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Theorem psrgrp 19612
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 2771 . 2 (𝜑 → (Base‘𝑆) = (Base‘𝑆))
2 eqidd 2771 . 2 (𝜑 → (+g𝑆) = (+g𝑆))
3 psrgrp.s . . 3 𝑆 = (𝐼 mPwSer 𝑅)
4 eqid 2770 . . 3 (Base‘𝑆) = (Base‘𝑆)
5 eqid 2770 . . 3 (+g𝑆) = (+g𝑆)
6 psrgrp.r . . . 4 (𝜑𝑅 ∈ Grp)
763ad2ant1 1126 . . 3 ((𝜑𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑅 ∈ Grp)
8 simp2 1130 . . 3 ((𝜑𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑥 ∈ (Base‘𝑆))
9 simp3 1131 . . 3 ((𝜑𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑦 ∈ (Base‘𝑆))
103, 4, 5, 7, 8, 9psraddcl 19597 . 2 ((𝜑𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥(+g𝑆)𝑦) ∈ (Base‘𝑆))
11 ovex 6822 . . . . . . 7 (ℕ0𝑚 𝐼) ∈ V
1211rabex 4943 . . . . . 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 2770 . . . . . 6 (Base‘𝑅) = (Base‘𝑅)
15 eqid 2770 . . . . . 6 {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}
16 simpr1 1232 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑥 ∈ (Base‘𝑆))
173, 14, 15, 4, 16psrelbas 19593 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑥:{𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
18 simpr2 1234 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑦 ∈ (Base‘𝑆))
193, 14, 15, 4, 18psrelbas 19593 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑦:{𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
20 simpr3 1236 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑧 ∈ (Base‘𝑆))
213, 14, 15, 4, 20psrelbas 19593 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑧:{𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
226adantr 466 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑅 ∈ Grp)
23 eqid 2770 . . . . . . 7 (+g𝑅) = (+g𝑅)
2414, 23grpass 17638 . . . . . 6 ((𝑅 ∈ Grp ∧ (𝑟 ∈ (Base‘𝑅) ∧ 𝑠 ∈ (Base‘𝑅) ∧ 𝑡 ∈ (Base‘𝑅))) → ((𝑟(+g𝑅)𝑠)(+g𝑅)𝑡) = (𝑟(+g𝑅)(𝑠(+g𝑅)𝑡)))
2522, 24sylan 561 . . . . 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 7077 . . . 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 19596 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑥(+g𝑆)𝑦) = (𝑥𝑓 (+g𝑅)𝑦))
2827oveq1d 6807 . . . 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 19596 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑦(+g𝑆)𝑧) = (𝑦𝑓 (+g𝑅)𝑧))
3029oveq2d 6808 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑥𝑓 (+g𝑅)(𝑦(+g𝑆)𝑧)) = (𝑥𝑓 (+g𝑅)(𝑦𝑓 (+g𝑅)𝑧)))
3126, 28, 303eqtr4d 2814 . . 3 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → ((𝑥(+g𝑆)𝑦) ∘𝑓 (+g𝑅)𝑧) = (𝑥𝑓 (+g𝑅)(𝑦(+g𝑆)𝑧)))
32103adant3r3 1198 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑥(+g𝑆)𝑦) ∈ (Base‘𝑆))
333, 4, 23, 5, 32, 20psradd 19596 . . 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 19597 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑦(+g𝑆)𝑧) ∈ (Base‘𝑆))
353, 4, 23, 5, 16, 34psradd 19596 . . 3 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑥(+g𝑆)(𝑦(+g𝑆)𝑧)) = (𝑥𝑓 (+g𝑅)(𝑦(+g𝑆)𝑧)))
3631, 33, 353eqtr4d 2814 . 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 2770 . . 3 (0g𝑅) = (0g𝑅)
393, 37, 6, 15, 38, 4psr0cl 19608 . 2 (𝜑 → ({𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} × {(0g𝑅)}) ∈ (Base‘𝑆))
4037adantr 466 . . 3 ((𝜑𝑥 ∈ (Base‘𝑆)) → 𝐼𝑉)
416adantr 466 . . 3 ((𝜑𝑥 ∈ (Base‘𝑆)) → 𝑅 ∈ Grp)
42 simpr 471 . . 3 ((𝜑𝑥 ∈ (Base‘𝑆)) → 𝑥 ∈ (Base‘𝑆))
433, 40, 41, 15, 38, 4, 5, 42psr0lid 19609 . 2 ((𝜑𝑥 ∈ (Base‘𝑆)) → (({𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} × {(0g𝑅)})(+g𝑆)𝑥) = 𝑥)
44 eqid 2770 . . 3 (invg𝑅) = (invg𝑅)
453, 40, 41, 15, 44, 4, 42psrnegcl 19610 . 2 ((𝜑𝑥 ∈ (Base‘𝑆)) → ((invg𝑅) ∘ 𝑥) ∈ (Base‘𝑆))
463, 40, 41, 15, 44, 4, 42, 38, 5psrlinv 19611 . 2 ((𝜑𝑥 ∈ (Base‘𝑆)) → (((invg𝑅) ∘ 𝑥)(+g𝑆)𝑥) = ({𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} × {(0g𝑅)}))
471, 2, 10, 36, 39, 43, 45, 46isgrpd 17651 1 (𝜑𝑆 ∈ Grp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1070   = wceq 1630  wcel 2144  {crab 3064  Vcvv 3349  {csn 4314   × cxp 5247  ccnv 5248  cima 5252  ccom 5253  cfv 6031  (class class class)co 6792  𝑓 cof 7041  𝑚 cmap 8008  Fincfn 8108  cn 11221  0cn0 11493  Basecbs 16063  +gcplusg 16148  0gc0g 16307  Grpcgrp 17629  invgcminusg 17630   mPwSer cmps 19565
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095  ax-cnex 10193  ax-resscn 10194  ax-1cn 10195  ax-icn 10196  ax-addcl 10197  ax-addrcl 10198  ax-mulcl 10199  ax-mulrcl 10200  ax-mulcom 10201  ax-addass 10202  ax-mulass 10203  ax-distr 10204  ax-i2m1 10205  ax-1ne0 10206  ax-1rid 10207  ax-rnegex 10208  ax-rrecex 10209  ax-cnre 10210  ax-pre-lttri 10211  ax-pre-lttrn 10212  ax-pre-ltadd 10213  ax-pre-mulgt0 10214
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-nel 3046  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-int 4610  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6753  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-of 7043  df-om 7212  df-1st 7314  df-2nd 7315  df-supp 7446  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-1o 7712  df-oadd 7716  df-er 7895  df-map 8010  df-en 8109  df-dom 8110  df-sdom 8111  df-fin 8112  df-fsupp 8431  df-pnf 10277  df-mnf 10278  df-xr 10279  df-ltxr 10280  df-le 10281  df-sub 10469  df-neg 10470  df-nn 11222  df-2 11280  df-3 11281  df-4 11282  df-5 11283  df-6 11284  df-7 11285  df-8 11286  df-9 11287  df-n0 11494  df-z 11579  df-uz 11888  df-fz 12533  df-struct 16065  df-ndx 16066  df-slot 16067  df-base 16069  df-plusg 16161  df-mulr 16162  df-sca 16164  df-vsca 16165  df-tset 16167  df-0g 16309  df-mgm 17449  df-sgrp 17491  df-mnd 17502  df-grp 17632  df-minusg 17633  df-psr 19570
This theorem is referenced by:  psr0  19613  psrneg  19614  psrlmod  19615  psrring  19625  mplsubglem  19648
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