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Theorem gsumsub 15542
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 2436 . . . 4  |-  ( +g  `  G )  =  ( +g  `  G )
4 gsumsub.g . . . . 5  |-  ( ph  ->  G  e.  Abel )
5 ablcmn 15418 . . . . 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 2436 . . . . . . 7  |-  ( inv g `  G )  =  ( inv g `  G )
10 ablgrp 15417 . . . . . . . 8  |-  ( G  e.  Abel  ->  G  e. 
Grp )
114, 10syl 16 . . . . . . 7  |-  ( ph  ->  G  e.  Grp )
121, 9, 11grpinvf1o 14861 . . . . . 6  |-  ( ph  ->  ( inv g `  G ) : B -1-1-onto-> B
)
13 f1of 5674 . . . . . 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 5600 . . . . 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 3469 . . . . . . . 8  |-  ( k  e.  ( A  \ 
( `' H "
( _V  \  {  .0.  } ) ) )  ->  k  e.  A
)
21 fvco3 5800 . . . . . . . 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 3367 . . . . . . . . . . 11  |-  ( `' H " ( _V 
\  {  .0.  }
) )  C_  ( `' H " ( _V 
\  {  .0.  }
) )
2423a1i 11 . . . . . . . . . 10  |-  ( ph  ->  ( `' H "
( _V  \  {  .0.  } ) )  C_  ( `' H " ( _V 
\  {  .0.  }
) ) )
2515, 24suppssr 5864 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( A  \  ( `' H " ( _V 
\  {  .0.  }
) ) ) )  ->  ( H `  k )  =  .0.  )
2625fveq2d 5732 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( A  \  ( `' H " ( _V 
\  {  .0.  }
) ) ) )  ->  ( ( inv g `  G ) `
 ( H `  k ) )  =  ( ( inv g `  G ) `  .0.  ) )
272, 9grpinvid 14856 . . . . . . . . . 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 2468 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( A  \  ( `' H " ( _V 
\  {  .0.  }
) ) ) )  ->  ( ( inv g `  G ) `
 ( H `  k ) )  =  .0.  )
3122, 30eqtrd 2468 . . . . . 6  |-  ( (
ph  /\  k  e.  ( A  \  ( `' H " ( _V 
\  {  .0.  }
) ) ) )  ->  ( ( ( inv g `  G
)  o.  H ) `
 k )  =  .0.  )
3217, 31suppss 5863 . . . . 5  |-  ( ph  ->  ( `' ( ( inv g `  G
)  o.  H )
" ( _V  \  {  .0.  } ) ) 
C_  ( `' H " ( _V  \  {  .0.  } ) ) )
33 ssfi 7329 . . . . 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 15528 . . 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 15541 . . . 4  |-  ( ph  ->  ( G  gsumg  ( ( inv g `  G )  o.  H
) )  =  ( ( inv g `  G ) `  ( G  gsumg  H ) ) )
3736oveq2d 6097 . . 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 2468 . 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 5870 . . . . . 6  |-  ( (
ph  /\  k  e.  A )  ->  ( F `  k )  e.  B )
4015ffvelrnda 5870 . . . . . 6  |-  ( (
ph  /\  k  e.  A )  ->  ( H `  k )  e.  B )
41 gsumsub.m . . . . . . 7  |-  .-  =  ( -g `  G )
421, 3, 9, 41grpsubval 14848 . . . . . 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 4295 . . . 4  |-  ( ph  ->  ( k  e.  A  |->  ( ( F `  k )  .-  ( H `  k )
) )  =  ( k  e.  A  |->  ( ( F `  k
) ( +g  `  G
) ( ( inv g `  G ) `
 ( H `  k ) ) ) ) )
458feqmptd 5779 . . . . 5  |-  ( ph  ->  F  =  ( k  e.  A  |->  ( F `
 k ) ) )
4615feqmptd 5779 . . . . 5  |-  ( ph  ->  H  =  ( k  e.  A  |->  ( H `
 k ) ) )
477, 39, 40, 45, 46offval2 6322 . . . 4  |-  ( ph  ->  ( F  o F 
.-  H )  =  ( k  e.  A  |->  ( ( F `  k )  .-  ( H `  k )
) ) )
48 fvex 5742 . . . . . 6  |-  ( ( inv g `  G
) `  ( H `  k ) )  e. 
_V
4948a1i 11 . . . . 5  |-  ( (
ph  /\  k  e.  A )  ->  (
( inv g `  G ) `  ( H `  k )
)  e.  _V )
5014feqmptd 5779 . . . . . 6  |-  ( ph  ->  ( inv g `  G )  =  ( x  e.  B  |->  ( ( inv g `  G ) `  x
) ) )
51 fveq2 5728 . . . . . 6  |-  ( x  =  ( H `  k )  ->  (
( inv g `  G ) `  x
)  =  ( ( inv g `  G
) `  ( H `  k ) ) )
5240, 46, 50, 51fmptco 5901 . . . . 5  |-  ( ph  ->  ( ( inv g `  G )  o.  H
)  =  ( k  e.  A  |->  ( ( inv g `  G
) `  ( H `  k ) ) ) )
537, 39, 49, 45, 52offval2 6322 . . . 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 2478 . . 3  |-  ( ph  ->  ( F  o F 
.-  H )  =  ( F  o F ( +g  `  G
) ( ( inv g `  G )  o.  H ) ) )
5554oveq2d 6097 . 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 15521 . . 3  |-  ( ph  ->  ( G  gsumg  F )  e.  B
)
571, 2, 6, 7, 15, 19gsumcl 15521 . . 3  |-  ( ph  ->  ( G  gsumg  H )  e.  B
)
581, 3, 9, 41grpsubval 14848 . . 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 2478 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 1652    e. wcel 1725   _Vcvv 2956    \ cdif 3317    C_ wss 3320   {csn 3814    e. cmpt 4266   `'ccnv 4877   "cima 4881    o. ccom 4882   -->wf 5450   -1-1-onto->wf1o 5453   ` cfv 5454  (class class class)co 6081    o Fcof 6303   Fincfn 7109   Basecbs 13469   +g cplusg 13529   0gc0g 13723    gsumg cgsu 13724   Grpcgrp 14685   inv gcminusg 14686   -gcsg 14688  CMndccmn 15412   Abelcabel 15413
This theorem is referenced by:  tsmsxplem2  18183
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-of 6305  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-oi 7479  df-card 7826  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-n0 10222  df-z 10283  df-uz 10489  df-fz 11044  df-fzo 11136  df-seq 11324  df-hash 11619  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-0g 13727  df-gsum 13728  df-mnd 14690  df-mhm 14738  df-submnd 14739  df-grp 14812  df-minusg 14813  df-sbg 14814  df-ghm 15004  df-cntz 15116  df-cmn 15414  df-abl 15415
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