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Theorem gsumpt 15533
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.)
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 "
( _V  \  {  .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 3935 . . . 4  |-  ( ph  ->  { X }  C_  A )
41, 3feqresmpt 5771 . . 3  |-  ( ph  ->  ( F  |`  { X } )  =  ( a  e.  { X }  |->  ( F `  a ) ) )
54oveq2d 6088 . 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 2435 . . 3  |-  (Cntz `  G )  =  (Cntz `  G )
9 gsumpt.g . . 3  |-  ( ph  ->  G  e.  Mnd )
10 gsumpt.a . . 3  |-  ( ph  ->  A  e.  V )
111, 2ffvelrnd 5862 . . . . . . . 8  |-  ( ph  ->  ( F `  X
)  e.  B )
12 eqidd 2436 . . . . . . . 8  |-  ( ph  ->  ( ( F `  X ) ( +g  `  G ) ( F `
 X ) )  =  ( ( F `
 X ) ( +g  `  G ) ( F `  X
) ) )
13 eqid 2435 . . . . . . . . . 10  |-  ( +g  `  G )  =  ( +g  `  G )
146, 13, 8elcntzsn 15112 . . . . . . . . 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 889 . . . . . . 7  |-  ( ph  ->  ( F `  X
)  e.  ( (Cntz `  G ) `  {
( F `  X
) } ) )
1716snssd 3935 . . . . . 6  |-  ( ph  ->  { ( F `  X ) }  C_  ( (Cntz `  G ) `  { ( F `  X ) } ) )
18 eqid 2435 . . . . . . 7  |-  (mrCls `  (SubMnd `  G ) )  =  (mrCls `  (SubMnd `  G ) )
19 eqid 2435 . . . . . . 7  |-  ( Gs  ( (mrCls `  (SubMnd `  G
) ) `  {
( F `  X
) } ) )  =  ( Gs  ( (mrCls `  (SubMnd `  G )
) `  { ( F `  X ) } ) )
208, 18, 19cntzspan 15448 . . . . . 6  |-  ( ( G  e.  Mnd  /\  { ( F `  X
) }  C_  (
(Cntz `  G ) `  { ( F `  X ) } ) )  ->  ( Gs  (
(mrCls `  (SubMnd `  G
) ) `  {
( F `  X
) } ) )  e. CMnd )
219, 17, 20syl2anc 643 . . . . 5  |-  ( ph  ->  ( Gs  ( (mrCls `  (SubMnd `  G ) ) `
 { ( F `
 X ) } ) )  e. CMnd )
226submacs 14753 . . . . . . . 8  |-  ( G  e.  Mnd  ->  (SubMnd `  G )  e.  (ACS
`  B ) )
23 acsmre 13865 . . . . . . . 8  |-  ( (SubMnd `  G )  e.  (ACS
`  B )  -> 
(SubMnd `  G )  e.  (Moore `  B )
)
249, 22, 233syl 19 . . . . . . 7  |-  ( ph  ->  (SubMnd `  G )  e.  (Moore `  B )
)
2511snssd 3935 . . . . . . 7  |-  ( ph  ->  { ( F `  X ) }  C_  B )
2618mrccl 13824 . . . . . . 7  |-  ( ( (SubMnd `  G )  e.  (Moore `  B )  /\  { ( F `  X ) }  C_  B )  ->  (
(mrCls `  (SubMnd `  G
) ) `  {
( F `  X
) } )  e.  (SubMnd `  G )
)
2724, 25, 26syl2anc 643 . . . . . 6  |-  ( ph  ->  ( (mrCls `  (SubMnd `  G ) ) `  { ( F `  X ) } )  e.  (SubMnd `  G
) )
2819, 8submcmn2 15446 . . . . . 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 202 . . . 4  |-  ( ph  ->  ( (mrCls `  (SubMnd `  G ) ) `  { ( F `  X ) } ) 
C_  ( (Cntz `  G ) `  (
(mrCls `  (SubMnd `  G
) ) `  {
( F `  X
) } ) ) )
31 ffn 5582 . . . . . . 7  |-  ( F : A --> B  ->  F  Fn  A )
321, 31syl 16 . . . . . 6  |-  ( ph  ->  F  Fn  A )
33 simpr 448 . . . . . . . . . 10  |-  ( ( ( ph  /\  a  e.  A )  /\  a  =  X )  ->  a  =  X )
3433fveq2d 5723 . . . . . . . . 9  |-  ( ( ( ph  /\  a  e.  A )  /\  a  =  X )  ->  ( F `  a )  =  ( F `  X ) )
3524, 18, 25mrcssidd 13838 . . . . . . . . . . 11  |-  ( ph  ->  { ( F `  X ) }  C_  ( (mrCls `  (SubMnd `  G
) ) `  {
( F `  X
) } ) )
36 fvex 5733 . . . . . . . . . . . 12  |-  ( F `
 X )  e. 
