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Theorem resspsrmul 21996
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 2737 . . . . . . . 8 {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}
21psrbaglefi 21946 . . . . . . 7 (𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} → {𝑦 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∣ 𝑦r𝑘} ∈ Fin)
32adantl 481 . . . . . 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 20577 . . . . . . . . 9 (𝑇 ∈ (SubRing‘𝑅) → 𝑇 ∈ (SubGrp‘𝑅))
64, 5syl 17 . . . . . . . 8 (𝜑𝑇 ∈ (SubGrp‘𝑅))
7 subgsubm 19166 . . . . . . . 8 (𝑇 ∈ (SubGrp‘𝑅) → 𝑇 ∈ (SubMnd‘𝑅))
86, 7syl 17 . . . . . . 7 (𝜑𝑇 ∈ (SubMnd‘𝑅))
98ad2antrr 726 . . . . . 6 (((𝜑 ∧ (𝑋𝐵𝑌𝐵)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → 𝑇 ∈ (SubMnd‘𝑅))
104ad3antrrr 730 . . . . . . . 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 2737 . . . . . . . . . . . 12 (Base‘𝐻) = (Base‘𝐻)
13 resspsr.b . . . . . . . . . . . 12 𝐵 = (Base‘𝑈)
14 simprl 771 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → 𝑋𝐵)
1511, 12, 1, 13, 14psrelbas 21954 . . . . . . . . . . 11 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → 𝑋:{𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝐻))
1615adantr 480 . . . . . . . . . 10 (((𝜑 ∧ (𝑋𝐵𝑌𝐵)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → 𝑋:{𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝐻))
17 elrabi 3687 . . . . . . . . . 10 (𝑥 ∈ {𝑦 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∣ 𝑦r𝑘} → 𝑥 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin})
18 ffvelcdm 7101 . . . . . . . . . 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 20581 . . . . . . . . . 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 2844 . . . . . . . 8 ((((𝜑 ∧ (𝑋𝐵𝑌𝐵)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑥 ∈ {𝑦 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∣ 𝑦r𝑘}) → (𝑋𝑥) ∈ 𝑇)
24 simprr 773 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → 𝑌𝐵)
2511, 12, 1, 13, 24psrelbas 21954 . . . . . . . . . . 11 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → 𝑌:{𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝐻))
2625ad2antrr 726 . . . . . . . . . 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 4080 . . . . . . . . . . 11 {𝑦 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∣ 𝑦r𝑘} ⊆ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}
28 simplr 769 . . . . . . . . . . . 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 484 . . . . . . . . . . . 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 2737 . . . . . . . . . . . . 13 {𝑦 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∣ 𝑦r𝑘} = {𝑦 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∣ 𝑦r𝑘}
311, 30psrbagconcl 21947 . . . . . . . . . . . 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 3981 . . . . . . . . . 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 7105 . . . . . . . . 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 2844 . . . . . . . 8 ((((𝜑 ∧ (𝑋𝐵𝑌𝐵)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑥 ∈ {𝑦 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∣ 𝑦r𝑘}) → (𝑌‘(𝑘f𝑥)) ∈ 𝑇)
36 eqid 2737 . . . . . . . . 9 (.r𝑅) = (.r𝑅)
3736subrgmcl 20584 . . . . . . . 8 ((𝑇 ∈ (SubRing‘𝑅) ∧ (𝑋𝑥) ∈ 𝑇 ∧ (𝑌‘(𝑘f𝑥)) ∈ 𝑇) → ((𝑋𝑥)(.r𝑅)(𝑌‘(𝑘f𝑥))) ∈ 𝑇)
3810, 23, 35, 37syl3anc 1373 . . . . . . 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 7135 . . . . . 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 18848 . . . . 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 17351 . . . . . . . . . 10 (𝑇 ∈ (SubRing‘𝑅) → (.r𝑅) = (.r𝐻))
424, 41syl 17 . . . . . . . . 9 (𝜑 → (.r𝑅) = (.r𝐻))
4342ad3antrrr 730 . . . . . . . 8 ((((𝜑 ∧ (𝑋𝐵𝑌𝐵)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑥 ∈ {𝑦 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∣ 𝑦r𝑘}) → (.r𝑅) = (.r𝐻))
