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Theorem resspsrmul 21293
Description: A restricted power series algebra has the same multiplication operation. (Contributed by Mario Carneiro, 3-Jul-2015.)
Hypotheses
Ref Expression
resspsr.s 𝑆 = (𝐼 mPwSer 𝑅)
resspsr.h 𝐻 = (𝑅s 𝑇)
resspsr.u 𝑈 = (𝐼 mPwSer 𝐻)
resspsr.b 𝐵 = (Base‘𝑈)
resspsr.p 𝑃 = (𝑆s 𝐵)
resspsr.2 (𝜑𝑇 ∈ (SubRing‘𝑅))
Assertion
Ref Expression
resspsrmul ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋(.r𝑈)𝑌) = (𝑋(.r𝑃)𝑌))

Proof of Theorem resspsrmul
Dummy variables 𝑥 𝑘 𝑓 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2736 . . . . . . . 8 {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}
21psrbaglefi 21242 . . . . . . 7 (𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} → {𝑦 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∣ 𝑦r𝑘} ∈ Fin)
32adantl 482 . . . . . 6 (((𝜑 ∧ (𝑋𝐵𝑌𝐵)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → {𝑦 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∣ 𝑦r𝑘} ∈ Fin)
4 resspsr.2 . . . . . . . . 9 (𝜑𝑇 ∈ (SubRing‘𝑅))
5 subrgsubg 20136 . . . . . . . . 9 (𝑇 ∈ (SubRing‘𝑅) → 𝑇 ∈ (SubGrp‘𝑅))
64, 5syl 17 . . . . . . . 8 (𝜑𝑇 ∈ (SubGrp‘𝑅))
7 subgsubm 18874 . . . . . . . 8 (𝑇 ∈ (SubGrp‘𝑅) → 𝑇 ∈ (SubMnd‘𝑅))
86, 7syl 17 . . . . . . 7 (𝜑𝑇 ∈ (SubMnd‘𝑅))
98ad2antrr 723 . . . . . 6 (((𝜑 ∧ (𝑋𝐵𝑌𝐵)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → 𝑇 ∈ (SubMnd‘𝑅))
104ad3antrrr 727 . . . . . . . 8 ((((𝜑 ∧ (𝑋𝐵𝑌𝐵)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑥 ∈ {𝑦 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∣ 𝑦r𝑘}) → 𝑇 ∈ (SubRing‘𝑅))
11 resspsr.u . . . . . . . . . . . 12 𝑈 = (𝐼 mPwSer 𝐻)
12 eqid 2736 . . . . . . . . . . . 12 (Base‘𝐻) = (Base‘𝐻)
13 resspsr.b . . . . . . . . . . . 12 𝐵 = (Base‘𝑈)
14 simprl 768 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → 𝑋𝐵)
1511, 12, 1, 13, 14psrelbas 21255 . . . . . . . . . . 11 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → 𝑋:{𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝐻))
1615adantr 481 . . . . . . . . . 10 (((𝜑 ∧ (𝑋𝐵𝑌𝐵)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → 𝑋:{𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝐻))
17 elrabi 3628 . . . . . . . . . 10 (𝑥 ∈ {𝑦 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∣ 𝑦r𝑘} → 𝑥 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin})
18 ffvelcdm 7016 . . . . . . . . . 10 ((𝑋:{𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝐻) ∧ 𝑥 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑋𝑥) ∈ (Base‘𝐻))
1916, 17, 18syl2an 596 . . . . . . . . 9 ((((𝜑 ∧ (𝑋𝐵𝑌𝐵)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑥 ∈ {𝑦 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∣ 𝑦r𝑘}) → (𝑋𝑥) ∈ (Base‘𝐻))
20 resspsr.h . . . . . . . . . . 11 𝐻 = (𝑅s 𝑇)
2120subrgbas 20139 . . . . . . . . . 10 (𝑇 ∈ (SubRing‘𝑅) → 𝑇 = (Base‘𝐻))
