MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  gsumptOLD Structured version   Unicode version

Theorem gsumptOLD 16454
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.) Obsolete version of gsumpt 16453 as of 6-Jun-2019. (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
gsumptOLD.b  |-  B  =  ( Base `  G
)
gsumptOLD.z  |-  .0.  =  ( 0g `  G )
gsumptOLD.g  |-  ( ph  ->  G  e.  Mnd )
gsumptOLD.a  |-  ( ph  ->  A  e.  V )
gsumptOLD.x  |-  ( ph  ->  X  e.  A )
gsumptOLD.f  |-  ( ph  ->  F : A --> B )
gsumptOLD.s  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  C_  { X } )
Assertion
Ref Expression
gsumptOLD  |-  ( ph  ->  ( G  gsumg  F )  =  ( F `  X ) )

Proof of Theorem gsumptOLD
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 gsumptOLD.f . . . 4  |-  ( ph  ->  F : A --> B )
2 gsumptOLD.x . . . . 5  |-  ( ph  ->  X  e.  A )
32snssd 4017 . . . 4  |-  ( ph  ->  { X }  C_  A )
41, 3feqresmpt 5744 . . 3  |-  ( ph  ->  ( F  |`  { X } )  =  ( a  e.  { X }  |->  ( F `  a ) ) )
54oveq2d 6106 . 2  |-  ( ph  ->  ( G  gsumg  ( F  |`  { X } ) )  =  ( G  gsumg  ( a  e.  { X }  |->  ( F `
 a ) ) ) )
6 gsumptOLD.b . . 3  |-  B  =  ( Base `  G
)
7 gsumptOLD.z . . 3  |-  .0.  =  ( 0g `  G )
8 eqid 2442 . . 3  |-  (Cntz `  G )  =  (Cntz `  G )
9 gsumptOLD.g . . 3  |-  ( ph  ->  G  e.  Mnd )
10 gsumptOLD.a . . 3  |-  ( ph  ->  A  e.  V )
111, 2ffvelrnd 5843 . . . . . . . 8  |-  ( ph  ->  ( F `  X
)  e.  B )
12 eqidd 2443 . . . . . . . 8  |-  ( ph  ->  ( ( F `  X ) ( +g  `  G ) ( F `
 X ) )  =  ( ( F `
 X ) ( +g  `  G ) ( F `  X
) ) )
13 eqid 2442 . . . . . . . . . 10  |-  ( +g  `  G )  =  ( +g  `  G )
146, 13, 8elcntzsn 15842 . . . . . . . . 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 16 . . . . . . . 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 913 . . . . . . 7  |-  ( ph  ->  ( F `  X
)  e.  ( (Cntz `  G ) `  {
( F `  X
) } ) )
1716snssd 4017 . . . . . 6  |-  ( ph  ->  { ( F `  X ) }  C_  ( (Cntz `  G ) `  { ( F `  X ) } ) )
18 eqid 2442 . . . . . . 7  |-  (mrCls `  (SubMnd `  G ) )  =  (mrCls `  (SubMnd `  G ) )
19 eqid 2442 . . . . . . 7  |-  ( Gs  ( (mrCls `  (SubMnd `  G
) ) `  {
( F `  X
) } ) )  =  ( Gs  ( (mrCls `  (SubMnd `  G )
) `  { ( F `  X ) } ) )
208, 18, 19cntzspan 16325 . . . . . 6  |-  ( ( G  e.  Mnd  /\  { ( F `  X
) }  C_  (
(Cntz `  G ) `  { ( F `  X ) } ) )  ->  ( Gs  (
(mrCls `  (SubMnd `  G
) ) `  {
( F `  X
) } ) )  e. CMnd )
219, 17, 20syl2anc 661 . . . . 5  |-  ( ph  ->  ( Gs  ( (mrCls `  (SubMnd `  G ) ) `
 { ( F `
 X ) } ) )  e. CMnd )
226submacs 15492 . . . . . . . 8  |-  ( G  e.  Mnd  ->  (SubMnd `  G )  e.  (ACS
`  B ) )
23 acsmre 14589 . . . . . . . 8  |-  ( (SubMnd `  G )  e.  (ACS
`  B )  -> 
(SubMnd `  G )  e.  (Moore `  B )
)
249, 22, 233syl 20 . . . . . . 7  |-  ( ph  ->  (SubMnd `  G )  e.  (Moore `  B )
)
2511snssd 4017 . . . . . . 7  |-  ( ph  ->  { ( F `  X ) }  C_  B )
2618mrccl 14548 . . . . . . 7  |-  ( ( (SubMnd `  G )  e.  (Moore `  B )  /\  { ( F `  X ) }  C_  B )  ->  (
(mrCls `  (SubMnd `  G
) ) `  {
( F `  X
) } )  e.  (SubMnd `  G )
)
2724, 25, 26syl2anc 661 . . . . . 6  |-  ( ph  ->  ( (mrCls `  (SubMnd `  G ) ) `  { ( F `  X ) } )  e.  (SubMnd `  G
