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Theorem gsumpt 17672
Description: Sum of a family that is nonzero at at most one point. (Contributed by Stefan O'Rear, 7-Feb-2015.) (Revised by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 6-Jun-2019.)
Hypotheses
Ref Expression
gsumpt.b  |-  B  =  ( Base `  G
)
gsumpt.z  |-  .0.  =  ( 0g `  G )
gsumpt.g  |-  ( ph  ->  G  e.  Mnd )
gsumpt.a  |-  ( ph  ->  A  e.  V )
gsumpt.x  |-  ( ph  ->  X  e.  A )
gsumpt.f  |-  ( ph  ->  F : A --> B )
gsumpt.s  |-  ( ph  ->  ( F supp  .0.  )  C_ 
{ X } )
Assertion
Ref Expression
gsumpt  |-  ( ph  ->  ( G  gsumg  F )  =  ( F `  X ) )

Proof of Theorem gsumpt
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 gsumpt.f . . . 4  |-  ( ph  ->  F : A --> B )
2 gsumpt.x . . . . 5  |-  ( ph  ->  X  e.  A )
32snssd 4108 . . . 4  |-  ( ph  ->  { X }  C_  A )
41, 3feqresmpt 5933 . . 3  |-  ( ph  ->  ( F  |`  { X } )  =  ( a  e.  { X }  |->  ( F `  a ) ) )
54oveq2d 6324 . 2  |-  ( ph  ->  ( G  gsumg  ( F  |`  { X } ) )  =  ( G  gsumg  ( a  e.  { X }  |->  ( F `
 a ) ) ) )
6 gsumpt.b . . 3  |-  B  =  ( Base `  G
)
7 gsumpt.z . . 3  |-  .0.  =  ( 0g `  G )
8 eqid 2471 . . 3  |-  (Cntz `  G )  =  (Cntz `  G )
9 gsumpt.g . . 3  |-  ( ph  ->  G  e.  Mnd )
10 gsumpt.a . . 3  |-  ( ph  ->  A  e.  V )
111, 2ffvelrnd 6038 . . . . . . . 8  |-  ( ph  ->  ( F `  X
)  e.  B )
12 eqidd 2472 . . . . . . . 8  |-  ( ph  ->  ( ( F `  X ) ( +g  `  G ) ( F `
 X ) )  =  ( ( F `
 X ) ( +g  `  G ) ( F `  X
) ) )
13 eqid 2471 . . . . . . . . . 10  |-  ( +g  `  G )  =  ( +g  `  G )
146, 13, 8elcntzsn 17057 . . . . . . . . 9  |-  ( ( F `  X )  e.  B  ->  (
( F `  X
)  e.  ( (Cntz `  G ) `  {
( F `  X
) } )  <->  ( ( F `  X )  e.  B  /\  (
( F `  X
) ( +g  `  G
) ( F `  X ) )  =  ( ( F `  X ) ( +g  `  G ) ( F `
 X ) ) ) ) )
1511, 14syl 17 . . . . . . . 8  |-  ( ph  ->  ( ( F `  X )  e.  ( (Cntz `  G ) `  { ( F `  X ) } )  <-> 
( ( F `  X )  e.  B  /\  ( ( F `  X ) ( +g  `  G ) ( F `
 X ) )  =  ( ( F `
 X ) ( +g  `  G ) ( F `  X
) ) ) ) )
1611, 12, 15mpbir2and 936 . . . . . . 7  |-  ( ph  ->  ( F `  X
)  e.  ( (Cntz `  G ) `  {
( F `  X
) } ) )
1716snssd 4108 . . . . . 6  |-  ( ph  ->  { ( F `  X ) }  C_  ( (Cntz `  G ) `  { ( F `  X ) } ) )
18 eqid 2471 . . . . . . 7  |-  (mrCls `  (SubMnd `  G ) )  =  (mrCls `  (SubMnd `  G ) )
