∈ V)
418 0zd 12577 . . . . . . . . 9 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → 0 ∈ ℤ)
419413ffund 6721 . . . . . . . . 9 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → Fun (𝑏𝑎))
420144psrbagfsupp 21784 . . . . . . . . . . 11 (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} → 𝑏 finSupp 0)
421420ad2antrl 725 . . . . . . . . . 10 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → 𝑏 finSupp 0)
42213psrbagfsupp 21784 . . . . . . . . . . 11 (𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} → 𝑎 finSupp 0)
423422ad2antll 726 . . . . . . . . . 10 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → 𝑎 finSupp 0)
424421, 423fsuppun 9388 . . . . . . . . 9 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → ((𝑏𝑎) supp 0) ∈ Fin)
425417, 418, 419, 424isfsuppd 9372 . . . . . . . 8 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → (𝑏𝑎) finSupp 0)
426 fcdmnn0fsuppg 12538 . . . . . . . . 9 (((𝑏𝑎) ∈ V ∧ (𝑏𝑎):𝐼⟶ℕ0) → ((𝑏𝑎) finSupp 0 ↔ ((𝑏𝑎) “ ℕ) ∈ Fin))
427417, 413, 426syl2anc 583 . . . . . . . 8 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → ((𝑏𝑎) finSupp 0 ↔ ((𝑏𝑎) “ ℕ) ∈ Fin))
428425, 427mpbid 231 . . . . . . 7 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → ((𝑏𝑎) “ ℕ) ∈ Fin)
4294adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → 𝐼𝑉)
430302psrbag 21781 . . . . . . . 8 (𝐼𝑉 → ((𝑏𝑎) ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↔ ((𝑏𝑎):𝐼⟶ℕ0 ∧ ((𝑏𝑎) “ ℕ) ∈ Fin)))
431429, 430syl 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)))
432413, 428, 431mpbir2and 710 . . . . . 6 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → (𝑏𝑎) ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin})
433 eqidd 2732 . . . . . 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)))
434 eqidd 2732 . . . . . 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)))))))
435 reseq1 5975 . . . . . . . . . . . 12 (𝑑 = (𝑏𝑎) → (𝑑𝐽) = ((𝑏𝑎) ↾ 𝐽))
436435fveq2d 6895 . . . . . . . . . . 11 (𝑑 = (𝑏𝑎) → ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑𝐽)) = ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘((𝑏𝑎) ↾ 𝐽)))
437 reseq1 5975 . . . . . . . . . . 11 (𝑑 = (𝑏𝑎) → (𝑑 ↾ (𝐼𝐽)) = ((𝑏𝑎) ↾ (𝐼𝐽)))
438436, 437fveq12d 6898 . . . . . . . . . 10 (𝑑 = (𝑏𝑎) → (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑𝐽))‘(𝑑 ↾ (𝐼𝐽))) = (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘((𝑏𝑎) ↾ 𝐽))‘((𝑏𝑎) ↾ (𝐼𝐽))))
439435fveq1d 6893 . . . . . . . . . . . . 13 (𝑑 = (𝑏𝑎) → ((𝑑𝐽)‘𝑗) = (((𝑏𝑎) ↾ 𝐽)‘𝑗))
440439oveq1d 7427 . . . . . . . . . . . 12 (𝑑 = (𝑏𝑎) → (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)) = ((((𝑏𝑎) ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))
441440mpteq2dv 5250 . . . . . . . . . . 11 (𝑑 = (𝑏𝑎) → (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))) = (𝑗𝐽 ↦ ((((𝑏𝑎) ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))
442441oveq2d 7428 . . . . . . . . . 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)))))
443438, 442oveq12d 7430 . . . . . . . . 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))))))
444437fveq1d 6893 . . . . . . . . . . . 12 (𝑑 = (𝑏𝑎) → ((𝑑 ↾ (𝐼𝐽))‘𝑘) = (((𝑏𝑎) ↾ (𝐼𝐽))‘𝑘))
445444oveq1d 7427 . . . . . . . . . . 11 (𝑑 = (𝑏𝑎) → (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)) = ((((𝑏𝑎) ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)))
