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Theorem psrgrp 19795
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 2778 . 2 (𝜑 → (Base‘𝑆) = (Base‘𝑆))
2 eqidd 2778 . 2 (𝜑 → (+g𝑆) = (+g𝑆))
3 psrgrp.s . . 3 𝑆 = (𝐼 mPwSer 𝑅)
4 eqid 2777 . . 3 (Base‘𝑆) = (Base‘𝑆)
5 eqid 2777 . . 3 (+g𝑆) = (+g𝑆)
6 psrgrp.r . . . 4 (𝜑𝑅 ∈ Grp)
763ad2ant1 1124 . . 3 ((𝜑𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑅 ∈ Grp)
8 simp2 1128 . . 3 ((𝜑𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑥 ∈ (Base‘𝑆))
9 simp3 1129 . . 3 ((𝜑𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑦 ∈ (Base‘𝑆))
103, 4, 5, 7, 8, 9psraddcl 19780 . 2 ((𝜑𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥(+g𝑆)𝑦) ∈ (Base‘𝑆))
11 ovex 6954 . . . . . . 7 (ℕ0𝑚 𝐼) ∈ V
1211rabex 5049 . . . . . 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 2777 . . . . . 6 (Base‘𝑅) = (Base‘𝑅)
15 eqid 2777 . . . . . 6 {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}
16 simpr1 1205 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑥 ∈ (Base‘𝑆))
173, 14, 15, 4, 16psrelbas 19776 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑥:{𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
18 simpr2 1207 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑦 ∈ (Base‘𝑆))
193, 14, 15, 4, 18psrelbas 19776 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑦:{𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
20 simpr3 1209 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑧 ∈ (Base‘𝑆))
213, 14, 15, 4, 20psrelbas 19776 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑧:{𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
226adantr 474 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑅 ∈ Grp)
23 eqid 2777 . . . . . . 7 (+g𝑅) = (+g𝑅)
2414, 23grpass 17818 . . . . . 6 ((𝑅 ∈ Grp ∧ (𝑟 ∈ (Base‘𝑅) ∧ 𝑠 ∈ (Base‘𝑅) ∧ 𝑡 ∈ (Base‘𝑅))) → ((𝑟(+g𝑅)𝑠)(+g𝑅)𝑡) = (𝑟(+g𝑅)(𝑠(+g𝑅)𝑡)))
2522, 24sylan 575 . . . . 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 7208 . . . 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 19779 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑥(+g𝑆)𝑦) = (𝑥𝑓 (+g𝑅)𝑦))
2827oveq1d 6937 . . . 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 19779 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑦(+g𝑆)𝑧) = (𝑦𝑓 (+g𝑅)𝑧))
3029oveq2d 6938 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑥𝑓 (+g𝑅)(𝑦(+g𝑆)𝑧)) = (𝑥𝑓 (+g𝑅)(𝑦𝑓 (+g𝑅)𝑧)))
3126, 28, 303eqtr4d 2823 . . 3 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → ((𝑥(+g𝑆)𝑦) ∘𝑓 (+g𝑅)𝑧) = (𝑥𝑓 (+g𝑅)(𝑦(+g𝑆)𝑧)))
32103adant3r3 1192 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑥(+g𝑆)𝑦) ∈ (Base‘𝑆))
333, 4, 23, 5, 32, 20psradd 19779 . . 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 19780 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑦(+g𝑆)𝑧) ∈ (Base‘𝑆))
353, 4, 23, 5, 16, 34psradd 19779 . . 3 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑥(+g𝑆)(𝑦(+g𝑆)𝑧)) = (𝑥𝑓 (+g𝑅)(𝑦(+g𝑆)𝑧)))