_V
3736snss 3918 . . . . . . . . . . 11  |-  ( ( F `  X )  e.  ( (mrCls `  (SubMnd `  G ) ) `
 { ( F `
 X ) } )  <->  { ( F `  X ) }  C_  ( (mrCls `  (SubMnd `  G
) ) `  {
( F `  X
) } ) )
3835, 37sylibr 204 . . . . . . . . . 10  |-  ( ph  ->  ( F `  X
)  e.  ( (mrCls `  (SubMnd `  G )
) `  { ( F `  X ) } ) )
3938ad2antrr 707 . . . . . . . . 9  |-  ( ( ( ph  /\  a  e.  A )  /\  a  =  X )  ->  ( F `  X )  e.  ( (mrCls `  (SubMnd `  G ) ) `  { ( F `  X ) } ) )
4034, 39eqeltrd 2509 . . . . . . . 8  |-  ( ( ( ph  /\  a  e.  A )  /\  a  =  X )  ->  ( F `  a )  e.  ( (mrCls `  (SubMnd `  G ) ) `  { ( F `  X ) } ) )
41 eldifsn 3919 . . . . . . . . . . 11  |-  ( a  e.  ( A  \  { X } )  <->  ( a  e.  A  /\  a  =/=  X ) )
42 gsumpt.s . . . . . . . . . . . 12  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  C_  { X } )
431, 42suppssr 5855 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  ( A  \  { X } ) )  -> 
( F `  a
)  =  .0.  )
4441, 43sylan2br 463 . . . . . . . . . 10  |-  ( (
ph  /\  ( a  e.  A  /\  a  =/=  X ) )  -> 
( F `  a
)  =  .0.  )
457subm0cl 14740 . . . . . . . . . . . 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 452 . . . . . . . . . 10  |-  ( (
ph  /\  ( a  e.  A  /\  a  =/=  X ) )  ->  .0.  e.  ( (mrCls `  (SubMnd `  G ) ) `
 { ( F `
 X ) } ) )
4844, 47eqeltrd 2509 . . . . . . . . 9  |-  ( (
ph  /\  ( a  e.  A  /\  a  =/=  X ) )  -> 
( F `  a
)  e.  ( (mrCls `  (SubMnd `  G )
) `  { ( F `  X ) } ) )
4948anassrs 630 . . . . . . . 8  |-  ( ( ( ph  /\  a  e.  A )  /\  a  =/=  X )  ->  ( F `  a )  e.  ( (mrCls `  (SubMnd `  G ) ) `  { ( F `  X ) } ) )
5040, 49pm2.61dane 2676 . . . . . . 7  |-  ( (
ph  /\  a  e.  A )  ->  ( F `  a )  e.  ( (mrCls `  (SubMnd `  G ) ) `  { ( F `  X ) } ) )
5150ralrimiva 2781 . . . . . 6  |-  ( ph  ->  A. a  e.  A  ( F `  a )  e.  ( (mrCls `  (SubMnd `  G ) ) `
 { ( F `
 X ) } ) )
52 ffnfv 5885 . . . . . 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 646 . . . . 5  |-  ( ph  ->  F : A --> ( (mrCls `  (SubMnd `  G )
) `  { ( F `  X ) } ) )
54 frn 5588 . . . . 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 15124 . . . 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 643 . . 3  |-  ( ph  ->  ran  F  C_  (
(Cntz `  G ) `  ran  F ) )
58 snfi 7178 . . . 4  |-  { X }  e.  Fin
59 ssfi 7320 . . . 4  |-  ( ( { X }  e.  Fin  /\  ( `' F " ( _V  \  {  .0.  } ) )  C_  { X } )  -> 
( `' F "
( _V  \  {  .0.  } ) )  e. 
