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Theorem gsumsub 15501
Description: The difference of two group sums. (Contributed by Mario Carneiro, 28-Dec-2014.)
Hypotheses
Ref Expression
gsumsub.b  |-  B  =  ( Base `  G
)
gsumsub.z  |-  .0.  =  ( 0g `  G )
gsumsub.m  |-  .-  =  ( -g `  G )
gsumsub.g  |-  ( ph  ->  G  e.  Abel )
gsumsub.a  |-  ( ph  ->  A  e.  V )
gsumsub.f  |-  ( ph  ->  F : A --> B )
gsumsub.h  |-  ( ph  ->  H : A --> B )
gsumsub.fn  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  e. 
Fin )
gsumsub.hn  |-  ( ph  ->  ( `' H "
( _V  \  {  .0.  } ) )  e. 
Fin )
Assertion
Ref Expression
gsumsub  |-  ( ph  ->  ( G  gsumg  ( F  o F 
.-  H ) )  =  ( ( G 
gsumg  F )  .-  ( G  gsumg  H ) ) )

Proof of Theorem gsumsub
Dummy variables  k  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 gsumsub.b . . . 4  |-  B  =  ( Base `  G
)
2 gsumsub.z . . . 4  |-  .0.  =  ( 0g `  G )
3 eqid 2408 . . . 4  |-  ( +g  `  G )  =  ( +g  `  G )
4 gsumsub.g . . . . 5  |-  ( ph  ->  G  e.  Abel )
5 ablcmn 15377 . . . . 5  |-  ( G  e.  Abel  ->  G  e. CMnd
)
64, 5syl 16 . . . 4  |-  ( ph  ->  G  e. CMnd )
7 gsumsub.a . . . 4  |-  ( ph  ->  A  e.  V )
8 gsumsub.f . . . 4  |-  ( ph  ->  F : A --> B )
9 eqid 2408 . . . . . . 7  |-  ( inv g `  G )  =  ( inv g `  G )
10 ablgrp 15376 . . . . . . . 8  |-  ( G  e.  Abel  ->  G  e. 
Grp )
114, 10syl 16 . . . . . . 7  |-  ( ph  ->  G  e.  Grp )
121, 9, 11grpinvf1o 14820 . . . . . 6  |-  ( ph  ->  ( inv g `  G ) : B -1-1-onto-> B
)
13 f1of 5637 . . . . . 6  |-  ( ( inv g `  G
) : B -1-1-onto-> B  -> 
( inv g `  G ) : B --> B )
1412, 13syl 16 . . . . 5  |-  ( ph  ->  ( inv g `  G ) : B --> B )
15 gsumsub.h . . . . 5  |-  ( ph  ->  H : A --> B )
16 fco 5563 . . . . 5  |-  ( ( ( inv g `  G ) : B --> B  /\  H : A --> B )  ->  (
( inv g `  G )  o.  H
) : A --> B )
1714, 15, 16syl2anc 643 . . . 4  |-  ( ph  ->  ( ( inv g `  G )  o.  H
) : A --> B )
18 gsumsub.fn . . . 4  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  e. 
Fin )
19 gsumsub.hn . . . . 5  |-  ( ph  ->  ( `' H "
( _V  \  {  .0.  } ) )  e. 
Fin )
20 eldifi 3433 . . . . . . . 8  |-  ( k  e.  ( A  \ 
( `' H "
( _V  \  {  .0.  } ) ) )  ->  k  e.  A
)
21 fvco3 5763 . . . . . . . 8  |-  ( ( H : A --> B  /\  k  e.  A )  ->  ( ( ( inv g `  G )  o.  H ) `  k )  =  ( ( inv g `  G ) `  ( H `  k )
) )
2215, 20, 21syl2an 464 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( A  \  ( `' H " ( _V 
\  {  .0.  }
) ) ) )  ->  ( ( ( inv g `  G
)  o.  H ) `
 k )  =  ( ( inv g `  G ) `  ( H `  k )
) )
23 ssid 3331 . . . . . . . . . . 11  |-  ( `' H " ( _V 
\  {  .0.  }
) )  C_  ( `' H " ( _V 
\  {  .0.  }
) )
2423a1i 11 . . . . . . . . . 10  |-  ( ph  ->  ( `' H "
( _V  \  {  .0.  } ) )  C_  ( `' H " ( _V 
\  {  .0.  }
) ) )
2515, 24suppssr 5827 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( A  \  ( `' H " ( _V 
\  {  .0.  }
) ) ) )  ->  ( H `  k )  =  .0.  )
2625fveq2d 5695 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( A  \  ( `' H " ( _V 
\  {  .0.  }
) ) ) )  ->  ( ( inv g `  G ) `
 ( H `  k ) )  =  ( ( inv g `  G ) `  .0.  ) )
272, 9grpinvid 14815 . . . . . . . . . 10  |-  ( G  e.  Grp  ->  (
( inv g `  G ) `  .0.  )  =  .0.  )
2811, 27syl 16 . . . . . . . . 9  |-  ( ph  ->  ( ( inv g `  G ) `  .0.  )  =  .0.  )
2928adantr 452 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( A  \  ( `' H " ( _V 
\  {  .0.  }
) ) ) )  ->  ( ( inv g `  G ) `
 .0.  )  =  .0.  )
3026, 29eqtrd 2440 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( A  \  ( `' H " ( _V 
\  {  .0.  }
) ) ) )  ->  ( ( inv g `  G ) `
 ( H `  k ) )  =  .0.  )
3122, 30eqtrd 2440 . . . . . 6  |-  ( (
ph  /\  k  e.  ( A  \  ( `' H " ( _V 
\  {  .0.  }
) ) ) )  ->  ( ( ( inv g `  G
)  o.  H ) `
 k )  =  .0.  )
3217, 31suppss 5826 . . . . 5  |-  ( ph  ->  ( `' ( ( inv g `  G
)  o.  H )
" ( _V  \  {  .0.  } ) ) 
C_  ( `' H " ( _V  \  {  .0.  } ) ) )
33 ssfi 7292 . . . . 5  |-  ( ( ( `' H "
( _V  \  {  .0.  } ) )  e. 
