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Theorem gsumzinv 16565
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 2454 . . 3  |-  (oppg `  G
)  =  (oppg `  G
)
5 gsumzinv.g . . . 4  |-  ( ph  ->  G  e.  Grp )
6 grpmnd 15670 . . . 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 15702 . . . . 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 5677 . . . 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 15995 . . . . . 6  |-  ( G  e.  Grp  ->  I  e.  ( G GrpIso  (oppg `  G
) ) )
16 gimghm 15912 . . . . . 6  |-  ( I  e.  ( G GrpIso  (oppg `  G
) )  ->  I  e.  ( G  GrpHom  (oppg `  G
) ) )
17 ghmmhm 15877 . . . . . 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 2454 . . . . . 6  |-  (Cntz `  (oppg `  G ) )  =  (Cntz `  (oppg
`  G ) )
213, 20cntzmhm2 15977 . . . . 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 5454 . . . 4  |-  ran  (
I  o.  F )  =  ( I " ran  F )
2423fveq2i 5803 . . . . 5  |-  ( Z `
 ran  ( I  o.  F ) )  =  ( Z `  (
I " ran  F
) )
254, 3oppgcntz 15999 . . . . 5  |-  ( Z `
 ( I " ran  F ) )  =  ( (Cntz `  (oppg `  G
) ) `  (
I " ran  F
) )
2624, 25eqtri 2483 . . . 4  |-  ( Z `
 ran  ( I  o.  F ) )  =  ( (Cntz `  (oppg `  G
) ) `  (
I " ran  F
) )
2722, 23, 263sstr4g 3506 . . 3  |-  ( ph  ->  ran  ( I  o.  F )  C_  ( Z `  ran  ( I  o.  F ) ) )
28 fvex 5810 . . . . . 6  |-  ( 0g
`  G )  e. 
_V
292, 28eqeltri 2538 . . . . 5  |-  .0.  e.  _V
3029a1i 11 . . . 4  |-  ( ph  ->  .0.  e.  _V )
31 fvex 5810 . . . . . 6  |-  ( Base `  G )  e.  _V
321, 31eqeltri 2538 . . . . 5  |-  B  e. 
_V
3332a1i 11 . . . 4  |-  ( ph  ->  B  e.  _V )
34 gsumzinv.n . . . 4  |-  ( ph  ->  F finSupp  .0.  )
352, 9grpinvid 15709 . . . . 5  |-  ( G  e.  Grp  ->  (
I `  .0.  )  =  .0.  )
365, 35syl 16 . . . 4  |-  ( ph  ->  ( I `  .0.  )  =  .0.  )
3730, 12, 11, 8, 33, 34, 36fsuppco2 7764 . . 3  |-  ( ph  ->  ( I  o.  F
) finSupp  .0.  )
381, 2, 3, 4, 7, 8, 14, 27, 37gsumzoppg 16563 . 2  |-  ( ph  ->  ( (oppg
`  G )  gsumg  ( I  o.  F ) )  =  ( G  gsumg  ( I  o.  F ) ) )
394oppgmnd 15989 . . . 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 16553 . 2  |-  ( ph  ->  ( (oppg
`  G )  gsumg  ( I  o.  F ) )  =  ( I `  ( G  gsumg  F ) ) )
4238, 41eqtr3d 2497 1  |-  ( ph  ->  ( G  gsumg  ( I  o.  F
) )  =  ( I `  ( G 
gsumg  F ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758   _Vcvv 3078    C_ wss 3437   class class class wbr 4401   ran crn 4950   "cima 4952    o. ccom 4953   -->wf 5523   ` cfv 5527  (class class class)co 6201   finSupp cfsupp 7732   Basecbs 14293   0gc0g 14498    gsumg cgsu 14499   Mndcmnd 15529   Grpcgrp 15530   invgcminusg 15531   MndHom cmhm 15582    GrpHom cghm 15864   GrpIso cgim 15905  Cntzccntz 15953  oppgcoppg 15980
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-int 4238  df-iun 4282  df-iin 4283  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-se 4789  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-isom 5536  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-om 6588  df-1st 6688  df-2nd 6689  df-supp 6802  df-tpos 6856  df-recs 6943  df-rdg 6977  df-1o 7031  df-oadd 7035  df-er 7212  df-map 7327  df-en 7422  df-dom 7423  df-sdom 7424  df-fin 7425  df-fsupp 7733  df-oi 7836  df-card 8221  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-nn 10435  df-2 10492  df-n0 10692  df-z 10759  df-uz 10974  df-fz 11556  df-fzo 11667  df-seq 11925  df-hash 12222  df-ndx 14296  df-slot 14297  df-base 14298  df-sets 14299  df-ress 14300  df-plusg 14371  df-0g 14500  df-gsum 14501  df-mre 14644  df-mrc 14645  df-acs 14647  df-mnd 15535  df-mhm 15584  df-submnd 15585  df-grp 15665  df-minusg 15666  df-ghm 15865  df-gim 15907  df-cntz 15955  df-oppg 15981  df-cmn 16401
This theorem is referenced by:  dprdfinv  16632
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