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Theorem psrgrp 17447
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 2442 . 2  |-  ( ph  ->  ( Base `  S
)  =  ( Base `  S ) )
2 eqidd 2442 . 2  |-  ( ph  ->  ( +g  `  S
)  =  ( +g  `  S ) )
3 psrgrp.s . . 3  |-  S  =  ( I mPwSer  R )
4 eqid 2441 . . 3  |-  ( Base `  S )  =  (
Base `  S )
5 eqid 2441 . . 3  |-  ( +g  `  S )  =  ( +g  `  S )
6 psrgrp.r . . . 4  |-  ( ph  ->  R  e.  Grp )
763ad2ant1 1004 . . 3  |-  ( (
ph  /\  x  e.  ( Base `  S )  /\  y  e.  ( Base `  S ) )  ->  R  e.  Grp )
8 simp2 984 . . 3  |-  ( (
ph  /\  x  e.  ( Base `  S )  /\  y  e.  ( Base `  S ) )  ->  x  e.  (
Base `  S )
)
9 simp3 985 . . 3  |-  ( (
ph  /\  x  e.  ( Base `  S )  /\  y  e.  ( Base `  S ) )  ->  y  e.  (
Base `  S )
)
103, 4, 5, 7, 8, 9psraddcl 17432 . 2  |-  ( (
ph  /\  x  e.  ( Base `  S )  /\  y  e.  ( Base `  S ) )  ->  ( x ( +g  `  S ) y )  e.  (
Base `  S )
)
11 ovex 6115 . . . . . . 7  |-  ( NN0 
^m  I )  e. 
_V
1211rabex 4440 . . . . . 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 2441 . . . . . 6  |-  ( Base `  R )  =  (
Base `  R )
15 eqid 2441 . . . . . 6  |-  { f  e.  ( NN0  ^m  I )  |  ( `' f " NN )  e.  Fin }  =  { f  e.  ( NN0  ^m  I )  |  ( `' f
" NN )  e. 
Fin }
16 simpr1 989 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  S
)  /\  y  e.  ( Base `  S )  /\  z  e.  ( Base `  S ) ) )  ->  x  e.  ( Base `  S )
)
173, 14, 15, 4, 16psrelbas 17428 . . . . 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 990 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  S
)  /\  y  e.  ( Base `  S )  /\  z  e.  ( Base `  S ) ) )  ->  y  e.  ( Base `  S )
)
193, 14, 15, 4, 18psrelbas 17428 . . . . 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 991 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  S
)  /\  y  e.  ( Base `  S )  /\  z  e.  ( Base `  S ) ) )  ->  z  e.  ( Base `  S )
)
213, 14, 15, 4, 20psrelbas 17428 . . . . 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 462 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  S
)  /\  y  e.  ( Base `  S )  /\  z  e.  ( Base `  S ) ) )  ->  R  e.  Grp )
23 eqid 2441 . . . . . . 7  |-  ( +g  `  R )  =  ( +g  `  R )
2414, 23grpass 15545 . . . . . 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 468 . . . . 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 6353 . . . 4  |-  ( (
ph  /\  ( x  e.  ( Base `  S
)  /\  y  e.  ( Base `  S )  /\  z  e.  ( Base `  S ) ) )  ->  ( (
x  oF ( +g  `  R ) y )  oF ( +g  `  R
) z )  =  ( x  oF ( +g  `  R
) ( y  oF ( +g  `  R
) z ) ) )
273, 4, 23, 5, 16, 18psradd 17431 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( Base `  S
)  /\  y  e.  ( Base `  S )  /\  z  e.  ( Base `  S ) ) )  ->  ( x
( +g  `  S ) y )  =  ( x  oF ( +g  `  R ) y ) )
2827oveq1d 6105 . . . 4  |-  ( (
ph  /\  ( x  e.  ( Base `  S
)  /\  y  e.  ( Base `  S )  /\  z  e.  ( Base `  S ) ) )  ->  ( (
x ( +g  `  S
) y )  oF ( +g  `  R
) z )  =  ( ( x  oF ( +g  `  R
) y )  oF ( +g  `  R
) z ) )
293, 4, 23, 5, 18, 20psradd 17431 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( Base `  S
)  /\  y  e.  ( Base `  S )  /\  z  e.  ( Base `  S ) ) )  ->  ( y
