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Theorem evlselv 42597
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 2737 . . . . . . . . . . . . 13 (Base‘𝑈) = (Base‘𝑈)
2 eqid 2737 . . . . . . . . . . . . 13 (.r𝑈) = (.r𝑈)
3 evlselv.u . . . . . . . . . . . . . . . 16 𝑈 = ((𝐼𝐽) mPoly 𝑅)
4 evlselv.i . . . . . . . . . . . . . . . . 17 (𝜑𝐼𝑉)
5 difssd 4137 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐼𝐽) ⊆ 𝐼)
64, 5ssexd 5324 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐼𝐽) ∈ V)
7 evlselv.r . . . . . . . . . . . . . . . 16 (𝜑𝑅 ∈ CRing)
83, 6, 7mplcrngd 42557 . . . . . . . . . . . . . . 15 (𝜑𝑈 ∈ CRing)
98crngringd 20243 . . . . . . . . . . . . . 14 (𝜑𝑈 ∈ Ring)
109ad2antrr 726 . . . . . . . . . . . . 13 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → 𝑈 ∈ Ring)
11 evlselv.t . . . . . . . . . . . . . . . 16 𝑇 = (𝐽 mPoly 𝑈)
12 eqid 2737 . . . . . . . . . . . . . . . 16 (Base‘𝑇) = (Base‘𝑇)
13 eqid 2737 . . . . . . . . . . . . . . . 16 {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} = {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}
14 evlselv.p . . . . . . . . . . . . . . . . 17 𝑃 = (𝐼 mPoly 𝑅)
15 evlselv.b . . . . . . . . . . . . . . . . 17 𝐵 = (Base‘𝑃)
16 evlselv.j . . . . . . . . . . . . . . . . 17 (𝜑𝐽𝐼)
17 evlselv.f . . . . . . . . . . . . . . . . 17 (𝜑𝐹𝐵)
1814, 15, 3, 11, 12, 7, 16, 17selvcl 42593 . . . . . . . . . . . . . . . 16 (𝜑 → (((𝐼 selectVars 𝑅)‘𝐽)‘𝐹) ∈ (Base‘𝑇))
1911, 1, 12, 13, 18mplelf 22018 . . . . . . . . . . . . . . 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 7104 . . . . . . . . . . . . 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 2737 . . . . . . . . . . . . . 14 (mulGrp‘𝑈) = (mulGrp‘𝑈)
23 eqid 2737 . . . . . . . . . . . . . 14 (.g‘(mulGrp‘𝑈)) = (.g‘(mulGrp‘𝑈))
244, 16ssexd 5324 . . . . . . . . . . . . . . 15 (𝜑𝐽 ∈ V)
2524ad2antrr 726 . . . . . . . . . . . . . 14 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → 𝐽 ∈ V)
268ad2antrr 726 . . . . . . . . . . . . . 14 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → 𝑈 ∈ CRing)
27 fvexd 6921 . . . . . . . . . . . . . . . . 17 (𝜑 → (Base‘𝑈) ∈ V)
28 evlselv.k . . . . . . . . . . . . . . . . . . 19 𝐾 = (Base‘𝑅)
2928fvexi 6920 . . . . . . . . . . . . . . . . . 18 𝐾 ∈ V
3029a1i 11 . . . . . . . . . . . . . . . . 17 (𝜑𝐾 ∈ V)
31 evlselv.l . . . . . . . . . . . . . . . . . 18 𝐿 = (algSc‘𝑈)
327crngringd 20243 . . . . . . . . . . . . . . . . . 18 (𝜑𝑅 ∈ Ring)
333, 1, 28, 31, 6, 32mplasclf 22089 . . . . . . . . . . . . . . . . 17 (𝜑𝐿:𝐾⟶(Base‘𝑈))
3427, 30, 33elmapdd 8881 . . . . . . . . . . . . . . . 16 (𝜑𝐿 ∈ ((Base‘𝑈) ↑m 𝐾))
35 evlselv.a . . . . . . . . . . . . . . . . 17 (𝜑𝐴 ∈ (𝐾m 𝐼))
3635, 16elmapssresd 42282 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐴𝐽) ∈ (𝐾m 𝐽))
3734, 36mapcod 42284 . . . . . . . . . . . . . . 15 (𝜑 → (𝐿 ∘ (𝐴𝐽)) ∈ ((Base‘𝑈) ↑m 𝐽))
3837ad2antrr 726 . . . . . . . . . . . . . 14 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → (𝐿 ∘ (𝐴𝐽)) ∈ ((Base‘𝑈) ↑m 𝐽))
39 simpr 484 . . . . . . . . . . . . . 14 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})
4013, 1, 22, 23, 25, 26, 38, 39evlsvvvallem 42571 . . . . . . . . . . . . 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 20257 . . . . . . . . . . . 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 2738 . . . . . . . . . . . 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 2738 . . . . . . . . . . . 12 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑢 ∈ (Base‘𝑈) ↦ (𝑢𝑐)) = (𝑢 ∈ (Base‘𝑈) ↦ (𝑢𝑐)))
44 fveq1 6905 . . . . . . . . . . . 12 (𝑢 = (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r𝑈)((mulGrp‘𝑈) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴𝐽))‘𝑗))))) → (𝑢𝑐) = ((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r𝑈)((mulGrp‘𝑈) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴𝐽))‘𝑗)))))‘𝑐))
4541, 42, 43, 44fmptco 7149 . . . . . . . . . . 11 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → ((𝑢 ∈ (Base‘𝑈) ↦ (𝑢𝑐)) ∘ (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r𝑈)((mulGrp‘𝑈) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴𝐽))‘𝑗))))))) = (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ ((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r𝑈)((mulGrp‘𝑈) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴𝐽))‘𝑗)))))‘𝑐)))
4633ad2antrr 726 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → 𝐿:𝐾⟶(Base‘𝑈))
47 eqid 2737 . . . . . . . . . . . . . . . . . . . . . 22 (mulGrp‘𝑅) = (mulGrp‘𝑅)
4847, 28mgpbas 20142 . . . . . . . . . . . . . . . . . . . . 21 𝐾 = (Base‘(mulGrp‘𝑅))
49 eqid 2737 . . . . . . . . . . . . . . . . . . . . 21 (.g‘(mulGrp‘𝑅)) = (.g‘(mulGrp‘𝑅))
5047ringmgp 20236 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑅 ∈ Ring → (mulGrp‘𝑅) ∈ Mnd)
5132, 50syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (mulGrp‘𝑅) ∈ Mnd)
5251ad3antrrr 730 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) ∧ 𝑗𝐽) → (mulGrp‘𝑅) ∈ Mnd)
5313psrbagf 21938 . . . . . . . . . . . . . . . . . . . . . . 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 7104 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) ∧ 𝑗𝐽) → (𝑒𝑗) ∈ ℕ0)
56 elmapi 8889 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐴 ∈ (𝐾m 𝐼) → 𝐴:𝐼𝐾)
5735, 56syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑𝐴:𝐼𝐾)
5857, 16fssresd 6775 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (𝐴𝐽):𝐽𝐾)
5958ad2antrr 726 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → (𝐴𝐽):𝐽𝐾)
6059ffvelcdmda 7104 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) ∧ 𝑗𝐽) → ((𝐴𝐽)‘𝑗) ∈ 𝐾)
6148, 49, 52, 55, 60mulgnn0cld 19113 . . . . . . . . . . . . . . . . . . . 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 7152 . . . . . . . . . . . . . . . . . . 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 22042 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝐼𝐽) ∈ V ∧ 𝑅 ∈ CRing) → 𝑈 ∈ AssAlg)
646, 7, 63syl2anc 584 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑𝑈 ∈ AssAlg)
65 eqid 2737 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (Scalar‘𝑈) = (Scalar‘𝑈)
6631, 65asclrhm 21910 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑈 ∈ AssAlg → 𝐿 ∈ ((Scalar‘𝑈) RingHom 𝑈))
6764, 66syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑𝐿 ∈ ((Scalar‘𝑈) RingHom 𝑈))
683, 6, 7mplsca 22033 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑𝑅 = (Scalar‘𝑈))
6968eqcomd 2743 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → (Scalar‘𝑈) = 𝑅)
7069oveq1d 7446 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → ((Scalar‘𝑈) RingHom 𝑈) = (𝑅 RingHom 𝑈))
7167, 70eleqtrd 2843 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑𝐿 ∈ (𝑅 RingHom 𝑈))
7247, 22rhmmhm 20479 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐿 ∈ (𝑅 RingHom 𝑈) → 𝐿 ∈ ((mulGrp‘𝑅) MndHom (mulGrp‘𝑈)))
7371, 72syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝐿 ∈ ((mulGrp‘𝑅) MndHom (mulGrp‘𝑈)))
7473ad3antrrr 730 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) ∧ 𝑗𝐽) → 𝐿 ∈ ((mulGrp‘𝑅) MndHom (mulGrp‘𝑈)))
7548, 49, 23mhmmulg 19133 . . . . . . . . . . . . . . . . . . . . . 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 1373 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) ∧ 𝑗𝐽) → (𝐿‘((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))) = ((𝑒𝑗)(.g‘(mulGrp‘𝑈))(𝐿‘((𝐴𝐽)‘𝑗))))
7758ad3antrrr 730 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) ∧ 𝑗𝐽) → (𝐴𝐽):𝐽𝐾)
78 simpr 484 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) ∧ 𝑗𝐽) → 𝑗𝐽)
7977, 78fvco3d 7009 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) ∧ 𝑗𝐽) → ((𝐿 ∘ (𝐴𝐽))‘𝑗) = (𝐿‘((𝐴𝐽)‘𝑗)))
8079oveq2d 7447 . . . . . . . . . . . . . . . . . . . . 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 2780 . . . . . . . . . . . . . . . . . . . 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 5242 . . . . . . . . . . . . . . . . . . 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 2777 . . . . . . . . . . . . . . . . . 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 7447 . . . . . . . . . . . . . . . . 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 2737 . . . . . . . . . . . . . . . . . 18 (Base‘(mulGrp‘(Scalar‘𝑈))) = (Base‘(mulGrp‘(Scalar‘𝑈)))
86 eqid 2737 . . . . . . . . . . . . . . . . . 18 (0g‘(mulGrp‘(Scalar‘𝑈))) = (0g‘(mulGrp‘(Scalar‘𝑈)))
8768, 7eqeltrrd 2842 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (Scalar‘𝑈) ∈ CRing)
88 eqid 2737 . . . . . . . . . . . . . . . . . . . . 21 (mulGrp‘(Scalar‘𝑈)) = (mulGrp‘(Scalar‘𝑈))
8988crngmgp 20238 . . . . . . . . . . . . . . . . . . . 20 ((Scalar‘𝑈) ∈ CRing → (mulGrp‘(Scalar‘𝑈)) ∈ CMnd)
9087, 89syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (mulGrp‘(Scalar‘𝑈)) ∈ CMnd)
9190ad2antrr 726 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → (mulGrp‘(Scalar‘𝑈)) ∈ CMnd)
9222ringmgp 20236 . . . . . . . . . . . . . . . . . . . 20 (𝑈 ∈ Ring → (mulGrp‘𝑈) ∈ Mnd)
939, 92syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (mulGrp‘𝑈) ∈ Mnd)
9493ad2antrr 726 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → (mulGrp‘𝑈) ∈ Mnd)
9588, 22rhmmhm 20479 . . . . . . . . . . . . . . . . . . . 20 (𝐿 ∈ ((Scalar‘𝑈) RingHom 𝑈) → 𝐿 ∈ ((mulGrp‘(Scalar‘𝑈)) MndHom (mulGrp‘𝑈)))
9667, 95syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐿 ∈ ((mulGrp‘(Scalar‘𝑈)) MndHom (mulGrp‘𝑈)))
9796ad2antrr 726 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → 𝐿 ∈ ((mulGrp‘(Scalar‘𝑈)) MndHom (mulGrp‘𝑈)))
9868fveq2d 6910 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (Base‘𝑅) = (Base‘(Scalar‘𝑈)))
9928, 98eqtrid 2789 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑𝐾 = (Base‘(Scalar‘𝑈)))
10099ad3antrrr 730 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) ∧ 𝑗𝐽) → 𝐾 = (Base‘(Scalar‘𝑈)))
10161, 100eleqtrd 2843 . . . . . . . . . . . . . . . . . . . 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 2737 . . . . . . . . . . . . . . . . . . . . 21 (Base‘(Scalar‘𝑈)) = (Base‘(Scalar‘𝑈))
10388, 102mgpbas 20142 . . . . . . . . . . . . . . . . . . . 20 (Base‘(Scalar‘𝑈)) = (Base‘(mulGrp‘(Scalar‘𝑈)))
104101, 103eleqtrdi 2851 . . . . . . . . . . . . . . . . . . 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 7135 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))):𝐽⟶(Base‘(mulGrp‘(Scalar‘𝑈))))
10654feqmptd 6977 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → 𝑒 = (𝑗𝐽 ↦ (𝑒𝑗)))
10713psrbagfsupp 21939 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} → 𝑒 finSupp 0)
108107adantl 481 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → 𝑒 finSupp 0)
109106, 108eqbrtrrd 5167 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → (𝑗𝐽 ↦ (𝑒𝑗)) finSupp 0)
110 eqid 2737 . . . . . . . . . . . . . . . . . . . . . . 23 (0g‘(mulGrp‘𝑅)) = (0g‘(mulGrp‘𝑅))
11148, 110, 49mulg0 19092 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘𝐾 → (0(.g‘(mulGrp‘𝑅))𝑘) = (0g‘(mulGrp‘𝑅)))