2210, 21syl 17 . . . . . . . . 9 ((((𝜑 ∧ (𝑋𝐵𝑌𝐵)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑥 ∈ {𝑦 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∣ 𝑦r𝑘}) → 𝑇 = (Base‘𝐻))
2319, 22eleqtrrd 2840 . . . . . . . 8 ((((𝜑 ∧ (𝑋𝐵𝑌𝐵)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑥 ∈ {𝑦 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∣ 𝑦r𝑘}) → (𝑋𝑥) ∈ 𝑇)
24 simprr 770 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → 𝑌𝐵)
2511, 12, 1, 13, 24psrelbas 21255 . . . . . . . . . . 11 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → 𝑌:{𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝐻))
2625ad2antrr 723 . . . . . . . . . 10 ((((𝜑 ∧ (𝑋𝐵𝑌𝐵)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑥 ∈ {𝑦 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∣ 𝑦r𝑘}) → 𝑌:{𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝐻))
27 ssrab2 4025 . . . . . . . . . . 11 {𝑦 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∣ 𝑦r𝑘} ⊆ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}
28 simplr 766 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑋𝐵𝑌𝐵)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑥 ∈ {𝑦 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∣ 𝑦r𝑘}) → 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin})
29 simpr 485 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑋𝐵𝑌𝐵)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑥 ∈ {𝑦 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∣ 𝑦r𝑘}) → 𝑥 ∈ {𝑦 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∣ 𝑦r𝑘})
30 eqid 2736 . . . . . . . . . . . . 13 {𝑦 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∣ 𝑦r𝑘} = {𝑦 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∣ 𝑦r𝑘}
311, 30psrbagconcl 21244 . . . . . . . . . . . 12 ((𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∧ 𝑥 ∈ {𝑦 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∣ 𝑦r𝑘}) → (𝑘f𝑥) ∈ {𝑦 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∣ 𝑦r𝑘})
3228, 29, 31syl2anc 584 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑋𝐵𝑌𝐵)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑥 ∈ {𝑦 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∣ 𝑦r𝑘}) → (𝑘f𝑥) ∈ {𝑦 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∣ 𝑦r𝑘})
3327, 32sselid 3930 . . . . . . . . . 10 ((((𝜑 ∧ (𝑋𝐵𝑌𝐵)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑥 ∈ {𝑦 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∣ 𝑦r𝑘}) → (𝑘f𝑥) ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin})
3426, 33ffvelcdmd 7019 . . . . . . . . 9 ((((𝜑 ∧ (𝑋𝐵𝑌𝐵)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑥 ∈ {𝑦 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∣ 𝑦r𝑘}) → (𝑌‘(𝑘f𝑥)) ∈ (Base‘𝐻))
3534, 22eleqtrrd 2840 . . . . . . . 8 ((((𝜑 ∧ (𝑋𝐵𝑌𝐵)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑥 ∈ {𝑦 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∣ 𝑦r𝑘}) → (𝑌‘(𝑘f𝑥)) ∈ 𝑇)
