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

Theorem gsumpt 16459
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 4023 . . . 4  |-  ( ph  ->  { X }  C_  A )
41, 3feqresmpt 5750 . . 3  |-  ( ph  ->  ( F  |`  { X } )  =  ( a  e.  { X }  |->  ( F `  a ) ) )
54oveq2d 6112 . 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 2443 . . 3  |-  (Cntz `  G )  =  (Cntz `  G )
9 gsumpt.g . . 3  |-  ( ph  ->  G  e.  Mnd )
10 gsumpt.a . . 3  |-  ( ph  ->  A  e.  V )
111, 2ffvelrnd 5849 . . . . . . . 8  |-  ( ph  ->  ( F `  X
)  e.  B )
12 eqidd 2444 . . . . . . . 8  |-  ( ph  ->  ( ( F `  X ) ( +g  `  G ) ( F `
 X ) )  =  ( ( F `
 X ) ( +g  `  G ) ( F `  X
) ) )
13 eqid 2443 . . . . . . . . . 10  |-  ( +g  `  G )  =  ( +g  `  G )
146, 13, 8elcntzsn 15848 . . . . . . . . 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 4023 . . . . . 6  |-  ( ph  ->  { ( F `  X ) }  C_  ( (Cntz `  G ) `  { ( F `  X ) } ) )
18 eqid 2443 . . . . . . 7  |-  (mrCls `  (SubMnd `  G ) )  =  (mrCls `  (SubMnd `  G ) )
19 eqid 2443 . . . . . . 7  |-  ( Gs  ( (mrCls `  (SubMnd `  G
) ) `  {
( F `  X
) } ) )  =  ( Gs  ( (mrCls `  (SubMnd `  G )
) `  { ( F `  X ) } ) )
208, 18, 19cntzspan 16331 . . . . . 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 15498 . . . . . . . 8  |-  ( G  e.  Mnd  ->  (SubMnd `  G )  e.  (ACS
`  B ) )
23 acsmre 14595 . . . . . . . 8  |-  ( (SubMnd `  G )  e.  (ACS
`  B )  -> 
(SubMnd `  G )  e.  (Moore `  B )
)
249, 22, 233syl 20 . . . . . . 7  |-  ( ph  ->  (SubMnd `  G )  e.  (Moore `  B )
)
2511snssd 4023 . . . . . . 7  |-  ( ph  ->  { ( F `  X ) }  C_  B )
2618mrccl 14554 . . . . . . 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 16328 . . . . . 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 5564 . . . . . . 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 5700 . . . . . . . . 9  |-  ( ( ( ph  /\  a  e.  A )  /\  a  =  X )  ->  ( F `  a )  =  ( F `  X ) )
3524, 18, 25mrcssidd 14568 . . . . . . . . . . 11  |-  ( ph  ->  { ( F `  X ) }  C_  ( (mrCls `  (SubMnd `  G
) ) `  {
( F `  X
) } ) )
36 fvex 5706 . . . . . . . . . . . 12  |-  ( F `
 X )  e. 
_V
3736snss 4004 . . . . . . . . . . 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 2517 . . . . . . . 8  |-  ( ( ( ph  /\  a  e.  A )  /\  a  =  X )  ->  ( F `  a )  e.  ( (mrCls `  (SubMnd `  G ) ) `  { ( F `  X ) } ) )
41 eldifsn 4005 . . . . . . . . . . 11  |-  ( a  e.  ( A  \  { X } )  <->  ( a  e.  A  /\  a  =/=  X ) )
42 gsumpt.s . . . . . . . . . . . 12  |-  ( ph  ->  ( F supp  .0.  )  C_ 
{ X } )
43 fvex 5706 . . . . . . . . . . . . . 14  |-  ( 0g
`  G )  e. 
_V
447, 43eqeltri 2513 . . . . . . . . . . . . 13  |-  .0.  e.  _V
4544a1i 11 . . . . . . . . . . . 12  |-  ( ph  ->  .0.  e.  _V )
461, 42, 10, 45suppssr 6725 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  ( A  \  { X } ) )  -> 
( F `  a
)  =  .0.  )
4741, 46sylan2br 476 . . . . . . . . . 10  |-  ( (
ph  /\  ( a  e.  A  /\  a  =/=  X ) )  -> 
( F `  a
)  =  .0.  )
487subm0cl 15485 . . . . . . . . . . . 12  |-  ( ( (mrCls `  (SubMnd `  G
) ) `  {
( F `  X
) } )  e.  (SubMnd `  G )  ->  .0.  e.  ( (mrCls `  (SubMnd `  G )
) `  { ( F `  X ) } ) )
4927, 48syl 16 . . . . . . . . . . 11  |-  ( ph  ->  .0.  e.  ( (mrCls `  (SubMnd `  G )
) `  { ( F `  X ) } ) )
5049adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  ( a  e.  A  /\  a  =/=  X ) )  ->  .0.  e.  ( (mrCls `  (SubMnd `  G ) ) `
 { ( F `
 X ) } ) )
5147, 50eqeltrd 2517 . . . . . . . . 9  |-  ( (
ph  /\  ( a  e.  A  /\  a  =/=  X ) )  -> 
( F `  a
)  e.  ( (mrCls `  (SubMnd `  G )
) `  { ( F `  X ) } ) )
5251anassrs 648 . . . . . . . 8  |-  ( ( ( ph  /\  a  e.  A )  /\  a  =/=  X )  ->  ( F `  a )  e.  ( (mrCls `  (SubMnd `  G ) ) `  { ( F `  X ) } ) )
5340, 52pm2.61dane 2694 . . . . . . 7  |-  ( (
ph  /\  a  e.  A )  ->  ( F `  a )  e.  ( (mrCls `  (SubMnd `  G ) ) `  { ( F `  X ) } ) )
