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Theorem resspsrmul 19180
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 reldmpsr 19124 . . . . . . . . . 10 Rel dom mPwSer
2 resspsr.u . . . . . . . . . 10 𝑈 = (𝐼 mPwSer 𝐻)
3 resspsr.b . . . . . . . . . 10 𝐵 = (Base‘𝑈)
41, 2, 3elbasov 15691 . . . . . . . . 9 (𝑋𝐵 → (𝐼 ∈ V ∧ 𝐻 ∈ V))
54ad2antrl 759 . . . . . . . 8 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝐼 ∈ V ∧ 𝐻 ∈ V))
65simpld 473 . . . . . . 7 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → 𝐼 ∈ V)
7 eqid 2605 . . . . . . . 8 {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}
87psrbaglefi 19135 . . . . . . 7 ((𝐼 ∈ V ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → {𝑦 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∣ 𝑦𝑟𝑘} ∈ Fin)
96, 8sylan 486 . . . . . 6 (((𝜑 ∧ (𝑋𝐵𝑌𝐵)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → {𝑦 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∣ 𝑦𝑟𝑘} ∈ Fin)
10 resspsr.2 . . . . . . . . 9 (𝜑𝑇 ∈ (SubRing‘𝑅))
11 subrgsubg 18551 . . . . . . . . 9 (𝑇 ∈ (SubRing‘𝑅) → 𝑇 ∈ (SubGrp‘𝑅))
1210, 11syl 17 . . . . . . . 8 (𝜑𝑇 ∈ (SubGrp‘𝑅))
13 subgsubm 17381 . . . . . . . 8 (𝑇 ∈ (SubGrp‘𝑅) → 𝑇 ∈ (SubMnd‘𝑅))
1412, 13syl 17 . . . . . . 7 (𝜑𝑇 ∈ (SubMnd‘𝑅))
1514ad2antrr 757 . . . . . 6 (((𝜑 ∧ (𝑋𝐵𝑌𝐵)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → 𝑇 ∈ (SubMnd‘𝑅))
1610ad3antrrr 761 . . . . . . . 8 ((((𝜑 ∧ (𝑋𝐵𝑌𝐵)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑥 ∈ {𝑦 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∣ 𝑦𝑟𝑘}) → 𝑇 ∈ (SubRing‘𝑅))
17 eqid 2605 . . . . . . . . . . . 12 (Base‘𝐻) = (Base‘𝐻)
18 simprl 789 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → 𝑋𝐵)
192, 17, 7, 3, 18psrelbas 19142 . . . . . . . . . . 11 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → 𝑋:{𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝐻))
2019adantr 479 . . . . . . . . . 10 (((𝜑 ∧ (𝑋𝐵𝑌𝐵)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → 𝑋:{𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝐻))
21 elrabi 3323 . . . . . . . . . 10 (𝑥 ∈ {𝑦 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∣ 𝑦𝑟𝑘} → 𝑥 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin})
22 ffvelrn 6246 . . . . . . . . . 10 ((𝑋:{𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝐻) ∧ 𝑥 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑋𝑥) ∈ (Base‘𝐻))
2320, 21, 22syl2an 492 . . . . . . . . 9 ((((𝜑 ∧ (𝑋𝐵𝑌𝐵)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑥 ∈ {𝑦 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∣ 𝑦𝑟𝑘}) → (𝑋𝑥) ∈ (Base‘𝐻))
24 resspsr.h . . . . . . . . . . 11 𝐻 = (𝑅s 𝑇)
2524subrgbas 18554 . . . . . . . . . 10 (𝑇 ∈ (SubRing‘𝑅) → 𝑇 = (Base‘𝐻))
2616, 25syl 17 . . . . . . . . 9 ((((𝜑 ∧ (𝑋𝐵𝑌𝐵)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑥 ∈ {𝑦 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∣ 𝑦𝑟𝑘}) → 𝑇 = (Base‘𝐻))
2723, 26eleqtrrd 2686 . . . . . . . 8 ((((𝜑 ∧ (𝑋𝐵𝑌𝐵)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑥 ∈ {𝑦 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∣ 𝑦𝑟𝑘}) → (𝑋𝑥) ∈ 𝑇)
28 simprr 791 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → 𝑌𝐵)
292, 17, 7, 3, 28psrelbas 19142 . . . . . . . . . . 11 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → 𝑌:{𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝐻))
3029ad2antrr 757 . . . . . . . . . 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))
31 ssrab2 3645 . . . . . . . . . . 11 {𝑦 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∣ 𝑦𝑟𝑘} ⊆ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}
326ad2antrr 757 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑋𝐵𝑌𝐵)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑥 ∈ {𝑦 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∣ 𝑦𝑟𝑘}) → 𝐼 ∈ V)
