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Theorem psrgrp 19161
Description: The ring of power series is a group. (Contributed by Mario Carneiro, 29-Dec-2014.)
Hypotheses
Ref Expression
psrgrp.s 𝑆 = (𝐼 mPwSer 𝑅)
psrgrp.i (𝜑𝐼𝑉)
psrgrp.r (𝜑𝑅 ∈ Grp)
Assertion
Ref Expression
psrgrp (𝜑𝑆 ∈ Grp)

Proof of Theorem psrgrp
Dummy variables 𝑥 𝑠 𝑟 𝑡 𝑦 𝑧 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqidd 2606 . 2 (𝜑 → (Base‘𝑆) = (Base‘𝑆))
2 eqidd 2606 . 2 (𝜑 → (+g𝑆) = (+g𝑆))
3 psrgrp.s . . 3 𝑆 = (𝐼 mPwSer 𝑅)
4 eqid 2605 . . 3 (Base‘𝑆) = (Base‘𝑆)
5 eqid 2605 . . 3 (+g𝑆) = (+g𝑆)
6 psrgrp.r . . . 4 (𝜑𝑅 ∈ Grp)
763ad2ant1 1074 . . 3 ((𝜑𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑅 ∈ Grp)
8 simp2 1054 . . 3 ((𝜑𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑥 ∈ (Base‘𝑆))
9 simp3 1055 . . 3 ((𝜑𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑦 ∈ (Base‘𝑆))
103, 4, 5, 7, 8, 9psraddcl 19146 . 2 ((𝜑𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥(+g𝑆)𝑦) ∈ (Base‘𝑆))
11 ovex 6551 . . . . . . 7 (ℕ0𝑚 𝐼) ∈ V
1211rabex 4731 . . . . . 6 {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∈ V
1312a1i 11 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∈ V)
14 eqid 2605 . . . . . 6 (Base‘𝑅) = (Base‘𝑅)
15 eqid 2605 . . . . . 6 {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}
16 simpr1 1059 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑥 ∈ (Base‘𝑆))
173, 14, 15, 4, 16psrelbas 19142 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑥:{𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
18 simpr2 1060 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑦 ∈ (Base‘𝑆))
193, 14, 15, 4, 18psrelbas 19142 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑦:{𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
20 simpr3 1061 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑧 ∈ (Base‘𝑆))
213, 14, 15, 4, 20psrelbas 19142 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑧:{𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
226adantr 479 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑅 ∈ Grp)
23 eqid 2605 . . . . . . 7 (+g𝑅) = (+g𝑅)
2414, 23grpass 17196 . . . . . 6 ((𝑅 ∈ Grp ∧ (𝑟 ∈ (Base‘𝑅) ∧ 𝑠 ∈ (Base‘𝑅) ∧ 𝑡 ∈ (Base‘𝑅))) → ((𝑟(+g𝑅)𝑠)(+g𝑅)𝑡) = (𝑟(+g𝑅)(𝑠(+g𝑅)𝑡)))
2522, 24sylan 486 . . . . 5 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) ∧ (𝑟 ∈ (Base‘𝑅) ∧ 𝑠 ∈ (Base‘𝑅) ∧ 𝑡 ∈ (Base‘𝑅))) → ((𝑟(+g𝑅)𝑠)(+g𝑅)𝑡) = (𝑟(+g𝑅)(𝑠(+g𝑅)𝑡)))
2613, 17, 19, 21, 25caofass 6802 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → ((𝑥𝑓 (+g𝑅)𝑦) ∘𝑓 (+g𝑅)𝑧) = (𝑥𝑓 (+g𝑅)(𝑦𝑓 (+g𝑅)𝑧)))
273, 4, 23, 5, 16, 18psradd 19145 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑥(+g𝑆)𝑦) = (𝑥𝑓 (+g𝑅)𝑦))
2827oveq1d 6538 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → ((𝑥(+g𝑆)𝑦) ∘𝑓 (+g𝑅)𝑧) = ((𝑥𝑓 (+g𝑅)𝑦) ∘𝑓 (+g𝑅)𝑧))
