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Theorem gsumzsubmcl 15249
Description: Closure of a group sum in a submonoid. (Contributed by Mario Carneiro, 24-Apr-2016.)
Hypotheses
Ref Expression
gsumzsubmcl.0  |-  .0.  =  ( 0g `  G )
gsumzsubmcl.z  |-  Z  =  (Cntz `  G )
gsumzsubmcl.g  |-  ( ph  ->  G  e.  Mnd )
gsumzsubmcl.a  |-  ( ph  ->  A  e.  V )
gsumzsubmcl.s  |-  ( ph  ->  S  e.  (SubMnd `  G ) )
gsumzsubmcl.f  |-  ( ph  ->  F : A --> S )
gsumzsubmcl.c  |-  ( ph  ->  ran  F  C_  ( Z `  ran  F ) )
gsumzsubmcl.w  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  e. 
Fin )
Assertion
Ref Expression
gsumzsubmcl  |-  ( ph  ->  ( G  gsumg  F )  e.  S
)

Proof of Theorem gsumzsubmcl
StepHypRef Expression
1 gsumzsubmcl.a . . 3  |-  ( ph  ->  A  e.  V )
2 gsumzsubmcl.s . . 3  |-  ( ph  ->  S  e.  (SubMnd `  G ) )
3 gsumzsubmcl.f . . 3  |-  ( ph  ->  F : A --> S )
4 eqid 2316 . . 3  |-  ( Gs  S )  =  ( Gs  S )
51, 2, 3, 4gsumsubm 14504 . 2  |-  ( ph  ->  ( G  gsumg  F )  =  ( ( Gs  S )  gsumg  F ) )
6 eqid 2316 . . . 4  |-  ( Base `  ( Gs  S ) )  =  ( Base `  ( Gs  S ) )
7 eqid 2316 . . . 4  |-  ( 0g
`  ( Gs  S ) )  =  ( 0g
`  ( Gs  S ) )
8 eqid 2316 . . . 4  |-  (Cntz `  ( Gs  S ) )  =  (Cntz `  ( Gs  S
) )
94submmnd 14480 . . . . 5  |-  ( S  e.  (SubMnd `  G
)  ->  ( Gs  S
)  e.  Mnd )
102, 9syl 15 . . . 4  |-  ( ph  ->  ( Gs  S )  e.  Mnd )
114submbas 14481 . . . . . . 7  |-  ( S  e.  (SubMnd `  G
)  ->  S  =  ( Base `  ( Gs  S
) ) )
122, 11syl 15 . . . . . 6  |-  ( ph  ->  S  =  ( Base `  ( Gs  S ) ) )
13 feq3 5414 . . . . . 6  |-  ( S  =  ( Base `  ( Gs  S ) )  -> 
( F : A --> S 
<->  F : A --> ( Base `  ( Gs  S ) ) ) )
1412, 13syl 15 . . . . 5  |-  ( ph  ->  ( F : A --> S 
<->  F : A --> ( Base `  ( Gs  S ) ) ) )
153, 14mpbid 201 . . . 4  |-  ( ph  ->  F : A --> ( Base `  ( Gs  S ) ) )
16 gsumzsubmcl.c . . . . . 6  |-  ( ph  ->  ran  F  C_  ( Z `  ran  F ) )
17 frn 5433 . . . . . . 7  |-  ( F : A --> S  ->  ran  F  C_  S )
183, 17syl 15 . . . . . 6  |-  ( ph  ->  ran  F  C_  S
)
1916, 18ssind 3427 . . . . 5  |-  ( ph  ->  ran  F  C_  (
( Z `  ran  F )  i^i  S ) )
20 gsumzsubmcl.z . . . . . . 7  |-  Z  =  (Cntz `  G )
214, 20, 8resscntz 14856 . . . . . 6  |-  ( ( S  e.  (SubMnd `  G )  /\  ran  F 
C_  S )  -> 
( (Cntz `  ( Gs  S ) ) `  ran  F )  =  ( ( Z `  ran  F )  i^i  S ) )
222, 18, 21syl2anc 642 . . . . 5  |-  ( ph  ->  ( (Cntz `  ( Gs  S ) ) `  ran  F )  =  ( ( Z `  ran  F )  i^i  S ) )
2319, 22sseqtr4d 3249 . . . 4  |-  ( ph  ->  ran  F  C_  (
(Cntz `  ( Gs  S
) ) `  ran  F ) )
24 gsumzsubmcl.0 . . . . . . . . . 10  |-  .0.  =  ( 0g `  G )
254, 24subm0 14482 . . . . . . . . 9  |-  ( S  e.  (SubMnd `  G
)  ->  .0.  =  ( 0g `  ( Gs  S ) ) )
262, 25syl 15 . . . . . . . 8  |-  ( ph  ->  .0.  =  ( 0g
`  ( Gs  S ) ) )
2726sneqd 3687 . . . . . . 7  |-  ( ph  ->  {  .0.  }  =  { ( 0g `  ( Gs  S ) ) } )
2827difeq2d 3328 . . . . . 6  |-  ( ph  ->  ( _V  \  {  .0.  } )  =  ( _V  \  { ( 0g `  ( Gs  S ) ) } ) )
2928imaeq2d 5049 . . . . 5  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  =  ( `' F "
( _V  \  {
( 0g `  ( Gs  S ) ) } ) ) )
30 gsumzsubmcl.w . . . . 5  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  e. 
Fin )
3129, 30eqeltrrd 2391 . . . 4  |-  ( ph  ->  ( `' F "
( _V  \  {
( 0g `  ( Gs  S ) ) } ) )  e.  Fin )
326, 7, 8, 10, 1, 15, 23, 31gsumzcl 15244 . . 3  |-  ( ph  ->  ( ( Gs  S ) 
gsumg  F )  e.  (
Base `  ( Gs  S
) ) )
3332, 12eleqtrrd 2393 . 2  |-  ( ph  ->  ( ( Gs  S ) 
gsumg  F )  e.  S
)
345, 33eqeltrd 2390 1  |-  ( ph  ->  ( G  gsumg  F )  e.  S
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1633    e. wcel 1701   _Vcvv 2822    \ cdif 3183    i^i cin 3185    C_ wss 3186   {csn 3674   `'ccnv 4725   ran crn 4727   "cima 4729   -->wf 5288   ` cfv 5292  (class class class)co 5900   Fincfn 6906   Basecbs 13195   ↾s cress 13196   0gc0g 13449    gsumg cgsu 13450   Mndcmnd 14410  SubMndcsubmnd 14463  Cntzccntz 14840
This theorem is referenced by:  gsumsubmcl  15250  gsumzadd  15253  dprdfadd  15304  dprdfeq0  15306  dprdlub  15310
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-int 3900  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-se 4390  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-isom 5301  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-riota 6346  df-recs 6430  df-rdg 6465  df-1o 6521  df-oadd 6525  df-er 6702  df-en 6907  df-dom 6908  df-sdom 6909  df-fin 6910  df-oi 7270  df-card 7617  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-nn 9792  df-2 9849  df-n0 10013  df-z 10072  df-uz 10278  df-fz 10830  df-fzo 10918  df-seq 11094  df-hash 11385  df-ndx 13198  df-slot 13199  df-base 13200  df-sets 13201  df-ress 13202  df-plusg 13268  df-0g 13453  df-gsum 13454  df-mnd 14416  df-submnd 14465  df-cntz 14842
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