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Theorem gsumpt 15222
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 3760 . . . 4  |-  ( ph  ->  { X }  C_  A )
41, 3feqresmpt 5576 . . 3  |-  ( ph  ->  ( F  |`  { X } )  =  ( a  e.  { X }  |->  ( F `  a ) ) )
54oveq2d 5874 . 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 2283 . . 3  |-  (Cntz `  G )  =  (Cntz `  G )
9 gsumpt.g . . 3  |-  ( ph  ->  G  e.  Mnd )
10 gsumpt.a . . 3  |-  ( ph  ->  A  e.  V )
11 ffvelrn 5663 . . . . . . . . 9  |-  ( ( F : A --> B  /\  X  e.  A )  ->  ( F `  X
)  e.  B )
121, 2, 11syl2anc 642 . . . . . . . 8  |-  ( ph  ->  ( F `  X
)  e.  B )
13 eqidd 2284 . . . . . . . 8  |-  ( ph  ->  ( ( F `  X ) ( +g  `  G ) ( F `
 X ) )  =  ( ( F `
 X ) ( +g  `  G ) ( F `  X
) ) )
14 eqid 2283 . . . . . . . . . 10  |-  ( +g  `  G )  =  ( +g  `  G )
156, 14, 8elcntzsn 14801 . . . . . . . . 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 ) ) ) ) )
1612, 15syl 15 . . . . . . . 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
) ) ) ) )
1712, 13, 16mpbir2and 888 . . . . . . 7  |-  ( ph  ->  ( F `  X
)  e.  ( (Cntz `  G ) `  {
( F `  X
) } ) )
1817snssd 3760 . . . . . 6  |-  ( ph  ->  { ( F `  X ) }  C_  ( (Cntz `  G ) `  { ( F `  X ) } ) )
19 eqid 2283 . . . . . . 7  |-  (mrCls `  (SubMnd `  G ) )  =  (mrCls `  (SubMnd `  G ) )
20 eqid 2283 . . . . . . 7  |-  ( Gs  ( (mrCls `  (SubMnd `  G
) ) `  {
( F `  X
) } ) )  =  ( Gs  ( (mrCls `  (SubMnd `  G )
) `  { ( F `  X ) } ) )
218, 19, 20cntzspan 15137 . . . . . 6  |-  ( ( G  e.  Mnd  /\  { ( F `  X
) }  C_  (
(Cntz `  G ) `  { ( F `  X ) } ) )  ->  ( Gs  (
(mrCls `  (SubMnd `  G
) ) `  {
( F `  X
) } ) )  e. CMnd )
229, 18, 21syl2anc 642 . . . . 5  |-  ( ph  ->  ( Gs  ( (mrCls `  (SubMnd `  G ) ) `
 { ( F `
 X ) } ) )  e. CMnd )
236submacs 14442 . . . . . . . 8  |-  ( G  e.  Mnd  ->  (SubMnd `  G )  e.  (ACS
`  B ) )
24 acsmre 13554 . . . . . . . 8  |-  ( (SubMnd `  G )  e.  (ACS
`  B )  -> 
(SubMnd `  G )  e.  (Moore `  B )
)
259, 23, 243syl 18 . . . . . . 7  |-  ( ph  ->  (SubMnd `  G )  e.  (Moore `  B )
)
2612snssd 3760 . . . . . . 7  |-  ( ph  ->  { ( F `  X ) }  C_  B )
2719mrccl 13513 . . . . . . 7  |-  ( ( (SubMnd `  G )  e.  (Moore `  B )  /\  { ( F `  X ) }  C_  B )  ->  (
(mrCls `  (SubMnd `  G
) ) `  {
( F `  X
) } )  e.  (SubMnd `  G )
)
2825, 26, 27syl2anc 642 . . . . . 6  |-  ( ph  ->  ( (mrCls `  (SubMnd `  G ) ) `  { ( F `  X ) } )  e.  (SubMnd `  G
) )
2920, 8submcmn2 15135 . . . . . 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 ) } ) ) ) )
3028, 29syl 15 . . . . 5  |-  ( ph  ->  ( ( Gs  ( (mrCls `  (SubMnd `  G )
) `  { ( F `  X ) } ) )  e. CMnd  <->  ( (mrCls `  (SubMnd `  G
) ) `  {
( F `  X
) } )  C_  ( (Cntz `  G ) `  ( (mrCls `  (SubMnd `  G ) ) `  { ( F `  X ) } ) ) ) )
