MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  gsumval3aOLD Structured version   Unicode version

Theorem gsumval3aOLD 16502
Description: Value of the group sum operation over an index set with finite support. (Contributed by Mario Carneiro, 7-Dec-2014.) Obsolete version of gsumval3a 16501 as of 29-May-2019. (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
gsumval3.b  |-  B  =  ( Base `  G
)
gsumval3.0  |-  .0.  =  ( 0g `  G )
gsumval3.p  |-  .+  =  ( +g  `  G )
gsumval3.z  |-  Z  =  (Cntz `  G )
gsumval3.g  |-  ( ph  ->  G  e.  Mnd )
gsumval3.a  |-  ( ph  ->  A  e.  V )
gsumval3.f  |-  ( ph  ->  F : A --> B )
gsumval3.c  |-  ( ph  ->  ran  F  C_  ( Z `  ran  F ) )
gsumval3a.t  |-  ( ph  ->  W  e.  Fin )
gsumval3a.n  |-  ( ph  ->  W  =/=  (/) )
gsumval3aOLD.w  |-  W  =  ( `' F "
( _V  \  {  .0.  } ) )
gsumval3aOLD.i  |-  ( ph  ->  -.  A  e.  ran  ... )
Assertion
Ref Expression
gsumval3aOLD  |-  ( ph  ->  ( G  gsumg  F )  =  ( iota x E. f
( f : ( 1 ... ( # `  W ) ) -1-1-onto-> W  /\  x  =  (  seq 1 (  .+  , 
( F  o.  f
) ) `  ( # `
 W ) ) ) ) )
Distinct variable groups:    x, f,  .+    A, f, x    ph, f, x    x,  .0.    f, G, x   
x, V    B, f, x    f, F, x    f, W, x
Allowed substitution hints:    V( f)    .0. ( f)    Z( x, f)

Proof of Theorem gsumval3aOLD
Dummy variables  m  n  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 gsumval3.b . . 3  |-  B  =  ( Base `  G
)
2 gsumval3.0 . . 3  |-  .0.  =  ( 0g `  G )
3 gsumval3.p . . 3  |-  .+  =  ( +g  `  G )
4 eqid 2454 . . 3  |-  { z  e.  B  |  A. y  e.  B  (
( z  .+  y
)  =  y  /\  ( y  .+  z
)  =  y ) }  =  { z  e.  B  |  A. y  e.  B  (
( z  .+  y
)  =  y  /\  ( y  .+  z
)  =  y ) }
5 gsumval3.g . . . . . . 7  |-  ( ph  ->  G  e.  Mnd )
61, 2, 3, 4gsumvallem2 15621 . . . . . . 7  |-  ( G  e.  Mnd  ->  { z  e.  B  |  A. y  e.  B  (
( z  .+  y
)  =  y  /\  ( y  .+  z
)  =  y ) }  =  {  .0.  } )
75, 6syl 16 . . . . . 6  |-  ( ph  ->  { z  e.  B  |  A. y  e.  B  ( ( z  .+  y )  =  y  /\  ( y  .+  z )  =  y ) }  =  {  .0.  } )
87difeq2d 3583 . . . . 5  |-  ( ph  ->  ( _V  \  {
z  e.  B  |  A. y  e.  B  ( ( z  .+  y )  =  y  /\  ( y  .+  z )  =  y ) } )  =  ( _V  \  {  .0.  } ) )
98imaeq2d 5278 . . . 4  |-  ( ph  ->  ( `' F "
( _V  \  {
z  e.  B  |  A. y  e.  B  ( ( z  .+  y )  =  y  /\  ( y  .+  z )  =  y ) } ) )  =  ( `' F " ( _V  \  {  .0.  } ) ) )
10 gsumval3aOLD.w . . . 4  |-  W  =  ( `' F "
( _V  \  {  .0.  } ) )
119, 10syl6reqr 2514 . . 3  |-  ( ph  ->  W  =  ( `' F " ( _V 
\  { z  e.  B  |  A. y  e.  B  ( (
z  .+  y )  =  y  /\  (
y  .+  z )  =  y ) } ) ) )
12 gsumval3.a . . 3  |-  ( ph  ->  A  e.  V )
13 gsumval3.f . . 3  |-  ( ph  ->  F : A --> B )
141, 2, 3, 4, 11, 5, 12, 13gsumval 15623 . 2  |-  ( ph  ->  ( G  gsumg  F )  =  if ( ran  F  C_  { z  e.  B  |  A. y  e.  B  ( ( z  .+  y )  =  y  /\  ( y  .+  z )  =  y ) } ,  .0.  ,  if ( A  e. 
