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Theorem gsumzinv 15233
Description: Inverse of a group sum. (Contributed by Mario Carneiro, 25-Apr-2016.)
Hypotheses
Ref Expression
gsumzinv.b  |-  B  =  ( Base `  G
)
gsumzinv.0  |-  .0.  =  ( 0g `  G )
gsumzinv.z  |-  Z  =  (Cntz `  G )
gsumzinv.p  |-  I  =  ( inv g `  G )
gsumzinv.g  |-  ( ph  ->  G  e.  Grp )
gsumzinv.a  |-  ( ph  ->  A  e.  V )
gsumzinv.f  |-  ( ph  ->  F : A --> B )
gsumzinv.c  |-  ( ph  ->  ran  F  C_  ( Z `  ran  F ) )
gsumzinv.n  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  e. 
Fin )
Assertion
Ref Expression
gsumzinv  |-  ( ph  ->  ( G  gsumg  ( I  o.  F
) )  =  ( I `  ( G 
gsumg  F ) ) )

Proof of Theorem gsumzinv
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 gsumzinv.b . . 3  |-  B  =  ( Base `  G
)
2 gsumzinv.0 . . 3  |-  .0.  =  ( 0g `  G )
3 gsumzinv.z . . 3  |-  Z  =  (Cntz `  G )
4 eqid 2296 . . 3  |-  (oppg `  G
)  =  (oppg `  G
)
5 gsumzinv.g . . . 4  |-  ( ph  ->  G  e.  Grp )
6 grpmnd 14510 . . . 4  |-  ( G  e.  Grp  ->  G  e.  Mnd )
75, 6syl 15 . . 3  |-  ( ph  ->  G  e.  Mnd )
8 gsumzinv.a . . 3  |-  ( ph  ->  A  e.  V )
9 gsumzinv.p . . . . . 6  |-  I  =  ( inv g `  G )
101, 9grpinvf 14542 . . . . 5  |-  ( G  e.  Grp  ->  I : B --> B )
115, 10syl 15 . . . 4  |-  ( ph  ->  I : B --> B )
12 gsumzinv.f . . . 4  |-  ( ph  ->  F : A --> B )
13 fco 5414 . . . 4  |-  ( ( I : B --> B  /\  F : A --> B )  ->  ( I  o.  F ) : A --> B )
1411, 12, 13syl2anc 642 . . 3  |-  ( ph  ->  ( I  o.  F
) : A --> B )
154, 9invoppggim 14849 . . . . . . 7  |-  ( G  e.  Grp  ->  I  e.  ( G GrpIso  (oppg `  G
) ) )
165, 15syl 15 . . . . . 6  |-  ( ph  ->  I  e.  ( G GrpIso 
(oppg `  G ) ) )
17 gimghm 14744 . . . . . 6  |-  ( I  e.  ( G GrpIso  (oppg `  G
) )  ->  I  e.  ( G  GrpHom  (oppg `  G
) ) )
18 ghmmhm 14709 . . . . . 6  |-  ( I  e.  ( G  GrpHom  (oppg `  G ) )  ->  I  e.  ( G MndHom  (oppg `  G ) ) )
1916, 17, 183syl 18 . . . . 5  |-  ( ph  ->  I  e.  ( G MndHom 
(oppg `  G ) ) )
20 gsumzinv.c . . . . 5  |-  ( ph  ->  ran  F  C_  ( Z `  ran  F ) )
21 eqid 2296 . . . . . 6  |-  (Cntz `  (oppg `  G ) )  =  (Cntz `  (oppg
`  G ) )
223, 21cntzmhm2 14831 . . . . 5  |-  ( ( I  e.  ( G MndHom 
(oppg `  G ) )  /\  ran  F  C_  ( Z `  ran  F ) )  ->  ( I " ran  F )  C_  (
(Cntz `  (oppg
`  G ) ) `
 ( I " ran  F ) ) )
2319, 20, 22syl2anc 642 . . . 4  |-  ( ph  ->  ( I " ran  F )  C_  ( (Cntz `  (oppg
`  G ) ) `
 ( I " ran  F ) ) )
24 rnco2 5196 . . . 4  |-  ran  (
I  o.  F )  =  ( I " ran  F )
2524fveq2i 5544 . . . . 5  |-  ( Z `
 ran  ( I  o.  F ) )  =  ( Z `  (
I " ran  F
) )
264, 3oppgcntz 14853 . . . . 5  |-  ( Z `
 ( I " ran  F ) )  =  ( (Cntz `  (oppg `  G
) ) `  (
I " ran  F
) )
2725, 26eqtri 2316 . . . 4  |-  ( Z `
 ran  ( I  o.  F ) )  =  ( (Cntz `  (oppg `  G
) ) `  (
I " ran  F
) )
2823, 24, 273sstr4g 3232 . . 3  |-  ( ph  ->  ran  ( I  o.  F )  C_  ( Z `  ran  ( I  o.  F ) ) )
29 gsumzinv.n . . . 4  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  e. 