446445mpteq2dv 5250 . . . . . . . . . 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))))
447446oveq2d 7428 . . . . . . . . 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)))))
448443, 447oveq12d 7430 . . . . . . . 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))))))
449416, 448csbie 3929 . . . . . . 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)))))
450402ffnd 6718 . . . . . . . . . . . . 13 (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} → 𝑏 Fn (𝐼𝐽))
451450ad2antrl 725 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → 𝑏 Fn (𝐼𝐽))
452405ffnd 6718 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → 𝑎 Fn 𝐽)
453 fnunres2 6662 . . . . . . . . . . . 12 ((𝑏 Fn (𝐼𝐽) ∧ 𝑎 Fn 𝐽 ∧ ((𝐼𝐽) ∩ 𝐽) = ∅) → ((𝑏𝑎) ↾ 𝐽) = 𝑎)
454451, 452, 407, 453syl3anc 1370 . . . . . . . . . . 11 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → ((𝑏𝑎) ↾ 𝐽) = 𝑎)
455454fveq2d 6895 . . . . . . . . . 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))
456 fnunres1 6661 . . . . . . . . . . 11 ((𝑏 Fn (𝐼𝐽) ∧ 𝑎 Fn 𝐽 ∧ ((𝐼𝐽) ∩ 𝐽) = ∅) → ((𝑏𝑎) ↾ (𝐼𝐽)) = 𝑏)
457451, 452, 407, 456syl3anc 1370 . . . . . . . . . 10 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → ((𝑏𝑎) ↾ (𝐼𝐽)) = 𝑏)
458455, 457fveq12d 6898 . . . . . . . . 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))
459454fveq1d 6893 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → (((𝑏𝑎) ↾ 𝐽)‘𝑗) = (𝑎𝑗))
460459oveq1d 7427 . . . . . . . . . . 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)))
461460mpteq2dv 5250 . . . . . . . . . 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))))
462461oveq2d 7428 . . . . . . . . 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)))))
463458, 462oveq12d 7430 . . . . . . . 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))))))
464457fveq1d 6893 . . . . . . . . . . 11 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → (((𝑏𝑎) ↾ (𝐼𝐽))‘𝑘) = (𝑏𝑘))
465464oveq1d 7427 . . . . . . . . . 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)))
466465mpteq2dv 5250 . . . . . . . . 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))))
467466oveq2d 7428 . . . . . . . 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)))))
468463, 467oveq12d 7430 . . . . . . 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))))))
469449, 468eqtrid 2783 . . . . . 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))))))
470432, 433, 434, 469fmpocos 41526 . . . . 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)))))))
471470oveq2d 7428 . . . 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))))))))
472 ovex 7445 . . . . . . 7 (ℕ0m (𝐼𝐽)) ∈ V
473472rabex 5332 . . . . . 6 {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∈ V
474473a1i 11 . . . . 5 (𝜑 → {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∈ V)
475185a1i 11 . . . . 5 (𝜑 → {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ∈ V)
47632adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → 𝑅 ∈ Ring)
47719ffvelcdmda 7086 . . . . . . . . . . . . 13 ((𝜑𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎) ∈ (Base‘𝑈))
4783, 28, 1, 144, 477mplelf 21869 . . . . . . . . . . . 12 ((𝜑𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎):{𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}⟶𝐾)