) )
2819, 8submcmn2 16322 . . . . . 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 16 . . . . 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 210 . . . 4  |-  ( ph  ->  ( (mrCls `  (SubMnd `  G ) ) `  { ( F `  X ) } ) 
C_  ( (Cntz `  G ) `  (
(mrCls `  (SubMnd `  G
) ) `  {
( F `  X
) } ) ) )
31 ffn 5558 . . . . . . 7  |-  ( F : A --> B  ->  F  Fn  A )
321, 31syl 16 . . . . . 6  |-  ( ph  ->  F  Fn  A )
33 simpr 461 . . . . . . . . . 10  |-  ( ( ( ph  /\  a  e.  A )  /\  a  =  X )  ->  a  =  X )
3433fveq2d 5694 . . . . . . . . 9  |-  ( ( ( ph  /\  a  e.  A )  /\  a  =  X )  ->  ( F `  a )  =  ( F `  X ) )
3524, 18, 25mrcssidd 14562 . . . . . . . . . . 11  |-  ( ph  ->  { ( F `  X ) }  C_  ( (mrCls `  (SubMnd `  G
) ) `  {
( F `  X
) } ) )
36 fvex 5700 . . . . . . . . . . . 12  |-  ( F `
 X )  e. 
_V
3736snss 3998 . . . . . . . . . . 11  |-  ( ( F `  X )  e.  ( (mrCls `  (SubMnd `  G ) ) `
 { ( F `
 X ) } )  <->  { ( F `  X ) }  C_  ( (mrCls `  (SubMnd `  G
) ) `  {
( F `  X
) } ) )
3835, 37sylibr 212 . . . . . . . . . 10  |-  ( ph  ->  ( F `  X
)  e.  ( (mrCls `  (SubMnd `  G )
) `  { ( F `  X ) } ) )
3938ad2antrr 725 . . . . . . . . 9  |-  ( ( ( ph  /\  a  e.  A )  /\  a  =  X )  ->  ( F `  X )  e.  ( (mrCls `  (SubMnd `  G ) ) `  { ( F `  X ) } ) )
4034, 39eqeltrd 2516 . . . . . . . 8  |-  ( ( ( ph  /\  a  e.  A )  /\  a  =  X )  ->  ( F `  a )  e.  ( (mrCls `  (SubMnd `  G ) ) `  { ( F `  X ) } ) )
41 eldifsn 3999 . . . . . . . . . . 11  |-  ( a  e.  ( A  \  { X } )  <->  ( a  e.  A  /\  a  =/=  X ) )
42 gsumptOLD.s . . . . . . . . . . . 12  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  C_  { X } )
431, 42suppssrOLD 5836 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  ( A  \  { X } ) )  -> 
( F `  a
)  =  .0.  )
4441, 43sylan2br 476 . . . . . . . . . 10  |-  ( (
ph  /\  ( a  e.  A  /\  a  =/=  X ) )  -> 
( F `  a
)  =  .0.  )
457subm0cl 15479 . . . . . . . . . . . 12  |-  ( ( (mrCls `  (SubMnd `  G
) ) `  {
( F `  X
) } )  e.  (SubMnd `  G )  ->  .0.  e.  ( (mrCls `  (SubMnd `  G )
) `  { ( F `  X ) } ) )
4627, 45syl 16 . . . . . . . . . . 11  |-  ( ph  ->  .0.  e.  ( (mrCls `  (SubMnd `  G )
) `  { ( F `  X ) } ) )
4746adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  ( a  e.  A  /\  a  =/=  X ) )  ->  .0.  e.  ( (mrCls `  (SubMnd `  G ) ) `
 { ( F `
 X ) } ) )
4844, 47eqeltrd 2516 . . . . . . . . 9  |-  ( (
ph  /\  ( a  e.  A  /\  a  =/=  X ) )  -> 
( F `  a
)  e.  ( (mrCls `  (SubMnd `  G )
) `  { ( F `  X ) } ) )
4948anassrs 648 . . . . . . . 8  |-  ( ( ( ph  /\  a  e.  A )  /\  a  =/=  X )  ->  ( F `  a )  e.  ( (mrCls `  (SubMnd `  G ) ) `  { ( F `  X ) } ) )
5040, 49pm2.61dane 2688 . . . . . . 7  |-  ( (
ph  /\  a  e.  A )  ->  ( F `  a )  e.  ( (mrCls `  (SubMnd `  G ) ) `  { ( F `  X ) } ) )
5150ralrimiva 2798 . . . . . 6  |-  ( ph  ->  A. a  e.  A  ( F `  a )  e.  ( (mrCls `  (SubMnd `  G ) ) `
 { ( F `
 X ) } ) )
52 ffnfv 5868 . . . . . 6  |-  ( F : A --> ( (mrCls `  (SubMnd `  G )
) `  { ( F `  X ) } )  <->  ( F  Fn  A  /\  A. a  e.  A  ( F `  a )  e.  ( (mrCls `  (SubMnd `  G
) ) `  {
( F `  X
) } ) ) )
5332, 51, 52sylanbrc 664 . . . . 5  |-  ( ph  ->  F : A --> ( (mrCls `  (SubMnd `  G )