( +g  `  S ) z )  =  ( y  oF ( +g  `  R ) z ) )
3029oveq2d 6106 . . . 4  |-  ( (
ph  /\  ( x  e.  ( Base `  S
)  /\  y  e.  ( Base `  S )  /\  z  e.  ( Base `  S ) ) )  ->  ( x  oF ( +g  `  R ) ( y ( +g  `  S
) z ) )  =  ( x  oF ( +g  `  R
) ( y  oF ( +g  `  R
) z ) ) )
3126, 28, 303eqtr4d 2483 . . 3  |-  ( (
ph  /\  ( x  e.  ( Base `  S
)  /\  y  e.  ( Base `  S )  /\  z  e.  ( Base `  S ) ) )  ->  ( (
x ( +g  `  S
) y )  oF ( +g  `  R
) z )  =  ( x  oF ( +g  `  R
) ( y ( +g  `  S ) z ) ) )
32103adant3r3 1193 . . . 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 17431 . . 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 )  oF ( +g  `  R ) z ) )
343, 4, 5, 22, 18, 20psraddcl 17432 . . . 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 17431 . . 3  |-  ( (
ph  /\  ( x  e.  ( Base `  S
)  /\  y  e.  ( Base `  S )  /\  z  e.  ( Base `  S ) ) )  ->  ( x
( +g  `  S ) ( y ( +g  `  S ) z ) )  =  ( x  oF ( +g  `  R ) ( y ( +g  `  S
) z ) ) )
3631, 33, 353eqtr4d 2483 . 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 2441 . . 3  |-  ( 0g
`  R )  =  ( 0g `  R
)
393, 37, 6, 15, 38, 4psr0cl 17443 . 2  |-  ( ph  ->  ( { f  e.  ( NN0  ^m  I
)  |  ( `' f " NN )  e.  Fin }  X.  { ( 0g `  R ) } )  e.  ( Base `  S
) )
4037adantr 462 . . 3  |-  ( (
ph  /\  x  e.  ( Base `  S )
)  ->  I  e.  V )
416adantr 462 . . 3  |-  ( (
ph  /\  x  e.  ( Base `  S )
)  ->  R  e.  Grp )
42 simpr 458 . . 3  |-  ( (
ph  /\  x  e.  ( Base `  S )
)  ->  x  e.  ( Base `  S )
)
433, 40, 41, 15, 38, 4, 5, 42psr0lid 17444 . 2  |-  ( (
ph  /\  x  e.  ( Base `  S )
)  ->  ( ( { f  e.  ( NN0  ^m  I )  |  ( `' f
" NN )  e. 
Fin }  X.  { ( 0g `  R ) } ) ( +g  `  S ) x )  =  x )
44 eqid 2441 . . 3  |-  ( invg `  R )  =  ( invg `  R )
453, 40, 41, 15, 44, 4, 42psrnegcl 17445 . 2  |-  ( (
ph  /\  x  e.  ( Base `  S )
)  ->  ( ( invg `  R )  o.  x )  e.  ( Base `  S
) )
463, 40, 41, 15, 44, 4, 42, 38, 5psrlinv 17446 . 2  |-  ( (
ph  /\  x  e.  ( Base `  S )
)  ->  ( (
( invg `  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 15556 1  |-  ( ph  ->  S  e.  Grp )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761   {crab 2717   _Vcvv 2970   {csn 3874    X. cxp 4834   `'ccnv 4835   "cima 4839    o. ccom 4840   ` cfv 5415  (class class class)co 6090    oFcof 6317    ^m cmap 7210   Fincfn 7306   NNcn 10318   NN0cn0 10575   Basecbs 14170   +g cplusg 14234   0gc0g 14374   Grpcgrp 15406   invgcminusg 15407   mPwSer cmps 17396
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-supp 6690  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-map 7212  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-fsupp 7617  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-n0 10576  df-z 10643  df-uz 10858  df-fz 11434  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-plusg 14247  df-mulr 14248  df-sca 14250  df-vsca 14251  df-tset 14253  df-0g 14376  df-mnd 15411  df-grp 15538  df-minusg 15539  df-psr 17401
This theorem is referenced by:  psr0  17448  psrneg  17449  psrlmod  17450  psrrng  17461  mplsubglem  17488  mplsubglemOLD  17490
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