19 eqid 2471 . . . . . . 7  |-  ( Gs  ( (mrCls `  (SubMnd `  G
) ) `  {
( F `  X
) } ) )  =  ( Gs  ( (mrCls `  (SubMnd `  G )
) `  { ( F `  X ) } ) )
208, 18, 19cntzspan 17560 . . . . . 6  |-  ( ( G  e.  Mnd  /\  { ( F `  X
) }  C_  (
(Cntz `  G ) `  { ( F `  X ) } ) )  ->  ( Gs  (
(mrCls `  (SubMnd `  G
) ) `  {
( F `  X
) } ) )  e. CMnd )
219, 17, 20syl2anc 673 . . . . 5  |-  ( ph  ->  ( Gs  ( (mrCls `  (SubMnd `  G ) ) `
 { ( F `
 X ) } ) )  e. CMnd )
226submacs 16690 . . . . . . . 8  |-  ( G  e.  Mnd  ->  (SubMnd `  G )  e.  (ACS
`  B ) )
23 acsmre 15636 . . . . . . . 8  |-  ( (SubMnd `  G )  e.  (ACS
`  B )  -> 
(SubMnd `  G )  e.  (Moore `  B )
)
249, 22, 233syl 18 . . . . . . 7  |-  ( ph  ->  (SubMnd `  G )  e.  (Moore `  B )
)
2511snssd 4108 . . . . . . 7  |-  ( ph  ->  { ( F `  X ) }  C_  B )
2618mrccl 15595 . . . . . . 7  |-  ( ( (SubMnd `  G )  e.  (Moore `  B )  /\  { ( F `  X ) }  C_  B )  ->  (
(mrCls `  (SubMnd `  G
) ) `  {
( F `  X
) } )  e.  (SubMnd `  G )
)
2724, 25, 26syl2anc 673 . . . . . 6  |-  ( ph  ->  ( (mrCls `  (SubMnd `  G ) ) `  { ( F `  X ) } )  e.  (SubMnd `  G
) )
2819, 8submcmn2 17557 . . . . . 6  |-  ( ( (mrCls `  (SubMnd `  G
) ) `  {
( F `  X
) } )  e.  (SubMnd `  G )  ->  ( ( Gs  ( (mrCls `  (SubMnd `  G )
) `  { ( F `  X ) } ) )  e. CMnd  <->  ( (mrCls `  (SubMnd `  G
) ) `  {
( F `  X
) } )  C_  ( (Cntz `  G ) `  ( (mrCls `  (SubMnd `  G ) ) `  { ( F `  X ) } ) ) ) )
2927, 28syl 17 . . . . 5  |-  ( ph  ->  ( ( Gs  ( (mrCls `  (SubMnd `  G )
) `  { ( F `  X ) } ) )  e. CMnd  <->  ( (mrCls `  (SubMnd `  G
) ) `  {
( F `  X
) } )  C_  ( (Cntz `  G ) `  ( (mrCls `  (SubMnd `  G ) ) `  { ( F `  X ) } ) ) ) )
3021, 29mpbid 215 . . . 4  |-  ( ph  ->  ( (mrCls `  (SubMnd `  G ) ) `  { ( F `  X ) } ) 
C_  ( (Cntz `  G ) `  (
(mrCls `  (SubMnd `  G
) ) `  {
( F `  X
) } ) ) )
31 ffn 5739 . . . . . . 7  |-  ( F : A --> B  ->  F  Fn  A )
321, 31syl 17 . . . . . 6  |-  ( ph  ->  F  Fn  A )
33 simpr 468 . . . . . . . . . 10  |-  ( ( ( ph  /\  a  e.  A )  /\  a  =  X )  ->  a  =  X )
3433fveq2d 5883 . . . . . . . . 9  |-  ( ( ( ph  /\  a  e.  A )  /\  a  =  X )  ->  ( F `  a )  =  ( F `  X ) )
3524, 18, 25mrcssidd 15609 . . . . . . . . . . 11  |-  ( ph  ->  { ( F `  X ) }  C_  ( (mrCls `  (SubMnd `  G
) ) `  {
( F `  X
) } ) )
36 fvex 5889 . . . . . . . . . . . 12  |-  ( F `
 X )  e. 
