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Theorem resspsrmul 19357
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 19301 . . . . . . . . . 10 Rel dom mPwSer
2 resspsr.u . . . . . . . . . 10 𝑈 = (𝐼 mPwSer 𝐻)
3 resspsr.b . . . . . . . . . 10 𝐵 = (Base‘𝑈)
41, 2, 3elbasov 15861 . . . . . . . . 9 (𝑋𝐵 → (𝐼 ∈ V ∧ 𝐻 ∈ V))
54ad2antrl 763 . . . . . . . 8 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝐼 ∈ V ∧ 𝐻 ∈ V))
65simpld 475 . . . . . . 7 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → 𝐼 ∈ V)
7 eqid 2621 . . . . . . . 8 {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}
87psrbaglefi 19312 . . . . . . 7 ((𝐼 ∈ V ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → {𝑦 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∣ 𝑦𝑟𝑘} ∈ Fin)
96, 8sylan 488 . . . . . 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 18726 . . . . . . . . 9 (𝑇 ∈ (SubRing‘𝑅) → 𝑇 ∈ (SubGrp‘𝑅))
1210, 11syl 17 . . . . . . . 8 (𝜑𝑇 ∈ (SubGrp‘𝑅))
13 subgsubm 17556 . . . . . . . 8 (𝑇 ∈ (SubGrp‘𝑅) → 𝑇 ∈ (SubMnd‘𝑅))
1412, 13syl 17 . . . . . . 7 (𝜑𝑇 ∈ (SubMnd‘𝑅))
1514ad2antrr 761 . . . . . 6 (((𝜑 ∧ (𝑋𝐵𝑌𝐵)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → 𝑇 ∈ (SubMnd‘𝑅))
1610ad3antrrr 765 . . . . . . . 8 ((((𝜑 ∧ (𝑋𝐵𝑌𝐵)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑥 ∈ {𝑦 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∣ 𝑦𝑟𝑘}) → 𝑇 ∈ (SubRing‘𝑅))
17 eqid 2621 . . . . . . . . . . . 12 (Base‘𝐻) = (Base‘𝐻)
18 simprl 793 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → 𝑋𝐵)
192, 17, 7, 3, 18psrelbas 19319 . . . . . . . . . . 11 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → 𝑋:{𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝐻))
2019adantr 481 . . . . . . . . . 10 (((𝜑 ∧ (𝑋𝐵𝑌𝐵)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → 𝑋:{𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝐻))
21 elrabi 3347 . . . . . . . . . 10 (𝑥 ∈ {𝑦 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∣ 𝑦𝑟𝑘} → 𝑥 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin})
22 ffvelrn 6323 . . . . . . . . . 10 ((𝑋:{𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝐻) ∧ 𝑥 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑋𝑥) ∈ (Base‘𝐻))
2320, 21, 22syl2an 494 . . . . . . . . 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 18729 . . . . . . . . . 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 2701 . . . . . . . 8 ((((𝜑 ∧ (𝑋𝐵𝑌𝐵)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑥 ∈ {𝑦 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∣ 𝑦𝑟𝑘}) → (𝑋𝑥) ∈ 𝑇)
28 simprr 795 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → 𝑌𝐵)
292, 17, 7, 3, 28psrelbas 19319 . . . . . . . . . . 11 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → 𝑌:{𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝐻))
3029ad2antrr 761 . . . . . . . . . 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 3672 . . . . . . . . . . 11 {𝑦 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∣ 𝑦𝑟𝑘} ⊆ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}
326ad2antrr 761 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑋𝐵𝑌𝐵)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑥 ∈ {𝑦 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∣ 𝑦𝑟𝑘}) → 𝐼 ∈ V)
33 simplr 791 . . . . . . . . . . . 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 477 . . . . . . . . . . . 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 2621 . . . . . . . . . . . . 13 {𝑦 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∣ 𝑦𝑟𝑘} = {𝑦 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∣ 𝑦𝑟𝑘}
367, 35psrbagconcl 19313 . . . . . . . . . . . 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 1323 . . . . . . . . . . 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 3586 . . . . . . . . . 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 6326 . . . . . . . . 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 2701 . . . . . . . 8 ((((𝜑 ∧ (𝑋𝐵𝑌𝐵)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑥 ∈ {𝑦 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∣ 𝑦𝑟𝑘}) → (𝑌‘(𝑘𝑓𝑥)) ∈ 𝑇)
41 eqid 2621 . . . . . . . . 9 (.r𝑅) = (.r𝑅)
4241subrgmcl 18732 . . . . . . . 8 ((𝑇 ∈ (SubRing‘𝑅) ∧ (𝑋𝑥) ∈ 𝑇 ∧ (𝑌‘(𝑘𝑓𝑥)) ∈ 𝑇) → ((𝑋𝑥)(.r𝑅)(𝑌‘(𝑘𝑓𝑥))) ∈ 𝑇)