Fin  /\  ( `' ( ( inv g `  G )  o.  H
) " ( _V 
\  {  .0.  }
) )  C_  ( `' H " ( _V 
\  {  .0.  }
) ) )  -> 
( `' ( ( inv g `  G
)  o.  H )
" ( _V  \  {  .0.  } ) )  e.  Fin )
3419, 32, 33syl2anc 643 . . . 4  |-  ( ph  ->  ( `' ( ( inv g `  G
)  o.  H )
" ( _V  \  {  .0.  } ) )  e.  Fin )
351, 2, 3, 6, 7, 8, 17, 18, 34gsumadd 15487 . . 3  |-  ( ph  ->  ( G  gsumg  ( F  o F ( +g  `  G
) ( ( inv g `  G )  o.  H ) ) )  =  ( ( G  gsumg  F ) ( +g  `  G ) ( G 
gsumg  ( ( inv g `  G )  o.  H
) ) ) )
361, 2, 9, 4, 7, 15, 19gsuminv 15500 . . . 4  |-  ( ph  ->  ( G  gsumg  ( ( inv g `  G )  o.  H
) )  =  ( ( inv g `  G ) `  ( G  gsumg  H ) ) )
3736oveq2d 6060 . . 3  |-  ( ph  ->  ( ( G  gsumg  F ) ( +g  `  G
) ( G  gsumg  ( ( inv g `  G
)  o.  H ) ) )  =  ( ( G  gsumg  F ) ( +g  `  G ) ( ( inv g `  G
) `  ( G  gsumg  H ) ) ) )
3835, 37eqtrd 2440 . 2  |-  ( ph  ->  ( G  gsumg  ( F  o F ( +g  `  G
) ( ( inv g `  G )  o.  H ) ) )  =  ( ( G  gsumg  F ) ( +g  `  G ) ( ( inv g `  G
) `  ( G  gsumg  H ) ) ) )
398ffvelrnda 5833 . . . . . 6  |-  ( (
ph  /\  k  e.  A )  ->  ( F `  k )  e.  B )
4015ffvelrnda 5833 . . . . . 6  |-  ( (
ph  /\  k  e.  A )  ->  ( H `  k )  e.  B )
41 gsumsub.m . . . . . . 7  |-  .-  =  ( -g `  G )
421, 3, 9, 41grpsubval 14807 . . . . . 6  |-  ( ( ( F `  k
)  e.  B  /\  ( H `  k )  e.  B )  -> 
( ( F `  k )  .-  ( H `  k )
)  =  ( ( F `  k ) ( +g  `  G
) ( ( inv g `  G ) `
 ( H `  k ) ) ) )
4339, 40, 42syl2anc 643 . . . . 5  |-  ( (
ph  /\  k  e.  A )  ->  (
( F `  k
)  .-  ( H `  k ) )  =  ( ( F `  k ) ( +g  `  G ) ( ( inv g `  G
) `  ( H `  k ) ) ) )
4443mpteq2dva 4259 . . . 4  |-  ( ph  ->  ( k  e.  A  |->  ( ( F `  k )  .-  ( H `  k )
) )  =  ( k  e.  A  |->  ( ( F `  k
) ( +g  `  G
) ( ( inv g `  G ) `
 ( H `  k ) ) ) ) )
458feqmptd 5742 . . . . 5  |-  ( ph  ->  F  =  ( k  e.  A  |->  ( F `
 k ) ) )
4615feqmptd 5742 . . . . 5  |-  ( ph  ->  H  =  ( k  e.  A  |->  ( H `
 k ) ) )
477, 39, 40, 45, 46offval2 6285 . . . 4  |-  ( ph  ->  ( F  o F 
.-  H )  =  ( k  e.  A  |->  ( ( F `  k )  .-  ( H `  k )
) ) )
48 fvex 5705 . . . . . 6  |-  ( ( inv g `  G
) `  ( H `  k ) )  e. 