112111adantl 481 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) ∧ 𝑘𝐾) → (0(.g‘(mulGrp‘𝑅))𝑘) = (0g‘(mulGrp‘𝑅)))
113 fvexd 6921 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → (0g‘(mulGrp‘𝑅)) ∈ V)
114109, 112, 55, 60, 113fsuppssov1 9424 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))) finSupp (0g‘(mulGrp‘𝑅)))
115 eqid 2737 . . . . . . . . . . . . . . . . . . . . 21 (1r𝑅) = (1r𝑅)
11647, 115ringidval 20180 . . . . . . . . . . . . . . . . . . . 20 (1r𝑅) = (0g‘(mulGrp‘𝑅))
117114, 116breqtrrdi 5185 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))) finSupp (1r𝑅))
11868fveq2d 6910 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (1r𝑅) = (1r‘(Scalar‘𝑈)))
119 eqid 2737 . . . . . . . . . . . . . . . . . . . . . 22 (1r‘(Scalar‘𝑈)) = (1r‘(Scalar‘𝑈))
12088, 119ringidval 20180 . . . . . . . . . . . . . . . . . . . . 21 (1r‘(Scalar‘𝑈)) = (0g‘(mulGrp‘(Scalar‘𝑈)))
121118, 120eqtrdi 2793 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (1r𝑅) = (0g‘(mulGrp‘(Scalar‘𝑈))))
122121ad2antrr 726 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → (1r𝑅) = (0g‘(mulGrp‘(Scalar‘𝑈))))
123117, 122breqtrd 5169 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))) finSupp (0g‘(mulGrp‘(Scalar‘𝑈))))
12485, 86, 91, 94, 25, 97, 105, 123gsummhm 19956 . . . . . . . . . . . . . . . . 17 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → ((mulGrp‘𝑈) Σg (𝐿 ∘ (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))) = (𝐿‘((mulGrp‘(Scalar‘𝑈)) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))))
12584, 124eqtr3d 2779 . . . . . . . . . . . . . . . 16 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → ((mulGrp‘𝑈) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴𝐽))‘𝑗)))) = (𝐿‘((mulGrp‘(Scalar‘𝑈)) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))))
126125oveq2d 7447 . . . . . . . . . . . . . . 15 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r𝑈)((mulGrp‘𝑈) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴𝐽))‘𝑗))))) = (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r𝑈)(𝐿‘((mulGrp‘(Scalar‘𝑈)) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))))
12764ad2antrr 726 . . . . . . . . . . . . . . . 16 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → 𝑈 ∈ AssAlg)
128101fmpttd 7135 . . . . . . . . . . . . . . . . 17 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))):𝐽⟶(Base‘(Scalar‘𝑈)))
129123, 120breqtrrdi 5185 . . . . . . . . . . . . . . . . 17 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))) finSupp (1r‘(Scalar‘𝑈)))
130103, 120, 91, 25, 128, 129gsumcl 19933 . . . . . . . . . . . . . . . 16 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → ((mulGrp‘(Scalar‘𝑈)) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))) ∈ (Base‘(Scalar‘𝑈)))
131 eqid 2737 . . . . . . . . . . . . . . . . 17 ( ·𝑠𝑈) = ( ·𝑠𝑈)
13231, 65, 102, 1, 2, 131asclmul2 21907 . . . . . . . . . . . . . . . 16 ((𝑈 ∈ AssAlg ∧ ((mulGrp‘(Scalar‘𝑈)) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))) ∈ (Base‘(Scalar‘𝑈)) ∧ ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒) ∈ (Base‘𝑈)) → (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r𝑈)(𝐿‘((mulGrp‘(Scalar‘𝑈)) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))) = (((mulGrp‘(Scalar‘𝑈)) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))( ·𝑠𝑈)((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)))
133127, 130, 21, 132syl3anc 1373 . . . . . . . . . . . . . . 15 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r𝑈)(𝐿‘((mulGrp‘(Scalar‘𝑈)) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))) = (((mulGrp‘(Scalar‘𝑈)) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))( ·𝑠𝑈)((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)))
134126, 133eqtrd 2777 . . . . . . . . . . . . . 14 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r𝑈)((mulGrp‘𝑈) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴𝐽))‘𝑗))))) = (((mulGrp‘(Scalar‘𝑈)) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))( ·𝑠𝑈)((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)))
135134fveq1d 6908 . . . . . . . . . . . . 13 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → ((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r𝑈)((mulGrp‘𝑈) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴𝐽))‘𝑗)))))‘𝑐) = ((((mulGrp‘(Scalar‘𝑈)) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))( ·𝑠𝑈)((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒))‘𝑐))
136 eqid 2737 . . . . . . . . . . . . . 14 (.r𝑅) = (.r𝑅)
137 eqid 2737 . . . . . . . . . . . . . 14 {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}
13899ad2antrr 726 . . . . . . . . . . . . . . 15 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → 𝐾 = (Base‘(Scalar‘𝑈)))
139130, 138eleqtrrd 2844 . . . . . . . . . . . . . 14 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → ((mulGrp‘(Scalar‘𝑈)) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))) ∈ 𝐾)
140 simplr 769 . . . . . . . . . . . . . 14 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → 𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin})
1413, 131, 28, 1, 136, 137, 139, 21, 140mplvscaval 22036 . . . . . . . . . . . . 13 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → ((((mulGrp‘(Scalar‘𝑈)) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))( ·𝑠𝑈)((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒))‘𝑐) = (((mulGrp‘(Scalar‘𝑈)) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))(.r𝑅)(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)))
142135, 141eqtrd 2777 . . . . . . . . . . . 12 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → ((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r𝑈)((mulGrp‘𝑈) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴𝐽))‘𝑗)))))‘𝑐) = (((mulGrp‘(Scalar‘𝑈)) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))(.r𝑅)(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)))
143142mpteq2dva 5242 . . . . . . . . . . 11 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ ((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r𝑈)((mulGrp‘𝑈) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴𝐽))‘𝑗)))))‘𝑐)) = (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((mulGrp‘(Scalar‘𝑈)) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))(.r𝑅)(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐))))
14445, 143eqtrd 2777 . . . . . . . . . 10 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → ((𝑢 ∈ (Base‘𝑈) ↦ (𝑢𝑐)) ∘ (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r𝑈)((mulGrp‘𝑈) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴𝐽))‘𝑗))))))) = (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((mulGrp‘(Scalar‘𝑈)) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))(.r𝑅)(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐))))
145144oveq2d 7447 . . . . . . . . 9 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑅 Σg ((𝑢 ∈ (Base‘𝑈) ↦ (𝑢𝑐)) ∘ (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r𝑈)((mulGrp‘𝑈) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴𝐽))‘𝑗)))))))) = (𝑅 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((mulGrp‘(Scalar‘𝑈)) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))(.r𝑅)(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)))))
14669fveq2d 6910 . . . . . . . . . . . . . . 15 (𝜑 → (mulGrp‘(Scalar‘𝑈)) = (mulGrp‘𝑅))
147146ad2antrr 726 . . . . . . . . . . . . . 14 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → (mulGrp‘(Scalar‘𝑈)) = (mulGrp‘𝑅))
148147oveq1d 7446 . . . . . . . . . . . . 13 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → ((mulGrp‘(Scalar‘𝑈)) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))) = ((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))
149148oveq1d 7446 . . . . . . . . . . . 12 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → (((mulGrp‘(Scalar‘𝑈)) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))(.r𝑅)(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)) = (((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))(.r𝑅)(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)))
1507ad2antrr 726 . . . . . . . . . . . . 13 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → 𝑅 ∈ CRing)
151148, 139eqeltrrd 2842 . . . . . . . . . . . . 13 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → ((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))) ∈ 𝐾)
15219ffvelcdmda 7104 . . . . . . . . . . . . . . . 16 ((𝜑𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒) ∈ (Base‘𝑈))
1533, 28, 1, 137, 152mplelf 22018 . . . . . . . . . . . . . . 15 ((𝜑𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒):{𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}⟶𝐾)
154153ffvelcdmda 7104 . . . . . . . . . . . . . 14 (((𝜑𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐) ∈ 𝐾)
155154an32s 652 . . . . . . . . . . . . 13 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐) ∈ 𝐾)
15628, 136, 150, 151, 155crngcomd 20252 . . . . . . . . . . . 12 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → (((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))(.r𝑅)(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)) = ((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))))
157149, 156eqtrd 2777 . . . . . . . . . . 11 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → (((mulGrp‘(Scalar‘𝑈)) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))(.r𝑅)(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)) = ((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))))
158157mpteq2dva 5242 . . . . . . . . . 10 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((mulGrp‘(Scalar‘𝑈)) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))(.r𝑅)(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐))) = (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ ((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))))
159158oveq2d 7447 . . . . . . . . 9 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑅 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((mulGrp‘(Scalar‘𝑈)) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))(.r𝑅)(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)))) = (𝑅 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ ((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))))))
160145, 159eqtrd 2777 . . . . . . . 8 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑅 Σg ((𝑢 ∈ (Base‘𝑈) ↦ (𝑢𝑐)) ∘ (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r𝑈)((mulGrp‘𝑈) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴𝐽))‘𝑗)))))))) = (𝑅 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ ((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))))))
161160oveq1d 7446 . . . . . . 7 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → ((𝑅 Σg ((𝑢 ∈ (Base‘𝑈) ↦ (𝑢𝑐)) ∘ (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r𝑈)((mulGrp‘𝑈) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴𝐽))‘𝑗))))))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑐𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))) = ((𝑅 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ ((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑐𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))))