479478ffvelcdmda 7086 . . . . . . . . . . 11 (((𝜑𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) ∧ 𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏) ∈ 𝐾)
480479an32s 649 . . . . . . . . . 10 (((𝜑𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏) ∈ 𝐾)
481480anasss 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)
48224adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → 𝐽 ∈ V)
4837adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → 𝑅 ∈ CRing)
48436adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → (𝐴𝐽) ∈ (𝐾m 𝐽))
485 simprr 770 . . . . . . . . . 10 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})
48613, 28, 47, 49, 482, 483, 484, 485evlsvvvallem 41599 . . . . . . . . 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)
48728, 143, 476, 481, 486ringcld 20158 . . . . . . . 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)
4886adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → (𝐼𝐽) ∈ V)
48935, 5elmapssresd 41536 . . . . . . . . . 10 (𝜑 → (𝐴 ↾ (𝐼𝐽)) ∈ (𝐾m (𝐼𝐽)))
490489adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → (𝐴 ↾ (𝐼𝐽)) ∈ (𝐾m (𝐼𝐽)))
491 simprl 768 . . . . . . . . 9 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → 𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin})
492144, 28, 47, 49, 488, 483, 490, 491evlsvvvallem 41599 . . . . . . . 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)
49328, 143, 476, 487, 492ringcld 20158 . . . . . . 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)
494493ralrimivva 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)
495287fmpo 8058 . . . . . 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)
496494, 495sylib 217 . . . . 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)
497 f1of1 6832 . . . . . . . 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})
498400, 497syl 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})
499398, 498, 371, 370fsuppco 9403 . . . . . 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))
500470, 499eqbrtrrd 5172 . . . . 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))
50128, 204, 296, 474, 475, 496, 500gsumxp 19892 . . . 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))))))
502401, 471, 5013eqtrd 2775 . . 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))))))
50328, 143, 300, 311, 342, 366ringassd 20157 . . . . . 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)))))))
50447, 143mgpplusg 20039 . . . . . . . . 9 (.r𝑅) = (+g‘(mulGrp‘𝑅))
50551ad2antrr 723 . . . . . . . . . . 11 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑖𝐼) → (mulGrp‘𝑅) ∈ Mnd)
506316ffvelcdmda 7086 . . . . . . . . . . 11 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑖𝐼) → (𝑑𝑖) ∈ ℕ0)
50757adantr 480 . . . . . . . . . . . 12 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → 𝐴:𝐼𝐾)
508507ffvelcdmda 7086 . . . . . . . . . . 11 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑖𝐼) → (𝐴𝑖) ∈ 𝐾)
50948, 49, 505, 506, 508mulgnn0cld 19018 . . . . . . . . . 10 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑖𝐼) → ((𝑑𝑖)(.g‘(mulGrp‘𝑅))(𝐴𝑖)) ∈ 𝐾)
510509fmpttd 7116 . . . . . . . . 9 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝑖𝐼 ↦ ((𝑑𝑖)(.g‘(mulGrp‘𝑅))(𝐴𝑖))):𝐼𝐾)
5114mptexd 7228 . . . . . . . . . . 11 (𝜑 → (𝑖𝐼 ↦ ((𝑑𝑖)(.g‘(mulGrp‘𝑅))(𝐴𝑖))) ∈ V)