Fin )
30 eldifi 3311 . . . . . . 7  |-  ( x  e.  ( A  \ 
( `' F "
( _V  \  {  .0.  } ) ) )  ->  x  e.  A
)
31 fvco3 5612 . . . . . . 7  |-  ( ( F : A --> B  /\  x  e.  A )  ->  ( ( I  o.  F ) `  x
)  =  ( I `
 ( F `  x ) ) )
3212, 30, 31syl2an 463 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A  \  ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( ( I  o.  F ) `  x )  =  ( I `  ( F `
 x ) ) )
33 ssid 3210 . . . . . . . . 9  |-  ( `' F " ( _V 
\  {  .0.  }
) )  C_  ( `' F " ( _V 
\  {  .0.  }
) )
3433a1i 10 . . . . . . . 8  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  C_  ( `' F " ( _V 
\  {  .0.  }
) ) )
3512, 34suppssr 5675 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A  \  ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( F `  x )  =  .0.  )
3635fveq2d 5545 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A  \  ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( I `  ( F `  x ) )  =  ( I `
 .0.  ) )
372, 9grpinvid 14549 . . . . . . . 8  |-  ( G  e.  Grp  ->  (
I `  .0.  )  =  .0.  )
385, 37syl 15 . . . . . . 7  |-  ( ph  ->  ( I `  .0.  )  =  .0.  )
3938adantr 451 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A  \  ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( I `  .0.  )  =  .0.  )
4032, 36, 393eqtrd 2332 . . . . 5  |-  ( (
ph  /\  x  e.  ( A  \  ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( ( I  o.  F ) `  x )  =  .0.  )
4114, 40suppss 5674 . . . 4  |-  ( ph  ->  ( `' ( I  o.  F ) "
( _V  \  {  .0.  } ) )  C_  ( `' F " ( _V 
\  {  .0.  }
) ) )
42 ssfi 7099 . . . 4  |-  ( ( ( `' F "
( _V  \  {  .0.  } ) )  e. 
Fin  /\  ( `' ( I  o.  F
) " ( _V 
\  {  .0.  }
) )  C_  ( `' F " ( _V 
\  {  .0.  }
) ) )  -> 
( `' ( I  o.  F ) "
( _V  \  {  .0.  } ) )  e. 
Fin )
4329, 41, 42syl2anc 642 . . 3  |-  ( ph  ->  ( `' ( I  o.  F ) "
( _V  \  {  .0.  } ) )  e. 
Fin )
441, 2, 3, 4, 7, 8, 14, 28, 43gsumzoppg 15232 . 2  |-  ( ph  ->  ( (oppg
`  G )  gsumg  ( I  o.  F ) )  =  ( G  gsumg  ( I  o.  F ) ) )
454oppgmnd 14843 . . . 4  |-  ( G  e.  Mnd  ->  (oppg `  G
)  e.  Mnd )
467, 45syl 15 . . 3  |-  ( ph  ->  (oppg
`  G )  e. 
Mnd )
471, 3, 7, 46, 8, 19, 12, 20, 2, 29gsumzmhm 15226 . 2  |-  ( ph  ->  ( (oppg
`  G )  gsumg  ( I  o.  F ) )  =  ( I `  ( G  gsumg  F ) ) )
4844, 47eqtr3d 2330 1  |-  ( ph  ->  ( G  gsumg  ( I  o.  F
) )  =  ( I `  ( G 
gsumg  F ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801    \ cdif 3162    C_ wss 3165   {csn 3653   `'ccnv 4704   ran crn 4706   "cima 4708    o. ccom 4709   -->wf 5267   ` cfv 5271  (class class class)co 5874   Fincfn 6879   Basecbs 13164   0gc0g 13416    gsumg cgsu 13417   Mndcmnd 14377   Grpcgrp 14378   inv gcminusg 14379   MndHom cmhm 14429    GrpHom cghm 14696   GrpIso cgim 14737  Cntzccntz 14807  oppgcoppg 14834
This theorem is referenced by:  dprdfinv  15270
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-iin 3924  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-se 4369  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-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-tpos 6250  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-oi 7241  df-card 7588  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-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-fzo 10887  df-seq 11063  df-hash 11354  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-0g 13420  df-gsum 13421  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-mhm 14431  df-submnd 14432  df-grp 14505  df-minusg 14506  df-ghm 14697  df-gim 14739  df-cntz 14809  df-oppg 14835  df-cmn 15107
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