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Theorem gsumsub 15219
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 2283 . . . 4  |-  ( +g  `  G )  =  ( +g  `  G )
4 gsumsub.g . . . . 5  |-  ( ph  ->  G  e.  Abel )
5 ablcmn 15095 . . . . 5  |-  ( G  e.  Abel  ->  G  e. CMnd
)
64, 5syl 15 . . . 4  |-  ( ph  ->  G  e. CMnd )
7 gsumsub.a . . . 4  |-  ( ph  ->  A  e.  V )
8 gsumsub.f . . . 4  |-  ( ph  ->  F : A --> B )
9 eqid 2283 . . . . . . 7  |-  ( inv g `  G )  =  ( inv g `  G )
10 ablgrp 15094 . . . . . . . 8  |-  ( G  e.  Abel  ->  G  e. 
Grp )
114, 10syl 15 . . . . . . 7  |-  ( ph  ->  G  e.  Grp )
121, 9, 11grpinvf1o 14538 . . . . . 6  |-  ( ph  ->  ( inv g `  G ) : B -1-1-onto-> B
)
13 f1of 5472 . . . . . 6  |-  ( ( inv g `  G
) : B -1-1-onto-> B  -> 
( inv g `  G ) : B --> B )
1412, 13syl 15 . . . . 5  |-  ( ph  ->  ( inv g `  G ) : B --> B )
15 gsumsub.h . . . . 5  |-  ( ph  ->  H : A --> B )
16 fco 5398 . . . . 5  |-  ( ( ( inv g `  G ) : B --> B  /\  H : A --> B )  ->  (
( inv g `  G )  o.  H
) : A --> B )
1714, 15, 16syl2anc 642 . . . 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 3298 . . . . . . . 8  |-  ( k  e.  ( A  \ 
( `' H "
( _V  \  {  .0.  } ) ) )  ->  k  e.  A
)
21 fvco3 5596 . . . . . . . 8  |-  ( ( H : A --> B  /\  k  e.  A )  ->  ( ( ( inv g `  G )  o.  H ) `  k )  =  ( ( inv g `  G ) `  ( H `  k )
) )
2215, 20, 21syl2an 463 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( A  \  ( `' H " ( _V 
\  {  .0.  }
) ) ) )  ->  ( ( ( inv g `  G
)  o.  H ) `
 k )  =  ( ( inv g `  G ) `  ( H `  k )
) )
23 ssid 3197 . . . . . . . . . . 11  |-  ( `' H " ( _V 
\  {  .0.  }
) )  C_  ( `' H " ( _V 
\  {  .0.  }
) )
2423a1i 10 . . . . . . . . . 10  |-  ( ph  ->  ( `' H "
( _V  \  {  .0.  } ) )  C_  ( `' H " ( _V 
\  {  .0.  }
) ) )
2515, 24suppssr 5659 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( A  \  ( `' H " ( _V 
\  {  .0.  }
) ) ) )  ->  ( H `  k )  =  .0.  )
2625fveq2d 5529 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( A  \  ( `' H " ( _V 
\  {  .0.  }
) ) ) )  ->  ( ( inv g `  G ) `
 ( H `  k ) )  =  ( ( inv g `  G ) `  .0.  ) )
272, 9grpinvid 14533 . . . . . . . . . 10  |-  ( G  e.  Grp  ->  (
( inv g `  G ) `  .0.  )  =  .0.  )
2811, 27syl 15 . . . . . . . . 9  |-  ( ph  ->  ( ( inv g `  G ) `  .0.  )  =  .0.  )
2928adantr 451 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( A  \  ( `' H " ( _V 
\  {  .0.  }
) ) ) )  ->  ( ( inv g `  G ) `
 .0.  )  =  .0.  )
3026, 29eqtrd 2315 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( A  \  ( `' H " ( _V 
\  {  .0.  }
) ) ) )  ->  ( ( inv g `  G ) `
 ( H `  k ) )  =  .0.  )
3122, 30eqtrd 2315 . . . . . 6  |-  ( (
ph  /\  k  e.  ( A  \  ( `' H " ( _V 
\  {  .0.  }
) ) ) )  ->  ( ( ( inv g `  G
)  o.  H ) `
 k )  =  .0.  )
3217, 31suppss 5658 . . . . 5  |-  ( ph  ->  ( `' ( ( inv g `  G
)  o.  H )
" ( _V  \  {  .0.  } ) ) 
C_  ( `' H " ( _V  \  {  .0.  } ) ) )
33 ssfi 7083 . . . . 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 642 . . . 4  |-  ( ph  ->  ( `' ( ( inv g `  G
)  o.  H )
" ( _V  \  {  .0.  } ) )  e.  Fin )
351, 2, 3, 6, 7, 8, 17, 18, 34gsumadd 15205 . . 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 15218 . . . 4  |-  ( ph  ->  ( G  gsumg  ( ( inv g `  G )  o.  H
) )  =  ( ( inv g `  G ) `  ( G  gsumg  H ) ) )
3736oveq2d 5874 . . 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 2315 . 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 ) ) ) )