_V
4948a1i 11 . . . . 5  |-  ( (
ph  /\  k  e.  A )  ->  (
( inv g `  G ) `  ( H `  k )
)  e.  _V )
5014feqmptd 5742 . . . . . 6  |-  ( ph  ->  ( inv g `  G )  =  ( x  e.  B  |->  ( ( inv g `  G ) `  x
) ) )
51 fveq2 5691 . . . . . 6  |-  ( x  =  ( H `  k )  ->  (
( inv g `  G ) `  x
)  =  ( ( inv g `  G
) `  ( H `  k ) ) )
5240, 46, 50, 51fmptco 5864 . . . . 5  |-  ( ph  ->  ( ( inv g `  G )  o.  H
)  =  ( k  e.  A  |->  ( ( inv g `  G
) `  ( H `  k ) ) ) )
537, 39, 49, 45, 52offval2 6285 . . . 4  |-  ( ph  ->  ( F  o F ( +g  `  G
) ( ( inv g `  G )  o.  H ) )  =  ( k  e.  A  |->  ( ( F `
 k ) ( +g  `  G ) ( ( inv g `  G ) `  ( H `  k )
) ) ) )
5444, 47, 533eqtr4d 2450 . . 3  |-  ( ph  ->  ( F  o F 
.-  H )  =  ( F  o F ( +g  `  G
) ( ( inv g `  G )  o.  H ) ) )
5554oveq2d 6060 . 2  |-  ( ph  ->  ( G  gsumg  ( F  o F 
.-  H ) )  =  ( G  gsumg  ( F  o F ( +g  `  G ) ( ( inv g `  G
)  o.  H ) ) ) )
561, 2, 6, 7, 8, 18gsumcl 15480 . . 3  |-  ( ph  ->  ( G  gsumg  F )  e.  B
)
571, 2, 6, 7, 15, 19gsumcl 15480 . . 3  |-  ( ph  ->  ( G  gsumg  H )  e.  B
)
581, 3, 9, 41grpsubval 14807 . . 3  |-  ( ( ( G  gsumg  F )  e.  B  /\  ( G  gsumg  H )  e.  B
)  ->  ( ( G  gsumg  F )  .-  ( G  gsumg  H ) )  =  ( ( G  gsumg  F ) ( +g  `  G
) ( ( inv g `  G ) `
 ( G  gsumg  H ) ) ) )
5956, 57, 58syl2anc 643 . 2  |-  ( ph  ->  ( ( G  gsumg  F ) 
.-  ( G  gsumg  H ) )  =  ( ( G  gsumg  F ) ( +g  `  G ) ( ( inv g `  G
) `  ( G  gsumg  H ) ) ) )
6038, 55, 593eqtr4d 2450 1  |-  ( ph  ->  ( G  gsumg  ( F  o F 
.-  H ) )  =  ( ( G 
gsumg  F )  .-  ( G  gsumg  H ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   _Vcvv 2920    \ cdif 3281    C_ wss 3284   {csn 3778    e. cmpt 4230   `'ccnv 4840   "cima 4844    o. ccom 4845   -->wf 5413   -1-1-onto->wf1o 5416   ` cfv 5417  (class class class)co 6044    o Fcof 6266   Fincfn 7072   Basecbs 13428   +g cplusg 13488   0gc0g 13682    gsumg cgsu 13683   Grpcgrp 14644   inv gcminusg 14645   -gcsg 14647  CMndccmn 15371   Abelcabel 15372
This theorem is referenced by:  tsmsxplem2  18140
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-cnex 9006  ax-resscn 9007  ax-1cn 9008  ax-icn 9009  ax-addcl 9010  ax-addrcl 9011  ax-mulcl 9012  ax-mulrcl 9013  ax-mulcom 9014  ax-addass 9015  ax-mulass 9016  ax-distr 9017  ax-i2m1 9018  ax-1ne0 9019  ax-1rid 9020  ax-rnegex 9021  ax-rrecex 9022  ax-cnre 9023  ax-pre-lttri 9024  ax-pre-lttrn 9025  ax-pre-ltadd 9026  ax-pre-mulgt0 9027
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-int 4015  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-se 4506  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-isom 5426  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-of 6268  df-1st 6312  df-2nd 6313  df-riota 6512  df-recs 6596  df-rdg 6631  df-1o 6687  df-oadd 6691  df-er 6868  df-map 6983  df-en 7073  df-dom 7074  df-sdom 7075  df-fin 7076  df-oi 7439  df-card 7786  df-pnf 9082  df-mnf 9083  df-xr 9084  df-ltxr 9085  df-le 9086  df-sub 9253  df-neg 9254  df-nn 9961  df-2 10018  df-n0 10182  df-z 10243  df-uz 10449  df-fz 11004  df-fzo 11095  df-seq 11283  df-hash 11578  df-ndx 13431  df-slot 13432  df-base 13433  df-sets 13434  df-ress 13435  df-plusg 13501  df-0g 13686  df-gsum 13687  df-mnd 14649  df-mhm 14697  df-submnd 14698  df-grp 14771  df-minusg 14772  df-sbg 14773  df-ghm 14963  df-cntz 15075  df-cmn 15373  df-abl 15374
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