512511adantr 480 . . . . . . . . . 10 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝑖𝐼 ↦ ((𝑑𝑖)(.g‘(mulGrp‘𝑅))(𝐴𝑖))) ∈ V)
513 funmpt 6586 . . . . . . . . . . 11 Fun (𝑖𝐼 ↦ ((𝑑𝑖)(.g‘(mulGrp‘𝑅))(𝐴𝑖)))
514513a1i 11 . . . . . . . . . 10 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → Fun (𝑖𝐼 ↦ ((𝑑𝑖)(.g‘(mulGrp‘𝑅))(𝐴𝑖))))
515316feqmptd 6960 . . . . . . . . . . 11 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → 𝑑 = (𝑖𝐼 ↦ (𝑑𝑖)))
516515, 329eqbrtrrd 5172 . . . . . . . . . 10 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝑖𝐼 ↦ (𝑑𝑖)) finSupp 0)
517 ssidd 4005 . . . . . . . . . . 11 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → ((𝑖𝐼 ↦ (𝑑𝑖)) supp 0) ⊆ ((𝑖𝐼 ↦ (𝑑𝑖)) supp 0))
518117adantl 481 . . . . . . . . . . 11 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑘𝐾) → (0(.g‘(mulGrp‘𝑅))𝑘) = (0g‘(mulGrp‘𝑅)))
519517, 518, 506, 508, 330suppssov1 8188 . . . . . . . . . 10 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → ((𝑖𝐼 ↦ ((𝑑𝑖)(.g‘(mulGrp‘𝑅))(𝐴𝑖))) supp (0g‘(mulGrp‘𝑅))) ⊆ ((𝑖𝐼 ↦ (𝑑𝑖)) supp 0))
520512, 325, 514, 516, 519fsuppsssuppgd 41534 . . . . . . . . 9 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝑖𝐼 ↦ ((𝑑𝑖)(.g‘(mulGrp‘𝑅))(𝐴𝑖))) finSupp (0g‘(mulGrp‘𝑅)))
521 disjdif 4471 . . . . . . . . . 10 (𝐽 ∩ (𝐼𝐽)) = ∅
522521a1i 11 . . . . . . . . 9 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝐽 ∩ (𝐼𝐽)) = ∅)
523 undif 4481 . . . . . . . . . . . 12 (𝐽𝐼 ↔ (𝐽 ∪ (𝐼𝐽)) = 𝐼)
52416, 523sylib 217 . . . . . . . . . . 11 (𝜑 → (𝐽 ∪ (𝐼𝐽)) = 𝐼)
525524eqcomd 2737 . . . . . . . . . 10 (𝜑𝐼 = (𝐽 ∪ (𝐼𝐽)))
526525adantr 480 . . . . . . . . 9 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → 𝐼 = (𝐽 ∪ (𝐼𝐽)))
52748, 116, 504, 312, 303, 510, 520, 522, 526gsumsplit 19844 . . . . . . . 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)))))
528304resmptd 6040 . . . . . . . . . . 11 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → ((𝑖𝐼 ↦ ((𝑑𝑖)(.g‘(mulGrp‘𝑅))(𝐴𝑖))) ↾ 𝐽) = (𝑖𝐽 ↦ ((𝑑𝑖)(.g‘(mulGrp‘𝑅))(𝐴𝑖))))
529 fveq2 6891 . . . . . . . . . . . . . 14 (𝑖 = 𝑗 → (𝑑𝑖) = (𝑑𝑗))
530 fveq2 6891 . . . . . . . . . . . . . 14 (𝑖 = 𝑗 → (𝐴𝑖) = (𝐴𝑗))
531529, 530oveq12d 7430 . . . . . . . . . . . . 13 (𝑖 = 𝑗 → ((𝑑𝑖)(.g‘(mulGrp‘𝑅))(𝐴𝑖)) = ((𝑑𝑗)(.g‘(mulGrp‘𝑅))(𝐴𝑗)))
532531cbvmptv 5261 . . . . . . . . . . . 12 (𝑖𝐽 ↦ ((𝑑𝑖)(.g‘(mulGrp‘𝑅))(𝐴𝑖))) = (𝑗𝐽 ↦ ((𝑑𝑗)(.g‘(mulGrp‘𝑅))(𝐴𝑗)))
533 simpr 484 . . . . . . . . . . . . . . . 16 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑗𝐽) → 𝑗𝐽)
534533fvresd 6911 . . . . . . . . . . . . . . 15 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑗𝐽) → ((𝑑𝐽)‘𝑗) = (𝑑𝑗))
535533fvresd 6911 . . . . . . . . . . . . . . 15 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑗𝐽) → ((𝐴𝐽)‘𝑗) = (𝐴𝑗))
536534, 535oveq12d 7430 . . . . . . . . . . . . . 14 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑗𝐽) → (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)) = ((𝑑𝑗)(.g‘(mulGrp‘𝑅))(𝐴𝑗)))
537536eqcomd 2737 . . . . . . . . . . . . 13 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑗𝐽) → ((𝑑𝑗)(.g‘(mulGrp‘𝑅))(𝐴𝑗)) = (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))
538537mpteq2dva 5248 . . . . . . . . . . . 12 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝑗𝐽 ↦ ((𝑑𝑗)(.g‘(mulGrp‘𝑅))(𝐴𝑗))) = (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))