4316, 27, 40, 42syl3anc 1323 . . . . . . 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 2621 . . . . . . 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 6351 . . . . . 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 17313 . . . . 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 15946 . . . . . . . . . 10 (𝑇 ∈ (SubRing‘𝑅) → (.r𝑅) = (.r𝐻))
4810, 47syl 17 . . . . . . . . 9 (𝜑 → (.r𝑅) = (.r𝐻))
4948ad3antrrr 765 . . . . . . . 8 ((((𝜑 ∧ (𝑋𝐵𝑌𝐵)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑥 ∈ {𝑦 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∣ 𝑦𝑟𝑘}) → (.r𝑅) = (.r𝐻))
5049oveqd 6632 . . . . . . 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 4714 . . . . . 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 6631 . . . . 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 2655 . . . 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 4714 . . 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 2621 . . . 4 (Base‘𝑆) = (Base‘𝑆)
57 eqid 2621 . . . 4 (.r𝑆) = (.r𝑆)
58 fvex 6168 . . . . . . . 8 (Base‘𝑅) ∈ V
5910, 25syl 17 . . . . . . . . 9 (𝜑𝑇 = (Base‘𝐻))
60 eqid 2621 . . . . . . . . . . 11 (Base‘𝑅) = (Base‘𝑅)
6160subrgss 18721 . . . . . . . . . 10 (𝑇 ∈ (SubRing‘𝑅) → 𝑇 ⊆ (Base‘𝑅))
6210, 61syl 17 . . . . . . . . 9 (𝜑𝑇 ⊆ (Base‘𝑅))
6359, 62eqsstr3d 3625 . . . . . . . 8 (𝜑 → (Base‘𝐻) ⊆ (Base‘𝑅))
64 mapss 7860 . . . . . . . 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 694 . . . . . . 7 (𝜑 → ((Base‘𝐻) ↑𝑚 {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ⊆ ((Base‘𝑅) ↑𝑚 {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}))
6665adantr 481 . . . . . 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 19318 . . . . . 6 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → 𝐵 = ((Base‘𝐻) ↑𝑚 {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}))
6855, 60, 7, 56, 6psrbas 19318 . . . . . 6 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (Base‘𝑆) = ((Base‘𝑅) ↑𝑚 {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}))
6966, 67, 683sstr4d 3633 . . . . 5 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → 𝐵 ⊆ (Base‘𝑆))
7069, 18sseldd 3589 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → 𝑋 ∈ (Base‘𝑆))
7169, 28sseldd 3589 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → 𝑌 ∈ (Base‘𝑆))
7255, 56, 41, 57, 7, 70, 71psrmulfval 19325 . . 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 2621 . . . 4 (.r𝐻) = (.r𝐻)
74 eqid 2621 . . . 4 (.r𝑈) = (.r𝑈)
752, 3, 73, 74, 7, 18, 28psrmulfval 19325 . . 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 2666 . 2 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋(.r𝑈)𝑌) = (𝑋(.r𝑆)𝑌))
77 fvex 6168 . . . . 5 (Base‘𝑈) ∈ V
783, 77eqeltri 2694 . . . 4 𝐵 ∈ V
79 resspsr.p . . . . 5 𝑃 = (𝑆s 𝐵)
8079, 57ressmulr 15946 . . . 4 (𝐵 ∈ V → (.r𝑆) = (.r𝑃))
8178, 80mp1i 13 . . 3 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (.r𝑆) = (.r𝑃))
8281oveqd 6632 . 2 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋(.r𝑆)𝑌) = (𝑋(.r𝑃)𝑌))
8376, 82eqtrd 2655 1 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋(.r𝑈)𝑌) = (𝑋(.r𝑃)𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  {crab 2912  Vcvv 3190  wss 3560   class class class wbr 4623  cmpt 4683  ccnv 5083  cima 5087  wf 5853  cfv 5857  (class class class)co 6615  𝑓 cof 6860  𝑟 cofr 6861  𝑚 cmap 7817  Fincfn 7915  cle 10035  cmin 10226  cn 10980  0cn0 11252  Basecbs 15800  s cress 15801  .rcmulr 15882   Σg cgsu 16041  SubMndcsubmnd 17274  SubGrpcsubg 17528  SubRingcsubrg 18716   mPwSer cmps 19291
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-of 6862  df-ofr 6863  df-om 7028  df-1st 7128  df-2nd 7129  df-supp 7256  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-2o 7521  df-oadd 7524  df-er 7702  df-map 7819  df-pm 7820  df-ixp 7869  df-en 7916  df-dom 7917  df-sdom 7918  df-fin 7919  df-fsupp 8236  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-nn 10981  df-2 11039  df-3 11040  df-4 11041  df-5 11042  df-6 11043  df-7 11044  df-8 11045  df-9 11046  df-n0 11253  df-z 11338  df-uz 11648  df-fz 12285  df-seq 12758  df-struct 15802  df-ndx 15803  df-slot 15804  df-base 15805  df-sets 15806  df-ress 15807  df-plusg 15894  df-mulr 15895  df-sca 15897  df-vsca 15898  df-tset 15900  df-0g 16042  df-gsum 16043  df-mgm 17182  df-sgrp 17224  df-mnd 17235  df-submnd 17276  df-grp 17365  df-minusg 17366  df-subg 17531  df-mgp 18430  df-ring 18489  df-subrg 18718  df-psr 19296
This theorem is referenced by:  subrgpsr  19359  ressmplmul  19398
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