_V
3736snss 4087 . . . . . . . . . . 11  |-  ( ( F `  X )  e.  ( (mrCls `  (SubMnd `  G ) ) `
 { ( F `
 X ) } )  <->  { ( F `  X ) }  C_  ( (mrCls `  (SubMnd `  G
) ) `  {
( F `  X
) } ) )
3835, 37sylibr 217 . . . . . . . . . 10  |-  ( ph  ->  ( F `  X
)  e.  ( (mrCls `  (SubMnd `  G )
) `  { ( F `  X ) } ) )
3938ad2antrr 740 . . . . . . . . 9  |-  ( ( ( ph  /\  a  e.  A )  /\  a  =  X )  ->  ( F `  X )  e.  ( (mrCls `  (SubMnd `  G ) ) `  { ( F `  X ) } ) )
4034, 39eqeltrd 2549 . . . . . . . 8  |-  ( ( ( ph  /\  a  e.  A )  /\  a  =  X )  ->  ( F `  a )  e.  ( (mrCls `  (SubMnd `  G ) ) `  { ( F `  X ) } ) )
41 eldifsn 4088 . . . . . . . . . . 11  |-  ( a  e.  ( A  \  { X } )  <->  ( a  e.  A  /\  a  =/=  X ) )
42 gsumpt.s . . . . . . . . . . . 12  |-  ( ph  ->  ( F supp  .0.  )  C_ 
{ X } )
43 fvex 5889 . . . . . . . . . . . . . 14  |-  ( 0g
`  G )  e. 
_V
447, 43eqeltri 2545 . . . . . . . . . . . . 13  |-  .0.  e.  _V
4544a1i 11 . . . . . . . . . . . 12  |-  ( ph  ->  .0.  e.  _V )
461, 42, 10, 45suppssr 6965 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  ( A  \  { X } ) )  -> 
( F `  a
)  =  .0.  )
4741, 46sylan2br 484 . . . . . . . . . 10  |-  ( (
ph  /\  ( a  e.  A  /\  a  =/=  X ) )  -> 
( F `  a
)  =  .0.  )
487subm0cl 16677 . . . . . . . . . . . 12  |-  ( ( (mrCls `  (SubMnd `  G
) ) `  {
( F `  X
) } )  e.  (SubMnd `  G )  ->  .0.  e.  ( (mrCls `  (SubMnd `  G )
) `  { ( F `  X ) } ) )
4927, 48syl 17 . . . . . . . . . . 11  |-  ( ph  ->  .0.  e.  ( (mrCls `  (SubMnd `  G )
) `  { ( F `  X ) } ) )
5049adantr 472 . . . . . . . . . 10  |-  ( (
ph  /\  ( a  e.  A  /\  a  =/=  X ) )  ->  .0.  e.  ( (mrCls `  (SubMnd `  G ) ) `
 { ( F `
 X ) } ) )
5147, 50eqeltrd 2549 . . . . . . . . 9  |-  ( (
ph  /\  ( a  e.  A  /\  a  =/=  X ) )  -> 
( F `  a
)  e.  ( (mrCls `  (SubMnd `  G )
) `  { ( F `  X ) } ) )
5251anassrs 660 . . . . . . . 8  |-  ( ( ( ph  /\  a  e.  A )  /\  a  =/=  X )  ->  ( F `  a )  e.  ( (mrCls `  (SubMnd `  G ) ) `  { ( F `  X ) } ) )
5340, 52pm2.61dane 2730 . . . . . . 7  |-  ( (
ph  /\  a  e.  A )  ->  ( F `  a )  e.  ( (mrCls `  (SubMnd `  G ) ) `  { ( F `  X ) } ) )
5453ralrimiva 2809 . . . . . 6  |-  ( ph  ->  A. a  e.  A  ( F `  a )  e.  ( (mrCls `  (SubMnd `  G ) ) `
 { ( F `
 X ) } ) )
55 ffnfv 6064 . . . . . 6  |-  ( F : A --> ( (mrCls `  (SubMnd `  G )
) `  { ( F `  X ) } )  <->  ( F  Fn  A  /\  A. a  e.  A  ( F `  a )  e.  ( (mrCls `  (SubMnd `  G
) ) `  {
( F `  X
) } ) ) )
5632, 54, 55sylanbrc 677 . . . . 5  |-  ( ph  ->  F : A --> ( (mrCls `  (SubMnd `  G )
) `  { ( F `  X ) } ) )
57 frn 5747 . . . . 5  |-  ( F : A --> ( (mrCls `  (SubMnd `  G )
) `  { ( F `  X ) } )  ->  ran  F 
C_  ( (mrCls `  (SubMnd `  G ) ) `
 { ( F `
 X ) } ) )
5856, 57syl 17 . . . 4  |-  ( ph  ->  ran  F  C_  (
(mrCls `  (SubMnd `  G
) ) `  {
( F `  X
) } ) )
