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Theorem gsumzinv 17182
Description: Inverse of a group sum. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 6-Jun-2019.)
Hypotheses
Ref Expression
gsumzinv.b  |-  B  =  ( Base `  G
)
gsumzinv.0  |-  .0.  =  ( 0g `  G )
gsumzinv.z  |-  Z  =  (Cntz `  G )
gsumzinv.i  |-  I  =  ( invg `  G )
gsumzinv.g  |-  ( ph  ->  G  e.  Grp )
gsumzinv.a  |-  ( ph  ->  A  e.  V )
gsumzinv.f  |-  ( ph  ->  F : A --> B )
gsumzinv.c  |-  ( ph  ->  ran  F  C_  ( Z `  ran  F ) )
gsumzinv.n  |-  ( ph  ->  F finSupp  .0.  )
Assertion
Ref Expression
gsumzinv  |-  ( ph  ->  ( G  gsumg  ( I  o.  F
) )  =  ( I `  ( G 
gsumg  F ) ) )

Proof of Theorem gsumzinv
StepHypRef Expression
1 gsumzinv.b . . 3  |-  B  =  ( Base `  G
)
2 gsumzinv.0 . . 3  |-  .0.  =  ( 0g `  G )
3 gsumzinv.z . . 3  |-  Z  =  (Cntz `  G )
4 eqid 2400 . . 3  |-  (oppg `  G
)  =  (oppg `  G
)
5 gsumzinv.g . . . 4  |-  ( ph  ->  G  e.  Grp )
6 grpmnd 16276 . . . 4  |-  ( G  e.  Grp  ->  G  e.  Mnd )
75, 6syl 17 . . 3  |-  ( ph  ->  G  e.  Mnd )
8 gsumzinv.a . . 3  |-  ( ph  ->  A  e.  V )
9 gsumzinv.i . . . . . 6  |-  I  =  ( invg `  G )
101, 9grpinvf 16308 . . . . 5  |-  ( G  e.  Grp  ->  I : B --> B )
115, 10syl 17 . . . 4  |-  ( ph  ->  I : B --> B )
12 gsumzinv.f . . . 4  |-  ( ph  ->  F : A --> B )
13 fco 5678 . . . 4  |-  ( ( I : B --> B  /\  F : A --> B )  ->  ( I  o.  F ) : A --> B )
1411, 12, 13syl2anc 659 . . 3  |-  ( ph  ->  ( I  o.  F
) : A --> B )
154, 9invoppggim 16609 . . . . . 6  |-  ( G  e.  Grp  ->  I  e.  ( G GrpIso  (oppg `  G
) ) )
16 gimghm 16526 . . . . . 6  |-  ( I  e.  ( G GrpIso  (oppg `  G
) )  ->  I  e.  ( G  GrpHom  (oppg `  G
) ) )
17 ghmmhm 16491 . . . . . 6  |-  ( I  e.  ( G  GrpHom  (oppg `  G ) )  ->  I  e.  ( G MndHom  (oppg `  G ) ) )
185, 15, 16, 174syl 21 . . . . 5  |-  ( ph  ->  I  e.  ( G MndHom 
(oppg `  G ) ) )
19 gsumzinv.c . . . . 5  |-  ( ph  ->  ran  F  C_  ( Z `  ran  F ) )
20 eqid 2400 . . . . . 6  |-  (Cntz `  (oppg `  G ) )  =  (Cntz `  (oppg
`  G ) )
213, 20cntzmhm2 16591 . . . . 5  |-  ( ( I  e.  ( G MndHom 
(oppg `  G ) )  /\  ran  F  C_  ( Z `  ran  F ) )  ->  ( I " ran  F )  C_  (
(Cntz `  (oppg
`  G ) ) `
 ( I " ran  F ) ) )
2218, 19, 21syl2anc 659 . . . 4  |-  ( ph  ->  ( I " ran  F )  C_  ( (Cntz `  (oppg
`  G ) ) `
 ( I " ran  F ) ) )
23 rnco2 5449 . . . 4  |-  ran  (
I  o.  F )  =  ( I " ran  F )
2423fveq2i 5806 . . . . 5  |-  ( Z `
 ran  ( I  o.  F ) )  =  ( Z `  (
I " ran  F
) )
254, 3oppgcntz 16613 . . . . 5  |-  ( Z `
 ( I " ran  F ) )  =  ( (Cntz `  (oppg `  G
) ) `  (
I " ran  F
) )
2624, 25eqtri 2429 . . . 4  |-  ( Z `
 ran  ( I  o.  F ) )  =  ( (Cntz `  (oppg `  G
) ) `  (
I " ran  F
) )
2722, 23, 263sstr4g 3480 . . 3  |-  ( ph  ->  ran  ( I  o.  F )  C_  ( Z `  ran  ( I  o.  F ) ) )
28 fvex 5813 . . . . . 6  |-  ( 0g
`  G )  e. 
