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Theorem psrgrp 17815
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 2461 . 2  |-  ( ph  ->  ( Base `  S
)  =  ( Base `  S ) )
2 eqidd 2461 . 2  |-  ( ph  ->  ( +g  `  S
)  =  ( +g  `  S ) )
3 psrgrp.s . . 3  |-  S  =  ( I mPwSer  R )
4 eqid 2460 . . 3  |-  ( Base `  S )  =  (
Base `  S )
5 eqid 2460 . . 3  |-  ( +g  `  S )  =  ( +g  `  S )
6 psrgrp.r . . . 4  |-  ( ph  ->  R  e.  Grp )
763ad2ant1 1012 . . 3  |-  ( (
ph  /\  x  e.  ( Base `  S )  /\  y  e.  ( Base `  S ) )  ->  R  e.  Grp )
8 simp2 992 . . 3  |-  ( (
ph  /\  x  e.  ( Base `  S )  /\  y  e.  ( Base `  S ) )  ->  x  e.  (
Base `  S )
)
9 simp3 993 . . 3  |-  ( (
ph  /\  x  e.  ( Base `  S )  /\  y  e.  ( Base `  S ) )  ->  y  e.  (
Base `  S )
)
103, 4, 5, 7, 8, 9psraddcl 17800 . 2  |-  ( (
ph  /\  x  e.  ( Base `  S )  /\  y  e.  ( Base `  S ) )  ->  ( x ( +g  `  S ) y )  e.  (
Base `  S )
)
11 ovex 6300 . . . . . . 7  |-  ( NN0 
^m  I )  e. 
_V
1211rabex 4591 . . . . . 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 2460 . . . . . 6  |-  ( Base `  R )  =  (
Base `  R )
15 eqid 2460 . . . . . 6  |-  { f  e.  ( NN0  ^m  I )  |  ( `' f " NN )  e.  Fin }  =  { f  e.  ( NN0  ^m  I )  |  ( `' f
" NN )  e. 
Fin }
16 simpr1 997 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  S
)  /\  y  e.  ( Base `  S )  /\  z  e.  ( Base `  S ) ) )  ->  x  e.  ( Base `  S )
)
173, 14, 15, 4, 16psrelbas 17796 . . . . 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 998 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  S
)  /\  y  e.  ( Base `  S )  /\  z  e.  ( Base `  S ) ) )  ->  y  e.  ( Base `  S )
)
193, 14, 15, 4, 18psrelbas 17796 . . . . 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 999 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  S
)  /\  y  e.  ( Base `  S )  /\  z  e.  ( Base `  S ) ) )  ->  z  e.  ( Base `  S )
)
213, 14, 15, 4, 20psrelbas 17796 . . . . 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 465 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  S
)  /\  y  e.  ( Base `  S )  /\  z  e.  ( Base `  S ) ) )  ->  R  e.  Grp )
23 eqid 2460 . . . . . . 7  |-  ( +g  `  R )  =  ( +g  `  R )
2414, 23grpass 15858 . . . . . 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 471 . . . . 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 6549 . . . 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 17799 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( Base `  S
)  /\  y  e.  ( Base `  S )  /\  z  e.  ( Base `  S ) ) )  ->  ( x
( +g  `  S ) y )  =  ( x  oF ( +g  `  R ) y ) )
2827oveq1d 6290 . . . 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 17799 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( Base `  S
)  /\  y  e.  ( Base `  S )  /\  z  e.  ( Base `  S ) ) )  ->  ( y
( +g  `  S ) z )  =  ( y  oF ( +g  `  R ) z ) )
3029oveq2d 6291 . . . 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 2511 . . 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 1202 . . . 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 17799 . . 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 17800 . . . 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 17799 . . 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 2511 . 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 2460 . . 3  |-  ( 0g
`  R )  =  ( 0g `  R
)
393, 37, 6, 15, 38, 4psr0cl 17811 . 2  |-  ( ph  ->  ( { f  e.  ( NN0  ^m  I
)  |  ( `' f " NN )  e.  Fin }  X.  { ( 0g `  R ) } )  e.  ( Base `  S
) )
4037adantr 465 . . 3  |-  ( (
ph  /\  x  e.  ( Base `  S )
)  ->  I  e.  V )
416adantr 465 . . 3  |-  ( (
ph  /\  x  e.  ( Base `  S )
)  ->  R  e.  Grp )
42 simpr 461 . . 3  |-  ( (
ph  /\  x  e.  ( Base `  S )
)  ->  x  e.  ( Base `  S )
)
433, 40, 41, 15, 38, 4, 5, 42psr0lid 17812 . 2  |-  ( (
ph  /\  x  e.  ( Base `  S )
)  ->  ( ( { f  e.  ( NN0  ^m  I )  |  ( `' f
" NN )  e. 
Fin }  X.  { ( 0g `  R ) } ) ( +g  `  S ) x )  =  x )
44 eqid 2460 . . 3  |-  ( invg `  R )  =  ( invg `  R )
453, 40, 41, 15, 44, 4, 42psrnegcl 17813 . 2  |-  ( (
ph  /\  x  e.  ( Base `  S )
)  ->  ( ( invg `  R )  o.  x )  e.  ( Base `  S
) )
463, 40, 41, 15, 44, 4, 42, 38, 5psrlinv 17814 . 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 15869 1  |-  ( ph  ->  S  e.  Grp )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   {crab 2811   _Vcvv 3106   {csn 4020    X. cxp 4990   `'ccnv 4991   "cima 4995    o. ccom 4996   ` cfv 5579  (class class class)co 6275    oFcof 6513    ^m cmap 7410   Fincfn 7506   NNcn 10525   NN0cn0 10784   Basecbs 14479   +g cplusg 14544   0gc0g 14684   Grpcgrp 15716   invgcminusg 15717   mPwSer cmps 17764
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 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-of 6515  df-om 6672  df-1st 6774  df-2nd 6775  df-supp 6892  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-er 7301  df-map 7412  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-fsupp 7819  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-2 10583  df-3 10584  df-4 10585  df-5 10586  df-6 10587  df-7 10588  df-8 10589  df-9 10590  df-n0 10785  df-z 10854  df-uz 11072  df-fz 11662  df-struct 14481  df-ndx 14482  df-slot 14483  df-base 14484  df-plusg 14557  df-mulr 14558  df-sca 14560  df-vsca 14561  df-tset 14563  df-0g 14686  df-mnd 15721  df-grp 15851  df-minusg 15852  df-psr 17769
This theorem is referenced by:  psr0  17816  psrneg  17817  psrlmod  17818  psrrng  17830  mplsubglem  17857  mplsubglemOLD  17859
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