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Theorem gsumsub 15318
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 2358 . . . 4  |-  ( +g  `  G )  =  ( +g  `  G )
4 gsumsub.g . . . . 5  |-  ( ph  ->  G  e.  Abel )
5 ablcmn 15194 . . . . 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 2358 . . . . . . 7  |-  ( inv g `  G )  =  ( inv g `  G )
10 ablgrp 15193 . . . . . . . 8  |-  ( G  e.  Abel  ->  G  e. 
Grp )
114, 10syl 15 . . . . . . 7  |-  ( ph  ->  G  e.  Grp )
121, 9, 11grpinvf1o 14637 . . . . . 6  |-  ( ph  ->  ( inv g `  G ) : B -1-1-onto-> B
)
13 f1of 5555 . . . . . 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 5481 . . . . 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 3374 . . . . . . . 8  |-  ( k  e.  ( A  \ 
( `' H "
( _V  \  {  .0.  } ) ) )  ->  k  e.  A
)
21 fvco3 5679 . . . . . . . 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 3273 . . . . . . . . . . 11  |-  ( `' H " ( _V 
\  {  .0.  }
) )  C_  ( `' H " ( _V 
\  {  .0.  }
) )
2423a1i 10 . . . . . . . . . 10  |-  ( ph  ->  ( `' H "
( _V  \  {  .0.  } ) )  C_  ( `' H " ( _V 
\  {  .0.  }
) ) )
2515, 24suppssr 5742 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( A  \  ( `' H " ( _V 
\  {  .0.  }
) ) ) )  ->  ( H `  k )  =  .0.  )
2625fveq2d 5612 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( A  \  ( `' H " ( _V 
\  {  .0.  }
) ) ) )  ->  ( ( inv g `  G ) `
 ( H `  k ) )  =  ( ( inv g `  G ) `  .0.  ) )
272, 9grpinvid 14632 . . . . . . . . . 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 2390 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( A  \  ( `' H " ( _V 
\  {  .0.  }
) ) ) )  ->  ( ( inv g `  G ) `
 ( H `  k ) )  =  .0.  )
3122, 30eqtrd 2390 . . . . . 6  |-  ( (
ph  /\  k  e.  ( A  \  ( `' H " ( _V 
\  {  .0.  }
) ) ) )  ->  ( ( ( inv g `  G
)  o.  H ) `
 k )  =  .0.  )
3217, 31suppss 5741 . . . . 5  |-  ( ph  ->  ( `' ( ( inv g `  G
)  o.  H )
" ( _V  \  {  .0.  } ) ) 
C_  ( `' H " ( _V  \  {  .0.  } ) ) )
33 ssfi 7171 . . . . 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 15304 . . 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 15317 . . . 4  |-  ( ph  ->  ( G  gsumg  ( ( inv g `  G )  o.  H
) )  =  ( ( inv g `  G ) `  ( G  gsumg  H ) ) )
3736oveq2d 5961 . . 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 2390 . 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 5746 . . . . . . 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 5746 . . . . . . 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 14624 . . . . . 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 4187 . . . 4  |-  ( ph  ->  ( k  e.  A  |->  ( ( F `  k )  .-  ( H `  k )
) )  =  ( k  e.  A  |->  ( ( F `  k
) ( +g  `  G
) ( ( inv g `  G ) `
 ( H `  k ) ) ) ) )
478feqmptd 5658 . . . . 5  |-  ( ph  ->  F  =  ( k  e.  A  |->  ( F `
 k ) ) )
4815feqmptd 5658 . . . . 5  |-  ( ph  ->  H  =  ( k  e.  A  |->  ( H `
 k ) ) )
497, 40, 42, 47, 48offval2 6182 . . . 4  |-  ( ph  ->  ( F  o F 
.-  H )  =  ( k  e.  A  |->  ( ( F `  k )  .-  ( H `  k )
) ) )
50 fvex 5622 . . . . . 6  |-  ( ( inv g `  G
) `  ( H `  k ) )  e. 
_V
5150a1i 10 . . . . 5  |-  ( (
ph  /\  k  e.  A )  ->  (
( inv g `  G ) `  ( H `  k )
)  e.  _V )
5214feqmptd 5658 . . . . . 6  |-  ( ph  ->  ( inv g `  G )  =  ( x  e.  B  |->  ( ( inv g `  G ) `  x
) ) )
53 fveq2 5608 . . . . . 6  |-  ( x  =  ( H `  k )  ->  (
( inv g `  G ) `  x
)  =  ( ( inv g `  G
) `  ( H `  k ) ) )
5442, 48, 52, 53fmptco 5774 . . . . 5  |-  ( ph  ->  ( ( inv g `  G )  o.  H
)  =  ( k  e.  A  |->  ( ( inv g `  G
) `  ( H `  k ) ) ) )
557, 40, 51, 47, 54offval2 6182 . . . 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 2400 . . 3  |-  ( ph  ->  ( F  o F 
.-  H )  =  ( F  o F ( +g  `  G
) ( ( inv g `  G )  o.  H ) ) )
5756oveq2d 5961 . 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 15297 . . 3  |-  ( ph  ->  ( G  gsumg  F )  e.  B
)
591, 2, 6, 7, 15, 19gsumcl 15297 . . 3  |-  ( ph  ->  ( G  gsumg  H )  e.  B
)
601, 3, 9, 43grpsubval 14624 . . 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 2400 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 1642    e. wcel 1710   _Vcvv 2864    \ cdif 3225    C_ wss 3228   {csn 3716    e. cmpt 4158   `'ccnv 4770   "cima 4774    o. ccom 4775   -->wf 5333   -1-1-onto->wf1o 5336   ` cfv 5337  (class class class)co 5945    o Fcof 6163   Fincfn 6951   Basecbs 13245   +g cplusg 13305   0gc0g 13499    gsumg cgsu 13500   Grpcgrp 14461   inv gcminusg 14462   -gcsg 14464  CMndccmn 15188   Abelcabel 15189
This theorem is referenced by:  tsmsxplem2  17938
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-cnex 8883  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903  ax-pre-mulgt0 8904
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-int 3944  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-se 4435  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-isom 5346  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-of 6165  df-1st 6209  df-2nd 6210  df-riota 6391  df-recs 6475  df-rdg 6510  df-1o 6566  df-oadd 6570  df-er 6747  df-map 6862  df-en 6952  df-dom 6953  df-sdom 6954  df-fin 6955  df-oi 7315  df-card 7662  df-pnf 8959  df-mnf 8960  df-xr 8961  df-ltxr 8962  df-le 8963  df-sub 9129  df-neg 9130  df-nn 9837  df-2 9894  df-n0 10058  df-z 10117  df-uz 10323  df-fz 10875  df-fzo 10963  df-seq 11139  df-hash 11431  df-ndx 13248  df-slot 13249  df-base 13250  df-sets 13251  df-ress 13252  df-plusg 13318  df-0g 13503  df-gsum 13504  df-mnd 14466  df-mhm 14514  df-submnd 14515  df-grp 14588  df-minusg 14589  df-sbg 14590  df-ghm 14780  df-cntz 14892  df-cmn 15190  df-abl 15191
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