162 eqid 2737 . . . . . . . . . 10 (𝑢 ∈ (Base‘𝑈) ↦ (𝑢𝑐)) = (𝑢 ∈ (Base‘𝑈) ↦ (𝑢𝑐))
163 fveq1 6905 . . . . . . . . . 10 (𝑢 = (𝑈 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r𝑈)((mulGrp‘𝑈) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴𝐽))‘𝑗))))))) → (𝑢𝑐) = ((𝑈 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r𝑈)((mulGrp‘𝑈) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴𝐽))‘𝑗)))))))‘𝑐))
164 eqid 2737 . . . . . . . . . . . . 13 (𝐽 eval 𝑈) = (𝐽 eval 𝑈)
165164, 11, 12, 13, 1, 22, 23, 2, 24, 8, 18, 37evlvvval 42583 . . . . . . . . . . . 12 (𝜑 → (((𝐽 eval 𝑈)‘(((𝐼 selectVars 𝑅)‘𝐽)‘𝐹))‘(𝐿 ∘ (𝐴𝐽))) = (𝑈 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r𝑈)((mulGrp‘𝑈) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴𝐽))‘𝑗))))))))
166164, 11, 12, 1, 24, 8, 18, 37evlcl 42582 . . . . . . . . . . . 12 (𝜑 → (((𝐽 eval 𝑈)‘(((𝐼 selectVars 𝑅)‘𝐽)‘𝐹))‘(𝐿 ∘ (𝐴𝐽))) ∈ (Base‘𝑈))
167165, 166eqeltrrd 2842 . . . . . . . . . . 11 (𝜑 → (𝑈 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r𝑈)((mulGrp‘𝑈) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴𝐽))‘𝑗))))))) ∈ (Base‘𝑈))
168167adantr 480 . . . . . . . . . 10 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑈 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r𝑈)((mulGrp‘𝑈) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴𝐽))‘𝑗))))))) ∈ (Base‘𝑈))
169 fvexd 6921 . . . . . . . . . 10 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → ((𝑈 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r𝑈)((mulGrp‘𝑈) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴𝐽))‘𝑗)))))))‘𝑐) ∈ V)
170162, 163, 168, 169fvmptd3 7039 . . . . . . . . 9 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → ((𝑢 ∈ (Base‘𝑈) ↦ (𝑢𝑐))‘(𝑈 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r𝑈)((mulGrp‘𝑈) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴𝐽))‘𝑗)))))))) = ((𝑈 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r𝑈)((mulGrp‘𝑈) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴𝐽))‘𝑗)))))))‘𝑐))
171 eqid 2737 . . . . . . . . . 10 (0g𝑈) = (0g𝑈)
1729ringcmnd 20281 . . . . . . . . . . 11 (𝜑𝑈 ∈ CMnd)
173172adantr 480 . . . . . . . . . 10 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → 𝑈 ∈ CMnd)
1747crnggrpd 20244 . . . . . . . . . . . 12 (𝜑𝑅 ∈ Grp)
175174grpmndd 18964 . . . . . . . . . . 11 (𝜑𝑅 ∈ Mnd)
176175adantr 480 . . . . . . . . . 10 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → 𝑅 ∈ Mnd)
177 ovex 7464 . . . . . . . . . . . 12 (ℕ0m 𝐽) ∈ V
178177rabex 5339 . . . . . . . . . . 11 {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ∈ V
179178a1i 11 . . . . . . . . . 10 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ∈ V)
1806adantr 480 . . . . . . . . . . . 12 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝐼𝐽) ∈ V)
181174adantr 480 . . . . . . . . . . . 12 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → 𝑅 ∈ Grp)
182 simpr 484 . . . . . . . . . . . 12 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → 𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin})
1833, 1, 137, 162, 180, 181, 182mplmapghm 42566 . . . . . . . . . . 11 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑢 ∈ (Base‘𝑈) ↦ (𝑢𝑐)) ∈ (𝑈 GrpHom 𝑅))
184 ghmmhm 19244 . . . . . . . . . . 11 ((𝑢 ∈ (Base‘𝑈) ↦ (𝑢𝑐)) ∈ (𝑈 GrpHom 𝑅) → (𝑢 ∈ (Base‘𝑈) ↦ (𝑢𝑐)) ∈ (𝑈 MndHom 𝑅))
185183, 184syl 17 . . . . . . . . . 10 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑢 ∈ (Base‘𝑈) ↦ (𝑢𝑐)) ∈ (𝑈 MndHom 𝑅))
18641fmpttd 7135 . . . . . . . . . 10 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r𝑈)((mulGrp‘𝑈) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴𝐽))‘𝑗)))))):{𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}⟶(Base‘𝑈))
18724adantr 480 . . . . . . . . . . 11 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → 𝐽 ∈ V)
1888adantr 480 . . . . . . . . . . 11 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → 𝑈 ∈ CRing)
18918adantr 480 . . . . . . . . . . 11 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (((𝐼 selectVars 𝑅)‘𝐽)‘𝐹) ∈ (Base‘𝑇))
19037adantr 480 . . . . . . . . . . 11 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝐿 ∘ (𝐴𝐽)) ∈ ((Base‘𝑈) ↑m 𝐽))
19113, 11, 12, 1, 22, 23, 2, 187, 188, 189, 190evlvvvallem 42584 . . . . . . . . . 10 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r𝑈)((mulGrp‘𝑈) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴𝐽))‘𝑗)))))) finSupp (0g𝑈))
1921, 171, 173, 176, 179, 185, 186, 191gsummhm 19956 . . . . . . . . 9 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑅 Σg ((𝑢 ∈ (Base‘𝑈) ↦ (𝑢𝑐)) ∘ (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r𝑈)((mulGrp‘𝑈) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴𝐽))‘𝑗)))))))) = ((𝑢 ∈ (Base‘𝑈) ↦ (𝑢𝑐))‘(𝑈 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r𝑈)((mulGrp‘𝑈) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴𝐽))‘𝑗)))))))))
193165adantr 480 . . . . . . . . . 10 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (((𝐽 eval 𝑈)‘(((𝐼 selectVars 𝑅)‘𝐽)‘𝐹))‘(𝐿 ∘ (𝐴𝐽))) = (𝑈 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r𝑈)((mulGrp‘𝑈) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴𝐽))‘𝑗))))))))
194193fveq1d 6908 . . . . . . . . 9 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → ((((𝐽 eval 𝑈)‘(((𝐼 selectVars 𝑅)‘𝐽)‘𝐹))‘(𝐿 ∘ (𝐴𝐽)))‘𝑐) = ((𝑈 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r𝑈)((mulGrp‘𝑈) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴𝐽))‘𝑗)))))))‘𝑐))
195170, 192, 1943eqtr4rd 2788 . . . . . . . 8 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → ((((𝐽 eval 𝑈)‘(((𝐼 selectVars 𝑅)‘𝐽)‘𝐹))‘(𝐿 ∘ (𝐴𝐽)))‘𝑐) = (𝑅 Σg ((𝑢 ∈ (Base‘𝑈) ↦ (𝑢𝑐)) ∘ (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r𝑈)((mulGrp‘𝑈) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴𝐽))‘𝑗)))))))))
196195oveq1d 7446 . . . . . . 7 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (((((𝐽 eval 𝑈)‘(((𝐼 selectVars 𝑅)‘𝐽)‘𝐹))‘(𝐿 ∘ (𝐴𝐽)))‘𝑐)(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑐𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))) = ((𝑅 Σg ((𝑢 ∈ (Base‘𝑈) ↦ (𝑢𝑐)) ∘ (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r𝑈)((mulGrp‘𝑈) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴𝐽))‘𝑗))))))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑐𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))))
197 eqid 2737 . . . . . . . 8 (0g𝑅) = (0g𝑅)
19832adantr 480 . . . . . . . 8 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → 𝑅 ∈ Ring)
19947crngmgp 20238 . . . . . . . . . . 11 (𝑅 ∈ CRing → (mulGrp‘𝑅) ∈ CMnd)
2007, 199syl 17 . . . . . . . . . 10 (𝜑 → (mulGrp‘𝑅) ∈ CMnd)
201200adantr 480 . . . . . . . . 9 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (mulGrp‘𝑅) ∈ CMnd)
20251ad2antrr 726 . . . . . . . . . . 11 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ (𝐼𝐽)) → (mulGrp‘𝑅) ∈ Mnd)
203137psrbagf 21938 . . . . . . . . . . . . 13 (𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} → 𝑐:(𝐼𝐽)⟶ℕ0)
204203adantl 481 . . . . . . . . . . . 12 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → 𝑐:(𝐼𝐽)⟶ℕ0)
205204ffvelcdmda 7104 . . . . . . . . . . 11 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ (𝐼𝐽)) → (𝑐𝑘) ∈ ℕ0)
20657, 5fssresd 6775 . . . . . . . . . . . . 13 (𝜑 → (𝐴 ↾ (𝐼𝐽)):(𝐼𝐽)⟶𝐾)
207206adantr 480 . . . . . . . . . . . 12 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝐴 ↾ (𝐼𝐽)):(𝐼𝐽)⟶𝐾)
208207ffvelcdmda 7104 . . . . . . . . . . 11 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ (𝐼𝐽)) → ((𝐴 ↾ (𝐼𝐽))‘𝑘) ∈ 𝐾)
20948, 49, 202, 205, 208mulgnn0cld 19113 . . . . . . . . . 10 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ (𝐼𝐽)) → ((𝑐𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)) ∈ 𝐾)
210209fmpttd 7135 . . . . . . . . 9 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑘 ∈ (𝐼𝐽) ↦ ((𝑐𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))):(𝐼𝐽)⟶𝐾)
211204feqmptd 6977 . . . . . . . . . . 11 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → 𝑐 = (𝑘 ∈ (𝐼𝐽) ↦ (𝑐𝑘)))
212137psrbagfsupp 21939 . . . . . . . . . . . 12 (𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} → 𝑐 finSupp 0)
213212adantl 481 . . . . . . . . . . 11 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → 𝑐 finSupp 0)
214211, 213eqbrtrrd 5167 . . . . . . . . . 10 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑘 ∈ (𝐼𝐽) ↦ (𝑐𝑘)) finSupp 0)
21548, 110, 49mulg0 19092 . . . . . . . . . . 11 (𝑣𝐾 → (0(.g‘(mulGrp‘𝑅))𝑣) = (0g‘(mulGrp‘𝑅)))
216215adantl 481 . . . . . . . . . 10 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑣𝐾) → (0(.g‘(mulGrp‘𝑅))𝑣) = (0g‘(mulGrp‘𝑅)))
217 fvexd 6921 . . . . . . . . . 10 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ (𝐼𝐽)) → (𝑐𝑘) ∈ V)
218 fvexd 6921 . . . . . . . . . 10 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (0g‘(mulGrp‘𝑅)) ∈ V)
219214, 216, 217, 208, 218fsuppssov1 9424 . . . . . . . . 9 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑘 ∈ (𝐼𝐽) ↦ ((𝑐𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))) finSupp (0g‘(mulGrp‘𝑅)))
22048, 110, 201, 180, 210, 219gsumcl 19933 . . . . . . . 8 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → ((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑐𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)))) ∈ 𝐾)
22132ad2antrr 726 . . . . . . . . 9 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → 𝑅 ∈ Ring)
22228, 136, 221, 155, 151ringcld 20257 . . . . . . . 8 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → ((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))) ∈ 𝐾)
223178mptex 7243 . . . . . . . . . . 11 (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)) ∈ V
224223a1i 11 . . . . . . . . . 10 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)) ∈ V)
225 fvexd 6921 . . . . . . . . . 10 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (0g𝑅) ∈ V)
226 funmpt 6604 . . . . . . . . . . 11 Fun (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐))
227226a1i 11 . . . . . . . . . 10 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → Fun (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)))
22811, 12, 171, 18mplelsfi 22015 . . . . . . . . . . 11 (𝜑 → (((𝐼 selectVars 𝑅)‘𝐽)‘𝐹) finSupp (0g𝑈))
229228adantr 480 . . . . . . . . . 10 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (((𝐼 selectVars 𝑅)‘𝐽)‘𝐹) finSupp (0g𝑈))
230 ssidd 4007 . . . . . . . . . . . . . 14 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹) supp (0g𝑈)) ⊆ ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹) supp (0g𝑈)))
231 fvexd 6921 . . . . . . . . . . . . . 14 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (0g𝑈) ∈ V)
