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Theorem psrlinv 17814
Description: The negative function in the ring of power series. (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 )
psrnegcl.d  |-  D  =  { f  e.  ( NN0  ^m  I )  |  ( `' f
" NN )  e. 
Fin }
psrnegcl.i  |-  N  =  ( invg `  R )
psrnegcl.b  |-  B  =  ( Base `  S
)
psrnegcl.z  |-  ( ph  ->  X  e.  B )
psrlinv.o  |-  .0.  =  ( 0g `  R )
psrlinv.p  |-  .+  =  ( +g  `  S )
Assertion
Ref Expression
psrlinv  |-  ( ph  ->  ( ( N  o.  X )  .+  X
)  =  ( D  X.  {  .0.  }
) )
Distinct variable group:    f, I
Allowed substitution hints:    ph( f)    B( f)    D( f)    .+ ( f)    R( f)    S( f)    N( f)    V( f)    X( f)    .0. ( f)

Proof of Theorem psrlinv
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 psrnegcl.d . . . . 5  |-  D  =  { f  e.  ( NN0  ^m  I )  |  ( `' f
" NN )  e. 
Fin }
2 ovex 6300 . . . . . 6  |-  ( NN0 
^m  I )  e. 
_V
32rabex 4591 . . . . 5  |-  { f  e.  ( NN0  ^m  I )  |  ( `' f " NN )  e.  Fin }  e.  _V
41, 3eqeltri 2544 . . . 4  |-  D  e. 
_V
54a1i 11 . . 3  |-  ( ph  ->  D  e.  _V )
6 fvex 5867 . . . 4  |-  ( N `
 ( X `  x ) )  e. 
_V
76a1i 11 . . 3  |-  ( (
ph  /\  x  e.  D )  ->  ( N `  ( X `  x ) )  e. 
_V )
8 psrgrp.s . . . . 5  |-  S  =  ( I mPwSer  R )
9 eqid 2460 . . . . 5  |-  ( Base `  R )  =  (
Base `  R )
10 psrnegcl.b . . . . 5  |-  B  =  ( Base `  S
)
11 psrnegcl.z . . . . 5  |-  ( ph  ->  X  e.  B )
128, 9, 1, 10, 11psrelbas 17796 . . . 4  |-  ( ph  ->  X : D --> ( Base `  R ) )
1312ffvelrnda 6012 . . 3  |-  ( (
ph  /\  x  e.  D )  ->  ( X `  x )  e.  ( Base `  R
) )
1412feqmptd 5911 . . . 4  |-  ( ph  ->  X  =  ( x  e.  D  |->  ( X `
 x ) ) )
15 psrnegcl.i . . . . . . 7  |-  N  =  ( invg `  R )
16 psrgrp.r . . . . . . 7  |-  ( ph  ->  R  e.  Grp )
179, 15, 16grpinvf1o 15902 . . . . . 6  |-  ( ph  ->  N : ( Base `  R ) -1-1-onto-> ( Base `  R
) )
18 f1of 5807 . . . . . 6  |-  ( N : ( Base `  R
)
-1-1-onto-> ( Base `  R )  ->  N : ( Base `  R ) --> ( Base `  R ) )
1917, 18syl 16 . . . . 5  |-  ( ph  ->  N : ( Base `  R ) --> ( Base `  R ) )
2019feqmptd 5911 . . . 4  |-  ( ph  ->  N  =  ( y  e.  ( Base `  R
)  |->  ( N `  y ) ) )
21 fveq2 5857 . . . 4  |-  ( y  =  ( X `  x )  ->  ( N `  y )  =  ( N `  ( X `  x ) ) )
2213, 14, 20, 21fmptco 6045 . . 3  |-  ( ph  ->  ( N  o.  X
)  =  ( x  e.  D  |->  ( N `
 ( X `  x ) ) ) )
235, 7, 13, 22, 14offval2 6531 . 2  |-  ( ph  ->  ( ( N  o.  X )  oF ( +g  `  R
) X )  =  ( x  e.  D  |->  ( ( N `  ( X `  x ) ) ( +g  `  R
) ( X `  x ) ) ) )
24 eqid 2460 . . 3  |-  ( +g  `  R )  =  ( +g  `  R )
25 psrlinv.p . . 3  |-  .+  =  ( +g  `  S )
26 psrgrp.i . . . 4  |-  ( ph  ->  I  e.  V )
278, 26, 16, 1, 15, 10, 11psrnegcl 17813 . . 3  |-  ( ph  ->  ( N  o.  X
)  e.  B )
288, 10, 24, 25, 27, 11psradd 17799 . 2  |-  ( ph  ->  ( ( N  o.  X )  .+  X
)  =  ( ( N  o.  X )  oF ( +g  `  R ) X ) )
29 psrlinv.o . . . . . . 7  |-  .0.  =  ( 0g `  R )
309, 24, 29, 15grplinv 15890 . . . . . 6  |-  ( ( R  e.  Grp  /\  ( X `  x )  e.  ( Base `  R
) )  ->  (
( N `  ( X `  x )
) ( +g  `  R
) ( X `  x ) )  =  .0.  )
3116, 30sylan 471 . . . . 5  |-  ( (
ph  /\  ( X `  x )  e.  (
Base `  R )
)  ->  ( ( N `  ( X `  x ) ) ( +g  `  R ) ( X `  x
) )  =  .0.  )
3213, 31syldan 470 . . . 4  |-  ( (
ph  /\  x  e.  D )  ->  (
( N `  ( X `  x )
) ( +g  `  R
) ( X `  x ) )  =  .0.  )
3332mpteq2dva 4526 . . 3  |-  ( ph  ->  ( x  e.  D  |->  ( ( N `  ( X `  x ) ) ( +g  `  R
) ( X `  x ) ) )  =  ( x  e.  D  |->  .0.  ) )
34 fconstmpt 5035 . . 3  |-  ( D  X.  {  .0.  }
)  =  ( x  e.  D  |->  .0.  )
3533, 34syl6reqr 2520 . 2  |-  ( ph  ->  ( D  X.  {  .0.  } )  =  ( x  e.  D  |->  ( ( N `  ( X `  x )
) ( +g  `  R
) ( X `  x ) ) ) )
3623, 28, 353eqtr4d 2511 1  |-  ( ph  ->  ( ( N  o.  X )  .+  X
)  =  ( D  X.  {  .0.  }
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762   {crab 2811   _Vcvv 3106   {csn 4020    |-> cmpt 4498    X. cxp 4990   `'ccnv 4991   "cima 4995    o. ccom 4996   -->wf 5575   -1-1-onto->wf1o 5578   ` 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:  psrgrp  17815  psrneg  17817
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