5453ralrimiva 2804 . . . . . 6  |-  ( ph  ->  A. a  e.  A  ( F `  a )  e.  ( (mrCls `  (SubMnd `  G ) ) `
 { ( F `
 X ) } ) )
55 ffnfv 5874 . . . . . 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 664 . . . . 5  |-  ( ph  ->  F : A --> ( (mrCls `  (SubMnd `  G )
) `  { ( F `  X ) } ) )
57 frn 5570 . . . . 5  |-  ( F : A --> ( (mrCls `  (SubMnd `  G )
) `  { ( F `  X ) } )  ->  ran  F 
C_  ( (mrCls `  (SubMnd `  G ) ) `
 { ( F `
 X ) } ) )
5856, 57syl 16 . . . 4  |-  ( ph  ->  ran  F  C_  (
(mrCls `  (SubMnd `  G
) ) `  {
( F `  X
) } ) )
598cntzidss 15860 . . . 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 661 . . 3  |-  ( ph  ->  ran  F  C_  (
(Cntz `  G ) `  ran  F ) )
61 ffun 5566 . . . . 5  |-  ( F : A --> B  ->  Fun  F )
621, 61syl 16 . . . 4  |-  ( ph  ->  Fun  F )
63 snfi 7395 . . . . 5  |-  { X }  e.  Fin
64 ssfi 7538 . . . . 5  |-  ( ( { X }  e.  Fin  /\  ( F supp  .0.  )  C_  { X }
)  ->  ( F supp  .0.  )  e.  Fin )
6563, 42, 64sylancr 663 . . . 4  |-  ( ph  ->  ( F supp  .0.  )  e.  Fin )
66 fex 5955 . . . . . 6  |-  ( ( F : A --> B  /\  A  e.  V )  ->  F  e.  _V )
671, 10, 66syl2anc 661 . . . . 5  |-  ( ph  ->  F  e.  _V )
68 isfsupp 7629 . . . . 5  |-  ( ( F  e.  _V  /\  .0.  e.  _V )  -> 
( F finSupp  .0.  <->  ( Fun  F  /\  ( F supp  .0.  )  e.  Fin )
) )
6967, 45, 68syl2anc 661 . . . 4  |-  ( ph  ->  ( F finSupp  .0.  <->  ( Fun  F  /\  ( F supp  .0.  )  e.  Fin )
) )
7062, 65, 69mpbir2and 913 . . 3  |-  ( ph  ->  F finSupp  .0.  )
716, 7, 8, 9, 10, 1, 60, 42, 70gsumzres 16393 . 2  |-  ( ph  ->  ( G  gsumg  ( F  |`  { X } ) )  =  ( G  gsumg  F ) )
72 fveq2 5696 . . . 4  |-  ( a  =  X  ->  ( F `  a )  =  ( F `  X ) )
736, 72gsumsn 16454 . . 3  |-  ( ( G  e.  Mnd  /\  X  e.  A  /\  ( F `  X )  e.  B )  -> 
( G  gsumg  ( a  e.  { X }  |->  ( F `
 a ) ) )  =  ( F `
 X ) )
749, 2, 11, 73syl3anc 1218 . 2  |-  ( ph  ->  ( G  gsumg  ( a  e.  { X }  |->  ( F `
 a ) ) )  =  ( F `
 X ) )
755, 71, 743eqtr3d 2483 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 2611   A.wral 2720   _Vcvv 2977    \ cdif 3330    C_ wss 3333   {csn 3882   class class class wbr 4297    e. cmpt 4355   ran crn 4846    |` cres 4847   Fun wfun 5417    Fn wfn 5418   -->wf 5419   ` cfv 5423  (class class class)co 6096   supp csupp 6695   Fincfn 7315   finSupp cfsupp 7625   Basecbs 14179   ↾s cress 14180   +g cplusg 14243   0gc0g 14383    gsumg cgsu 14384  Moorecmre 14525  mrClscmrc 14526  ACScacs 14528   Mndcmnd 15414  SubMndcsubmnd 15468  Cntzccntz 15838  CMndccmn 16282
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 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-inf2 7852  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
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 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-iin 4179  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-se 4685  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-isom 5432  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-supp 6696  df-recs 6837  df-rdg 6871  df-1o 6925  df-oadd 6929  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-fsupp 7626  df-oi 7729  df-card 8114  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-2 10385  df-n0 10585  df-z 10652  df-uz 10867  df-fz 11443  df-fzo 11554  df-seq 11812  df-hash 12109  df-ndx 14182  df-slot 14183  df-base 14184  df-sets 14185  df-ress 14186  df-plusg 14256  df-0g 14385  df-gsum 14386  df-mre 14529  df-mrc 14530  df-acs 14532  df-mnd 15420  df-submnd 15470  df-mulg 15553  df-cntz 15840  df-cmn 16284
This theorem is referenced by:  gsummpt1n0  16461  dprdfid  16512  evlslem3  17605  evlslem1  17606  coe1tmmul2  17734  coe1tmmul  17735  uvcresum  18223  frlmup2  18232  mamulid  18309  mamurid  18310  coe1mul3  21576  tayl0  21832  jensen  22387  linc1  30964
  Copyright terms: Public domain W3C validator