23220, 230, 179, 231suppssr 8220 . . . . . . . . . . . . 13 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ ({𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ∖ ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹) supp (0g𝑈)))) → ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒) = (0g𝑈))
233232fveq1d 6908 . . . . . . . . . . . 12 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ ({𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ∖ ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹) supp (0g𝑈)))) → (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐) = ((0g𝑈)‘𝑐))
2343, 137, 197, 171, 6, 174mpl0 22026 . . . . . . . . . . . . . . . 16 (𝜑 → (0g𝑈) = ({𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} × {(0g𝑅)}))
235234adantr 480 . . . . . . . . . . . . . . 15 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (0g𝑈) = ({𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} × {(0g𝑅)}))
236235fveq1d 6908 . . . . . . . . . . . . . 14 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → ((0g𝑈)‘𝑐) = (({𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} × {(0g𝑅)})‘𝑐))
237 fvex 6919 . . . . . . . . . . . . . . . 16 (0g𝑅) ∈ V
238237fvconst2 7224 . . . . . . . . . . . . . . 15 (𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} → (({𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} × {(0g𝑅)})‘𝑐) = (0g𝑅))
239238adantl 481 . . . . . . . . . . . . . 14 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (({𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} × {(0g𝑅)})‘𝑐) = (0g𝑅))
240236, 239eqtrd 2777 . . . . . . . . . . . . 13 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → ((0g𝑈)‘𝑐) = (0g𝑅))
241240adantr 480 . . . . . . . . . . . 12 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ ({𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ∖ ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹) supp (0g𝑈)))) → ((0g𝑈)‘𝑐) = (0g𝑅))
242233, 241eqtrd 2777 . . . . . . . . . . 11 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ ({𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ∖ ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹) supp (0g𝑈)))) → (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐) = (0g𝑅))
243242, 179suppss2 8225 . . . . . . . . . 10 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → ((𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)) supp (0g𝑅)) ⊆ ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹) supp (0g𝑈)))
244224, 225, 227, 229, 243fsuppsssuppgd 9422 . . . . . . . . 9 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)) finSupp (0g𝑅))
24532ad2antrr 726 . . . . . . . . . 10 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑣𝐾) → 𝑅 ∈ Ring)
246 simpr 484 . . . . . . . . . 10 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑣𝐾) → 𝑣𝐾)
24728, 136, 197, 245, 246ringlzd 20292 . . . . . . . . 9 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑣𝐾) → ((0g𝑅)(.r𝑅)𝑣) = (0g𝑅))
248244, 247, 155, 151, 225fsuppssov1 9424 . . . . . . . 8 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ ((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))) finSupp (0g𝑅))
24928, 197, 136, 198, 179, 220, 222, 248gsummulc1 20313 . . . . . . 7 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑅 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑐𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))))) = ((𝑅 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ ((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑐𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))))
250161, 196, 2493eqtr4d 2787 . . . . . 6 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (((((𝐽 eval 𝑈)‘(((𝐼 selectVars 𝑅)‘𝐽)‘𝐹))‘(𝐿 ∘ (𝐴𝐽)))‘𝑐)(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑐𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))) = (𝑅 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑐𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))))))
251 fveq2 6906 . . . . . . . . . . . . . 14 (𝑎 = 𝑒 → ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎) = ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒))
252251adantl 481 . . . . . . . . . . . . 13 ((𝑏 = 𝑐𝑎 = 𝑒) → ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎) = ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒))
253 simpl 482 . . . . . . . . . . . . 13 ((𝑏 = 𝑐𝑎 = 𝑒) → 𝑏 = 𝑐)
254252, 253fveq12d 6913 . . . . . . . . . . . 12 ((𝑏 = 𝑐𝑎 = 𝑒) → (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏) = (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐))
255 fveq1 6905 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑒 → (𝑎𝑗) = (𝑒𝑗))
256255adantl 481 . . . . . . . . . . . . . . 15 ((𝑏 = 𝑐𝑎 = 𝑒) → (𝑎𝑗) = (𝑒𝑗))
257256oveq1d 7446 . . . . . . . . . . . . . 14 ((𝑏 = 𝑐𝑎 = 𝑒) → ((𝑎𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)) = ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))
258257mpteq2dv 5244 . . . . . . . . . . . . 13 ((𝑏 = 𝑐𝑎 = 𝑒) → (𝑗𝐽 ↦ ((𝑎𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))) = (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))
259258oveq2d 7447 . . . . . . . . . . . 12 ((𝑏 = 𝑐𝑎 = 𝑒) → ((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑎𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))) = ((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))
260254, 259oveq12d 7449 . . . . . . . . . . 11 ((𝑏 = 𝑐𝑎 = 𝑒) → ((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑎𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))) = ((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))))
261 fveq1 6905 . . . . . . . . . . . . . . 15 (𝑏 = 𝑐 → (𝑏𝑘) = (𝑐𝑘))
262261adantr 480 . . . . . . . . . . . . . 14 ((𝑏 = 𝑐𝑎 = 𝑒) → (𝑏𝑘) = (𝑐𝑘))
263262oveq1d 7446 . . . . . . . . . . . . 13 ((𝑏 = 𝑐𝑎 = 𝑒) → ((𝑏𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)) = ((𝑐𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)))
264263mpteq2dv 5244 . . . . . . . . . . . 12 ((𝑏 = 𝑐𝑎 = 𝑒) → (𝑘 ∈ (𝐼𝐽) ↦ ((𝑏𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))) = (𝑘 ∈ (𝐼𝐽) ↦ ((𝑐𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))
265264oveq2d 7447 . . . . . . . . . . 11 ((𝑏 = 𝑐𝑎 = 𝑒) → ((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑏𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)))) = ((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑐𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)))))
266260, 265oveq12d 7449 . . . . . . . . . 10 ((𝑏 = 𝑐𝑎 = 𝑒) → (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑎𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑏𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))) = (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑐𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))))
267 eqid 2737 . . . . . . . . . 10 (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑎𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑏𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)))))) = (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑎𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑏𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))))
268 ovex 7464 . . . . . . . . . 10 (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑐𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))) ∈ V
269266, 267, 268ovmpoa 7588 . . . . . . . . 9 ((𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → (𝑐(𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑎𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑏𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))))𝑒) = (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑐𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))))
270269adantll 714 . . . . . . . 8 (((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → (𝑐(𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑎𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑏𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))))𝑒) = (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑐𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))))
271270mpteq2dva 5242 . . . . . . 7 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (𝑐(𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑎𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑏𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))))𝑒)) = (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑐𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)))))))
272271oveq2d 7447 . . . . . 6 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑅 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (𝑐(𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑎𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑏𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))))𝑒))) = (𝑅 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑒𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑐𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))))))
273250, 272eqtr4d 2780 . . . . 5 ((𝜑𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (((((𝐽 eval 𝑈)‘(((𝐼 selectVars 𝑅)‘𝐽)‘𝐹))‘(𝐿 ∘ (𝐴𝐽)))‘𝑐)(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑐𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))) = (𝑅 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (𝑐(𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑎𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑏𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))))𝑒))))
274273mpteq2dva 5242 . . . 4 (𝜑 → (𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ↦ (((((𝐽 eval 𝑈)‘(((𝐼 selectVars 𝑅)‘𝐽)‘𝐹))‘(𝐿 ∘ (𝐴𝐽)))‘𝑐)(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑐𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)))))) = (𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ↦ (𝑅 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (𝑐(𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑎𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑏𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))))𝑒)))))
275274oveq2d 7447 . . 3 (𝜑 → (𝑅 Σg (𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ↦ (((((𝐽 eval 𝑈)‘(((𝐼 selectVars 𝑅)‘𝐽)‘𝐹))‘(𝐿 ∘ (𝐴𝐽)))‘𝑐)(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑐𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))))) = (𝑅 Σg (𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ↦ (𝑅 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (𝑐(𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑎𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑏𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))))𝑒))))))
27632ringcmnd 20281 . . . . 5 (𝜑𝑅 ∈ CMnd)
277 ovex 7464 . . . . . . 7 (ℕ0m 𝐼) ∈ V
278277rabex 5339 . . . . . 6 { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∈ V
279278a1i 11 . . . . 5 (𝜑 → { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∈ V)
28032adantr 480 . . . . . . 7 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → 𝑅 ∈ Ring)