_V
292, 28eqeltri 2484 . . . . 5  |-  .0.  e.  _V
3029a1i 11 . . . 4  |-  ( ph  ->  .0.  e.  _V )
31 fvex 5813 . . . . . 6  |-  ( Base `  G )  e.  _V
321, 31eqeltri 2484 . . . . 5  |-  B  e. 
_V
3332a1i 11 . . . 4  |-  ( ph  ->  B  e.  _V )
34 gsumzinv.n . . . 4  |-  ( ph  ->  F finSupp  .0.  )
352, 9grpinvid 16315 . . . . 5  |-  ( G  e.  Grp  ->  (
I `  .0.  )  =  .0.  )
365, 35syl 17 . . . 4  |-  ( ph  ->  ( I `  .0.  )  =  .0.  )
3730, 12, 11, 8, 33, 34, 36fsuppco2 7814 . . 3  |-  ( ph  ->  ( I  o.  F
) finSupp  .0.  )
381, 2, 3, 4, 7, 8, 14, 27, 37gsumzoppg 17180 . 2  |-  ( ph  ->  ( (oppg
`  G )  gsumg  ( I  o.  F ) )  =  ( G  gsumg  ( I  o.  F ) ) )
394oppgmnd 16603 . . . 4  |-  ( G  e.  Mnd  ->  (oppg `  G
)  e.  Mnd )
407, 39syl 17 . . 3  |-  ( ph  ->  (oppg
`  G )  e. 
Mnd )
411, 3, 7, 40, 8, 18, 12, 19, 2, 34gsumzmhm 17170 . 2  |-  ( ph  ->  ( (oppg
`  G )  gsumg  ( I  o.  F ) )  =  ( I `  ( G  gsumg  F ) ) )
4238, 41eqtr3d 2443 1  |-  ( ph  ->  ( G  gsumg  ( I  o.  F
) )  =  ( I `  ( G 
gsumg  F ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1403    e. wcel 1840   _Vcvv 3056    C_ wss 3411   class class class wbr 4392   ran crn 4941   "cima 4943    o. ccom 4944   -->wf 5519   ` cfv 5523  (class class class)co 6232   finSupp cfsupp 7781   Basecbs 14731   0gc0g 14944    gsumg cgsu 14945   Mndcmnd 16133   MndHom cmhm 16178   Grpcgrp 16267   invgcminusg 16268    GrpHom cghm 16478   GrpIso cgim 16519  Cntzccntz 16567  oppgcoppg 16594
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528  ax-cnex 9496  ax-resscn 9497  ax-1cn 9498  ax-icn 9499  ax-addcl 9500  ax-addrcl 9501  ax-mulcl 9502  ax-mulrcl 9503  ax-mulcom 9504  ax-addass 9505  ax-mulass 9506  ax-distr 9507  ax-i2m1 9508  ax-1ne0 9509  ax-1rid 9510  ax-rnegex 9511  ax-rrecex 9512  ax-cnre 9513  ax-pre-lttri 9514  ax-pre-lttrn 9515  ax-pre-ltadd 9516  ax-pre-mulgt0 9517
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 973  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-nel 2599  df-ral 2756  df-rex 2757  df-reu 2758  df-rmo 2759  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-pss 3427  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-tp 3974  df-op 3976  df-uni 4189  df-int 4225  df-iun 4270  df-iin 4271  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4487  df-eprel 4731  df-id 4735  df-po 4741  df-so 4742  df-fr 4779  df-se 4780  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-isom 5532  df-riota 6194  df-ov 6235  df-oprab 6236  df-mpt2 6237  df-om 6637  df-1st 6736  df-2nd 6737  df-supp 6855  df-tpos 6910  df-recs 6997  df-rdg 7031  df-1o 7085  df-oadd 7089  df-er 7266  df-map 7377  df-en 7473  df-dom 7474  df-sdom 7475  df-fin 7476  df-fsupp 7782  df-oi 7887  df-card 8270  df-pnf 9578  df-mnf 9579  df-xr 9580  df-ltxr 9581  df-le 9582  df-sub 9761  df-neg 9762  df-nn 10495  df-2 10553  df-n0 10755  df-z 10824  df-uz 11044  df-fz 11642  df-fzo 11766  df-seq 12060  df-hash 12358  df-ndx 14734  df-slot 14735  df-base 14736  df-sets 14737  df-ress 14738  df-plusg 14812  df-0g 14946  df-gsum 14947  df-mre 15090  df-mrc 15091  df-acs 15093  df-mgm 16086  df-sgrp 16125  df-mnd 16135  df-mhm 16180  df-submnd 16181  df-grp 16271  df-minusg 16272  df-ghm 16479  df-gim 16521  df-cntz 16569  df-oppg 16595  df-cmn 17014
This theorem is referenced by:  dprdfinv  17269
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