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Theorem psrlinv 16158
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  =  ( inv g `  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 5899 . . . . . 6  |-  ( NN0 
^m  I )  e. 
_V
32rabex 4181 . . . . 5  |-  { f  e.  ( NN0  ^m  I )  |  ( `' f " NN )  e.  Fin }  e.  _V
41, 3eqeltri 2366 . . . 4  |-  D  e. 
_V
54a1i 10 . . 3  |-  ( ph  ->  D  e.  _V )
6 fvex 5555 . . . 4  |-  ( N `
 ( X `  x ) )  e. 
_V
76a1i 10 . . 3  |-  ( (
ph  /\  x  e.  D )  ->  ( N `  ( X `  x ) )  e. 
_V )
8 psrgrp.s . . . . 5  |-  S  =  ( I mPwSer  R )
9 eqid 2296 . . . . 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 16141 . . . 4  |-  ( ph  ->  X : D --> ( Base `  R ) )
13 ffvelrn 5679 . . . 4  |-  ( ( X : D --> ( Base `  R )  /\  x  e.  D )  ->  ( X `  x )  e.  ( Base `  R
) )
1412, 13sylan 457 . . 3  |-  ( (
ph  /\  x  e.  D )  ->  ( X `  x )  e.  ( Base `  R
) )
1512feqmptd 5591 . . . 4  |-  ( ph  ->  X  =  ( x  e.  D  |->  ( X `
 x ) ) )
16 psrnegcl.i . . . . . . 7  |-  N  =  ( inv g `  R )
17 psrgrp.r . . . . . . 7  |-  ( ph  ->  R  e.  Grp )
189, 16, 17grpinvf1o 14554 . . . . . 6  |-  ( ph  ->  N : ( Base `  R ) -1-1-onto-> ( Base `  R
) )
19 f1of 5488 . . . . . 6  |-  ( N : ( Base `  R
)
-1-1-onto-> ( Base `  R )  ->  N : ( Base `  R ) --> ( Base `  R ) )
2018, 19syl 15 . . . . 5  |-  ( ph  ->  N : ( Base `  R ) --> ( Base `  R ) )
2120feqmptd 5591 . . . 4  |-  ( ph  ->  N  =  ( y  e.  ( Base `  R
)  |->  ( N `  y ) ) )
22 fveq2 5541 . . . 4  |-  ( y  =  ( X `  x )  ->  ( N `  y )  =  ( N `  ( X `  x ) ) )
2314, 15, 21, 22fmptco 5707 . . 3  |-  ( ph  ->  ( N  o.  X
)  =  ( x  e.  D  |->  ( N `
 ( X `  x ) ) ) )
245, 7, 14, 23, 15offval2 6111 . 2  |-  ( ph  ->  ( ( N  o.  X )  o F ( +g  `  R
) X )  =  ( x  e.  D  |->  ( ( N `  ( X `  x ) ) ( +g  `  R
) ( X `  x ) ) ) )
25 eqid 2296 . . 3  |-  ( +g  `  R )  =  ( +g  `  R )
26 psrlinv.p . . 3  |-  .+  =  ( +g  `  S )
27 psrgrp.i . . . 4  |-  ( ph  ->  I  e.  V )
288, 27, 17, 1, 16, 10, 11psrnegcl 16157 . . 3  |-  ( ph  ->  ( N  o.  X
)  e.  B )
298, 10, 25, 26, 28, 11psradd 16143 . 2  |-  ( ph  ->  ( ( N  o.  X )  .+  X
)  =  ( ( N  o.  X )  o F ( +g  `  R ) X ) )
30 psrlinv.o . . . . . . 7  |-  .0.  =  ( 0g `  R )
319, 25, 30, 16grplinv 14544 . . . . . 6  |-  ( ( R  e.  Grp  /\  ( X `  x )  e.  ( Base `  R
) )  ->  (
( N `  ( X `  x )
) ( +g  `  R
) ( X `  x ) )  =  .0.  )
3217, 31sylan 457 . . . . 5  |-  ( (
ph  /\  ( X `  x )  e.  (
Base `  R )
)  ->  ( ( N `  ( X `  x ) ) ( +g  `  R ) ( X `  x
) )  =  .0.  )
3314, 32syldan 456 . . . 4  |-  ( (
ph  /\  x  e.  D )  ->  (
( N `  ( X `  x )
) ( +g  `  R
) ( X `  x ) )  =  .0.  )
3433mpteq2dva 4122 . . 3  |-  ( ph  ->  ( x  e.  D  |->  ( ( N `  ( X `  x ) ) ( +g  `  R
) ( X `  x ) ) )  =  ( x  e.  D  |->  .0.  ) )
35 fconstmpt 4748 . . 3  |-  ( D  X.  {  .0.  }
)  =  ( x  e.  D  |->  .0.  )
3634, 35syl6reqr 2347 . 2  |-  ( ph  ->  ( D  X.  {  .0.  } )  =  ( x  e.  D  |->  ( ( N `  ( X `  x )
) ( +g  `  R
) ( X `  x ) ) ) )
3724, 29, 363eqtr4d 2338 1  |-  ( ph  ->  ( ( N  o.  X )  .+  X
)  =  ( D  X.  {  .0.  }
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   {crab 2560   _Vcvv 2801   {csn 3653    e. cmpt 4093    X. cxp 4703   `'ccnv 4704   "cima 4708    o. ccom 4709   -->wf 5267   -1-1-onto->wf1o 5270   ` 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:  psrgrp  16159  psrneg  16161
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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