598cntzidss 17069 . . . 4  |-  ( ( ( (mrCls `  (SubMnd `  G ) ) `  { ( F `  X ) } ) 
C_  ( (Cntz `  G ) `  (
(mrCls `  (SubMnd `  G
) ) `  {
( F `  X
) } ) )  /\  ran  F  C_  ( (mrCls `  (SubMnd `  G
) ) `  {
( F `  X
) } ) )  ->  ran  F  C_  (
(Cntz `  G ) `  ran  F ) )
6030, 58, 59syl2anc 673 . . 3  |-  ( ph  ->  ran  F  C_  (
(Cntz `  G ) `  ran  F ) )
61 ffun 5742 . . . . 5  |-  ( F : A --> B  ->  Fun  F )
621, 61syl 17 . . . 4  |-  ( ph  ->  Fun  F )
63 snfi 7668 . . . . 5  |-  { X }  e.  Fin
64 ssfi 7810 . . . . 5  |-  ( ( { X }  e.  Fin  /\  ( F supp  .0.  )  C_  { X }
)  ->  ( F supp  .0.  )  e.  Fin )
6563, 42, 64sylancr 676 . . . 4  |-  ( ph  ->  ( F supp  .0.  )  e.  Fin )
66 fex 6155 . . . . . 6  |-  ( ( F : A --> B  /\  A  e.  V )  ->  F  e.  _V )
671, 10, 66syl2anc 673 . . . . 5  |-  ( ph  ->  F  e.  _V )
68 isfsupp 7905 . . . . 5  |-  ( ( F  e.  _V  /\  .0.  e.  _V )  -> 
( F finSupp  .0.  <->  ( Fun  F  /\  ( F supp  .0.  )  e.  Fin )
) )
6967, 45, 68syl2anc 673 . . . 4  |-  ( ph  ->  ( F finSupp  .0.  <->  ( Fun  F  /\  ( F supp  .0.  )  e.  Fin )
) )
7062, 65, 69mpbir2and 936 . . 3  |-  ( ph  ->  F finSupp  .0.  )
716, 7, 8, 9, 10, 1, 60, 42, 70gsumzres 17621 . 2  |-  ( ph  ->  ( G  gsumg  ( F  |`  { X } ) )  =  ( G  gsumg  F ) )
72 fveq2 5879 . . . 4  |-  ( a  =  X  ->  ( F `  a )  =  ( F `  X ) )
736, 72gsumsn 17665 . . 3  |-  ( ( G  e.  Mnd  /\  X  e.  A  /\  ( F `  X )  e.  B )  -> 
( G  gsumg  ( a  e.  { X }  |->  ( F `
 a ) ) )  =  ( F `
 X ) )
749, 2, 11, 73syl3anc 1292 . 2  |-  ( ph  ->  ( G  gsumg  ( a  e.  { X }  |->  ( F `
 a ) ) )  =  ( F `
 X ) )
755, 71, 743eqtr3d 2513 1  |-  ( ph  ->  ( G  gsumg  F )  =  ( F `  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    = wceq 1452    e. wcel 1904    =/= wne 2641   A.wral 2756   _Vcvv 3031    \ cdif 3387    C_ wss 3390   {csn 3959   class class class wbr 4395    |-> cmpt 4454   ran crn 4840    |` cres 4841   Fun wfun 5583    Fn wfn 5584   -->wf 5585   ` cfv 5589  (class class class)co 6308   supp csupp 6933   Fincfn 7587   finSupp cfsupp 7901   Basecbs 15199   ↾s cress 15200   +g cplusg 15268   0gc0g 15416    gsumg cgsu 15417  Moorecmre 15566  mrClscmrc 15567  ACScacs 15569   Mndcmnd 16613  SubMndcsubmnd 16659  Cntzccntz 17047  CMndccmn 17508
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-inf2 8164  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-iin 4272  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-mre 15570  df-mrc 15571  df-acs 15573  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-submnd 16661  df-mulg 16754  df-cntz 17049  df-cmn 17510
This theorem is referenced by:  gsummpt1n0  17675  dprdfid  17728  evlslem3  18814  evlslem1  18815  coe1tmmul2  18946  coe1tmmul  18947  uvcresum  19428  frlmup2  19434  mamulid  19543  mamurid  19544  coe1mul3  23127  tayl0  23396  jensen  23993  linc1  40726
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