28119adantr 480 . . . . . . . . . . 11 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (((𝐼 selectVars 𝑅)‘𝐽)‘𝐹):{𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}⟶(Base‘𝑈))
282 eqid 2737 . . . . . . . . . . . 12 { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} = { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}
2834adantr 480 . . . . . . . . . . . 12 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → 𝐼𝑉)
28416adantr 480 . . . . . . . . . . . 12 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → 𝐽𝐼)
285 simpr 484 . . . . . . . . . . . 12 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → 𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin})
286282, 13, 283, 284, 285psrbagres 42556 . . . . . . . . . . 11 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝑑𝐽) ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})
287281, 286ffvelcdmd 7105 . . . . . . . . . 10 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑𝐽)) ∈ (Base‘𝑈))
2883, 28, 1, 137, 287mplelf 22018 . . . . . . . . 9 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑𝐽)):{𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}⟶𝐾)
289 difssd 4137 . . . . . . . . . 10 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝐼𝐽) ⊆ 𝐼)
290282, 137, 283, 289, 285psrbagres 42556 . . . . . . . . 9 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝑑 ↾ (𝐼𝐽)) ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin})
291288, 290ffvelcdmd 7105 . . . . . . . 8 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑𝐽))‘(𝑑 ↾ (𝐼𝐽))) ∈ 𝐾)
292200adantr 480 . . . . . . . . 9 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (mulGrp‘𝑅) ∈ CMnd)
29324adantr 480 . . . . . . . . 9 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → 𝐽 ∈ V)
29451ad2antrr 726 . . . . . . . . . . 11 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑗𝐽) → (mulGrp‘𝑅) ∈ Mnd)
295282psrbagf 21938 . . . . . . . . . . . . . 14 (𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} → 𝑑:𝐼⟶ℕ0)
296295adantl 481 . . . . . . . . . . . . 13 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → 𝑑:𝐼⟶ℕ0)
297296, 284fssresd 6775 . . . . . . . . . . . 12 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝑑𝐽):𝐽⟶ℕ0)
298297ffvelcdmda 7104 . . . . . . . . . . 11 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑗𝐽) → ((𝑑𝐽)‘𝑗) ∈ ℕ0)
29958ffvelcdmda 7104 . . . . . . . . . . . 12 ((𝜑𝑗𝐽) → ((𝐴𝐽)‘𝑗) ∈ 𝐾)
300299adantlr 715 . . . . . . . . . . 11 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑗𝐽) → ((𝐴𝐽)‘𝑗) ∈ 𝐾)
30148, 49, 294, 298, 300mulgnn0cld 19113 . . . . . . . . . 10 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑗𝐽) → (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)) ∈ 𝐾)
302301fmpttd 7135 . . . . . . . . 9 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))):𝐽𝐾)
30324mptexd 7244 . . . . . . . . . . 11 (𝜑 → (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))) ∈ V)
304303adantr 480 . . . . . . . . . 10 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))) ∈ V)
305 fvexd 6921 . . . . . . . . . 10 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (0g‘(mulGrp‘𝑅)) ∈ V)
306 funmpt 6604 . . . . . . . . . . 11 Fun (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))
307306a1i 11 . . . . . . . . . 10 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → Fun (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))
308282psrbagfsupp 21939 . . . . . . . . . . . 12 (𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} → 𝑑 finSupp 0)
309308adantl 481 . . . . . . . . . . 11 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → 𝑑 finSupp 0)
310 0zd 12625 . . . . . . . . . . 11 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → 0 ∈ ℤ)
311309, 310fsuppres 9433 . . . . . . . . . 10 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝑑𝐽) finSupp 0)
312 ssidd 4007 . . . . . . . . . . . . . 14 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → ((𝑑𝐽) supp 0) ⊆ ((𝑑𝐽) supp 0))
313297, 312, 293, 310suppssr 8220 . . . . . . . . . . . . 13 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑗 ∈ (𝐽 ∖ ((𝑑𝐽) supp 0))) → ((𝑑𝐽)‘𝑗) = 0)
314313oveq1d 7446 . . . . . . . . . . . 12 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑗 ∈ (𝐽 ∖ ((𝑑𝐽) supp 0))) → (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)) = (0(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))
315 eldifi 4131 . . . . . . . . . . . . 13 (𝑗 ∈ (𝐽 ∖ ((𝑑𝐽) supp 0)) → 𝑗𝐽)
31648, 110, 49mulg0 19092 . . . . . . . . . . . . . 14 (((𝐴𝐽)‘𝑗) ∈ 𝐾 → (0(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)) = (0g‘(mulGrp‘𝑅)))
317300, 316syl 17 . . . . . . . . . . . . 13 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑗𝐽) → (0(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)) = (0g‘(mulGrp‘𝑅)))
318315, 317sylan2 593 . . . . . . . . . . . 12 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑗 ∈ (𝐽 ∖ ((𝑑𝐽) supp 0))) → (0(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)) = (0g‘(mulGrp‘𝑅)))
319314, 318eqtrd 2777 . . . . . . . . . . 11 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑗 ∈ (𝐽 ∖ ((𝑑𝐽) supp 0))) → (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)) = (0g‘(mulGrp‘𝑅)))
320319, 293suppss2 8225 . . . . . . . . . 10 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → ((𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))) supp (0g‘(mulGrp‘𝑅))) ⊆ ((𝑑𝐽) supp 0))
321304, 305, 307, 311, 320fsuppsssuppgd 9422 . . . . . . . . 9 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))) finSupp (0g‘(mulGrp‘𝑅)))
32248, 110, 292, 293, 302, 321gsumcl 19933 . . . . . . . 8 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → ((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))) ∈ 𝐾)
32328, 136, 280, 291, 322ringcld 20257 . . . . . . 7 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → ((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑𝐽))‘(𝑑 ↾ (𝐼𝐽)))(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))) ∈ 𝐾)
3246adantr 480 . . . . . . . 8 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝐼𝐽) ∈ V)
32551ad2antrr 726 . . . . . . . . . 10 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ (𝐼𝐽)) → (mulGrp‘𝑅) ∈ Mnd)
326296, 289fssresd 6775 . . . . . . . . . . 11 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝑑 ↾ (𝐼𝐽)):(𝐼𝐽)⟶ℕ0)
327326ffvelcdmda 7104 . . . . . . . . . 10 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ (𝐼𝐽)) → ((𝑑 ↾ (𝐼𝐽))‘𝑘) ∈ ℕ0)
328206ffvelcdmda 7104 . . . . . . . . . . 11 ((𝜑𝑘 ∈ (𝐼𝐽)) → ((𝐴 ↾ (𝐼𝐽))‘𝑘) ∈ 𝐾)
329328adantlr 715 . . . . . . . . . 10 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ (𝐼𝐽)) → ((𝐴 ↾ (𝐼𝐽))‘𝑘) ∈ 𝐾)
33048, 49, 325, 327, 329mulgnn0cld 19113 . . . . . . . . 9 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ (𝐼𝐽)) → (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)) ∈ 𝐾)
331330fmpttd 7135 . . . . . . . 8 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝑘 ∈ (𝐼𝐽) ↦ (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))):(𝐼𝐽)⟶𝐾)
332324mptexd 7244 . . . . . . . . 9 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝑘 ∈ (𝐼𝐽) ↦ (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))) ∈ V)
333 funmpt 6604 . . . . . . . . . 10 Fun (𝑘 ∈ (𝐼𝐽) ↦ (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)))
334333a1i 11 . . . . . . . . 9 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → Fun (𝑘 ∈ (𝐼𝐽) ↦ (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))
335309, 310fsuppres 9433 . . . . . . . . 9 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝑑 ↾ (𝐼𝐽)) finSupp 0)
336 ssidd 4007 . . . . . . . . . . . . 13 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → ((𝑑 ↾ (𝐼𝐽)) supp 0) ⊆ ((𝑑 ↾ (𝐼𝐽)) supp 0))
337326, 336, 324, 310suppssr 8220 . . . . . . . . . . . 12 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ ((𝐼𝐽) ∖ ((𝑑 ↾ (𝐼𝐽)) supp 0))) → ((𝑑 ↾ (𝐼𝐽))‘𝑘) = 0)
338337oveq1d 7446 . . . . . . . . . . 11 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ ((𝐼𝐽) ∖ ((𝑑 ↾ (𝐼𝐽)) supp 0))) → (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)) = (0(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)))
339 eldifi 4131 . . . . . . . . . . . . 13 (𝑘 ∈ ((𝐼𝐽) ∖ ((𝑑 ↾ (𝐼𝐽)) supp 0)) → 𝑘 ∈ (𝐼𝐽))
340339, 329sylan2 593 . . . . . . . . . . . 12 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ ((𝐼𝐽) ∖ ((𝑑 ↾ (𝐼𝐽)) supp 0))) → ((𝐴 ↾ (𝐼𝐽))‘𝑘) ∈ 𝐾)
34148, 110, 49mulg0 19092 . . . . . . . . . . . 12 (((𝐴 ↾ (𝐼𝐽))‘𝑘) ∈ 𝐾 → (0(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)) = (0g‘(mulGrp‘𝑅)))
342340, 341syl 17 . . . . . . . . . . 11 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ ((𝐼𝐽) ∖ ((𝑑 ↾ (𝐼𝐽)) supp 0))) → (0(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)) = (0g‘(mulGrp‘𝑅)))
343338, 342eqtrd 2777 . . . . . . . . . 10 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ ((𝐼𝐽) ∖ ((𝑑 ↾ (𝐼𝐽)) supp 0))) → (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)) = (0g‘(mulGrp‘𝑅)))
344343, 324suppss2 8225 . . . . . . . . 9 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → ((𝑘 ∈ (𝐼𝐽) ↦ (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))) supp (0g‘(mulGrp‘𝑅))) ⊆ ((𝑑 ↾ (𝐼𝐽)) supp 0))
345332, 305, 334, 335, 344fsuppsssuppgd 9422 . . . . . . . 8 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝑘 ∈ (𝐼𝐽) ↦ (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))) finSupp (0g‘(mulGrp‘𝑅)))
34648, 110, 292, 324, 331, 345gsumcl 19933 . . . . . . 7 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → ((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)))) ∈ 𝐾)
34728, 136, 280, 323, 346ringcld 20257 . . . . . 6 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑𝐽))‘(𝑑 ↾ (𝐼𝐽)))(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))) ∈ 𝐾)
348347fmpttd 7135 . . . . 5 (𝜑 → (𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑𝐽))‘(𝑑 ↾ (𝐼𝐽)))(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)))))):{ ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}⟶𝐾)
3497adantr 480 . . . . . . . . . 10 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → 𝑅 ∈ CRing)
35017adantr 480 . . . . . . . . . 10 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → 𝐹𝐵)
351282, 14, 15, 349, 284, 350, 285selvvvval 42595 . . . . . . . . 9 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑𝐽))‘(𝑑 ↾ (𝐼𝐽))) = (𝐹𝑑))
352351mpteq2dva 5242 . . . . . . . 8 (𝜑 → (𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑𝐽))‘(𝑑 ↾ (𝐼𝐽)))) = (𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (𝐹𝑑)))
353 eqid 2737 . . . . . . . . . . 11 (Base‘𝑅) = (Base‘𝑅)
35414, 353, 15, 282, 17mplelf 22018 . . . . . . . . . 10 (𝜑𝐹:{ ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}⟶(Base‘𝑅))
355354feqmptd 6977 . . . . . . . . 9 (𝜑𝐹 = (𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (𝐹𝑑)))
35614, 15, 197, 17mplelsfi 22015 . . . . . . . . 9 (𝜑𝐹 finSupp (0g𝑅))