293, 4, 23, 5, 18, 20psradd 19145 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑦(+g𝑆)𝑧) = (𝑦𝑓 (+g𝑅)𝑧))
3029oveq2d 6539 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑥𝑓 (+g𝑅)(𝑦(+g𝑆)𝑧)) = (𝑥𝑓 (+g𝑅)(𝑦𝑓 (+g𝑅)𝑧)))
3126, 28, 303eqtr4d 2649 . . 3 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → ((𝑥(+g𝑆)𝑦) ∘𝑓 (+g𝑅)𝑧) = (𝑥𝑓 (+g𝑅)(𝑦(+g𝑆)𝑧)))
32103adant3r3 1267 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑥(+g𝑆)𝑦) ∈ (Base‘𝑆))
333, 4, 23, 5, 32, 20psradd 19145 . . 3 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → ((𝑥(+g𝑆)𝑦)(+g𝑆)𝑧) = ((𝑥(+g𝑆)𝑦) ∘𝑓 (+g𝑅)𝑧))
343, 4, 5, 22, 18, 20psraddcl 19146 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑦(+g𝑆)𝑧) ∈ (Base‘𝑆))
353, 4, 23, 5, 16, 34psradd 19145 . . 3 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑥(+g𝑆)(𝑦(+g𝑆)𝑧)) = (𝑥𝑓 (+g𝑅)(𝑦(+g𝑆)𝑧)))
3631, 33, 353eqtr4d 2649 . 2 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → ((𝑥(+g𝑆)𝑦)(+g𝑆)𝑧) = (𝑥(+g𝑆)(𝑦(+g𝑆)𝑧)))
37 psrgrp.i . . 3 (𝜑𝐼𝑉)
38 eqid 2605 . . 3 (0g𝑅) = (0g𝑅)
393, 37, 6, 15, 38, 4psr0cl 19157 . 2 (𝜑 → ({𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} × {(0g𝑅)}) ∈ (Base‘𝑆))
4037adantr 479 . . 3 ((𝜑𝑥 ∈ (Base‘𝑆)) → 𝐼𝑉)
416adantr 479 . . 3 ((𝜑𝑥 ∈ (Base‘𝑆)) → 𝑅 ∈ Grp)
42 simpr 475 . . 3 ((𝜑𝑥 ∈ (Base‘𝑆)) → 𝑥 ∈ (Base‘𝑆))
433, 40, 41, 15, 38, 4, 5, 42psr0lid 19158 . 2 ((𝜑𝑥 ∈ (Base‘𝑆)) → (({𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} × {(0g𝑅)})(+g𝑆)𝑥) = 𝑥)
44 eqid 2605 . . 3 (invg𝑅) = (invg𝑅)
453, 40, 41, 15, 44, 4, 42psrnegcl 19159 . 2 ((𝜑𝑥 ∈ (Base‘𝑆)) → ((invg𝑅) ∘ 𝑥) ∈ (Base‘𝑆))
463, 40, 41, 15, 44, 4, 42, 38, 5psrlinv 19160 . 2 ((𝜑𝑥 ∈ (Base‘𝑆)) → (((invg𝑅) ∘ 𝑥)(+g𝑆)𝑥) = ({𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} × {(0g𝑅)}))
471, 2, 10, 36, 39, 43, 45, 46isgrpd 17209 1 (𝜑𝑆 ∈ Grp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1030   = wceq 1474  wcel 1975  {crab 2895  Vcvv 3168  {csn 4120   × cxp 5022  ccnv 5023  cima 5027  ccom 5028  cfv 5786  (class class class)co 6523  𝑓 cof 6766  𝑚 cmap 7717  Fincfn 7814  cn 10863  0cn0 11135  Basecbs 15637  +gcplusg 15710  0gc0g 15865  Grpcgrp 17187  invgcminusg 17188   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-om 6931  df-1st 7032  df-2nd 7033  df-supp 7156  df-wrecs 7267  df-recs 7328  df-rdg 7366  df-1o 7420  df-oadd 7424  df-er 7602  df-map 7719  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-struct 15639  df-ndx 15640  df-slot 15641  df-base 15642  df-plusg 15723  df-mulr 15724  df-sca 15726  df-vsca 15727  df-tset 15729  df-0g 15867  df-mgm 17007  df-sgrp 17049  df-mnd 17060  df-grp 17190  df-minusg 17191  df-psr 19119
This theorem is referenced by:  psr0  19162  psrneg  19163  psrlmod  19164  psrring  19174  mplsubglem  19197
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