539532, 538eqtrid 2783 . . . . . . . . . . 11 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝑖𝐽 ↦ ((𝑑𝑖)(.g‘(mulGrp‘𝑅))(𝐴𝑖))) = (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))
540528, 539eqtrd 2771 . . . . . . . . . 10 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → ((𝑖𝐼 ↦ ((𝑑𝑖)(.g‘(mulGrp‘𝑅))(𝐴𝑖))) ↾ 𝐽) = (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))
541540oveq2d 7428 . . . . . . . . 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)))))
542309resmptd 6040 . . . . . . . . . . 11 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → ((𝑖𝐼 ↦ ((𝑑𝑖)(.g‘(mulGrp‘𝑅))(𝐴𝑖))) ↾ (𝐼𝐽)) = (𝑖 ∈ (𝐼𝐽) ↦ ((𝑑𝑖)(.g‘(mulGrp‘𝑅))(𝐴𝑖))))
543 fveq2 6891 . . . . . . . . . . . . . 14 (𝑖 = 𝑘 → (𝑑𝑖) = (𝑑𝑘))
544 fveq2 6891 . . . . . . . . . . . . . 14 (𝑖 = 𝑘 → (𝐴𝑖) = (𝐴𝑘))
545543, 544oveq12d 7430 . . . . . . . . . . . . 13 (𝑖 = 𝑘 → ((𝑑𝑖)(.g‘(mulGrp‘𝑅))(𝐴𝑖)) = ((𝑑𝑘)(.g‘(mulGrp‘𝑅))(𝐴𝑘)))
546545cbvmptv 5261 . . . . . . . . . . . 12 (𝑖 ∈ (𝐼𝐽) ↦ ((𝑑𝑖)(.g‘(mulGrp‘𝑅))(𝐴𝑖))) = (𝑘 ∈ (𝐼𝐽) ↦ ((𝑑𝑘)(.g‘(mulGrp‘𝑅))(𝐴𝑘)))
547 simpr 484 . . . . . . . . . . . . . . . 16 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ (𝐼𝐽)) → 𝑘 ∈ (𝐼𝐽))
548547fvresd 6911 . . . . . . . . . . . . . . 15 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ (𝐼𝐽)) → ((𝑑 ↾ (𝐼𝐽))‘𝑘) = (𝑑𝑘))
549547fvresd 6911 . . . . . . . . . . . . . . 15 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ (𝐼𝐽)) → ((𝐴 ↾ (𝐼𝐽))‘𝑘) = (𝐴𝑘))
550548, 549oveq12d 7430 . . . . . . . . . . . . . 14 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ (𝐼𝐽)) → (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)) = ((𝑑𝑘)(.g‘(mulGrp‘𝑅))(𝐴𝑘)))
551550eqcomd 2737 . . . . . . . . . . . . 13 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ (𝐼𝐽)) → ((𝑑𝑘)(.g‘(mulGrp‘𝑅))(𝐴𝑘)) = (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)))
552551mpteq2dva 5248 . . . . . . . . . . . 12 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝑘 ∈ (𝐼𝐽) ↦ ((𝑑𝑘)(.g‘(mulGrp‘𝑅))(𝐴𝑘))) = (𝑘 ∈ (𝐼𝐽) ↦ (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))
553546, 552eqtrid 2783 . . . . . . . . . . 11 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝑖 ∈ (𝐼𝐽) ↦ ((𝑑𝑖)(.g‘(mulGrp‘𝑅))(𝐴𝑖))) = (𝑘 ∈ (𝐼𝐽) ↦ (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))
554542, 553eqtrd 2771 . . . . . . . . . 10 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → ((𝑖𝐼 ↦ ((𝑑𝑖)(.g‘(mulGrp‘𝑅))(𝐴𝑖))) ↾ (𝐼𝐽)) = (𝑘 ∈ (𝐼𝐽) ↦ (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))
555554oveq2d 7428 . . . . . . . . 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)))))
556541, 555oveq12d 7430 . . . . . . . 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))))))
557527, 556eqtr2d 2772 . . . . . . 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)))))
558380, 557oveq12d 7430 . . . . . 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))))))
559503, 558eqtrd 2771 . . . . 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))))))
560559mpteq2dva 5248 . . . 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)))))))
561560oveq2d 7428 . . 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))))))))
562295, 502, 5613eqtr2d 2777 . 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))))))))
563 eqid 2731 . . 3 ((𝐼𝐽) eval 𝑅) = ((𝐼𝐽) eval 𝑅)