33 simplr 787 . . . . . . . . . . . 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})
34 simpr 475 . . . . . . . . . . . 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})
35 eqid 2605 . . . . . . . . . . . . 13 {𝑦 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∣ 𝑦𝑟𝑘} = {𝑦 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∣ 𝑦𝑟𝑘}
367, 35psrbagconcl 19136 . . . . . . . . . . . 12 ((𝐼 ∈ V ∧ 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})
3732, 33, 34, 36syl3anc 1317 . . . . . . . . . . 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})
3831, 37sseldi 3561 . . . . . . . . . 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})
3930, 38ffvelrnd 6249 . . . . . . . . 9 ((((𝜑 ∧ (𝑋𝐵𝑌𝐵)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑥 ∈ {𝑦 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∣ 𝑦𝑟𝑘}) → (𝑌‘(𝑘𝑓𝑥)) ∈ (Base‘𝐻))
4039, 26eleqtrrd 2686 . . . . . . . 8 ((((𝜑 ∧ (𝑋𝐵𝑌𝐵)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑥 ∈ {𝑦 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∣ 𝑦𝑟𝑘}) → (𝑌‘(𝑘𝑓𝑥)) ∈ 𝑇)
41 eqid 2605 . . . . . . . . 9 (.r𝑅) = (.r𝑅)
4241subrgmcl 18557 . . . . . . . 8 ((𝑇 ∈ (SubRing‘𝑅) ∧ (𝑋𝑥) ∈ 𝑇 ∧ (𝑌‘(𝑘𝑓𝑥)) ∈ 𝑇) → ((𝑋𝑥)(.r𝑅)(𝑌‘(𝑘𝑓𝑥))) ∈ 𝑇)
4316, 27, 40, 42syl3anc 1317 . . . . . . 7 ((((𝜑 ∧ (𝑋𝐵𝑌𝐵)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑥 ∈ {𝑦 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∣ 𝑦𝑟𝑘}) → ((𝑋𝑥)(.r𝑅)(𝑌‘(𝑘𝑓𝑥))) ∈ 𝑇)
44 eqid 2605 . . . . . . 7 (𝑥 ∈ {𝑦 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∣ 𝑦𝑟𝑘} ↦ ((𝑋𝑥)(.r𝑅)(𝑌‘(𝑘𝑓𝑥)))) = (𝑥 ∈ {𝑦 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∣ 𝑦𝑟𝑘} ↦ ((𝑋𝑥)(.r𝑅)(𝑌‘(𝑘𝑓𝑥))))
4543, 44fmptd 6273 . . . . . 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)
469, 15, 45, 24gsumsubm 17138 . . . . 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))))))
4724, 41ressmulr 15771 . . . . . . . . . 10 (𝑇 ∈ (SubRing‘𝑅) → (.r𝑅) = (.r𝐻))
4810, 47syl 17 . . . . . . . . 9 (𝜑 → (.r𝑅) = (.r𝐻))
4948ad3antrrr 761 . . . . . . . 8 ((((𝜑 ∧ (𝑋𝐵𝑌𝐵)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑥 ∈ {𝑦 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∣ 𝑦𝑟𝑘}) → (.r𝑅) = (.r𝐻))
5049oveqd 6540 . . . . . . 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))))
5150mpteq2dva 4662 . . . . . 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)))))
5251oveq2d 6539 . . . . 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))))))
5346, 52eqtrd 2639 . . . 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))))))
5453mpteq2dva 4662 . . 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)))))))
55 resspsr.s . . . 4 𝑆 = (𝐼 mPwSer 𝑅)
56 eqid 2605 . . . 4 (Base‘𝑆) = (Base‘𝑆)
57 eqid 2605 . . . 4 (.r𝑆) = (.r𝑆)
58 fvex 6094 . . . . . . . 8 (Base‘𝑅) ∈ V
5910, 25syl 17 . . . . . . . . 9 (𝜑𝑇 = (Base‘𝐻))
60 eqid 2605 . . . . . . . . . . 11 (Base‘𝑅) = (Base‘𝑅)
6160subrgss 18546 . . . . . . . . . 10 (𝑇 ∈ (SubRing‘𝑅) → 𝑇 ⊆ (Base‘𝑅))
6210, 61syl 17 . . . . . . . . 9 (𝜑𝑇 ⊆ (Base‘𝑅))
6359, 62eqsstr3d 3598 . . . . . . . 8 (𝜑 → (Base‘𝐻) ⊆ (Base‘𝑅))
64 mapss 7759 . . . . . . . 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}))
6558, 63, 64sylancr 693 . . . . . . 7 (𝜑 → ((Base‘𝐻) ↑𝑚 {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ⊆ ((Base‘𝑅) ↑𝑚 {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}))
6665adantr 479 . . . . . 6 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → ((Base‘𝐻) ↑𝑚 {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ⊆ ((Base‘𝑅) ↑𝑚 {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}))
672, 17, 7, 3, 6psrbas 19141 . . . . . 6 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → 𝐵 = ((Base‘𝐻) ↑𝑚 {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}))
6855, 60, 7, 56, 6psrbas 19141 . . . . . 6 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (Base‘𝑆) = ((Base‘𝑅) ↑𝑚 {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}))
