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Theorem gsumzsubmcl 15525
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 eqid 2438 . . 3  |-  ( Base `  ( Gs  S ) )  =  ( Base `  ( Gs  S ) )
2 eqid 2438 . . 3  |-  ( 0g
`  ( Gs  S ) )  =  ( 0g
`  ( Gs  S ) )
3 eqid 2438 . . 3  |-  (Cntz `  ( Gs  S ) )  =  (Cntz `  ( Gs  S
) )
4 gsumzsubmcl.s . . . 4  |-  ( ph  ->  S  e.  (SubMnd `  G ) )
5 eqid 2438 . . . . 5  |-  ( Gs  S )  =  ( Gs  S )
65submmnd 14756 . . . 4  |-  ( S  e.  (SubMnd `  G
)  ->  ( Gs  S
)  e.  Mnd )
74, 6syl 16 . . 3  |-  ( ph  ->  ( Gs  S )  e.  Mnd )
8 gsumzsubmcl.a . . 3  |-  ( ph  ->  A  e.  V )
9 gsumzsubmcl.f . . . 4  |-  ( ph  ->  F : A --> S )
105submbas 14757 . . . . . 6  |-  ( S  e.  (SubMnd `  G
)  ->  S  =  ( Base `  ( Gs  S
) ) )
114, 10syl 16 . . . . 5  |-  ( ph  ->  S  =  ( Base `  ( Gs  S ) ) )
12 feq3 5580 . . . . 5  |-  ( S  =  ( Base `  ( Gs  S ) )  -> 
( F : A --> S 
<->  F : A --> ( Base `  ( Gs  S ) ) ) )
1311, 12syl 16 . . . 4  |-  ( ph  ->  ( F : A --> S 
<->  F : A --> ( Base `  ( Gs  S ) ) ) )
149, 13mpbid 203 . . 3  |-  ( ph  ->  F : A --> ( Base `  ( Gs  S ) ) )
15 gsumzsubmcl.c . . . . 5  |-  ( ph  ->  ran  F  C_  ( Z `  ran  F ) )
16 frn 5599 . . . . . 6  |-  ( F : A --> S  ->  ran  F  C_  S )
179, 16syl 16 . . . . 5  |-  ( ph  ->  ran  F  C_  S
)
1815, 17ssind 3567 . . . 4  |-  ( ph  ->  ran  F  C_  (
( Z `  ran  F )  i^i  S ) )
19 gsumzsubmcl.z . . . . . 6  |-  Z  =  (Cntz `  G )
205, 19, 3resscntz 15132 . . . . 5  |-  ( ( S  e.  (SubMnd `  G )  /\  ran  F 
C_  S )  -> 
( (Cntz `  ( Gs  S ) ) `  ran  F )  =  ( ( Z `  ran  F )  i^i  S ) )
214, 17, 20syl2anc 644 . . . 4  |-  ( ph  ->  ( (Cntz `  ( Gs  S ) ) `  ran  F )  =  ( ( Z `  ran  F )  i^i  S ) )
2218, 21sseqtr4d 3387 . . 3  |-  ( ph  ->  ran  F  C_  (
(Cntz `  ( Gs  S
) ) `  ran  F ) )
23 gsumzsubmcl.0 . . . . . . . . 9  |-  .0.  =  ( 0g `  G )
245, 23subm0 14758 . . . . . . . 8  |-  ( S  e.  (SubMnd `  G
)  ->  .0.  =  ( 0g `  ( Gs  S ) ) )
254, 24syl 16 . . . . . . 7  |-  ( ph  ->  .0.  =  ( 0g
`  ( Gs  S ) ) )
2625sneqd 3829 . . . . . 6  |-  ( ph  ->  {  .0.  }  =  { ( 0g `  ( Gs  S ) ) } )
2726difeq2d 3467 . . . . 5  |-  ( ph  ->  ( _V  \  {  .0.  } )  =  ( _V  \  { ( 0g `  ( Gs  S ) ) } ) )
2827imaeq2d 5205 . . . 4  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  =  ( `' F "
( _V  \  {
( 0g `  ( Gs  S ) ) } ) ) )
29 gsumzsubmcl.w . . . 4  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  e. 
Fin )
3028, 29eqeltrrd 2513 . . 3  |-  ( ph  ->  ( `' F "
( _V  \  {
( 0g `  ( Gs  S ) ) } ) )  e.  Fin )
311, 2, 3, 7, 8, 14, 22, 30gsumzcl 15520 . 2  |-  ( ph  ->  ( ( Gs  S ) 
gsumg  F )  e.  (
Base `  ( Gs  S
) ) )
328, 4, 9, 5gsumsubm 14780 . 2  |-  ( ph  ->  ( G  gsumg  F )  =  ( ( Gs  S )  gsumg  F ) )
3331, 32, 113eltr4d 2519 1  |-  ( ph  ->  ( G  gsumg  F )  e.  S
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    = wceq 1653    e. wcel 1726   _Vcvv 2958    \ cdif 3319    i^i cin 3321    C_ wss 3322   {csn 3816   `'ccnv 4879   ran crn 4881   "cima 4883   -->wf 5452   ` cfv 5456  (class class class)co 6083   Fincfn 7111   Basecbs 13471   ↾s cress 13472   0gc0g 13725    gsumg cgsu 13726   Mndcmnd 14686  SubMndcsubmnd 14739  Cntzccntz 15116
This theorem is referenced by:  gsumsubmcl  15526  gsumzadd  15529  dprdfadd  15580  dprdfeq0  15582  dprdlub  15586
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-se 4544  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-isom 5465  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-oadd 6730  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-oi 7481  df-card 7828  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-nn 10003  df-2 10060  df-n0 10224  df-z 10285  df-uz 10491  df-fz 11046  df-fzo 11138  df-seq 11326  df-hash 11621  df-ndx 13474  df-slot 13475  df-base 13476  df-sets 13477  df-ress 13478  df-plusg 13544  df-0g 13729  df-gsum 13730  df-mnd 14692  df-submnd 14741  df-cntz 15118
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