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Theorem psrgrp 16159
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 2297 . 2  |-  ( ph  ->  ( Base `  S
)  =  ( Base `  S ) )
2 eqidd 2297 . 2  |-  ( ph  ->  ( +g  `  S
)  =  ( +g  `  S ) )
3 psrgrp.s . . 3  |-  S  =  ( I mPwSer  R )
4 eqid 2296 . . 3  |-  ( Base `  S )  =  (
Base `  S )
5 eqid 2296 . . 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 16144 . 2  |-  ( (
ph  /\  x  e.  ( Base `  S )  /\  y  e.  ( Base `  S ) )  ->  ( x ( +g  `  S ) y )  e.  (
Base `  S )
)
11 ovex 5899 . . . . . . 7  |-  ( NN0 
^m  I )  e. 
_V
1211rabex 4181 . . . . . 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 2296 . . . . . 6  |-  ( Base `  R )  =  (
Base `  R )
15 eqid 2296 . . . . . 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 16141 . . . . 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 16141 . . . . 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 16141 . . . . 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 2296 . . . . . . 7  |-  ( +g  `  R )  =  ( +g  `  R )
2414, 23grpass 14512 . . . . . 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 6127 . . . 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 16143 . . . . 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 5889 . . . 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 16143 . . . . 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 5890 . . . 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 2338 . . 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 16143 . . 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 16144 . . . 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 16143 . . 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 2338 . 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 2296 . . 3  |-  ( 0g
`  R )  =  ( 0g `  R
)
393, 37, 6, 15, 38, 4psr0cl 16155 . 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 16156 . 2  |-  ( (
ph  /\  x  e.  ( Base `  S )
)  ->  ( ( { f  e.  ( NN0  ^m  I )  |  ( `' f
" NN )  e. 
Fin }  X.  { ( 0g `  R ) } ) ( +g  `  S ) x )  =  x )
44 eqid 2296 . . 3  |-  ( inv g `  R )  =  ( inv g `  R )
453, 40, 41, 15, 44, 4, 42psrnegcl 16157 . 2  |-  ( (
ph  /\  x  e.  ( Base `  S )
)  ->  ( ( inv g `  R )  o.  x )  e.  ( Base `  S
) )
463, 40, 41, 15, 44, 4, 42, 38, 5psrlinv 16158 . 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 14523 1  |-  ( ph  ->  S  e.  Grp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   {crab 2560   _Vcvv 2801   {csn 3653    X. cxp 4703   `'ccnv 4704   "cima 4708    o. ccom 4709   ` cfv 5271  (class class class)co 5874    o Fcof 6092    ^m cmap 6788   Fincfn 6879   NNcn 9762   NN0cn0 9981   Basecbs 13164   +g cplusg 13224   0gc0g 13416   Grpcgrp 14378   inv gcminusg 14379   mPwSer cmps 16103
This theorem is referenced by:  psr0  16160  psrneg  16161  psrlmod  16162  psrrng  16171  mplsubglem  16195
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-plusg 13237  df-mulr 13238  df-sca 13240  df-vsca 13241  df-tset 13243  df-0g 13420  df-mnd 14383  df-grp 14505  df-minusg 14506  df-psr 16114
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