36 eqid 2736 . . . . . . . . 9 (.r𝑅) = (.r𝑅)
3736subrgmcl 20142 . . . . . . . 8 ((𝑇 ∈ (SubRing‘𝑅) ∧ (𝑋𝑥) ∈ 𝑇 ∧ (𝑌‘(𝑘f𝑥)) ∈ 𝑇) → ((𝑋𝑥)(.r𝑅)(𝑌‘(𝑘f𝑥))) ∈ 𝑇)
3810, 23, 35, 37syl3anc 1370 . . . . . . 7 ((((𝜑 ∧ (𝑋𝐵𝑌𝐵)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑥 ∈ {𝑦 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∣ 𝑦r𝑘}) → ((𝑋𝑥)(.r𝑅)(𝑌‘(𝑘f𝑥))) ∈ 𝑇)
3938fmpttd 7046 . . . . . 6 (((𝜑 ∧ (𝑋𝐵𝑌𝐵)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑥 ∈ {𝑦 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∣ 𝑦r𝑘} ↦ ((𝑋𝑥)(.r𝑅)(𝑌‘(𝑘f𝑥)))):{𝑦 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∣ 𝑦r𝑘}⟶𝑇)
403, 9, 39, 20gsumsubm 18571 . . . . 5 (((𝜑 ∧ (𝑋𝐵𝑌𝐵)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑅 Σg (𝑥 ∈ {𝑦 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∣ 𝑦r𝑘} ↦ ((𝑋𝑥)(.r𝑅)(𝑌‘(𝑘f𝑥))))) = (𝐻 Σg (𝑥 ∈ {𝑦 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∣ 𝑦r𝑘} ↦ ((𝑋𝑥)(.r𝑅)(𝑌‘(𝑘f𝑥))))))
4120, 36ressmulr 17115 . . . . . . . . . 10 (𝑇 ∈ (SubRing‘𝑅) → (.r𝑅) = (.r𝐻))
424, 41syl 17 . . . . . . . . 9 (𝜑 → (.r𝑅) = (.r𝐻))
4342ad3antrrr 727 . . . . . . . 8 ((((𝜑 ∧ (𝑋𝐵𝑌𝐵)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑥 ∈ {𝑦 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∣ 𝑦r𝑘}) → (.r𝑅) = (.r𝐻))
4443oveqd 7355 . . . . . . 7 ((((𝜑 ∧ (𝑋𝐵𝑌𝐵)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑥 ∈ {𝑦 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∣ 𝑦r𝑘}) → ((𝑋𝑥)(.r𝑅)(𝑌‘(𝑘f𝑥))) = ((𝑋𝑥)(.r𝐻)(𝑌‘(𝑘f𝑥))))
4544mpteq2dva 5193 . . . . . 6 (((𝜑 ∧ (𝑋𝐵𝑌𝐵)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑥 ∈ {𝑦 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∣ 𝑦r𝑘} ↦ ((𝑋𝑥)(.r𝑅)(𝑌‘(𝑘f𝑥)))) = (𝑥 ∈ {𝑦 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∣ 𝑦r𝑘} ↦ ((𝑋𝑥)(.r𝐻)(𝑌‘(𝑘f𝑥)))))
4645oveq2d 7354 . . . . 5 (((𝜑 ∧ (𝑋𝐵𝑌𝐵)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝐻 Σg (𝑥 ∈ {𝑦 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∣ 𝑦r𝑘} ↦ ((𝑋𝑥)(.r𝑅)(𝑌‘(𝑘f𝑥))))) = (𝐻 Σg (𝑥 ∈ {𝑦 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∣ 𝑦r𝑘} ↦ ((𝑋𝑥)(.r𝐻)(𝑌‘(𝑘f𝑥))))))
4740, 46eqtrd 2776 . . . 4 (((𝜑 ∧ (𝑋𝐵𝑌𝐵)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑅 Σg (𝑥 ∈ {𝑦 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∣ 𝑦r𝑘} ↦ ((𝑋𝑥)(.r𝑅)(𝑌‘(𝑘f𝑥))))) = (𝐻 Σg (𝑥 ∈ {𝑦 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∣ 𝑦r𝑘} ↦ ((𝑋𝑥)(.r𝐻)(𝑌‘(𝑘f𝑥))))))
4847mpteq2dva 5193 . . 3 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ↦ (𝑅 Σg (𝑥 ∈ {𝑦 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∣ 𝑦r𝑘} ↦ ((𝑋𝑥)(.r𝑅)(𝑌‘(𝑘f𝑥)))))) = (𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ↦ (𝐻 Σg (𝑥 ∈ {𝑦 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∣ 𝑦r𝑘} ↦ ((𝑋𝑥)(.r𝐻)(𝑌‘(𝑘f𝑥)))))))
49 resspsr.s . . . 4 𝑆 = (𝐼 mPwSer 𝑅)
50 eqid 2736 . . . 4 (Base‘𝑆) = (Base‘𝑆)
51 eqid 2736 . . . 4 (.r𝑆) = (.r𝑆)
52 fvex 6839 . . . . . . . 8 (Base‘𝑅) ∈ V
