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Theorem gsumzinvOLD 16557
Description: Inverse of a group sum. (Contributed by Mario Carneiro, 25-Apr-2016.) Obsolete version of gsumzinv 16556 as of 6-Jun-2019. (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
gsumzinvOLD.b  |-  B  =  ( Base `  G
)
gsumzinvOLD.0  |-  .0.  =  ( 0g `  G )
gsumzinvOLD.z  |-  Z  =  (Cntz `  G )
gsumzinvOLD.p  |-  I  =  ( invg `  G )
gsumzinvOLD.g  |-  ( ph  ->  G  e.  Grp )
gsumzinvOLD.a  |-  ( ph  ->  A  e.  V )
gsumzinvOLD.f  |-  ( ph  ->  F : A --> B )
gsumzinvOLD.c  |-  ( ph  ->  ran  F  C_  ( Z `  ran  F ) )
gsumzinvOLD.n  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  e. 
Fin )
Assertion
Ref Expression
gsumzinvOLD  |-  ( ph  ->  ( G  gsumg  ( I  o.  F
) )  =  ( I `  ( G 
gsumg  F ) ) )

Proof of Theorem gsumzinvOLD
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 gsumzinvOLD.b . . 3  |-  B  =  ( Base `  G
)
2 gsumzinvOLD.0 . . 3  |-  .0.  =  ( 0g `  G )
3 gsumzinvOLD.z . . 3  |-  Z  =  (Cntz `  G )
4 eqid 2451 . . 3  |-  (oppg `  G
)  =  (oppg `  G
)
5 gsumzinvOLD.g . . . 4  |-  ( ph  ->  G  e.  Grp )
6 grpmnd 15661 . . . 4  |-  ( G  e.  Grp  ->  G  e.  Mnd )
75, 6syl 16 . . 3  |-  ( ph  ->  G  e.  Mnd )
8 gsumzinvOLD.a . . 3  |-  ( ph  ->  A  e.  V )
9 gsumzinvOLD.p . . . . . 6  |-  I  =  ( invg `  G )
101, 9grpinvf 15693 . . . . 5  |-  ( G  e.  Grp  ->  I : B --> B )
115, 10syl 16 . . . 4  |-  ( ph  ->  I : B --> B )
12 gsumzinvOLD.f . . . 4  |-  ( ph  ->  F : A --> B )
13 fco 5669 . . . 4  |-  ( ( I : B --> B  /\  F : A --> B )  ->  ( I  o.  F ) : A --> B )
1411, 12, 13syl2anc 661 . . 3  |-  ( ph  ->  ( I  o.  F
) : A --> B )
154, 9invoppggim 15986 . . . . . 6  |-  ( G  e.  Grp  ->  I  e.  ( G GrpIso  (oppg `  G
) ) )
16 gimghm 15903 . . . . . 6  |-  ( I  e.  ( G GrpIso  (oppg `  G
) )  ->  I  e.  ( G  GrpHom  (oppg `  G
) ) )
17 ghmmhm 15868 . . . . . 6  |-  ( I  e.  ( G  GrpHom  (oppg `  G ) )  ->  I  e.  ( G MndHom  (oppg `  G ) ) )
185, 15, 16, 174syl 21 . . . . 5  |-  ( ph  ->  I  e.  ( G MndHom 
(oppg `  G ) ) )
19 gsumzinvOLD.c . . . . 5  |-  ( ph  ->  ran  F  C_  ( Z `  ran  F ) )
20 eqid 2451 . . . . . 6  |-  (Cntz `  (oppg `  G ) )  =  (Cntz `  (oppg
`  G ) )
213, 20cntzmhm2 15968 . . . . 5  |-  ( ( I  e.  ( G MndHom 
(oppg `  G ) )  /\  ran  F  C_  ( Z `  ran  F ) )  ->  ( I " ran  F )  C_  (
(Cntz `  (oppg
`  G ) ) `
 ( I " ran  F ) ) )
2218, 19, 21syl2anc 661 . . . 4  |-  ( ph  ->  ( I " ran  F )  C_  ( (Cntz `  (oppg
`  G ) ) `
 ( I " ran  F ) ) )
23 rnco2 5446 . . . 4  |-  ran  (
I  o.  F )  =  ( I " ran  F )
2423fveq2i 5795 . . . . 5  |-  ( Z `
 ran  ( I  o.  F ) )  =  ( Z `  (
I " ran  F
) )
254, 3oppgcntz 15990 . . . . 5  |-  ( Z `
 ( I " ran  F ) )  =  ( (Cntz `  (oppg `  G
) ) `  (
I " ran  F
) )
2624, 25eqtri 2480 . . . 4  |-  ( Z `
 ran  ( I  o.  F ) )  =  ( (Cntz `  (oppg `  G
) ) `  (
I " ran  F
) )
2722, 23, 263sstr4g 3498 . . 3  |-  ( ph  ->  ran  ( I  o.  F )  C_  ( Z `  ran  ( I  o.  F ) ) )
28 gsumzinvOLD.n . . . 4  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  e. 