6966, 67, 683sstr4d 3606 . . . . 5 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → 𝐵 ⊆ (Base‘𝑆))
7069, 18sseldd 3564 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → 𝑋 ∈ (Base‘𝑆))
7169, 28sseldd 3564 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → 𝑌 ∈ (Base‘𝑆))
7255, 56, 41, 57, 7, 70, 71psrmulfval 19148 . . 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)))))))
73 eqid 2605 . . . 4 (.r𝐻) = (.r𝐻)
74 eqid 2605 . . . 4 (.r𝑈) = (.r𝑈)
752, 3, 73, 74, 7, 18, 28psrmulfval 19148 . . 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)))))))
7654, 72, 753eqtr4rd 2650 . 2 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋(.r𝑈)𝑌) = (𝑋(.r𝑆)𝑌))
77 fvex 6094 . . . . 5 (Base‘𝑈) ∈ V
783, 77eqeltri 2679 . . . 4 𝐵 ∈ V
79 resspsr.p . . . . 5 𝑃 = (𝑆s 𝐵)
8079, 57ressmulr 15771 . . . 4 (𝐵 ∈ V → (.r𝑆) = (.r𝑃))
8178, 80mp1i 13 . . 3 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (.r𝑆) = (.r𝑃))
8281oveqd 6540 . 2 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋(.r𝑆)𝑌) = (𝑋(.r𝑃)𝑌))
8376, 82eqtrd 2639 1 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋(.r𝑈)𝑌) = (𝑋(.r𝑃)𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1975  {crab 2895  Vcvv 3168  wss 3535   class class class wbr 4573  cmpt 4633  ccnv 5023  cima 5027  wf 5782  cfv 5786  (class class class)co 6523  𝑓 cof 6766  𝑟 cofr 6767  𝑚 cmap 7717  Fincfn 7814  cle 9927  cmin 10113  cn 10863  0cn0 11135  Basecbs 15637  s cress 15638  .rcmulr 15711   Σg cgsu 15866  SubMndcsubmnd 17099  SubGrpcsubg 17353  SubRingcsubrg 18541   mPwSer cmps 19114
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-rep 4689  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820  ax-cnex 9844  ax-resscn 9845  ax-1cn 9846  ax-icn 9847  ax-addcl 9848  ax-addrcl 9849  ax-mulcl 9850  ax-mulrcl 9851  ax-mulcom 9852  ax-addass 9853  ax-mulass 9854  ax-distr 9855  ax-i2m1 9856  ax-1ne0 9857  ax-1rid 9858  ax-rnegex 9859  ax-rrecex 9860  ax-cnre 9861  ax-pre-lttri 9862  ax-pre-lttrn 9863  ax-pre-ltadd 9864  ax-pre-mulgt0 9865
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-nel 2778  df-ral 2896  df-rex 2897  df-reu 2898  df-rmo 2899  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-pss 3551  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-tp 4125  df-op 4127  df-uni 4363  df-int 4401  df-iun 4447  df-br 4574  df-opab 4634  df-mpt 4635  df-tr 4671  df-eprel 4935  df-id 4939  df-po 4945  df-so 4946  df-fr 4983  df-we 4985  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-pred 5579  df-ord 5625  df-on 5626  df-lim 5627  df-suc 5628  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-riota 6485  df-ov 6526  df-oprab 6527  df-mpt2 6528  df-of 6768  df-ofr 6769  df-om 6931  df-1st 7032  df-2nd 7033  df-supp 7156  df-wrecs 7267  df-recs 7328  df-rdg 7366  df-1o 7420  df-2o 7421  df-oadd 7424  df-er 7602  df-map 7719  df-pm 7720  df-ixp 7768  df-en 7815  df-dom 7816  df-sdom 7817  df-fin 7818  df-fsupp 8132  df-pnf 9928  df-mnf 9929  df-xr 9930  df-ltxr 9931  df-le 9932  df-sub 10115  df-neg 10116  df-nn 10864  df-2 10922  df-3 10923  df-4 10924  df-5 10925  df-6 10926  df-7 10927  df-8 10928  df-9 10929  df-n0 11136  df-z 11207  df-uz 11516  df-fz 12149  df-seq 12615  df-struct 15639  df-ndx 15640  df-slot 15641  df-base 15642  df-sets 15643  df-ress 15644  df-plusg 15723  df-mulr 15724  df-sca 15726  df-vsca 15727  df-tset 15729  df-0g 15867  df-gsum 15868  df-mgm 17007  df-sgrp 17049  df-mnd 17060  df-submnd 17101  df-grp 17190  df-minusg 17191  df-subg 17356  df-mgp 18255  df-ring 18314  df-subrg 18543  df-psr 19119
This theorem is referenced by:  subrgpsr  19182  ressmplmul  19221
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