357355, 356eqbrtrrd 5167 . . . . . . . 8 (𝜑 → (𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (𝐹𝑑)) finSupp (0g𝑅))
358352, 357eqbrtrd 5165 . . . . . . 7 (𝜑 → (𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑𝐽))‘(𝑑 ↾ (𝐼𝐽)))) finSupp (0g𝑅))
35932adantr 480 . . . . . . . 8 ((𝜑𝑣𝐾) → 𝑅 ∈ Ring)
360 simpr 484 . . . . . . . 8 ((𝜑𝑣𝐾) → 𝑣𝐾)
36128, 136, 197, 359, 360ringlzd 20292 . . . . . . 7 ((𝜑𝑣𝐾) → ((0g𝑅)(.r𝑅)𝑣) = (0g𝑅))
362 fvexd 6921 . . . . . . 7 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑𝐽))‘(𝑑 ↾ (𝐼𝐽))) ∈ V)
363 fvexd 6921 . . . . . . 7 (𝜑 → (0g𝑅) ∈ V)
364358, 361, 362, 322, 363fsuppssov1 9424 . . . . . 6 (𝜑 → (𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ ((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑𝐽))‘(𝑑 ↾ (𝐼𝐽)))(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))) finSupp (0g𝑅))
365 ovexd 7466 . . . . . 6 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → ((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑𝐽))‘(𝑑 ↾ (𝐼𝐽)))(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))) ∈ V)
366364, 361, 365, 346, 363fsuppssov1 9424 . . . . 5 (𝜑 → (𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑𝐽))‘(𝑑 ↾ (𝐼𝐽)))(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)))))) finSupp (0g𝑅))
367 eqid 2737 . . . . . 6 (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (𝑏𝑎)) = (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (𝑏𝑎))
368282, 13, 137, 367, 4, 16evlselvlem 42596 . . . . 5 (𝜑 → (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (𝑏𝑎)):({𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} × {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})–1-1-onto→{ ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin})
36928, 197, 276, 279, 348, 366, 368gsumf1o 19934 . . . 4 (𝜑 → (𝑅 Σg (𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑𝐽))‘(𝑑 ↾ (𝐼𝐽)))(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))))) = (𝑅 Σg ((𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑𝐽))‘(𝑑 ↾ (𝐼𝐽)))(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)))))) ∘ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (𝑏𝑎)))))
370137psrbagf 21938 . . . . . . . . . 10 (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} → 𝑏:(𝐼𝐽)⟶ℕ0)
371370ad2antrl 728 . . . . . . . . 9 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → 𝑏:(𝐼𝐽)⟶ℕ0)
37213psrbagf 21938 . . . . . . . . . 10 (𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} → 𝑎:𝐽⟶ℕ0)
373372ad2antll 729 . . . . . . . . 9 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → 𝑎:𝐽⟶ℕ0)
374 disjdifr 4473 . . . . . . . . . 10 ((𝐼𝐽) ∩ 𝐽) = ∅
375374a1i 11 . . . . . . . . 9 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → ((𝐼𝐽) ∩ 𝐽) = ∅)
376371, 373, 375fun2d 6772 . . . . . . . 8 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → (𝑏𝑎):((𝐼𝐽) ∪ 𝐽)⟶ℕ0)
377 undifr 4483 . . . . . . . . . . 11 (𝐽𝐼 ↔ ((𝐼𝐽) ∪ 𝐽) = 𝐼)
37816, 377sylib 218 . . . . . . . . . 10 (𝜑 → ((𝐼𝐽) ∪ 𝐽) = 𝐼)
379378adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → ((𝐼𝐽) ∪ 𝐽) = 𝐼)
380379feq2d 6722 . . . . . . . 8 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → ((𝑏𝑎):((𝐼𝐽) ∪ 𝐽)⟶ℕ0 ↔ (𝑏𝑎):𝐼⟶ℕ0))
381376, 380mpbid 232 . . . . . . 7 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → (𝑏𝑎):𝐼⟶ℕ0)
382 vex 3484 . . . . . . . . . . 11 𝑏 ∈ V
383 vex 3484 . . . . . . . . . . 11 𝑎 ∈ V
384382, 383unex 7764 . . . . . . . . . 10 (𝑏𝑎) ∈ V
385384a1i 11 . . . . . . . . 9 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → (𝑏𝑎) ∈ V)
386 0zd 12625 . . . . . . . . 9 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → 0 ∈ ℤ)
387381ffund 6740 . . . . . . . . 9 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → Fun (𝑏𝑎))
388137psrbagfsupp 21939 . . . . . . . . . . 11 (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} → 𝑏 finSupp 0)
389388ad2antrl 728 . . . . . . . . . 10 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → 𝑏 finSupp 0)
39013psrbagfsupp 21939 . . . . . . . . . . 11 (𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} → 𝑎 finSupp 0)
391390ad2antll 729 . . . . . . . . . 10 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → 𝑎 finSupp 0)
392389, 391fsuppun 9427 . . . . . . . . 9 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → ((𝑏𝑎) supp 0) ∈ Fin)
393385, 386, 387, 392isfsuppd 9406 . . . . . . . 8 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → (𝑏𝑎) finSupp 0)
394 fcdmnn0fsuppg 12586 . . . . . . . . 9 (((𝑏𝑎) ∈ V ∧ (𝑏𝑎):𝐼⟶ℕ0) → ((𝑏𝑎) finSupp 0 ↔ ((𝑏𝑎) “ ℕ) ∈ Fin))
395385, 381, 394syl2anc 584 . . . . . . . 8 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → ((𝑏𝑎) finSupp 0 ↔ ((𝑏𝑎) “ ℕ) ∈ Fin))
396393, 395mpbid 232 . . . . . . 7 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → ((𝑏𝑎) “ ℕ) ∈ Fin)
3974adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → 𝐼𝑉)
398282psrbag 21937 . . . . . . . 8 (𝐼𝑉 → ((𝑏𝑎) ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↔ ((𝑏𝑎):𝐼⟶ℕ0 ∧ ((𝑏𝑎) “ ℕ) ∈ Fin)))
399397, 398syl 17 . . . . . . 7 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → ((𝑏𝑎) ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↔ ((𝑏𝑎):𝐼⟶ℕ0 ∧ ((𝑏𝑎) “ ℕ) ∈ Fin)))
400381, 396, 399mpbir2and 713 . . . . . 6 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → (𝑏𝑎) ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin})
401 eqidd 2738 . . . . . 6 (𝜑 → (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (𝑏𝑎)) = (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (𝑏𝑎)))
402 eqidd 2738 . . . . . 6 (𝜑 → (𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑𝐽))‘(𝑑 ↾ (𝐼𝐽)))(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)))))) = (𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑𝐽))‘(𝑑 ↾ (𝐼𝐽)))(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)))))))
403 reseq1 5991 . . . . . . . . . . . 12 (𝑑 = (𝑏𝑎) → (𝑑𝐽) = ((𝑏𝑎) ↾ 𝐽))
404403fveq2d 6910 . . . . . . . . . . 11 (𝑑 = (𝑏𝑎) → ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑𝐽)) = ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘((𝑏𝑎) ↾ 𝐽)))
405 reseq1 5991 . . . . . . . . . . 11 (𝑑 = (𝑏𝑎) → (𝑑 ↾ (𝐼𝐽)) = ((𝑏𝑎) ↾ (𝐼𝐽)))
406404, 405fveq12d 6913 . . . . . . . . . 10 (𝑑 = (𝑏𝑎) → (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑𝐽))‘(𝑑 ↾ (𝐼𝐽))) = (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘((𝑏𝑎) ↾ 𝐽))‘((𝑏𝑎) ↾ (𝐼𝐽))))
407403fveq1d 6908 . . . . . . . . . . . . 13 (𝑑 = (𝑏𝑎) → ((𝑑𝐽)‘𝑗) = (((𝑏𝑎) ↾ 𝐽)‘𝑗))
408407oveq1d 7446 . . . . . . . . . . . 12 (𝑑 = (𝑏𝑎) → (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)) = ((((𝑏𝑎) ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))
409408mpteq2dv 5244 . . . . . . . . . . 11 (𝑑 = (𝑏𝑎) → (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))) = (𝑗𝐽 ↦ ((((𝑏𝑎) ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))
410409oveq2d 7447 . . . . . . . . . 10 (𝑑 = (𝑏𝑎) → ((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))) = ((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((((𝑏𝑎) ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))
411406, 410oveq12d 7449 . . . . . . . . 9 (𝑑 = (𝑏𝑎) → ((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑𝐽))‘(𝑑 ↾ (𝐼𝐽)))(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))) = ((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘((𝑏𝑎) ↾ 𝐽))‘((𝑏𝑎) ↾ (𝐼𝐽)))(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((((𝑏𝑎) ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))))
412405fveq1d 6908 . . . . . . . . . . . 12 (𝑑 = (𝑏𝑎) → ((𝑑 ↾ (𝐼𝐽))‘𝑘) = (((𝑏𝑎) ↾ (𝐼𝐽))‘𝑘))
413412oveq1d 7446 . . . . . . . . . . 11 (𝑑 = (𝑏𝑎) → (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)) = ((((𝑏𝑎) ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)))
414413mpteq2dv 5244 . . . . . . . . . 10 (𝑑 = (𝑏𝑎) → (𝑘 ∈ (𝐼𝐽) ↦ (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))) = (𝑘 ∈ (𝐼𝐽) ↦ ((((𝑏𝑎) ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))
415414oveq2d 7447 . . . . . . . . 9 (𝑑 = (𝑏𝑎) → ((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)))) = ((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((((𝑏𝑎) ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)))))
416411, 415oveq12d 7449 . . . . . . . 8 (𝑑 = (𝑏𝑎) → (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑𝐽))‘(𝑑 ↾ (𝐼𝐽)))(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))) = (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘((𝑏𝑎) ↾ 𝐽))‘((𝑏𝑎) ↾ (𝐼𝐽)))(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((((𝑏𝑎) ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((((𝑏𝑎) ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))))
417384, 416csbie 3934 . . . . . . 7 (𝑏𝑎) / 𝑑(((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑𝐽))‘(𝑑 ↾ (𝐼𝐽)))(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))) = (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘((𝑏𝑎) ↾ 𝐽))‘((𝑏𝑎) ↾ (𝐼𝐽)))(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((((𝑏𝑎) ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((((𝑏𝑎) ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)))))
418370ffnd 6737 . . . . . . . . . . . . 13 (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} → 𝑏 Fn (𝐼𝐽))
419418ad2antrl 728 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → 𝑏 Fn (𝐼𝐽))
420373ffnd 6737 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → 𝑎 Fn 𝐽)
421 fnunres2 6681 . . . . . . . . . . . 12 ((𝑏 Fn (𝐼𝐽) ∧ 𝑎 Fn 𝐽 ∧ ((𝐼𝐽) ∩ 𝐽) = ∅) → ((𝑏𝑎) ↾ 𝐽) = 𝑎)
422419, 420, 375, 421syl3anc 1373 . . . . . . . . . . 11 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → ((𝑏𝑎) ↾ 𝐽) = 𝑎)
423422fveq2d 6910 . . . . . . . . . 10 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘((𝑏𝑎) ↾ 𝐽)) = ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎))
424 fnunres1 6680 . . . . . . . . . . 11 ((𝑏 Fn (𝐼𝐽) ∧ 𝑎 Fn 𝐽 ∧ ((𝐼𝐽) ∩ 𝐽) = ∅) → ((𝑏𝑎) ↾ (𝐼𝐽)) = 𝑏)
425419, 420, 375, 424syl3anc 1373 . . . . . . . . . 10 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → ((𝑏𝑎) ↾ (𝐼𝐽)) = 𝑏)
426423, 425fveq12d 6913 . . . . . . . . 9 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘((𝑏𝑎) ↾ 𝐽))‘((𝑏𝑎) ↾ (𝐼𝐽))) = (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏))
427422fveq1d 6908 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → (((𝑏𝑎) ↾ 𝐽)‘𝑗) = (𝑎𝑗))
428427oveq1d 7446 . . . . . . . . . . 11 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → ((((𝑏𝑎) ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)) = ((𝑎𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))
429428mpteq2dv 5244 . . . . . . . . . 10 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → (𝑗𝐽 ↦ ((((𝑏𝑎) ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))) = (𝑗𝐽 ↦ ((𝑎𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))
430429oveq2d 7447 . . . . . . . . 9 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → ((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((((𝑏𝑎) ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))) = ((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑎𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))