Fin )
6058, 42, 59sylancr 645 . . 3  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  e. 
Fin )
616, 7, 8, 9, 10, 1, 57, 42, 60gsumzres 15505 . 2  |-  ( ph  ->  ( G  gsumg  ( F  |`  { X } ) )  =  ( G  gsumg  F ) )
62 fveq2 5719 . . . 4  |-  ( a  =  X  ->  ( F `  a )  =  ( F `  X ) )
636, 62gsumsn 15531 . . 3  |-  ( ( G  e.  Mnd  /\  X  e.  A  /\  ( F `  X )  e.  B )  -> 
( G  gsumg  ( a  e.  { X }  |->  ( F `
 a ) ) )  =  ( F `
 X ) )
649, 2, 11, 63syl3anc 1184 . 2  |-  ( ph  ->  ( G  gsumg  ( a  e.  { X }  |->  ( F `
 a ) ) )  =  ( F `
 X ) )
655, 61, 643eqtr3d 2475 1  |-  ( ph  ->  ( G  gsumg  F )  =  ( F `  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   _Vcvv 2948    \ cdif 3309    C_ wss 3312   {csn 3806    e. cmpt 4258   `'ccnv 4868   ran crn 4870    |` cres 4871   "cima 4872    Fn wfn 5440   -->wf 5441   ` cfv 5445  (class class class)co 6072   Fincfn 7100   Basecbs 13457   ↾s cress 13458   +g cplusg 13517   0gc0g 13711    gsumg cgsu 13712  Moorecmre 13795  mrClscmrc 13796  ACScacs 13798   Mndcmnd 14672  SubMndcsubmnd 14725  Cntzccntz 15102  CMndccmn 15400
This theorem is referenced by:  dprdfid  15563  coe1tmmul2  16656  coe1tmmul  16657  evlslem3  19923  evlslem1  19924  coe1mul3  20010  tayl0  20266  jensen  20815  uvcresum  27157  frlmup2  27166  mamulid  27373  mamurid  27374
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692  ax-inf2 7585  ax-cnex 9035  ax-resscn 9036  ax-1cn 9037  ax-icn 9038  ax-addcl 9039  ax-addrcl 9040  ax-mulcl 9041  ax-mulrcl 9042  ax-mulcom 9043  ax-addass 9044  ax-mulass 9045  ax-distr 9046  ax-i2m1 9047  ax-1ne0 9048  ax-1rid 9049  ax-rnegex 9050  ax-rrecex 9051  ax-cnre 9052  ax-pre-lttri 9053  ax-pre-lttrn 9054  ax-pre-ltadd 9055  ax-pre-mulgt0 9056
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4837  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-isom 5454  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-1st 6340  df-2nd 6341  df-riota 6540  df-recs 6624  df-rdg 6659  df-1o 6715  df-oadd 6719  df-er 6896  df-en 7101  df-dom 7102  df-sdom 7103  df-fin 7104  df-oi 7468  df-card 7815  df-pnf 9111  df-mnf 9112  df-xr 9113  df-ltxr 9114  df-le 9115  df-sub 9282  df-neg 9283  df-nn 9990  df-2 10047  df-n0 10211  df-z 10272  df-uz 10478  df-fz 11033  df-fzo 11124  df-seq 11312  df-hash 11607  df-ndx 13460  df-slot 13461  df-base 13462  df-sets 13463  df-ress 13464  df-plusg 13530  df-0g 13715  df-gsum 13716  df-mre 13799  df-mrc 13800  df-acs 13802  df-mnd 14678  df-submnd 14727  df-mulg 14803  df-cntz 15104  df-cmn 15402
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