564563, 3, 1, 144, 28, 47, 49, 143, 6, 7, 173, 489evlvvval 41611 . 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))))))))
565 eqid 2731 . . 3 (𝐼 eval 𝑅) = (𝐼 eval 𝑅)
566565, 14, 15, 302, 28, 47, 49, 143, 4, 7, 17, 35evlvvval 41611 . 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))))))))
567562, 564, 5663eqtr4d 2781 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 205  wa 395   = wceq 1540  wcel 2105  wral 3060  {crab 3431  Vcvv 3473  csb 3893  cdif 3945  cun 3946  cin 3947  wss 3948  c0 4322  {csn 4628   class class class wbr 5148  cmpt 5231   × cxp 5674  ccnv 5675  cres 5678  cima 5679  ccom 5680  Fun wfun 6537   Fn wfn 6538  wf 6539  1-1wf1 6540  1-1-ontowf1o 6542  cfv 6543  (class class class)co 7412  cmpo 7414   supp csupp 8151  m cmap 8826  Fincfn 8945   finSupp cfsupp 9367  0cc0 11116  cn 12219  0cn0 12479  cz 12565  Basecbs 17151  .rcmulr 17205  Scalarcsca 17207   ·𝑠 cvsca 17208  0gc0g 17392   Σg cgsu 17393  Mndcmnd 18665   MndHom cmhm 18709  Grpcgrp 18861  .gcmg 18993   GrpHom cghm 19134  CMndccmn 19696  mulGrpcmgp 20035  1rcur 20082  Ringcrg 20134  CRingccrg 20135   RingHom crh 20367  AssAlgcasa 21716  algSccascl 21718   mPoly cmpl 21770   eval cevl 21946   selectVars cslv 21983
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-cnex 11172  ax-resscn 11173  ax-1cn 11174  ax-icn 11175  ax-addcl 11176  ax-addrcl 11177  ax-mulcl 11178  ax-mulrcl 11179  ax-mulcom 11180  ax-addass 11181  ax-mulass 11182  ax-distr 11183  ax-i2m1 11184  ax-1ne0 11185  ax-1rid 11186  ax-rnegex 11187  ax-rrecex 11188  ax-cnre 11189  ax-pre-lttri 11190  ax-pre-lttrn 11191  ax-pre-ltadd 11192  ax-pre-mulgt0 11193
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-of 7674  df-ofr 7675  df-om 7860  df-1st 7979  df-2nd 7980  df-supp 8152  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-rdg 8416  df-1o 8472  df-er 8709  df-map 8828  df-pm 8829  df-ixp 8898  df-en 8946  df-dom 8947  df-sdom 8948  df-fin 8949  df-fsupp 9368  df-sup 9443  df-oi 9511  df-card 9940  df-pnf 11257  df-mnf 11258  df-xr 11259  df-ltxr 11260  df-le 11261  df-sub 11453  df-neg 11454  df-nn 12220  df-2 12282  df-3 12283  df-4 12284  df-5 12285  df-6 12286  df-7 12287  df-8 12288  df-9 12289  df-n0 12480  df-z 12566  df-dec 12685  df-uz 12830  df-fz 13492  df-fzo 13635  df-seq 13974  df-hash 14298  df-struct 17087  df-sets 17104  df-slot 17122  df-ndx 17134  df-base 17152  df-ress 17181  df-plusg 17217  df-mulr 17218  df-sca 17220  df-vsca 17221  df-ip 17222  df-tset 17223  df-ple 17224  df-ds 17226  df-hom 17228  df-cco 17229  df-0g 17394  df-gsum 17395  df-prds 17400  df-pws 17402  df-mre 17537  df-mrc 17538  df-acs 17540  df-mgm 18571  df-sgrp 18650  df-mnd 18666  df-mhm 18711  df-submnd 18712  df-grp 18864  df-minusg 18865  df-sbg 18866  df-mulg 18994  df-subg 19046  df-ghm 19135  df-cntz 19229  df-cmn 19698  df-abl 19699  df-mgp 20036  df-rng 20054  df-ur 20083  df-srg 20088  df-ring 20136  df-cring 20137  df-rhm 20370  df-subrng 20442  df-subrg 20467  df-lmod 20704  df-lss 20775  df-lsp 20815  df-assa 21719  df-asp 21720  df-ascl 21721  df-psr 21773  df-mvr 21774  df-mpl 21775  df-evls 21947  df-evl 21948  df-selv 21987
This theorem is referenced by: (None)
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