534, 21syl 17 . . . . . . . . 9 (𝜑𝑇 = (Base‘𝐻))
54 eqid 2736 . . . . . . . . . . 11 (Base‘𝑅) = (Base‘𝑅)
5554subrgss 20131 . . . . . . . . . 10 (𝑇 ∈ (SubRing‘𝑅) → 𝑇 ⊆ (Base‘𝑅))
564, 55syl 17 . . . . . . . . 9 (𝜑𝑇 ⊆ (Base‘𝑅))
5753, 56eqsstrrd 3971 . . . . . . . 8 (𝜑 → (Base‘𝐻) ⊆ (Base‘𝑅))
58 mapss 8749 . . . . . . . 8 (((Base‘𝑅) ∈ V ∧ (Base‘𝐻) ⊆ (Base‘𝑅)) → ((Base‘𝐻) ↑m {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ⊆ ((Base‘𝑅) ↑m {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}))
5952, 57, 58sylancr 587 . . . . . . 7 (𝜑 → ((Base‘𝐻) ↑m {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ⊆ ((Base‘𝑅) ↑m {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}))
6059adantr 481 . . . . . 6 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → ((Base‘𝐻) ↑m {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ⊆ ((Base‘𝑅) ↑m {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}))
61 reldmpsr 21224 . . . . . . . . . 10 Rel dom mPwSer
6261, 11, 13elbasov 17017 . . . . . . . . 9 (𝑋𝐵 → (𝐼 ∈ V ∧ 𝐻 ∈ V))
6362ad2antrl 725 . . . . . . . 8 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝐼 ∈ V ∧ 𝐻 ∈ V))
6463simpld 495 . . . . . . 7 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → 𝐼 ∈ V)
6511, 12, 1, 13, 64psrbas 21254 . . . . . 6 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → 𝐵 = ((Base‘𝐻) ↑m {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}))
6649, 54, 1, 50, 64psrbas 21254 . . . . . 6 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (Base‘𝑆) = ((Base‘𝑅) ↑m {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}))
6760, 65, 663sstr4d 3979 . . . . 5 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → 𝐵 ⊆ (Base‘𝑆))
6867, 14sseldd 3933 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → 𝑋 ∈ (Base‘𝑆))
6967, 24sseldd 3933 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → 𝑌 ∈ (Base‘𝑆))
7049, 50, 36, 51, 1, 68, 69psrmulfval 21261 . . 3 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋(.r𝑆)𝑌) = (𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ↦ (𝑅 Σg (𝑥 ∈ {𝑦 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∣ 𝑦r𝑘} ↦ ((𝑋𝑥)(.r𝑅)(𝑌‘(𝑘f𝑥)))))))
71 eqid 2736 . . . 4 (.r𝐻) = (.r𝐻)
72 eqid 2736 . . . 4 (.r𝑈) = (.r𝑈)
7311, 13, 71, 72, 1, 14, 24psrmulfval 21261 . . 3 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋(.r𝑈)𝑌) = (𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ↦ (𝐻 Σg (𝑥 ∈ {𝑦 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∣ 𝑦r𝑘} ↦ ((𝑋𝑥)(.r𝐻)(𝑌‘(𝑘f𝑥)))))))
7448, 70, 733eqtr4rd 2787 . 2 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋(.r𝑈)𝑌) = (𝑋(.r𝑆)𝑌))
7513fvexi 6840 . . . 4 𝐵 ∈ V
76 resspsr.p . . . . 5 𝑃 = (𝑆s 𝐵)
7776, 51ressmulr 17115 . . . 4 (𝐵 ∈ V → (.r𝑆) = (.r𝑃))
7875, 77mp1i 13 . . 3 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (.r𝑆) = (.r𝑃))
7978oveqd 7355 . 2 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋(.r𝑆)𝑌) = (𝑋(.r𝑃)𝑌))