3122, 30mpbid 201 . . . 4  |-  ( ph  ->  ( (mrCls `  (SubMnd `  G ) ) `  { ( F `  X ) } ) 
C_  ( (Cntz `  G ) `  (
(mrCls `  (SubMnd `  G
) ) `  {
( F `  X
) } ) ) )
32 ffn 5389 . . . . . . 7  |-  ( F : A --> B  ->  F  Fn  A )
331, 32syl 15 . . . . . 6  |-  ( ph  ->  F  Fn  A )
34 simpr 447 . . . . . . . . . 10  |-  ( ( ( ph  /\  a  e.  A )  /\  a  =  X )  ->  a  =  X )
3534fveq2d 5529 . . . . . . . . 9  |-  ( ( ( ph  /\  a  e.  A )  /\  a  =  X )  ->  ( F `  a )  =  ( F `  X ) )
3619mrcssid 13519 . . . . . . . . . . . 12  |-  ( ( (SubMnd `  G )  e.  (Moore `  B )  /\  { ( F `  X ) }  C_  B )  ->  { ( F `  X ) }  C_  ( (mrCls `  (SubMnd `  G )
) `  { ( F `  X ) } ) )
3725, 26, 36syl2anc 642 . . . . . . . . . . 11  |-  ( ph  ->  { ( F `  X ) }  C_  ( (mrCls `  (SubMnd `  G
) ) `  {
( F `  X
) } ) )
38 fvex 5539 . . . . . . . . . . . 12  |-  ( F `
 X )  e. 
_V
3938snss 3748 . . . . . . . . . . 11  |-  ( ( F `  X )  e.  ( (mrCls `  (SubMnd `  G ) ) `
 { ( F `
 X ) } )  <->  { ( F `  X ) }  C_  ( (mrCls `  (SubMnd `  G
) ) `  {
( F `  X
) } ) )
4037, 39sylibr 203 . . . . . . . . . 10  |-  ( ph  ->  ( F `  X
)  e.  ( (mrCls `  (SubMnd `  G )
) `  { ( F `  X ) } ) )
4140ad2antrr 706 . . . . . . . . 9  |-  ( ( ( ph  /\  a  e.  A )  /\  a  =  X )  ->  ( F `  X )  e.  ( (mrCls `  (SubMnd `  G ) ) `  { ( F `  X ) } ) )
4235, 41eqeltrd 2357 . . . . . . . 8  |-  ( ( ( ph  /\  a  e.  A )  /\  a  =  X )  ->  ( F `  a )  e.  ( (mrCls `  (SubMnd `  G ) ) `  { ( F `  X ) } ) )
43 eldifsn 3749 . . . . . . . . . . 11  |-  ( a  e.  ( A  \  { X } )  <->  ( a  e.  A  /\  a  =/=  X ) )
44 gsumpt.s . . . . . . . . . . . 12  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  C_  { X } )
451, 44suppssr 5659 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  ( A  \  { X } ) )  -> 
( F `  a
)  =  .0.  )
4643, 45sylan2br 462 . . . . . . . . . 10  |-  ( (
ph  /\  ( a  e.  A  /\  a  =/=  X ) )  -> 
( F `  a
)  =  .0.  )
477subm0cl 14429 . . . . . . . . . . . 12  |-  ( ( (mrCls `  (SubMnd `  G
) ) `  {
( F `  X
) } )  e.  (SubMnd `  G )  ->  .0.  e.  ( (mrCls `  (SubMnd `  G )
) `  { ( F `  X ) } ) )
4828, 47syl 15 . . . . . . . . . . 11  |-  ( ph  ->  .0.  e.  ( (mrCls `  (SubMnd `  G )
) `  { ( F `  X ) } ) )
4948adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  ( a  e.  A  /\  a  =/=  X ) )  ->  .0.  e.  ( (mrCls `  (SubMnd `  G ) ) `
 { ( F `
 X ) } ) )
5046, 49eqeltrd 2357 . . . . . . . . 9  |-  ( (
ph  /\  ( a  e.  A  /\  a  =/=  X ) )  -> 
( F `  a
)  e.  ( (mrCls `  (SubMnd `  G )
) `  { ( F `  X ) } ) )
5150anassrs 629 . . . . . . . 8  |-  ( ( ( ph  /\  a  e.  A )  /\  a  =/=  X )  ->  ( F `  a )  e.  ( (mrCls `  (SubMnd `  G ) ) `  { ( F `  X ) } ) )
5242, 51pm2.61dane 2524 . . . . . . 7  |-  ( (
