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Theorem psrgrp 16462
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  |-  ( ph  ->  I  e.  V )
psrgrp.r  |-  ( ph  ->  R  e.  Grp )
Assertion
Ref Expression
psrgrp  |-  ( ph  ->  S  e.  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 2437 . 2  |-  ( ph  ->  ( Base `  S
)  =  ( Base `  S ) )
2 eqidd 2437 . 2  |-  ( ph  ->  ( +g  `  S
)  =  ( +g  `  S ) )
3 psrgrp.s . . 3  |-  S  =  ( I mPwSer  R )
4 eqid 2436 . . 3  |-  ( Base `  S )  =  (
Base `  S )
5 eqid 2436 . . 3  |-  ( +g  `  S )  =  ( +g  `  S )
6 psrgrp.r . . . 4  |-  ( ph  ->  R  e.  Grp )
763ad2ant1 978 . . 3  |-  ( (
ph  /\  x  e.  ( Base `  S )  /\  y  e.  ( Base `  S ) )  ->  R  e.  Grp )
8 simp2 958 . . 3  |-  ( (
ph  /\  x  e.  ( Base `  S )  /\  y  e.  ( Base `  S ) )  ->  x  e.  (
Base `  S )
)
9 simp3 959 . . 3  |-  ( (
ph  /\  x  e.  ( Base `  S )  /\  y  e.  ( Base `  S ) )  ->  y  e.  (
Base `  S )
)
103, 4, 5, 7, 8, 9psraddcl 16447 . 2  |-  ( (
ph  /\  x  e.  ( Base `  S )  /\  y  e.  ( Base `  S ) )  ->  ( x ( +g  `  S ) y )  e.  (
Base `  S )
)
11 ovex 6106 . . . . . . 7  |-  ( NN0 
^m  I )  e. 
_V
1211rabex 4354 . . . . . 6  |-  { f  e.  ( NN0  ^m  I )  |  ( `' f " NN )  e.  Fin }  e.  _V
1312a1i 11 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( Base `  S
)  /\  y  e.  ( Base `  S )  /\  z  e.  ( Base `  S ) ) )  ->  { f  e.  ( NN0  ^m  I
)  |  ( `' f " NN )  e.  Fin }  e.  _V )
14 eqid 2436 . . . . . 6  |-  ( Base `  R )  =  (
Base `  R )
15 eqid 2436 . . . . . 6  |-  { f  e.  ( NN0  ^m  I )  |  ( `' f " NN )  e.  Fin }  =  { f  e.  ( NN0  ^m  I )  |  ( `' f
" NN )  e. 
Fin }
16 simpr1 963 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  S
)  /\  y  e.  ( Base `  S )  /\  z  e.  ( Base `  S ) ) )  ->  x  e.  ( Base `  S )
)
173, 14, 15, 4, 16psrelbas 16444 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( Base `  S
)  /\  y  e.  ( Base `  S )  /\  z  e.  ( Base `  S ) ) )  ->  x : { f  e.  ( NN0  ^m  I )  |  ( `' f
" NN )  e. 
Fin } --> ( Base `  R
) )
18 simpr2 964 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  S
)  /\  y  e.  ( Base `  S )  /\  z  e.  ( Base `  S ) ) )  ->  y  e.  ( Base `  S )
)
193, 14, 15, 4, 18psrelbas 16444 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( Base `  S
)  /\  y  e.  ( Base `  S )  /\  z  e.  ( Base `  S ) ) )  ->  y : { f  e.  ( NN0  ^m  I )  |  ( `' f
" NN )  e. 
Fin } --> ( Base `  R
) )
20 simpr3 965 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  S
)  /\  y  e.  ( Base `  S )  /\  z  e.  ( Base `  S ) ) )  ->  z  e.  ( Base `  S )
)
213, 14, 15, 4, 20psrelbas 16444 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( Base `  S
)  /\  y  e.  ( Base `  S )  /\  z  e.  ( Base `  S ) ) )  ->  z : { f  e.  ( NN0  ^m  I )  |  ( `' f
" NN )  e. 