431426, 430oveq12d 7449 . . . . . . . 8 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → ((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘((𝑏𝑎) ↾ 𝐽))‘((𝑏𝑎) ↾ (𝐼𝐽)))(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((((𝑏𝑎) ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))) = ((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑎𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))))
432425fveq1d 6908 . . . . . . . . . . 11 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → (((𝑏𝑎) ↾ (𝐼𝐽))‘𝑘) = (𝑏𝑘))
433432oveq1d 7446 . . . . . . . . . 10 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → ((((𝑏𝑎) ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)) = ((𝑏𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)))
434433mpteq2dv 5244 . . . . . . . . 9 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → (𝑘 ∈ (𝐼𝐽) ↦ ((((𝑏𝑎) ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))) = (𝑘 ∈ (𝐼𝐽) ↦ ((𝑏𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))
435434oveq2d 7447 . . . . . . . 8 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → ((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((((𝑏𝑎) ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)))) = ((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑏𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)))))
436431, 435oveq12d 7449 . . . . . . 7 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘((𝑏𝑎) ↾ 𝐽))‘((𝑏𝑎) ↾ (𝐼𝐽)))(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((((𝑏𝑎) ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((((𝑏𝑎) ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))) = (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑎𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑏𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))))
437417, 436eqtrid 2789 . . . . . 6 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → (𝑏𝑎) / 𝑑(((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑𝐽))‘(𝑑 ↾ (𝐼𝐽)))(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))) = (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑎𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑏𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))))
438400, 401, 402, 437fmpocos 42275 . . . . 5 (𝜑 → ((𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑𝐽))‘(𝑑 ↾ (𝐼𝐽)))(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)))))) ∘ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (𝑏𝑎))) = (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑎𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑏𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)))))))
439438oveq2d 7447 . . . 4 (𝜑 → (𝑅 Σg ((𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑𝐽))‘(𝑑 ↾ (𝐼𝐽)))(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)))))) ∘ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (𝑏𝑎)))) = (𝑅 Σg (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑎𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑏𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))))))
440 ovex 7464 . . . . . . 7 (ℕ0m (𝐼𝐽)) ∈ V
441440rabex 5339 . . . . . 6 {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∈ V
442441a1i 11 . . . . 5 (𝜑 → {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∈ V)
443178a1i 11 . . . . 5 (𝜑 → {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ∈ V)
44432adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → 𝑅 ∈ Ring)
44519ffvelcdmda 7104 . . . . . . . . . . . . 13 ((𝜑𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎) ∈ (Base‘𝑈))
4463, 28, 1, 137, 445mplelf 22018 . . . . . . . . . . . 12 ((𝜑𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎):{𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}⟶𝐾)
447446ffvelcdmda 7104 . . . . . . . . . . 11 (((𝜑𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) ∧ 𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) → (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏) ∈ 𝐾)
448447an32s 652 . . . . . . . . . 10 (((𝜑𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin}) → (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏) ∈ 𝐾)
449448anasss 466 . . . . . . . . 9 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏) ∈ 𝐾)
45024adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → 𝐽 ∈ V)
4517adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → 𝑅 ∈ CRing)
45236adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → (𝐴𝐽) ∈ (𝐾m 𝐽))
453 simprr 773 . . . . . . . . . 10 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})
45413, 28, 47, 49, 450, 451, 452, 453evlsvvvallem 42571 . . . . . . . . 9 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → ((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑎𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))) ∈ 𝐾)
45528, 136, 444, 449, 454ringcld 20257 . . . . . . . 8 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → ((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑎𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))) ∈ 𝐾)
4566adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → (𝐼𝐽) ∈ V)
45735, 5elmapssresd 42282 . . . . . . . . . 10 (𝜑 → (𝐴 ↾ (𝐼𝐽)) ∈ (𝐾m (𝐼𝐽)))
458457adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → (𝐴 ↾ (𝐼𝐽)) ∈ (𝐾m (𝐼𝐽)))
459 simprl 771 . . . . . . . . 9 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → 𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin})
460137, 28, 47, 49, 456, 451, 458, 459evlsvvvallem 42571 . . . . . . . 8 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → ((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑏𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)))) ∈ 𝐾)
46128, 136, 444, 455, 460ringcld 20257 . . . . . . 7 ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})) → (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑎𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑏𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))) ∈ 𝐾)
462461ralrimivva 3202 . . . . . 6 (𝜑 → ∀𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}∀𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑎𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑏𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))) ∈ 𝐾)
463267fmpo 8093 . . . . . 6 (∀𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}∀𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑎𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑏𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))) ∈ 𝐾 ↔ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑎𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑏𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)))))):({𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} × {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})⟶𝐾)
464462, 463sylib 218 . . . . 5 (𝜑 → (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑎𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑏𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)))))):({𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} × {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})⟶𝐾)
465 f1of1 6847 . . . . . . . 8 ((𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (𝑏𝑎)):({𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} × {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})–1-1-onto→{ ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} → (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (𝑏𝑎)):({𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} × {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})–1-1→{ ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin})
466368, 465syl 17 . . . . . . 7 (𝜑 → (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (𝑏𝑎)):({𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} × {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin})–1-1→{ ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin})
467278mptex 7243 . . . . . . . 8 (𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑𝐽))‘(𝑑 ↾ (𝐼𝐽)))(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)))))) ∈ V
468467a1i 11 . . . . . . 7 (𝜑 → (𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑𝐽))‘(𝑑 ↾ (𝐼𝐽)))(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)))))) ∈ V)
469366, 466, 363, 468fsuppco 9442 . . . . . 6 (𝜑 → ((𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑𝐽))‘(𝑑 ↾ (𝐼𝐽)))(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)))))) ∘ (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (𝑏𝑎))) finSupp (0g𝑅))
470438, 469eqbrtrrd 5167 . . . . 5 (𝜑 → (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑎𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑏𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)))))) finSupp (0g𝑅))
47128, 197, 276, 442, 443, 464, 470gsumxp 19994 . . . 4 (𝜑 → (𝑅 Σg (𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑎𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑏𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))))) = (𝑅 Σg (𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ↦ (𝑅 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (𝑐(𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑎𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑏𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))))𝑒))))))
472369, 439, 4713eqtrd 2781 . . 3 (𝜑 → (𝑅 Σg (𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑𝐽))‘(𝑑 ↾ (𝐼𝐽)))(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))))) = (𝑅 Σg (𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ↦ (𝑅 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (𝑐(𝑏 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0m 𝐽) ∣ (𝑔 “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ ((𝑎𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑏𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))))𝑒))))))
47328, 136, 280, 291, 322, 346ringassd 20254 . . . . . 6 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑𝐽))‘(𝑑 ↾ (𝐼𝐽)))(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))) = ((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑𝐽))‘(𝑑 ↾ (𝐼𝐽)))(.r𝑅)(((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)))))))
47447, 136mgpplusg 20141 . . . . . . . . 9 (.r𝑅) = (+g‘(mulGrp‘𝑅))
47551ad2antrr 726 . . . . . . . . . . 11 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑖𝐼) → (mulGrp‘𝑅) ∈ Mnd)
476296ffvelcdmda 7104 . . . . . . . . . . 11 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑖𝐼) → (𝑑𝑖) ∈ ℕ0)
47757adantr 480 . . . . . . . . . . . 12 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → 𝐴:𝐼𝐾)
478477ffvelcdmda 7104 . . . . . . . . . . 11 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑖𝐼) → (𝐴𝑖) ∈ 𝐾)
47948, 49, 475, 476, 478mulgnn0cld 19113 . . . . . . . . . 10 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑖𝐼) → ((𝑑𝑖)(.g‘(mulGrp‘𝑅))(𝐴𝑖)) ∈ 𝐾)