39 ffvelrn 5663 . . . . . . 7  |-  ( ( F : A --> B  /\  k  e.  A )  ->  ( F `  k
)  e.  B )
408, 39sylan 457 . . . . . 6  |-  ( (
ph  /\  k  e.  A )  ->  ( F `  k )  e.  B )
41 ffvelrn 5663 . . . . . . 7  |-  ( ( H : A --> B  /\  k  e.  A )  ->  ( H `  k
)  e.  B )
4215, 41sylan 457 . . . . . 6  |-  ( (
ph  /\  k  e.  A )  ->  ( H `  k )  e.  B )
43 gsumsub.m . . . . . . 7  |-  .-  =  ( -g `  G )
441, 3, 9, 43grpsubval 14525 . . . . . 6  |-  ( ( ( F `  k
)  e.  B  /\  ( H `  k )  e.  B )  -> 
( ( F `  k )  .-  ( H `  k )
)  =  ( ( F `  k ) ( +g  `  G
) ( ( inv g `  G ) `
 ( H `  k ) ) ) )
4540, 42, 44syl2anc 642 . . . . 5  |-  ( (
ph  /\  k  e.  A )  ->  (
( F `  k
)  .-  ( H `  k ) )  =  ( ( F `  k ) ( +g  `  G ) ( ( inv g `  G
) `  ( H `  k ) ) ) )
4645mpteq2dva 4106 . . . 4  |-  ( ph  ->  ( k  e.  A  |->  ( ( F `  k )  .-  ( H `  k )
) )  =  ( k  e.  A  |->  ( ( F `  k
) ( +g  `  G
) ( ( inv g `  G ) `
 ( H `  k ) ) ) ) )
478feqmptd 5575 . . . . 5  |-  ( ph  ->  F  =  ( k  e.  A  |->  ( F `
 k ) ) )
4815feqmptd 5575 . . . . 5  |-  ( ph  ->  H  =  ( k  e.  A  |->  ( H `
 k ) ) )
497, 40, 42, 47, 48offval2 6095 . . . 4  |-  ( ph  ->  ( F  o F 
.-  H )  =  ( k  e.  A  |->  ( ( F `  k )  .-  ( H `  k )
) ) )
50 fvex 5539 . . . . . 6  |-  ( ( inv g `  G
) `  ( H `  k ) )  e. 
_V
5150a1i 10 . . . . 5  |-  ( (
ph  /\  k  e.  A )  ->  (
( inv g `  G ) `  ( H `  k )
)  e.  _V )
5214feqmptd 5575 . . . . . 6  |-  ( ph  ->  ( inv g `  G )  =  ( x  e.  B  |->  ( ( inv g `  G ) `  x
) ) )
53 fveq2 5525 . . . . . 6  |-  ( x  =  ( H `  k )  ->  (
( inv g `  G ) `  x
)  =  ( ( inv g `  G
) `  ( H `  k ) ) )
5442, 48, 52, 53fmptco 5691 . . . . 5  |-  ( ph  ->  ( ( inv g `  G )  o.  H
)  =  ( k  e.  A  |->  ( ( inv g `  G
) `  ( H `  k ) ) ) )
557, 40, 51, 47, 54offval2 6095 . . . 4  |-  ( ph  ->  ( F  o F ( +g  `  G
) ( ( inv g `  G )  o.  H ) )  =  ( k  e.  A  |->  ( ( F `
 k ) ( +g  `  G ) ( ( inv g `  G ) `  ( H `  k )
) ) ) )
5646, 49, 553eqtr4d 2325 . . 3  |-  ( ph  ->  ( F  o F 
.-  H )  =  ( F  o F ( +g  `  G
) ( ( inv g `  G )  o.  H ) ) )
5756oveq2d 5874 . 2  |-  ( ph  ->  ( G  gsumg  ( F  o F 
.-  H ) )  =  ( G  gsumg  ( F  o F ( +g  `  G ) ( ( inv g `  G
)  o.  H ) ) ) )
581, 2, 6, 7, 8, 18gsumcl 15198 . . 3  |-  ( ph  ->  ( G  gsumg  F )  e.  B
)
591, 2, 6, 7, 15, 19gsumcl 15198 . . 3  |-  ( ph  ->  ( G  gsumg  H )  e.  B
)
601, 3, 9, 43grpsubval 14525 . . 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 ) ) ) )
6158, 59, 60syl2anc 642 . 2  |-  ( ph  ->  ( ( G  gsumg  F ) 
.-  ( G  gsumg  H ) )  =  ( ( G  gsumg  F ) ( +g  `  G ) ( ( inv g `  G
) `  ( G  gsumg  H ) ) ) )
6238, 57, 613eqtr4d 2325 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 358    = wceq 1623    e. wcel 1684   _Vcvv 2788    \ cdif 3149    C_ wss 3152   {csn 3640    e. cmpt 4077   `'ccnv 4688   "cima 4692    o. ccom 4693   -->wf 5251   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5858    o Fcof 6076   Fincfn 6863   Basecbs 13148   +g cplusg 13208   0gc0g 13400    gsumg cgsu 13401   Grpcgrp 14362   inv gcminusg 14363   -gcsg 14365  CMndccmn 15089   Abelcabel 15090
This theorem is referenced by:  tsmsxplem2  17836
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-fzo 10871  df-seq 11047  df-hash 11338  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-0g 13404  df-gsum 13405  df-mnd 14367  df-mhm 14415  df-submnd 14416  df-grp 14489  df-minusg 14490  df-sbg 14491  df-ghm 14681  df-cntz 14793  df-cmn 15091  df-abl 15092
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