3631, 33, 353eqtr4d 2823 . 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 2777 . . 3 (0g𝑅) = (0g𝑅)
393, 37, 6, 15, 38, 4psr0cl 19791 . 2 (𝜑 → ({𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} × {(0g𝑅)}) ∈ (Base‘𝑆))
4037adantr 474 . . 3 ((𝜑𝑥 ∈ (Base‘𝑆)) → 𝐼𝑉)
416adantr 474 . . 3 ((𝜑𝑥 ∈ (Base‘𝑆)) → 𝑅 ∈ Grp)
42 simpr 479 . . 3 ((𝜑𝑥 ∈ (Base‘𝑆)) → 𝑥 ∈ (Base‘𝑆))
433, 40, 41, 15, 38, 4, 5, 42psr0lid 19792 . 2 ((𝜑𝑥 ∈ (Base‘𝑆)) → (({𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} × {(0g𝑅)})(+g𝑆)𝑥) = 𝑥)
44 eqid 2777 . . 3 (invg𝑅) = (invg𝑅)
453, 40, 41, 15, 44, 4, 42psrnegcl 19793 . 2 ((𝜑𝑥 ∈ (Base‘𝑆)) → ((invg𝑅) ∘ 𝑥) ∈ (Base‘𝑆))
463, 40, 41, 15, 44, 4, 42, 38, 5psrlinv 19794 . 2 ((𝜑𝑥 ∈ (Base‘𝑆)) → (((invg𝑅) ∘ 𝑥)(+g𝑆)𝑥) = ({𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} × {(0g𝑅)}))
471, 2, 10, 36, 39, 43, 45, 46isgrpd 17831 1 (𝜑𝑆 ∈ Grp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  w3a 1071   = wceq 1601  wcel 2106  {crab 3093  Vcvv 3397  {csn 4397   × cxp 5353  ccnv 5354  cima 5358  ccom 5359  cfv 6135  (class class class)co 6922  𝑓 cof 7172  𝑚 cmap 8140  Fincfn 8241  cn 11374  0cn0 11642  Basecbs 16255  +gcplusg 16338  0gc0g 16486  Grpcgrp 17809  invgcminusg 17810   mPwSer cmps 19748
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-cnex 10328  ax-resscn 10329  ax-1cn 10330  ax-icn 10331  ax-addcl 10332  ax-addrcl 10333  ax-mulcl 10334  ax-mulrcl 10335  ax-mulcom 10336  ax-addass 10337  ax-mulass 10338  ax-distr 10339  ax-i2m1 10340  ax-1ne0 10341  ax-1rid 10342  ax-rnegex 10343  ax-rrecex 10344  ax-cnre 10345  ax-pre-lttri 10346  ax-pre-lttrn 10347  ax-pre-ltadd 10348  ax-pre-mulgt0 10349
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-nel 3075  df-ral 3094  df-rex 3095  df-reu 3096  df-rmo 3097  df-rab 3098  df-v 3399  df-sbc 3652  df-csb 3751  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-pss 3807  df-nul 4141  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4672  df-int 4711  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-of 7174  df-om 7344  df-1st 7445  df-2nd 7446  df-supp 7577  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-1o 7843  df-oadd 7847  df-er 8026  df-map 8142  df-en 8242  df-dom 8243  df-sdom 8244  df-fin 8245  df-fsupp 8564  df-pnf 10413  df-mnf 10414  df-xr 10415  df-ltxr 10416  df-le 10417  df-sub 10608  df-neg 10609  df-nn 11375  df-2 11438  df-3 11439  df-4 11440  df-5 11441  df-6 11442  df-7 11443  df-8 11444  df-9 11445  df-n0 11643  df-z 11729  df-uz 11993  df-fz 12644  df-struct 16257  df-ndx 16258  df-slot 16259  df-base 16261  df-plusg 16351  df-mulr 16352  df-sca 16354  df-vsca 16355  df-tset 16357  df-0g 16488  df-mgm 17628  df-sgrp 17670  df-mnd 17681  df-grp 17812  df-minusg 17813  df-psr 19753
This theorem is referenced by:  psr0  19796  psrneg  19797  psrlmod  19798  psrring  19808  mplsubglem  19831
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