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Theorem gsumzsubmcl 17629
Description: Closure of a group sum in a submonoid. (Contributed by Mario Carneiro, 24-Apr-2016.) (Revised by AV, 3-Jun-2019.)
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 finSupp  .0.  )
Assertion
Ref Expression
gsumzsubmcl  |-  ( ph  ->  ( G  gsumg  F )  e.  S
)

Proof of Theorem gsumzsubmcl
StepHypRef Expression
1 eqid 2471 . . 3  |-  ( Base `  ( Gs  S ) )  =  ( Base `  ( Gs  S ) )
2 eqid 2471 . . 3  |-  ( 0g
`  ( Gs  S ) )  =  ( 0g
`  ( Gs  S ) )
3 eqid 2471 . . 3  |-  (Cntz `  ( Gs  S ) )  =  (Cntz `  ( Gs  S
) )
4 gsumzsubmcl.s . . . 4  |-  ( ph  ->  S  e.  (SubMnd `  G ) )
5 eqid 2471 . . . . 5  |-  ( Gs  S )  =  ( Gs  S )
65submmnd 16679 . . . 4  |-  ( S  e.  (SubMnd `  G
)  ->  ( Gs  S
)  e.  Mnd )
74, 6syl 17 . . 3  |-  ( ph  ->  ( Gs  S )  e.  Mnd )
8 gsumzsubmcl.a . . 3  |-  ( ph  ->  A  e.  V )
9 gsumzsubmcl.f . . . 4  |-  ( ph  ->  F : A --> S )
105submbas 16680 . . . . . 6  |-  ( S  e.  (SubMnd `  G
)  ->  S  =  ( Base `  ( Gs  S
) ) )
114, 10syl 17 . . . . 5  |-  ( ph  ->  S  =  ( Base `  ( Gs  S ) ) )
1211feq3d 5726 . . . 4  |-  ( ph  ->  ( F : A --> S 
<->  F : A --> ( Base `  ( Gs  S ) ) ) )
139, 12mpbid 215 . . 3  |-  ( ph  ->  F : A --> ( Base `  ( Gs  S ) ) )
14 gsumzsubmcl.c . . . . 5  |-  ( ph  ->  ran  F  C_  ( Z `  ran  F ) )
15 frn 5747 . . . . . 6  |-  ( F : A --> S  ->  ran  F  C_  S )
169, 15syl 17 . . . . 5  |-  ( ph  ->  ran  F  C_  S
)
1714, 16ssind 3647 . . . 4  |-  ( ph  ->  ran  F  C_  (
( Z `  ran  F )  i^i  S ) )
18 gsumzsubmcl.z . . . . . 6  |-  Z  =  (Cntz `  G )
195, 18, 3resscntz 17063 . . . . 5  |-  ( ( S  e.  (SubMnd `  G )  /\  ran  F 
C_  S )  -> 
( (Cntz `  ( Gs  S ) ) `  ran  F )  =  ( ( Z `  ran  F )  i^i  S ) )
204, 16, 19syl2anc 673 . . . 4  |-  ( ph  ->  ( (Cntz `  ( Gs  S ) ) `  ran  F )  =  ( ( Z `  ran  F )  i^i  S ) )
2117, 20sseqtr4d 3455 . . 3  |-  ( ph  ->  ran  F  C_  (
(Cntz `  ( Gs  S
) ) `  ran  F ) )
22 gsumzsubmcl.w . . . 4  |-  ( ph  ->  F finSupp  .0.  )
23 gsumzsubmcl.0 . . . . . 6  |-  .0.  =  ( 0g `  G )
245, 23subm0 16681 . . . . 5  |-  ( S  e.  (SubMnd `  G
)  ->  .0.  =  ( 0g `  ( Gs  S ) ) )
254, 24syl 17 . . . 4  |-  ( ph  ->  .0.  =  ( 0g
`  ( Gs  S ) ) )
2622, 25breqtrd 4420 . . 3  |-  ( ph  ->  F finSupp  ( 0g `  ( Gs  S ) ) )
271, 2, 3, 7, 8, 13, 21, 26gsumzcl 17623 . 2  |-  ( ph  ->  ( ( Gs  S ) 
gsumg  F )  e.  (
Base `  ( Gs  S
) ) )
288, 4, 9, 5gsumsubm 16698 . 2  |-  ( ph  ->  ( G  gsumg  F )  =  ( ( Gs  S )  gsumg  F ) )
2927, 28, 113eltr4d 2564 1  |-  ( ph  ->  ( G  gsumg  F )  e.  S
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1452    e. wcel 1904    i^i cin 3389    C_ wss 3390   class class class wbr 4395   ran crn 4840   -->wf 5585   ` cfv 5589  (class class class)co 6308   finSupp cfsupp 7901   Basecbs 15199   ↾s cress 15200   0gc0g 15416    gsumg cgsu 15417   Mndcmnd 16613  SubMndcsubmnd 16659  Cntzccntz 17047
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-oi 8043  df-card 8391  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-fzo 11943  df-seq 12252  df-hash 12554  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-0g 15418  df-gsum 15419  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-submnd 16661  df-cntz 17049
This theorem is referenced by:  gsumsubmcl  17630  gsumzadd  17633  dprdfadd  17731  dprdfeq0  17733  dprdlub  17737
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