ran  ... ,  ( iota
x E. m E. n  e.  ( ZZ>= `  m ) ( A  =  ( m ... n )  /\  x  =  (  seq m
(  .+  ,  F
) `  n )
) ) ,  ( iota x E. f
( f : ( 1 ... ( # `  W ) ) -1-1-onto-> W  /\  x  =  (  seq 1 (  .+  , 
( F  o.  f
) ) `  ( # `
 W ) ) ) ) ) ) )
15 gsumval3a.n . . . 4  |-  ( ph  ->  W  =/=  (/) )
167sseq2d 3493 . . . . . 6  |-  ( ph  ->  ( ran  F  C_  { z  e.  B  |  A. y  e.  B  ( ( z  .+  y )  =  y  /\  ( y  .+  z )  =  y ) }  <->  ran  F  C_  {  .0.  } ) )
17 ffn 5668 . . . . . . . . . . . 12  |-  ( F : A --> B  ->  F  Fn  A )
1813, 17syl 16 . . . . . . . . . . 11  |-  ( ph  ->  F  Fn  A )
1918adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  ran  F  C_  {  .0.  } )  ->  F  Fn  A )
20 simpr 461 . . . . . . . . . 10  |-  ( (
ph  /\  ran  F  C_  {  .0.  } )  ->  ran  F  C_  {  .0.  } )
21 df-f 5531 . . . . . . . . . 10  |-  ( F : A --> {  .0.  }  <-> 
( F  Fn  A  /\  ran  F  C_  {  .0.  } ) )
2219, 20, 21sylanbrc 664 . . . . . . . . 9  |-  ( (
ph  /\  ran  F  C_  {  .0.  } )  ->  F : A --> {  .0.  } )
23 disjdif 3860 . . . . . . . . 9  |-  ( {  .0.  }  i^i  ( _V  \  {  .0.  }
) )  =  (/)
24 fimacnvdisj 5698 . . . . . . . . 9  |-  ( ( F : A --> {  .0.  }  /\  ( {  .0.  }  i^i  ( _V  \  {  .0.  } ) )  =  (/) )  ->  ( `' F " ( _V 
\  {  .0.  }
) )  =  (/) )
2522, 23, 24sylancl 662 . . . . . . . 8  |-  ( (
ph  /\  ran  F  C_  {  .0.  } )  -> 
( `' F "
( _V  \  {  .0.  } ) )  =  (/) )
2610, 25syl5eq 2507 . . . . . . 7  |-  ( (
ph  /\  ran  F  C_  {  .0.  } )  ->  W  =  (/) )
2726ex 434 . . . . . 6  |-  ( ph  ->  ( ran  F  C_  {  .0.  }  ->  W  =  (/) ) )
2816, 27sylbid 215 . . . . 5  |-  ( ph  ->  ( ran  F  C_  { z  e.  B  |  A. y  e.  B  ( ( z  .+  y )  =  y  /\  ( y  .+  z )  =  y ) }  ->  W  =  (/) ) )
2928necon3ad 2662 . . . 4  |-  ( ph  ->  ( W  =/=  (/)  ->  -.  ran  F  C_  { z  e.  B  |  A. y  e.  B  (
( z  .+  y
)  =  y  /\  ( y  .+  z
)  =  y ) } ) )
3015, 29mpd 15 . . 3  |-  ( ph  ->  -.  ran  F  C_  { z  e.  B  |  A. y  e.  B  ( ( z  .+  y )  =  y  /\  ( y  .+  z )  =  y ) } )
31 iffalse 3908 . . 3  |-  ( -. 