4443oveqd 7448 . . . . . . 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 5242 . . . . . 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 7447 . . . . 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 2777 . . . 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 5242 . . 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 2737 . . . 4 (Base‘𝑆) = (Base‘𝑆)
51 eqid 2737 . . . 4 (.r𝑆) = (.r𝑆)
52 fvex 6919 . . . . . . . 8 (Base‘𝑅) ∈ V
534, 21syl 17 . . . . . . . . 9 (𝜑𝑇 = (Base‘𝐻))
54 eqid 2737 . . . . . . . . . . 11 (Base‘𝑅) = (Base‘𝑅)
5554subrgss 20572 . . . . . . . . . 10 (𝑇 ∈ (SubRing‘𝑅) → 𝑇 ⊆ (Base‘𝑅))
564, 55syl 17 . . . . . . . . 9 (𝜑𝑇 ⊆ (Base‘𝑅))
5753, 56eqsstrrd 4019 . . . . . . . 8 (𝜑 → (Base‘𝐻) ⊆ (Base‘𝑅))
58 mapss 8929 . . . . . . . 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 480 . . . . . 6 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → ((Base‘𝐻) ↑m {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ⊆ ((Base‘𝑅) ↑m {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}))
61 reldmpsr 21934 . . . . . . . . . 10 Rel dom mPwSer
6261, 11, 13elbasov 17254 . . . . . . . . 9 (𝑋𝐵 → (𝐼 ∈ V ∧ 𝐻 ∈ V))
6362ad2antrl 728 . . . . . . . 8 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝐼 ∈ V ∧ 𝐻 ∈ V))
6463simpld 494 . . . . . . 7 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → 𝐼 ∈ V)
6511, 12, 1, 13, 64psrbas 21953 . . . . . 6 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → 𝐵 = ((Base‘𝐻) ↑m {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}))
6649, 54, 1, 50, 64psrbas 21953 . . . . . 6 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (Base‘𝑆) = ((Base‘𝑅) ↑m {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}))
6760, 65, 663sstr4d 4039 . . . . 5 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → 𝐵 ⊆ (Base‘𝑆))
6867, 14sseldd 3984 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → 𝑋 ∈ (Base‘𝑆))
6967, 24sseldd 3984 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → 𝑌 ∈ (Base‘𝑆))
7049, 50, 36, 51, 1, 68, 69psrmulfval 21963 . . 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 2737 . . . 4 (.r𝐻) = (.r𝐻)
72 eqid 2737 . . . 4 (.r𝑈) = (.r𝑈)
7311, 13, 71, 72, 1, 14, 24psrmulfval 21963 . . 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 2788 . 2 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋(.r𝑈)𝑌) = (𝑋(.r𝑆)𝑌))
7513fvexi 6920 . . . 4 𝐵 ∈ V
76 resspsr.p . . . . 5 𝑃 = (𝑆s 𝐵)
7776, 51ressmulr 17351 . . . 4 (𝐵 ∈ V → (.r𝑆) = (.r𝑃))
7875, 77mp1i 13 . . 3 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (.r𝑆) = (.r𝑃))
7978oveqd 7448 . 2 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋(.r𝑆)𝑌) = (𝑋(.r𝑃)𝑌))
8074, 79eqtrd 2777 1 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋(.r𝑈)𝑌) = (𝑋(.r𝑃)𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  {crab 3436  Vcvv 3480  wss 3951   class class class wbr 5143  cmpt 5225  ccnv 5684  cima 5688  wf 6557  cfv 6561  (class class class)co 7431  f cof 7695  r cofr 7696  m cmap 8866  Fincfn 8985  cle 11296  cmin 11492  cn 12266  0cn0 12526  Basecbs 17247  s cress 17274  .rcmulr 17298   Σg cgsu 17485  SubMndcsubmnd 18795  SubGrpcsubg 19138  SubRingcsubrg 20569   mPwSer cmps 21924
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-iun 4993  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-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-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-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-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-uz 12879  df-fz 13548  df-seq 14043  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-tset 17316  df-0g 17486  df-gsum 17487  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-submnd 18797  df-grp 18954  df-minusg 18955  df-subg 19141  df-cmn 19800  df-abl 19801  df-mgp 20138  df-rng 20150  df-ur 20179  df-ring 20232  df-subrng 20546  df-subrg 20570  df-psr 21929
This theorem is referenced by:  subrgpsr  21998  ressmplmul  22048
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