ph  /\  a  e.  A )  ->  ( F `  a )  e.  ( (mrCls `  (SubMnd `  G ) ) `  { ( F `  X ) } ) )
5352ralrimiva 2626 . . . . . 6  |-  ( ph  ->  A. a  e.  A  ( F `  a )  e.  ( (mrCls `  (SubMnd `  G ) ) `
 { ( F `
 X ) } ) )
54 ffnfv 5685 . . . . . 6  |-  ( F : A --> ( (mrCls `  (SubMnd `  G )
) `  { ( F `  X ) } )  <->  ( F  Fn  A  /\  A. a  e.  A  ( F `  a )  e.  ( (mrCls `  (SubMnd `  G
) ) `  {
( F `  X
) } ) ) )
5533, 53, 54sylanbrc 645 . . . . 5  |-  ( ph  ->  F : A --> ( (mrCls `  (SubMnd `  G )
) `  { ( F `  X ) } ) )
56 frn 5395 . . . . 5  |-  ( F : A --> ( (mrCls `  (SubMnd `  G )
) `  { ( F `  X ) } )  ->  ran  F 
C_  ( (mrCls `  (SubMnd `  G ) ) `
 { ( F `
 X ) } ) )
5755, 56syl 15 . . . 4  |-  ( ph  ->  ran  F  C_  (
(mrCls `  (SubMnd `  G
) ) `  {
( F `  X
) } ) )
588cntzidss 14813 . . . 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 ) )
5931, 57, 58syl2anc 642 . . 3  |-  ( ph  ->  ran  F  C_  (
(Cntz `  G ) `  ran  F ) )
60 snfi 6941 . . . 4  |-  { X }  e.  Fin
61 ssfi 7083 . . . 4  |-  ( ( { X }  e.  Fin  /\  ( `' F " ( _V  \  {  .0.  } ) )  C_  { X } )  -> 
( `' F "
( _V  \  {  .0.  } ) )  e. 
Fin )
6260, 44, 61sylancr 644 . . 3  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  e. 
Fin )
636, 7, 8, 9, 10, 1, 59, 44, 62gsumzres 15194 . 2  |-  ( ph  ->  ( G  gsumg  ( F  |`  { X } ) )  =  ( G  gsumg  F ) )
64 fveq2 5525 . . . 4  |-  ( a  =  X  ->  ( F `  a )  =  ( F `  X ) )
656, 64gsumsn 15220 . . 3  |-  ( ( G  e.  Mnd  /\  X  e.  A  /\  ( F `  X )  e.  B )  -> 
( G  gsumg  ( a  e.  { X }  |->  ( F `
 a ) ) )  =  ( F `
 X ) )
669, 2, 12, 65syl3anc 1182 . 2  |-  ( ph  ->  ( G  gsumg  ( a  e.  { X }  |->  ( F `
 a ) ) )  =  ( F `
 X ) )
675, 63, 663eqtr3d 2323 1  |-  ( ph  ->  ( G  gsumg  F )  =  ( F `  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   _Vcvv 2788    \ cdif 3149    C_ wss 3152   {csn 3640    e. cmpt 4077   `'ccnv 4688   ran crn 4690    |` cres 4691   "cima 4692    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   Fincfn 6863   Basecbs 13148   ↾s cress 13149   +g cplusg 13208   0gc0g 13400    gsumg cgsu 13401  Moorecmre 13484  mrClscmrc 13485  ACScacs 13487   Mndcmnd 14361  SubMndcsubmnd 14414  Cntzccntz 14791  CMndccmn 15089
This theorem is referenced by:  dprdfid  15252  coe1tmmul2  16352  coe1tmmul  16353  evlslem3  19398  evlslem1  19399  coe1mul3  19485  tayl0  19741  jensen  20283  uvcresum  27242  frlmup2  27251  mamulid  27458  mamurid  27459
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-fzo 10871  df-seq 11047  df-hash 11338  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-0g 13404  df-gsum 13405  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-mulg 14492  df-cntz 14793  df-cmn 15091
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