Fin )
29 eldifi 3579 . . . . . . 7  |-  ( x  e.  ( A  \ 
( `' F "
( _V  \  {  .0.  } ) ) )  ->  x  e.  A
)
30 fvco3 5870 . . . . . . 7  |-  ( ( F : A --> B  /\  x  e.  A )  ->  ( ( I  o.  F ) `  x
)  =  ( I `
 ( F `  x ) ) )
3112, 29, 30syl2an 477 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A  \  ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( ( I  o.  F ) `  x )  =  ( I `  ( F `
 x ) ) )
32 ssid 3476 . . . . . . . . 9  |-  ( `' F " ( _V 
\  {  .0.  }
) )  C_  ( `' F " ( _V 
\  {  .0.  }
) )
3332a1i 11 . . . . . . . 8  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  C_  ( `' F " ( _V 
\  {  .0.  }
) ) )
3412, 33suppssrOLD 5939 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A  \  ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( F `  x )  =  .0.  )
3534fveq2d 5796 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A  \  ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( I `  ( F `  x ) )  =  ( I `
 .0.  ) )
362, 9grpinvid 15700 . . . . . . . 8  |-  ( G  e.  Grp  ->  (
I `  .0.  )  =  .0.  )
375, 36syl 16 . . . . . . 7  |-  ( ph  ->  ( I `  .0.  )  =  .0.  )
3837adantr 465 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A  \  ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( I `  .0.  )  =  .0.  )
3931, 35, 383eqtrd 2496 . . . . 5  |-  ( (
ph  /\  x  e.  ( A  \  ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( ( I  o.  F ) `  x )  =  .0.  )
4014, 39suppssOLD 5938 . . . 4  |-  ( ph  ->  ( `' ( I  o.  F ) "
( _V  \  {  .0.  } ) )  C_  ( `' F " ( _V 
\  {  .0.  }
) ) )
41 ssfi 7637 . . . 4  |-  ( ( ( `' F "
( _V  \  {  .0.  } ) )  e. 
Fin  /\  ( `' ( I  o.  F
) " ( _V 
\  {  .0.  }
) )  C_  ( `' F " ( _V 
\  {  .0.  }
) ) )  -> 
( `' ( I  o.  F ) "
( _V  \  {  .0.  } ) )  e. 
Fin )
4228, 40, 41syl2anc 661 . . 3  |-  ( ph  ->  ( `' ( I  o.  F ) "
( _V  \  {  .0.  } ) )  e. 
Fin )
431, 2, 3, 4, 7, 8, 14, 27, 42gsumzoppgOLD 16555 . 2  |-  ( ph  ->  ( (oppg
`  G )  gsumg  ( I  o.  F ) )  =  ( G  gsumg  ( I  o.  F ) ) )
444oppgmnd 15980 . . . 4  |-  ( G  e.  Mnd  ->  (oppg `  G
)  e.  Mnd )
457, 44syl 16 . . 3  |-  ( ph  ->  (oppg
`  G )  e. 
Mnd )
461, 3, 7, 45, 8, 18, 12, 19, 2, 28gsumzmhmOLD 16545 . 2  |-  ( ph  ->  ( (oppg
`  G )  gsumg  ( I  o.  F ) )  =  ( I `  ( G  gsumg  F ) ) )
4743, 46eqtr3d 2494 1  |-  ( ph  ->  ( G  gsumg  ( I  o.  F
) )  =  ( I `  ( G 
gsumg  F ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   _Vcvv 3071    \ cdif 3426    C_ wss 3429   {csn 3978   `'ccnv 4940   ran crn 4942   "cima 4944    o. ccom 4945   -->wf 5515   ` cfv 5519  (class class class)co 6193   Fincfn 7413   Basecbs 14285   0gc0g 14489    gsumg cgsu 14490   Mndcmnd 15520   Grpcgrp 15521   invgcminusg 15522   MndHom cmhm 15573    GrpHom cghm 15855   GrpIso cgim 15896  Cntzccntz 15944  oppgcoppg 15971
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-int 4230  df-iun 4274  df-iin 4275  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-se 4781  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-isom 5528  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-om 6580  df-1st 6680  df-2nd 6681  df-tpos 6848  df-recs 6935  df-rdg 6969  df-1o 7023  df-oadd 7027  df-er 7204  df-map 7319  df-en 7414  df-dom 7415  df-sdom 7416  df-fin 7417  df-oi 7828  df-card 8213  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-nn 10427  df-2 10484  df-n0 10684  df-z 10751  df-uz 10966  df-fz 11548  df-fzo 11659  df-seq 11917  df-hash 12214  df-ndx 14288  df-slot 14289  df-base 14290  df-sets 14291  df-ress 14292  df-plusg 14362  df-0g 14491  df-gsum 14492  df-mre 14635  df-mrc 14636  df-acs 14638  df-mnd 15526  df-mhm 15575  df-submnd 15576  df-grp 15656  df-minusg 15657  df-ghm 15856  df-gim 15898  df-cntz 15946  df-oppg 15972  df-cmn 16392
This theorem is referenced by:  dprdfinvOLD  16630
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