Fin } --> ( Base `  R
) )
226adantr 452 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  S
)  /\  y  e.  ( Base `  S )  /\  z  e.  ( Base `  S ) ) )  ->  R  e.  Grp )
23 eqid 2436 . . . . . . 7  |-  ( +g  `  R )  =  ( +g  `  R )
2414, 23grpass 14819 . . . . . 6  |-  ( ( R  e.  Grp  /\  ( r  e.  (
Base `  R )  /\  s  e.  ( Base `  R )  /\  t  e.  ( Base `  R ) ) )  ->  ( ( r ( +g  `  R
) s ) ( +g  `  R ) t )  =  ( r ( +g  `  R
) ( s ( +g  `  R ) t ) ) )
2522, 24sylan 458 . . . . 5  |-  ( ( ( ph  /\  (
x  e.  ( Base `  S )  /\  y  e.  ( Base `  S
)  /\  z  e.  ( Base `  S )
) )  /\  (
r  e.  ( Base `  R )  /\  s  e.  ( Base `  R
)  /\  t  e.  ( Base `  R )
) )  ->  (
( r ( +g  `  R ) s ) ( +g  `  R
) t )  =  ( r ( +g  `  R ) ( s ( +g  `  R
) t ) ) )
2613, 17, 19, 21, 25caofass 6338 . . . 4  |-  ( (
ph  /\  ( x  e.  ( Base `  S
)  /\  y  e.  ( Base `  S )  /\  z  e.  ( Base `  S ) ) )  ->  ( (
x  o F ( +g  `  R ) y )  o F ( +g  `  R
) z )  =  ( x  o F ( +g  `  R
) ( y  o F ( +g  `  R
) z ) ) )
273, 4, 23, 5, 16, 18psradd 16446 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( Base `  S
)  /\  y  e.  ( Base `  S )  /\  z  e.  ( Base `  S ) ) )  ->  ( x
( +g  `  S ) y )  =  ( x  o F ( +g  `  R ) y ) )
2827oveq1d 6096 . . . 4  |-  ( (
ph  /\  ( x  e.  ( Base `  S
)  /\  y  e.  ( Base `  S )  /\  z  e.  ( Base `  S ) ) )  ->  ( (
x ( +g  `  S
) y )  o F ( +g  `  R
) z )  =  ( ( x  o F ( +g  `  R
) y )  o F ( +g  `  R
) z ) )
293, 4, 23, 5, 18, 20psradd 16446 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( Base `  S
)  /\  y  e.  ( Base `  S )  /\  z  e.  ( Base `  S ) ) )  ->  ( y
( +g  `  S ) z )  =  ( y  o F ( +g  `  R ) z ) )
3029oveq2d 6097 . . . 4  |-  ( (
ph  /\  ( x  e.  ( Base `  S
)  /\  y  e.  ( Base `  S )  /\  z  e.  ( Base `  S ) ) )  ->  ( x  o F ( +g  `  R
) ( y ( +g  `  S ) z ) )  =  ( x  o F ( +g  `  R
) ( y  o F ( +g  `  R
) z ) ) )
3126, 28, 303eqtr4d 2478 . . 3  |-  ( (
ph  /\  ( x  e.  ( Base `  S
)  /\  y  e.  ( Base `  S )  /\  z  e.  ( Base `  S ) ) )  ->  ( (
x ( +g  `  S
) y )  o F ( +g  `  R
) z )  =  ( x  o F ( +g  `  R
) ( y ( +g  `  S ) z ) ) )
32103adant3r3 1164 . . . 4  |-  ( (
ph  /\  ( x  e.  ( Base `  S
)  /\  y  e.  ( Base `  S )  /\  z  e.  ( Base `  S ) ) )  ->  ( x
( +g  `  S ) y )  e.  (
Base `  S )
)
333, 4, 23, 5, 32, 20psradd 16446 . . 3  |-  ( (
ph  /\  ( x  e.  ( Base `  S
)  /\  y  e.  ( Base `  S )  /\  z  e.  ( Base `  S ) ) )  ->  ( (
x ( +g  `  S
) y ) ( +g  `  S ) z )  =  ( ( x ( +g  `  S ) y )  o F ( +g  `  R ) z ) )
343, 4, 5, 22, 18, 20psraddcl 16447 . . . 4  |-  ( (
ph  /\  ( x  e.  ( Base `  S
)  /\  y  e.  ( Base `  S )  /\  z  e.  ( Base `  S ) ) )  ->  ( y
