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Theorem gsumptOLD 16858
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 16857 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 4156 . . . 4  |-  ( ph  ->  { X }  C_  A )
41, 3feqresmpt 5908 . . 3  |-  ( ph  ->  ( F  |`  { X } )  =  ( a  e.  { X }  |->  ( F `  a ) ) )
54oveq2d 6293 . 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 2441 . . 3  |-  (Cntz `  G )  =  (Cntz `  G )
9 gsumptOLD.g . . 3  |-  ( ph  ->  G  e.  Mnd )
10 gsumptOLD.a . . 3  |-  ( ph  ->  A  e.  V )
111, 2ffvelrnd 6013 . . . . . . . 8  |-  ( ph  ->  ( F `  X
)  e.  B )
12 eqidd 2442 . . . . . . . 8  |-  ( ph  ->  ( ( F `  X ) ( +g  `  G ) ( F `
 X ) )  =  ( ( F `
 X ) ( +g  `  G ) ( F `  X
) ) )
13 eqid 2441 . . . . . . . . . 10  |-  ( +g  `  G )  =  ( +g  `  G )
146, 13, 8elcntzsn 16232 . . . . . . . . 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 920 . . . . . . 7  |-  ( ph  ->  ( F `  X
)  e.  ( (Cntz `  G ) `  {
( F `  X
) } ) )
1716snssd 4156 . . . . . 6  |-  ( ph  ->  { ( F `  X ) }  C_  ( (Cntz `  G ) `  { ( F `  X ) } ) )
18 eqid 2441 . . . . . . 7  |-  (mrCls `  (SubMnd `  G ) )  =  (mrCls `  (SubMnd `  G ) )
19 eqid 2441 . . . . . . 7  |-  ( Gs  ( (mrCls `  (SubMnd `  G
) ) `  {
( F `  X
) } ) )  =  ( Gs  ( (mrCls `  (SubMnd `  G )
) `  { ( F `  X ) } ) )
208, 18, 19cntzspan 16719 . . . . . 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 15865 . . . . . . . 8  |-  ( G  e.  Mnd  ->  (SubMnd `  G )  e.  (ACS
`  B ) )
23 acsmre 14921 . . . . . . . 8  |-  ( (SubMnd `  G )  e.  (ACS
`  B )  -> 
(SubMnd `  G )  e.  (Moore `  B )
)
249, 22, 233syl 20 . . . . . . 7  |-  ( ph  ->  (SubMnd `  G )  e.  (Moore `  B )
)
2511snssd 4156 . . . . . . 7  |-  ( ph  ->  { ( F `  X ) }  C_  B )
2618mrccl 14880 . . . . . . 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 16716 . . . . . 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 5717 . . . . . . 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 5856 . . . . . . . . 9  |-  ( ( ( ph  /\  a  e.  A )  /\  a  =  X )  ->  ( F `  a )  =  ( F `  X ) )
3524, 18, 25mrcssidd 14894 . . . . . . . . . . 11  |-  ( ph  ->  { ( F `  X ) }  C_  ( (mrCls `  (SubMnd `  G
) ) `  {
( F `  X
) } ) )
36 fvex 5862 . . . . . . . . . . . 12  |-  ( F `
 X )  e. 
_V
3736snss 4135 . . . . . . . . . . 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 2529 . . . . . . . 8  |-  ( ( ( ph  /\  a  e.  A )  /\  a  =  X )  ->  ( F `  a )  e.  ( (mrCls `  (SubMnd `  G ) ) `  { ( F `  X ) } ) )
41 eldifsn 4136 . . . . . . . . . . 11  |-  ( a  e.  ( A  \  { X } )  <->  ( a  e.  A  /\  a  =/=  X ) )
42 gsumptOLD.s . . . . . . . . . . . 12  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  C_  { X } )
431, 42suppssrOLD 6002 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  ( A  \  { X } ) )  -> 
( F `  a
)  =  .0.  )
4441, 43sylan2br 476 . . . . . . . . . 10  |-  ( (
ph  /\  ( a  e.  A  /\  a  =/=  X ) )  -> 
( F `  a
)  =  .0.  )
457subm0cl 15852 . . . . . . . . . . . 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 2529 . . . . . . . . 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 2759 . . . . . . 7  |-  ( (
ph  /\  a  e.  A )  ->  ( F `  a )  e.  ( (mrCls `  (SubMnd `  G ) ) `  { ( F `  X ) } ) )
5150ralrimiva 2855 . . . . . 6  |-  ( ph  ->  A. a  e.  A  ( F `  a )  e.  ( (mrCls `  (SubMnd `  G ) ) `
 { ( F `
 X ) } ) )
52 ffnfv 6038 . . . . . 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 5723 . . . . 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 16244 . . . 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 7594 . . . 4  |-  { X }  e.  Fin
59 ssfi 7738 . . . 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 16787 . 2  |-  ( ph  ->  ( G  gsumg  ( F  |`  { X } ) )  =  ( G  gsumg  F ) )
62 fveq2 5852 . . . 4  |-  ( a  =  X  ->  ( F `  a )  =  ( F `  X ) )
636, 62gsumsn 16850 . . 3  |-  ( ( G  e.  Mnd  /\  X  e.  A  /\  ( F `  X )  e.  B )  -> 
( G  gsumg  ( a  e.  { X }  |->  ( F `
 a ) ) )  =  ( F `
 X ) )
649, 2, 11, 63syl3anc 1227 . 2  |-  ( ph  ->  ( G  gsumg  ( a  e.  { X }  |->  ( F `
 a ) ) )  =  ( F `
 X ) )
655, 61, 643eqtr3d 2490 1  |-  ( ph  ->  ( G  gsumg  F )  =  ( F `  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1381    e. wcel 1802    =/= wne 2636   A.wral 2791   _Vcvv 3093    \ cdif 3455    C_ wss 3458   {csn 4010    |-> cmpt 4491   `'ccnv 4984   ran crn 4986    |` cres 4987   "cima 4988    Fn wfn 5569   -->wf 5570   ` cfv 5574  (class class class)co 6277   Fincfn 7514   Basecbs 14504   ↾s cress 14505   +g cplusg 14569   0gc0g 14709    gsumg cgsu 14710  Moorecmre 14851  mrClscmrc 14852  ACScacs 14854   Mndcmnd 15788  SubMndcsubmnd 15834  Cntzccntz 16222  CMndccmn 16667
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-inf2 8056  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-int 4268  df-iun 4313  df-iin 4314  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-se 4825  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-isom 5583  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-1st 6781  df-2nd 6782  df-supp 6900  df-recs 7040  df-rdg 7074  df-1o 7128  df-oadd 7132  df-er 7309  df-en 7515  df-dom 7516  df-sdom 7517  df-fin 7518  df-oi 7933  df-card 8318  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-nn 10538  df-2 10595  df-n0 10797  df-z 10866  df-uz 11086  df-fz 11677  df-fzo 11799  df-seq 12082  df-hash 12380  df-ndx 14507  df-slot 14508  df-base 14509  df-sets 14510  df-ress 14511  df-plusg 14582  df-0g 14711  df-gsum 14712  df-mre 14855  df-mrc 14856  df-acs 14858  df-mgm 15741  df-sgrp 15780  df-mnd 15790  df-submnd 15836  df-mulg 15929  df-cntz 16224  df-cmn 16669
This theorem is referenced by:  dprdfidOLD  16932
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