MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  gsumsub Structured version   Unicode version

Theorem gsumsub 17101
Description: The difference of two group sums. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by AV, 6-Jun-2019.)
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 finSupp  .0.  )
gsumsub.hn  |-  ( ph  ->  H finSupp  .0.  )
Assertion
Ref Expression
gsumsub  |-  ( ph  ->  ( G  gsumg  ( F  oF 
.-  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 2457 . . . 4  |-  ( +g  `  G )  =  ( +g  `  G )
4 gsumsub.g . . . . 5  |-  ( ph  ->  G  e.  Abel )
5 ablcmn 16931 . . . . 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 2457 . . . . . . 7  |-  ( invg `  G )  =  ( invg `  G )
10 ablgrp 16930 . . . . . . . 8  |-  ( G  e.  Abel  ->  G  e. 
Grp )
114, 10syl 16 . . . . . . 7  |-  ( ph  ->  G  e.  Grp )
121, 9, 11grpinvf1o 16235 . . . . . 6  |-  ( ph  ->  ( invg `  G ) : B -1-1-onto-> B
)
13 f1of 5822 . . . . . 6  |-  ( ( invg `  G
) : B -1-1-onto-> B  -> 
( invg `  G ) : B --> B )
1412, 13syl 16 . . . . 5  |-  ( ph  ->  ( invg `  G ) : B --> B )
15 gsumsub.h . . . . 5  |-  ( ph  ->  H : A --> B )
16 fco 5747 . . . . 5  |-  ( ( ( invg `  G ) : B --> B  /\  H : A --> B )  ->  (
( invg `  G )  o.  H
) : A --> B )
1714, 15, 16syl2anc 661 . . . 4  |-  ( ph  ->  ( ( invg `  G )  o.  H
) : A --> B )
18 gsumsub.fn . . . 4  |-  ( ph  ->  F finSupp  .0.  )
19 fvex 5882 . . . . . . 7  |-  ( 0g
`  G )  e. 
_V
202, 19eqeltri 2541 . . . . . 6  |-  .0.  e.  _V
2120a1i 11 . . . . 5  |-  ( ph  ->  .0.  e.  _V )
22 fvex 5882 . . . . . . 7  |-  ( Base `  G )  e.  _V
231, 22eqeltri 2541 . . . . . 6  |-  B  e. 
_V
2423a1i 11 . . . . 5  |-  ( ph  ->  B  e.  _V )
25 gsumsub.hn . . . . 5  |-  ( ph  ->  H finSupp  .0.  )
262, 9grpinvid 16228 . . . . . 6  |-  ( G  e.  Grp  ->  (
( invg `  G ) `  .0.  )  =  .0.  )
2711, 26syl 16 . . . . 5  |-  ( ph  ->  ( ( invg `  G ) `  .0.  )  =  .0.  )
2821, 15, 14, 7, 24, 25, 27fsuppco2 7880 . . . 4  |-  ( ph  ->  ( ( invg `  G )  o.  H
) finSupp  .0.  )
291, 2, 3, 6, 7, 8, 17, 18, 28gsumadd 17065 . . 3  |-  ( ph  ->  ( G  gsumg  ( F  oF ( +g  `  G
) ( ( invg `  G )  o.  H ) ) )  =  ( ( G  gsumg  F ) ( +g  `  G ) ( G 
gsumg  ( ( invg `  G )  o.  H
) ) ) )
301, 2, 9, 4, 7, 15, 25gsuminv 17098 . . . 4  |-  ( ph  ->  ( G  gsumg  ( ( invg `  G )  o.  H
) )  =  ( ( invg `  G ) `  ( G  gsumg  H ) ) )
3130oveq2d 6312 . . 3  |-  ( ph  ->  ( ( G  gsumg  F ) ( +g  `  G
) ( G  gsumg  ( ( invg `  G
)  o.  H ) ) )  =  ( ( G  gsumg  F ) ( +g  `  G ) ( ( invg `  G
) `  ( G  gsumg  H ) ) ) )
3229, 31eqtrd 2498 . 2  |-  ( ph  ->  ( G  gsumg  ( F  oF ( +g  `  G
) ( ( invg `  G )  o.  H ) ) )  =  ( ( G  gsumg  F ) ( +g  `  G ) ( ( invg `  G
) `  ( G  gsumg  H ) ) ) )
338ffvelrnda 6032 . . . . . 6  |-  ( (
ph  /\  k  e.  A )  ->  ( F `  k )  e.  B )
3415ffvelrnda 6032 . . . . . 6  |-  ( (
ph  /\  k  e.  A )  ->  ( H `  k )  e.  B )
35 gsumsub.m . . . . . . 7  |-  .-  =  ( -g `  G )
361, 3, 9, 35grpsubval 16220 . . . . . 6  |-  ( ( ( F `  k
)  e.  B  /\  ( H `  k )  e.  B )  -> 
( ( F `  k )  .-  ( H `  k )
)  =  ( ( F `  k ) ( +g  `  G
) ( ( invg `  G ) `
 ( H `  k ) ) ) )
3733, 34, 36syl2anc 661 . . . . 5  |-  ( (
ph  /\  k  e.  A )  ->  (
( F `  k
)  .-  ( H `  k ) )  =  ( ( F `  k ) ( +g  `  G ) ( ( invg `  G
) `  ( H `  k ) ) ) )
3837mpteq2dva 4543 . . . 4  |-  ( ph  ->  ( k  e.  A  |->  ( ( F `  k )  .-  ( H `  k )
) )  =  ( k  e.  A  |->  ( ( F `  k
) ( +g  `  G
) ( ( invg `  G ) `
 ( H `  k ) ) ) ) )
398feqmptd 5926 . . . . 5  |-  ( ph  ->  F  =  ( k  e.  A  |->  ( F `
 k ) ) )
4015feqmptd 5926 . . . . 5  |-  ( ph  ->  H  =  ( k  e.  A  |->  ( H `
 k ) ) )
417, 33, 34, 39, 40offval2 6555 . . . 4  |-  ( ph  ->  ( F  oF 
.-  H )  =  ( k  e.  A  |->  ( ( F `  k )  .-  ( H `  k )
) ) )
42 fvex 5882 . . . . . 6  |-  ( ( invg `  G
) `  ( H `  k ) )  e. 
