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Theorem gsumzinv 16948
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 2443 . . 3  |-  (oppg `  G
)  =  (oppg `  G
)
5 gsumzinv.g . . . 4  |-  ( ph  ->  G  e.  Grp )
6 grpmnd 16041 . . . 4  |-  ( G  e.  Grp  ->  G  e.  Mnd )
75, 6syl 16 . . 3  |-  ( ph  ->  G  e.  Mnd )
8 gsumzinv.a . . 3  |-  ( ph  ->  A  e.  V )
9 gsumzinv.i . . . . . 6  |-  I  =  ( invg `  G )
101, 9grpinvf 16073 . . . . 5  |-  ( G  e.  Grp  ->  I : B --> B )
115, 10syl 16 . . . 4  |-  ( ph  ->  I : B --> B )
12 gsumzinv.f . . . 4  |-  ( ph  ->  F : A --> B )
13 fco 5731 . . . 4  |-  ( ( I : B --> B  /\  F : A --> B )  ->  ( I  o.  F ) : A --> B )
1411, 12, 13syl2anc 661 . . 3  |-  ( ph  ->  ( I  o.  F
) : A --> B )
154, 9invoppggim 16374 . . . . . 6  |-  ( G  e.  Grp  ->  I  e.  ( G GrpIso  (oppg `  G
) ) )
16 gimghm 16291 . . . . . 6  |-  ( I  e.  ( G GrpIso  (oppg `  G
) )  ->  I  e.  ( G  GrpHom  (oppg `  G
) ) )
17 ghmmhm 16256 . . . . . 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 2443 . . . . . 6  |-  (Cntz `  (oppg `  G ) )  =  (Cntz `  (oppg
`  G ) )
213, 20cntzmhm2 16356 . . . . 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 661 . . . 4  |-  ( ph  ->  ( I " ran  F )  C_  ( (Cntz `  (oppg
`  G ) ) `
 ( I " ran  F ) ) )
23 rnco2 5504 . . . 4  |-  ran  (
I  o.  F )  =  ( I " ran  F )
2423fveq2i 5859 . . . . 5  |-  ( Z `
 ran  ( I  o.  F ) )  =  ( Z `  (
I " ran  F
) )
254, 3oppgcntz 16378 . . . . 5  |-  ( Z `
 ( I " ran  F ) )  =  ( (Cntz `  (oppg `  G
) ) `  (
I " ran  F
) )
2624, 25eqtri 2472 . . . 4  |-  ( Z `
 ran  ( I  o.  F ) )  =  ( (Cntz `  (oppg `  G
) ) `  (
I " ran  F
) )
2722, 23, 263sstr4g 3530 . . 3  |-  ( ph  ->  ran  ( I  o.  F )  C_  ( Z `  ran  ( I  o.  F ) ) )
28 fvex 5866 . . . . . 6  |-  ( 0g
`  G )  e. 
_V
292, 28eqeltri 2527 . . . . 5  |-  .0.  e.  _V
3029a1i 11 . . . 4  |-  ( ph  ->  .0.  e.  _V )
31 fvex 5866 . . . . . 6  |-  ( Base `  G )  e.  _V
321, 31eqeltri 2527 . . . . 5  |-  B  e. 
_V
3332a1i 11 . . . 4  |-  ( ph  ->  B  e.  _V )
34 gsumzinv.n . . . 4  |-  ( ph  ->  F finSupp  .0.  )
352, 9grpinvid 16080 . . . . 5  |-  ( G  e.  Grp  ->  (
I `  .0.  )  =  .0.  )
365, 35syl 16 . . . 4  |-  ( ph  ->  ( I `  .0.  )  =  .0.  )
3730, 12, 11, 8, 33, 34, 36fsuppco2 7864 . . 3  |-  ( ph  ->  ( I  o.  F
) finSupp  .0.  )
381, 2, 3, 4, 7, 8, 14, 27, 37gsumzoppg 16946 . 2  |-  ( ph  ->  ( (oppg
`  G )  gsumg  ( I  o.  F ) )  =  ( G  gsumg  ( I  o.  F ) ) )
394oppgmnd 16368 . . . 4  |-  ( G  e.  Mnd  ->  (oppg `  G
)  e.  Mnd )
407, 39syl 16 . . 3  |-  ( ph  ->  (oppg
`  G )  e. 
Mnd )
411, 3, 7, 40, 8, 18, 12, 19, 2, 34gsumzmhm 16936 . 2  |-  ( ph  ->  ( (oppg
`  G )  gsumg  ( I  o.  F ) )  =  ( I `  ( G  gsumg  F ) ) )
4238, 41eqtr3d 2486 1  |-  ( ph  ->  ( G  gsumg  ( I  o.  F
) )  =  ( I `  ( G 
gsumg  F ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1383    e. wcel 1804   _Vcvv 3095    C_ wss 3461   class class class wbr 4437   ran crn 4990   "cima 4992    o. ccom 4993   -->wf 5574   ` cfv 5578  (class class class)co 6281   finSupp cfsupp 7831   Basecbs 14614   0gc0g 14819    gsumg cgsu 14820   Mndcmnd 15898   MndHom cmhm 15943   Grpcgrp 16032   invgcminusg 16033    GrpHom cghm 16243   GrpIso cgim 16284  Cntzccntz 16332  oppgcoppg 16359
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-iin 4318  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-supp 6904  df-tpos 6957  df-recs 7044  df-rdg 7078  df-1o 7132  df-oadd 7136  df-er 7313  df-map 7424  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-fsupp 7832  df-oi 7938  df-card 8323  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10544  df-2 10601  df-n0 10803  df-z 10872  df-uz 11093  df-fz 11684  df-fzo 11807  df-seq 12090  df-hash 12388  df-ndx 14617  df-slot 14618  df-base 14619  df-sets 14620  df-ress 14621  df-plusg 14692  df-0g 14821  df-gsum 14822  df-mre 14965  df-mrc 14966  df-acs 14968  df-mgm 15851  df-sgrp 15890  df-mnd 15900  df-mhm 15945  df-submnd 15946  df-grp 16036  df-minusg 16037  df-ghm 16244  df-gim 16286  df-cntz 16334  df-oppg 16360  df-cmn 16779
This theorem is referenced by:  dprdfinv  17038
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