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Theorem psrgrp 16143
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 2284 . 2  |-  ( ph  ->  ( Base `  S
)  =  ( Base `  S ) )
2 eqidd 2284 . 2  |-  ( ph  ->  ( +g  `  S
)  =  ( +g  `  S ) )
3 psrgrp.s . . 3  |-  S  =  ( I mPwSer  R )
4 eqid 2283 . . 3  |-  ( Base `  S )  =  (
Base `  S )
5 eqid 2283 . . 3  |-  ( +g  `  S )  =  ( +g  `  S )
6 psrgrp.r . . . 4  |-  ( ph  ->  R  e.  Grp )
763ad2ant1 976 . . 3  |-  ( (
ph  /\  x  e.  ( Base `  S )  /\  y  e.  ( Base `  S ) )  ->  R  e.  Grp )
8 simp2 956 . . 3  |-  ( (
ph  /\  x  e.  ( Base `  S )  /\  y  e.  ( Base `  S ) )  ->  x  e.  (
Base `  S )
)
9 simp3 957 . . 3  |-  ( (
ph  /\  x  e.  ( Base `  S )  /\  y  e.  ( Base `  S ) )  ->  y  e.  (
Base `  S )
)
103, 4, 5, 7, 8, 9psraddcl 16128 . 2  |-  ( (
ph  /\  x  e.  ( Base `  S )  /\  y  e.  ( Base `  S ) )  ->  ( x ( +g  `  S ) y )  e.  (
Base `  S )
)
11 ovex 5883 . . . . . . 7  |-  ( NN0 
^m  I )  e. 
_V
1211rabex 4165 . . . . . 6  |-  { f  e.  ( NN0  ^m  I )  |  ( `' f " NN )  e.  Fin }  e.  _V
1312a1i 10 . . . . 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 2283 . . . . . 6  |-  ( Base `  R )  =  (
Base `  R )
15 eqid 2283 . . . . . 6  |-  { f  e.  ( NN0  ^m  I )  |  ( `' f " NN )  e.  Fin }  =  { f  e.  ( NN0  ^m  I )  |  ( `' f
" NN )  e. 
Fin }
16 simpr1 961 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  S
)  /\  y  e.  ( Base `  S )  /\  z  e.  ( Base `  S ) ) )  ->  x  e.  ( Base `  S )
)
173, 14, 15, 4, 16psrelbas 16125 . . . . 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 962 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  S
)  /\  y  e.  ( Base `  S )  /\  z  e.  ( Base `  S ) ) )  ->  y  e.  ( Base `  S )
)
193, 14, 15, 4, 18psrelbas 16125 . . . . 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 963 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  S
)  /\  y  e.  ( Base `  S )  /\  z  e.  ( Base `  S ) ) )  ->  z  e.  ( Base `  S )
)
213, 14, 15, 4, 20psrelbas 16125 . . . . 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 451 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  S
)  /\  y  e.  ( Base `  S )  /\  z  e.  ( Base `  S ) ) )  ->  R  e.  Grp )
23 eqid 2283 . . . . . . 7  |-  ( +g  `  R )  =  ( +g  `  R )
2414, 23grpass 14496 . . . . . 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 457 . . . . 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 6111 . . . 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 16127 . . . . 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 5873 . . . 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 16127 . . . . 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 5874 . . . 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 2325 . . 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 1162 . . . 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 16127 . . 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 16128 . . . 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 16127 . . 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 2325 . 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 2283 . . 3  |-  ( 0g
`  R )  =  ( 0g `  R
)
393, 37, 6, 15, 38, 4psr0cl 16139 . 2  |-  ( ph  ->  ( { f  e.  ( NN0  ^m  I
)  |  ( `' f " NN )  e.  Fin }  X.  { ( 0g `  R ) } )  e.  ( Base `  S
) )
4037adantr 451 . . 3  |-  ( (
ph  /\  x  e.  ( Base `  S )
)  ->  I  e.  V )
416adantr 451 . . 3  |-  ( (
ph  /\  x  e.  ( Base `  S )
)  ->  R  e.  Grp )
42 simpr 447 . . 3  |-  ( (
ph  /\  x  e.  ( Base `  S )
)  ->  x  e.  ( Base `  S )
)
433, 40, 41, 15, 38, 4, 5, 42psr0lid 16140 . 2  |-  ( (
ph  /\  x  e.  ( Base `  S )
)  ->  ( ( { f  e.  ( NN0  ^m  I )  |  ( `' f
" NN )  e. 
Fin }  X.  { ( 0g `  R ) } ) ( +g  `  S ) x )  =  x )
44 eqid 2283 . . 3  |-  ( inv g `  R )  =  ( inv g `  R )
453, 40, 41, 15, 44, 4, 42psrnegcl 16141 . 2  |-  ( (
ph  /\  x  e.  ( Base `  S )
)  ->  ( ( inv g `  R )  o.  x )  e.  ( Base `  S
) )
463, 40, 41, 15, 44, 4, 42, 38, 5psrlinv 16142 . 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 14507 1  |-  ( ph  ->  S  e.  Grp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   {crab 2547   _Vcvv 2788   {csn 3640    X. cxp 4687   `'ccnv 4688   "cima 4692    o. ccom 4693   ` cfv 5255  (class class class)co 5858    o Fcof 6076    ^m cmap 6772   Fincfn 6863   NNcn 9746   NN0cn0 9965   Basecbs 13148   +g cplusg 13208   0gc0g 13400   Grpcgrp 14362   inv gcminusg 14363   mPwSer cmps 16087
This theorem is referenced by:  psr0  16144  psrneg  16145  psrlmod  16146  psrrng  16155  mplsubglem  16179
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-plusg 13221  df-mulr 13222  df-sca 13224  df-vsca 13225  df-tset 13227  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-psr 16098
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