480479fmpttd 7135 . . . . . . . . 9 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝑖𝐼 ↦ ((𝑑𝑖)(.g‘(mulGrp‘𝑅))(𝐴𝑖))):𝐼𝐾)
481296feqmptd 6977 . . . . . . . . . . 11 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → 𝑑 = (𝑖𝐼 ↦ (𝑑𝑖)))
482481, 309eqbrtrrd 5167 . . . . . . . . . 10 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝑖𝐼 ↦ (𝑑𝑖)) finSupp 0)
483111adantl 481 . . . . . . . . . 10 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑘𝐾) → (0(.g‘(mulGrp‘𝑅))𝑘) = (0g‘(mulGrp‘𝑅)))
484482, 483, 476, 478, 305fsuppssov1 9424 . . . . . . . . 9 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝑖𝐼 ↦ ((𝑑𝑖)(.g‘(mulGrp‘𝑅))(𝐴𝑖))) finSupp (0g‘(mulGrp‘𝑅)))
485 disjdif 4472 . . . . . . . . . 10 (𝐽 ∩ (𝐼𝐽)) = ∅
486485a1i 11 . . . . . . . . 9 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝐽 ∩ (𝐼𝐽)) = ∅)
487 undif 4482 . . . . . . . . . . . 12 (𝐽𝐼 ↔ (𝐽 ∪ (𝐼𝐽)) = 𝐼)
48816, 487sylib 218 . . . . . . . . . . 11 (𝜑 → (𝐽 ∪ (𝐼𝐽)) = 𝐼)
489488eqcomd 2743 . . . . . . . . . 10 (𝜑𝐼 = (𝐽 ∪ (𝐼𝐽)))
490489adantr 480 . . . . . . . . 9 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → 𝐼 = (𝐽 ∪ (𝐼𝐽)))
49148, 110, 474, 292, 283, 480, 484, 486, 490gsumsplit 19946 . . . . . . . 8 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → ((mulGrp‘𝑅) Σg (𝑖𝐼 ↦ ((𝑑𝑖)(.g‘(mulGrp‘𝑅))(𝐴𝑖)))) = (((mulGrp‘𝑅) Σg ((𝑖𝐼 ↦ ((𝑑𝑖)(.g‘(mulGrp‘𝑅))(𝐴𝑖))) ↾ 𝐽))(.r𝑅)((mulGrp‘𝑅) Σg ((𝑖𝐼 ↦ ((𝑑𝑖)(.g‘(mulGrp‘𝑅))(𝐴𝑖))) ↾ (𝐼𝐽)))))
492284resmptd 6058 . . . . . . . . . . 11 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → ((𝑖𝐼 ↦ ((𝑑𝑖)(.g‘(mulGrp‘𝑅))(𝐴𝑖))) ↾ 𝐽) = (𝑖𝐽 ↦ ((𝑑𝑖)(.g‘(mulGrp‘𝑅))(𝐴𝑖))))
493 fveq2 6906 . . . . . . . . . . . . . 14 (𝑖 = 𝑗 → (𝑑𝑖) = (𝑑𝑗))
494 fveq2 6906 . . . . . . . . . . . . . 14 (𝑖 = 𝑗 → (𝐴𝑖) = (𝐴𝑗))
495493, 494oveq12d 7449 . . . . . . . . . . . . 13 (𝑖 = 𝑗 → ((𝑑𝑖)(.g‘(mulGrp‘𝑅))(𝐴𝑖)) = ((𝑑𝑗)(.g‘(mulGrp‘𝑅))(𝐴𝑗)))
496495cbvmptv 5255 . . . . . . . . . . . 12 (𝑖𝐽 ↦ ((𝑑𝑖)(.g‘(mulGrp‘𝑅))(𝐴𝑖))) = (𝑗𝐽 ↦ ((𝑑𝑗)(.g‘(mulGrp‘𝑅))(𝐴𝑗)))
497 simpr 484 . . . . . . . . . . . . . . . 16 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑗𝐽) → 𝑗𝐽)
498497fvresd 6926 . . . . . . . . . . . . . . 15 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑗𝐽) → ((𝑑𝐽)‘𝑗) = (𝑑𝑗))
499497fvresd 6926 . . . . . . . . . . . . . . 15 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑗𝐽) → ((𝐴𝐽)‘𝑗) = (𝐴𝑗))
500498, 499oveq12d 7449 . . . . . . . . . . . . . 14 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑗𝐽) → (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)) = ((𝑑𝑗)(.g‘(mulGrp‘𝑅))(𝐴𝑗)))
501500eqcomd 2743 . . . . . . . . . . . . 13 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑗𝐽) → ((𝑑𝑗)(.g‘(mulGrp‘𝑅))(𝐴𝑗)) = (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))
502501mpteq2dva 5242 . . . . . . . . . . . 12 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝑗𝐽 ↦ ((𝑑𝑗)(.g‘(mulGrp‘𝑅))(𝐴𝑗))) = (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))
503496, 502eqtrid 2789 . . . . . . . . . . 11 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝑖𝐽 ↦ ((𝑑𝑖)(.g‘(mulGrp‘𝑅))(𝐴𝑖))) = (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))
504492, 503eqtrd 2777 . . . . . . . . . 10 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → ((𝑖𝐼 ↦ ((𝑑𝑖)(.g‘(mulGrp‘𝑅))(𝐴𝑖))) ↾ 𝐽) = (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))
505504oveq2d 7447 . . . . . . . . 9 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → ((mulGrp‘𝑅) Σg ((𝑖𝐼 ↦ ((𝑑𝑖)(.g‘(mulGrp‘𝑅))(𝐴𝑖))) ↾ 𝐽)) = ((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))
506289resmptd 6058 . . . . . . . . . . 11 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → ((𝑖𝐼 ↦ ((𝑑𝑖)(.g‘(mulGrp‘𝑅))(𝐴𝑖))) ↾ (𝐼𝐽)) = (𝑖 ∈ (𝐼𝐽) ↦ ((𝑑𝑖)(.g‘(mulGrp‘𝑅))(𝐴𝑖))))
507 fveq2 6906 . . . . . . . . . . . . . 14 (𝑖 = 𝑘 → (𝑑𝑖) = (𝑑𝑘))
508 fveq2 6906 . . . . . . . . . . . . . 14 (𝑖 = 𝑘 → (𝐴𝑖) = (𝐴𝑘))
509507, 508oveq12d 7449 . . . . . . . . . . . . 13 (𝑖 = 𝑘 → ((𝑑𝑖)(.g‘(mulGrp‘𝑅))(𝐴𝑖)) = ((𝑑𝑘)(.g‘(mulGrp‘𝑅))(𝐴𝑘)))
510509cbvmptv 5255 . . . . . . . . . . . 12 (𝑖 ∈ (𝐼𝐽) ↦ ((𝑑𝑖)(.g‘(mulGrp‘𝑅))(𝐴𝑖))) = (𝑘 ∈ (𝐼𝐽) ↦ ((𝑑𝑘)(.g‘(mulGrp‘𝑅))(𝐴𝑘)))
511 simpr 484 . . . . . . . . . . . . . . . 16 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ (𝐼𝐽)) → 𝑘 ∈ (𝐼𝐽))
512511fvresd 6926 . . . . . . . . . . . . . . 15 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ (𝐼𝐽)) → ((𝑑 ↾ (𝐼𝐽))‘𝑘) = (𝑑𝑘))
513511fvresd 6926 . . . . . . . . . . . . . . 15 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ (𝐼𝐽)) → ((𝐴 ↾ (𝐼𝐽))‘𝑘) = (𝐴𝑘))
514512, 513oveq12d 7449 . . . . . . . . . . . . . 14 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ (𝐼𝐽)) → (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)) = ((𝑑𝑘)(.g‘(mulGrp‘𝑅))(𝐴𝑘)))
515514eqcomd 2743 . . . . . . . . . . . . 13 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ (𝐼𝐽)) → ((𝑑𝑘)(.g‘(mulGrp‘𝑅))(𝐴𝑘)) = (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)))
516515mpteq2dva 5242 . . . . . . . . . . . 12 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝑘 ∈ (𝐼𝐽) ↦ ((𝑑𝑘)(.g‘(mulGrp‘𝑅))(𝐴𝑘))) = (𝑘 ∈ (𝐼𝐽) ↦ (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))
517510, 516eqtrid 2789 . . . . . . . . . . 11 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝑖 ∈ (𝐼𝐽) ↦ ((𝑑𝑖)(.g‘(mulGrp‘𝑅))(𝐴𝑖))) = (𝑘 ∈ (𝐼𝐽) ↦ (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))
518506, 517eqtrd 2777 . . . . . . . . . 10 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → ((𝑖𝐼 ↦ ((𝑑𝑖)(.g‘(mulGrp‘𝑅))(𝐴𝑖))) ↾ (𝐼𝐽)) = (𝑘 ∈ (𝐼𝐽) ↦ (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))
519518oveq2d 7447 . . . . . . . . 9 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → ((mulGrp‘𝑅) Σg ((𝑖𝐼 ↦ ((𝑑𝑖)(.g‘(mulGrp‘𝑅))(𝐴𝑖))) ↾ (𝐼𝐽))) = ((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)))))
520505, 519oveq12d 7449 . . . . . . . 8 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (((mulGrp‘𝑅) Σg ((𝑖𝐼 ↦ ((𝑑𝑖)(.g‘(mulGrp‘𝑅))(𝐴𝑖))) ↾ 𝐽))(.r𝑅)((mulGrp‘𝑅) Σg ((𝑖𝐼 ↦ ((𝑑𝑖)(.g‘(mulGrp‘𝑅))(𝐴𝑖))) ↾ (𝐼𝐽)))) = (((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))))
521491, 520eqtr2d 2778 . . . . . . 7 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))) = ((mulGrp‘𝑅) Σg (𝑖𝐼 ↦ ((𝑑𝑖)(.g‘(mulGrp‘𝑅))(𝐴𝑖)))))
522351, 521oveq12d 7449 . . . . . 6 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → ((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑𝐽))‘(𝑑 ↾ (𝐼𝐽)))(.r𝑅)(((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)))))) = ((𝐹𝑑)(.r𝑅)((mulGrp‘𝑅) Σg (𝑖𝐼 ↦ ((𝑑𝑖)(.g‘(mulGrp‘𝑅))(𝐴𝑖))))))
523473, 522eqtrd 2777 . . . . 5 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑𝐽))‘(𝑑 ↾ (𝐼𝐽)))(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))) = ((𝐹𝑑)(.r𝑅)((mulGrp‘𝑅) Σg (𝑖𝐼 ↦ ((𝑑𝑖)(.g‘(mulGrp‘𝑅))(𝐴𝑖))))))
524523mpteq2dva 5242 . . . 4 (𝜑 → (𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑𝐽))‘(𝑑 ↾ (𝐼𝐽)))(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘)))))) = (𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ ((𝐹𝑑)(.r𝑅)((mulGrp‘𝑅) Σg (𝑖𝐼 ↦ ((𝑑𝑖)(.g‘(mulGrp‘𝑅))(𝐴𝑖)))))))
525524oveq2d 7447 . . 3 (𝜑 → (𝑅 Σg (𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑𝐽))‘(𝑑 ↾ (𝐼𝐽)))(.r𝑅)((mulGrp‘𝑅) Σg (𝑗𝐽 ↦ (((𝑑𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴𝐽)‘𝑗)))))(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ (((𝑑 ↾ (𝐼𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))))) = (𝑅 Σg (𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ ((𝐹𝑑)(.r𝑅)((mulGrp‘𝑅) Σg (𝑖𝐼 ↦ ((𝑑𝑖)(.g‘(mulGrp‘𝑅))(𝐴𝑖))))))))
526275, 472, 5253eqtr2d 2783 . 2 (𝜑 → (𝑅 Σg (𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ↦ (((((𝐽 eval 𝑈)‘(((𝐼 selectVars 𝑅)‘𝐽)‘𝐹))‘(𝐿 ∘ (𝐴𝐽)))‘𝑐)(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑐𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))))) = (𝑅 Σg (𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ ((𝐹𝑑)(.r𝑅)((mulGrp‘𝑅) Σg (𝑖𝐼 ↦ ((𝑑𝑖)(.g‘(mulGrp‘𝑅))(𝐴𝑖))))))))
527 eqid 2737 . . 3 ((𝐼𝐽) eval 𝑅) = ((𝐼𝐽) eval 𝑅)
528527, 3, 1, 137, 28, 47, 49, 136, 6, 7, 166, 457evlvvval 42583 . 2 (𝜑 → ((((𝐼𝐽) eval 𝑅)‘(((𝐽 eval 𝑈)‘(((𝐼 selectVars 𝑅)‘𝐽)‘𝐹))‘(𝐿 ∘ (𝐴𝐽))))‘(𝐴 ↾ (𝐼𝐽))) = (𝑅 Σg (𝑐 ∈ {𝑓 ∈ (ℕ0m (𝐼𝐽)) ∣ (𝑓 “ ℕ) ∈ Fin} ↦ (((((𝐽 eval 𝑈)‘(((𝐼 selectVars 𝑅)‘𝐽)‘𝐹))‘(𝐿 ∘ (𝐴𝐽)))‘𝑐)(.r𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼𝐽) ↦ ((𝑐𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼𝐽))‘𝑘))))))))
529 eqid 2737 . . 3 (𝐼 eval 𝑅) = (𝐼 eval 𝑅)
530529, 14, 15, 282, 28, 47, 49, 136, 4, 7, 17, 35evlvvval 42583 . 2 (𝜑 → (((𝐼 eval 𝑅)‘𝐹)‘𝐴) = (𝑅 Σg (𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ ((𝐹𝑑)(.r𝑅)((mulGrp‘𝑅) Σg (𝑖𝐼 ↦ ((𝑑𝑖)(.g‘(mulGrp‘𝑅))(𝐴𝑖))))))))
531526, 528, 5303eqtr4d 2787 1 (𝜑 → ((((𝐼𝐽) eval 𝑅)‘(((𝐽 eval 𝑈)‘(((𝐼 selectVars 𝑅)‘𝐽)‘𝐹))‘(𝐿 ∘ (𝐴𝐽))))‘(𝐴 ↾ (𝐼𝐽))) = (((𝐼 eval 𝑅)‘𝐹)‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wral 3061  {crab 3436  Vcvv 3480  csb 3899  cdif 3948  cun 3949  cin 3950  wss 3951  c0 4333  {csn 4626   class class class wbr 5143  cmpt 5225   × cxp 5683  ccnv 5684  cres 5687  cima 5688  ccom 5689  Fun wfun 6555   Fn wfn 6556  wf 6557  1-1wf1 6558  1-1-ontowf1o 6560  cfv 6561  (class class class)co 7431  cmpo 7433   supp csupp 8185  m cmap 8866  Fincfn 8985   finSupp cfsupp 9401  0cc0 11155  cn 12266  0cn0 12526  cz 12613  Basecbs 17247  .rcmulr 17298  Scalarcsca 17300   ·𝑠 cvsca 17301  0gc0g 17484   Σg cgsu 17485  Mndcmnd 18747   MndHom cmhm 18794  Grpcgrp 18951  .gcmg 19085   GrpHom cghm 19230  CMndccmn 19798  mulGrpcmgp 20137  1rcur 20178  Ringcrg 20230  CRingccrg 20231   RingHom crh 20469  AssAlgcasa 21870  algSccascl 21872   mPoly cmpl 21926   eval cevl 22097   selectVars cslv 22132
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-ofr 7698  df-om 7888  df-1st 8014  df-2nd 8015  df-supp 8186  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-map 8868  df-pm 8869  df-ixp 8938  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fsupp 9402  df-sup 9482  df-oi 9550  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-uz 12879  df-fz 13548  df-fzo 13695  df-seq 14043  df-hash 14370  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-mulr 17311  df-sca 17313  df-vsca 17314  df-ip 17315  df-tset 17316  df-ple 17317  df-ds 17319  df-hom 17321  df-cco 17322  df-0g 17486  df-gsum 17487  df-prds 17492  df-pws 17494  df-mre 17629  df-mrc 17630  df-acs 17632  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-mhm 18796  df-submnd 18797  df-grp 18954  df-minusg 18955  df-sbg 18956  df-mulg 19086  df-subg 19141  df-ghm 19231  df-cntz 19335  df-cmn 19800  df-abl 19801  df-mgp 20138  df-rng 20150  df-ur 20179  df-srg 20184  df-ring 20232  df-cring 20233  df-rhm 20472  df-subrng 20546  df-subrg 20570  df-lmod 20860  df-lss 20930  df-lsp 20970  df-assa 21873  df-asp 21874  df-ascl 21875  df-psr 21929  df-mvr 21930  df-mpl 21931  df-evls 22098  df-evl 22099  df-selv 22136
This theorem is referenced by: (None)
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