_V
4342a1i 11 . . . . 5  |-  ( (
ph  /\  k  e.  A )  ->  (
( invg `  G ) `  ( H `  k )
)  e.  _V )
4414feqmptd 5926 . . . . . 6  |-  ( ph  ->  ( invg `  G )  =  ( x  e.  B  |->  ( ( invg `  G ) `  x
) ) )
45 fveq2 5872 . . . . . 6  |-  ( x  =  ( H `  k )  ->  (
( invg `  G ) `  x
)  =  ( ( invg `  G
) `  ( H `  k ) ) )
4634, 40, 44, 45fmptco 6065 . . . . 5  |-  ( ph  ->  ( ( invg `  G )  o.  H
)  =  ( k  e.  A  |->  ( ( invg `  G
) `  ( H `  k ) ) ) )
477, 33, 43, 39, 46offval2 6555 . . . 4  |-  ( ph  ->  ( F  oF ( +g  `  G
) ( ( invg `  G )  o.  H ) )  =  ( k  e.  A  |->  ( ( F `
 k ) ( +g  `  G ) ( ( invg `  G ) `  ( H `  k )
) ) ) )
4838, 41, 473eqtr4d 2508 . . 3  |-  ( ph  ->  ( F  oF 
.-  H )  =  ( F  oF ( +g  `  G
) ( ( invg `  G )  o.  H ) ) )
4948oveq2d 6312 . 2  |-  ( ph  ->  ( G  gsumg  ( F  oF 
.-  H ) )  =  ( G  gsumg  ( F  oF ( +g  `  G ) ( ( invg `  G
)  o.  H ) ) ) )
501, 2, 6, 7, 8, 18gsumcl 17050 . . 3  |-  ( ph  ->  ( G  gsumg  F )  e.  B
)
511, 2, 6, 7, 15, 25gsumcl 17050 . . 3  |-  ( ph  ->  ( G  gsumg  H )  e.  B
)
521, 3, 9, 35grpsubval 16220 . . 3  |-  ( ( ( G  gsumg  F )  e.  B  /\  ( G  gsumg  H )  e.  B
)  ->  ( ( G  gsumg  F )  .-  ( G  gsumg  H ) )  =  ( ( G  gsumg  F ) ( +g  `  G
) ( ( invg `  G ) `
 ( G  gsumg  H ) ) ) )
5350, 51, 52syl2anc 661 . 2  |-  ( ph  ->  ( ( G  gsumg  F ) 
.-  ( G  gsumg  H ) )  =  ( ( G  gsumg  F ) ( +g  `  G ) ( ( invg `  G
) `  ( G  gsumg  H ) ) ) )
5432, 49, 533eqtr4d 2508 1  |-  ( ph  ->  ( G  gsumg  ( F  oF 
.-  H ) )  =  ( ( G 
gsumg  F )  .-  ( G  gsumg  H ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819   _Vcvv 3109   class class class wbr 4456    |-> cmpt 4515    o. ccom 5012   -->wf 5590   -1-1-onto->wf1o 5593   ` cfv 5594  (class class class)co 6296    oFcof 6537   finSupp cfsupp 7847   Basecbs 14644   +g cplusg 14712   0gc0g 14857    gsumg cgsu 14858   Grpcgrp 16180   invgcminusg 16181   -gcsg 16182  CMndccmn 16925   Abelcabl 16926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-om 6700  df-1st 6799  df-2nd 6800  df-supp 6918  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fsupp 7848  df-oi 7953  df-card 8337  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-fzo 11822  df-seq 12111  df-hash 12409  df-ndx 14647  df-slot 14648  df-base 14649  df-sets 14650  df-ress 14651  df-plusg 14725  df-0g 14859  df-gsum 14860  df-mgm 15999  df-sgrp 16038  df-mnd 16048  df-mhm 16093  df-submnd 16094  df-grp 16184  df-minusg 16185  df-sbg 16186  df-ghm 16392  df-cntz 16482  df-cmn 16927  df-abl 16928
This theorem is referenced by:  gsummptfssub  17103  tsmsxplem2  20782
  Copyright terms: Public domain W3C validator