( +g  `  S ) z )  e.  (
Base `  S )
)
353, 4, 23, 5, 16, 34psradd 16446 . . 3  |-  ( (
ph  /\  ( x  e.  ( Base `  S
)  /\  y  e.  ( Base `  S )  /\  z  e.  ( Base `  S ) ) )  ->  ( x
( +g  `  S ) ( y ( +g  `  S ) z ) )  =  ( x  o F ( +g  `  R ) ( y ( +g  `  S
) z ) ) )
3631, 33, 353eqtr4d 2478 . 2  |-  ( (
ph  /\  ( x  e.  ( Base `  S
)  /\  y  e.  ( Base `  S )  /\  z  e.  ( Base `  S ) ) )  ->  ( (
x ( +g  `  S
) y ) ( +g  `  S ) z )  =  ( x ( +g  `  S
) ( y ( +g  `  S ) z ) ) )
37 psrgrp.i . . 3  |-  ( ph  ->  I  e.  V )
38 eqid 2436 . . 3  |-  ( 0g
`  R )  =  ( 0g `  R
)
393, 37, 6, 15, 38, 4psr0cl 16458 . 2  |-  ( ph  ->  ( { f  e.  ( NN0  ^m  I
)  |  ( `' f " NN )  e.  Fin }  X.  { ( 0g `  R ) } )  e.  ( Base `  S
) )
4037adantr 452 . . 3  |-  ( (
ph  /\  x  e.  ( Base `  S )
)  ->  I  e.  V )
416adantr 452 . . 3  |-  ( (
ph  /\  x  e.  ( Base `  S )
)  ->  R  e.  Grp )
42 simpr 448 . . 3  |-  ( (
ph  /\  x  e.  ( Base `  S )
)  ->  x  e.  ( Base `  S )
)
433, 40, 41, 15, 38, 4, 5, 42psr0lid 16459 . 2  |-  ( (
ph  /\  x  e.  ( Base `  S )
)  ->  ( ( { f  e.  ( NN0  ^m  I )  |  ( `' f
" NN )  e. 
Fin }  X.  { ( 0g `  R ) } ) ( +g  `  S ) x )  =  x )
44 eqid 2436 . . 3  |-  ( inv g `  R )  =  ( inv g `  R )
453, 40, 41, 15, 44, 4, 42psrnegcl 16460 . 2  |-  ( (
ph  /\  x  e.  ( Base `  S )
)  ->  ( ( inv g `  R )  o.  x )  e.  ( Base `  S
) )
463, 40, 41, 15, 44, 4, 42, 38, 5psrlinv 16461 . 2  |-  ( (
ph  /\  x  e.  ( Base `  S )
)  ->  ( (
( inv g `  R )  o.  x
) ( +g  `  S
) x )  =  ( { f  e.  ( NN0  ^m  I
)  |  ( `' f " NN )  e.  Fin }  X.  { ( 0g `  R ) } ) )
471, 2, 10, 36, 39, 43, 45, 46isgrpd 14830 1  |-  ( ph  ->  S  e.  Grp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   {crab 2709   _Vcvv 2956   {csn 3814    X. cxp 4876   `'ccnv 4877   "cima 4881    o. ccom 4882   ` cfv 5454  (class class class)co 6081    o Fcof 6303    ^m cmap 7018   Fincfn 7109   NNcn 10000   NN0cn0 10221   Basecbs 13469   +g cplusg 13529   0gc0g 13723   Grpcgrp 14685   inv gcminusg 14686   mPwSer cmps 16406
This theorem is referenced by:  psr0  16463  psrneg  16464  psrlmod  16465  psrrng  16474  mplsubglem  16498
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-of 6305  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-5 10061  df-6 10062  df-7 10063  df-8 10064  df-9 10065  df-n0 10222  df-z 10283  df-uz 10489  df-fz 11044  df-struct 13471  df-ndx 13472  df-slot 13473  df-base 13474  df-plusg 13542  df-mulr 13543  df-sca 13545  df-vsca 13546  df-tset 13548  df-0g 13727  df-mnd 14690  df-grp 14812  df-minusg 14813  df-psr 16417
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