) `  { ( F `  X ) } ) )
54 frn 5564 . . . . 5  |-  ( F : A --> ( (mrCls `  (SubMnd `  G )
) `  { ( F `  X ) } )  ->  ran  F 
C_  ( (mrCls `  (SubMnd `  G ) ) `
 { ( F `
 X ) } ) )
5553, 54syl 16 . . . 4  |-  ( ph  ->  ran  F  C_  (
(mrCls `  (SubMnd `  G
) ) `  {
( F `  X
) } ) )
568cntzidss 15854 . . . 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 ) )
5730, 55, 56syl2anc 661 . . 3  |-  ( ph  ->  ran  F  C_  (
(Cntz `  G ) `  ran  F ) )
58 snfi 7389 . . . 4  |-  { X }  e.  Fin
59 ssfi 7532 . . . 4  |-  ( ( { X }  e.  Fin  /\  ( `' F " ( _V  \  {  .0.  } ) )  C_  { X } )  -> 
( `' F "
( _V  \  {  .0.  } ) )  e. 
Fin )
6058, 42, 59sylancr 663 . . 3  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  e. 
Fin )
616, 7, 8, 9, 10, 1, 57, 42, 60gsumzresOLD 16391 . 2  |-  ( ph  ->  ( G  gsumg  ( F  |`  { X } ) )  =  ( G  gsumg  F ) )
62 fveq2 5690 . . . 4  |-  ( a  =  X  ->  ( F `  a )  =  ( F `  X ) )
636, 62gsumsn 16448 . . 3  |-  ( ( G  e.  Mnd  /\  X  e.  A  /\  ( F `  X )  e.  B )  -> 
( G  gsumg  ( a  e.  { X }  |->  ( F `
 a ) ) )  =  ( F `
 X ) )
649, 2, 11, 63syl3anc 1218 . 2  |-  ( ph  ->  ( G  gsumg  ( a  e.  { X }  |->  ( F `
 a ) ) )  =  ( F `
 X ) )
655, 61, 643eqtr3d 2482 1  |-  ( ph  ->  ( G  gsumg  F )  =  ( F `  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2605   A.wral 2714   _Vcvv 2971    \ cdif 3324    C_ wss 3327   {csn 3876    e. cmpt 4349   `'ccnv 4838   ran crn 4840    |` cres 4841   "cima 4842    Fn wfn 5412   -->wf 5413   ` cfv 5417  (class class class)co 6090   Fincfn 7309   Basecbs 14173   ↾s cress 14174   +g cplusg 14237   0gc0g 14377    gsumg cgsu 14378  Moorecmre 14519  mrClscmrc 14520  ACScacs 14522   Mndcmnd 15408  SubMndcsubmnd 15462  Cntzccntz 15832  CMndccmn 16276
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-inf2 7846  ax-cnex 9337  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-mulcom 9345  ax-addass 9346  ax-mulass 9347  ax-distr 9348  ax-i2m1 9349  ax-1ne0 9350  ax-1rid 9351  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-pre-ltadd 9357  ax-pre-mulgt0 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-int 4128  df-iun 4172  df-iin 4173  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-se 4679  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  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-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-isom 5426  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-supp 6690  df-recs 6831  df-rdg 6865  df-1o 6919  df-oadd 6923  df-er 7100  df-en 7310  df-dom 7311  df-sdom 7312  df-fin 7313  df-oi 7723  df-card 8108  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423  df-sub 9596  df-neg 9597  df-nn 10322  df-2 10379  df-n0 10579  df-z 10646  df-uz 10861  df-fz 11437  df-fzo 11548  df-seq 11806  df-hash 12103  df-ndx 14176  df-slot 14177  df-base 14178  df-sets 14179  df-ress 14180  df-plusg 14250  df-0g 14379  df-gsum 14380  df-mre 14523  df-mrc 14524  df-acs 14526  df-mnd 15414  df-submnd 15464  df-mulg 15547  df-cntz 15834  df-cmn 16278
This theorem is referenced by:  dprdfidOLD  16513
  Copyright terms: Public domain W3C validator