8074, 79eqtrd 2776 1 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋(.r𝑈)𝑌) = (𝑋(.r𝑃)𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1540  wcel 2105  {crab 3403  Vcvv 3441  wss 3898   class class class wbr 5093  cmpt 5176  ccnv 5620  cima 5624  wf 6476  cfv 6480  (class class class)co 7338  f cof 7594  r cofr 7595  m cmap 8687  Fincfn 8805  cle 11112  cmin 11307  cn 12075  0cn0 12335  Basecbs 17010  s cress 17039  .rcmulr 17061   Σg cgsu 17249  SubMndcsubmnd 18527  SubGrpcsubg 18846  SubRingcsubrg 20126   mPwSer cmps 21214
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 2707  ax-rep 5230  ax-sep 5244  ax-nul 5251  ax-pow 5309  ax-pr 5373  ax-un 7651  ax-cnex 11029  ax-resscn 11030  ax-1cn 11031  ax-icn 11032  ax-addcl 11033  ax-addrcl 11034  ax-mulcl 11035  ax-mulrcl 11036  ax-mulcom 11037  ax-addass 11038  ax-mulass 11039  ax-distr 11040  ax-i2m1 11041  ax-1ne0 11042  ax-1rid 11043  ax-rnegex 11044  ax-rrecex 11045  ax-cnre 11046  ax-pre-lttri 11047  ax-pre-lttrn 11048  ax-pre-ltadd 11049  ax-pre-mulgt0 11050
This theorem depends on definitions:  df-bi 206  df-an 397  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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3349  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3917  df-nul 4271  df-if 4475  df-pw 4550  df-sn 4575  df-pr 4577  df-tp 4579  df-op 4581  df-uni 4854  df-iun 4944  df-br 5094  df-opab 5156  df-mpt 5177  df-tr 5211  df-id 5519  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5576  df-we 5578  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6239  df-ord 6306  df-on 6307  df-lim 6308  df-suc 6309  df-iota 6432  df-fun 6482  df-fn 6483  df-f 6484  df-f1 6485  df-fo 6486  df-f1o 6487  df-fv 6488  df-riota 7294  df-ov 7341  df-oprab 7342  df-mpo 7343  df-of 7596  df-ofr 7597  df-om 7782  df-1st 7900  df-2nd 7901  df-supp 8049  df-frecs 8168  df-wrecs 8199  df-recs 8273  df-rdg 8312  df-1o 8368  df-er 8570  df-map 8689  df-pm 8690  df-ixp 8758  df-en 8806  df-dom 8807  df-sdom 8808  df-fin 8809  df-fsupp 9228  df-pnf 11113  df-mnf 11114  df-xr 11115  df-ltxr 11116  df-le 11117  df-sub 11309  df-neg 11310  df-nn 12076  df-2 12138  df-3 12139  df-4 12140  df-5 12141  df-6 12142  df-7 12143  df-8 12144  df-9 12145  df-n0 12336  df-z 12422  df-uz 12685  df-fz 13342  df-seq 13824  df-struct 16946  df-sets 16963  df-slot 16981  df-ndx 16993  df-base 17011  df-ress 17040  df-plusg 17073  df-mulr 17074  df-sca 17076  df-vsca 17077  df-tset 17079  df-0g 17250  df-gsum 17251  df-mgm 18424  df-sgrp 18473  df-mnd 18484  df-submnd 18529  df-grp 18677  df-minusg 18678  df-subg 18849  df-mgp 19817  df-ring 19881  df-subrg 20128  df-psr 21219
This theorem is referenced by:  subrgpsr  21295  ressmplmul  21338
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