ran  F  C_  { z  e.  B  |  A. y  e.  B  (
( z  .+  y
)  =  y  /\  ( y  .+  z
)  =  y ) }  ->  if ( ran  F  C_  { z  e.  B  |  A. y  e.  B  (
( z  .+  y
)  =  y  /\  ( y  .+  z
)  =  y ) } ,  .0.  ,  if ( A  e.  ran  ...
,  ( iota x E. m E. n  e.  ( ZZ>= `  m )
( A  =  ( m ... n )  /\  x  =  (  seq m (  .+  ,  F ) `  n
) ) ) ,  ( iota x E. f ( f : ( 1 ... ( # `
 W ) ) -1-1-onto-> W  /\  x  =  (  seq 1 (  .+  ,  ( F  o.  f ) ) `  ( # `  W ) ) ) ) ) )  =  if ( A  e.  ran  ... ,  ( iota x E. m E. n  e.  (
ZZ>= `  m ) ( A  =  ( m ... n )  /\  x  =  (  seq m (  .+  ,  F ) `  n
) ) ) ,  ( iota x E. f ( f : ( 1 ... ( # `
 W ) ) -1-1-onto-> W  /\  x  =  (  seq 1 (  .+  ,  ( F  o.  f ) ) `  ( # `  W ) ) ) ) ) )
3230, 31syl 16 . 2  |-  ( ph  ->  if ( ran  F  C_ 
{ z  e.  B  |  A. y  e.  B  ( ( z  .+  y )  =  y  /\  ( y  .+  z )  =  y ) } ,  .0.  ,  if ( A  e. 
ran  ... ,  ( iota
x E. m E. n  e.  ( ZZ>= `  m ) ( A  =  ( m ... n )  /\  x  =  (  seq m
(  .+  ,  F
) `  n )
) ) ,  ( iota x E. f
( f : ( 1 ... ( # `  W ) ) -1-1-onto-> W  /\  x  =  (  seq 1 (  .+  , 
( F  o.  f
) ) `  ( # `
 W ) ) ) ) ) )  =  if ( A  e.  ran  ... , 
( iota x E. m E. n  e.  ( ZZ>=
`  m ) ( A  =  ( m ... n )  /\  x  =  (  seq m (  .+  ,  F ) `  n
) ) ) ,  ( iota x E. f ( f : ( 1 ... ( # `
 W ) ) -1-1-onto-> W  /\  x  =  (  seq 1 (  .+  ,  ( F  o.  f ) ) `  ( # `  W ) ) ) ) ) )
33 gsumval3aOLD.i . . 3  |-  ( ph  ->  -.  A  e.  ran  ... )
34 iffalse 3908 . . 3  |-  ( -.  A  e.  ran  ...  ->  if ( A  e. 
ran  ... ,  ( iota
x E. m E. n  e.  ( ZZ>= `  m ) ( A  =  ( m ... n )  /\  x  =  (  seq m
(  .+  ,  F
) `  n )
) ) ,  ( iota x E. f
( f : ( 1 ... ( # `  W ) ) -1-1-onto-> W  /\  x  =  (  seq 1 (  .+  , 
( F  o.  f
) ) `  ( # `
 W ) ) ) ) )  =  ( iota x E. f ( f : ( 1 ... ( # `
 W ) ) -1-1-onto-> W  /\  x  =  (  seq 1 (  .+  ,  ( F  o.  f ) ) `  ( # `  W ) ) ) ) )
3533, 34syl 16 . 2  |-  ( ph  ->  if ( A  e. 
ran  ... ,  ( iota
x E. m E. n  e.  ( ZZ>= `  m ) ( A  =  ( m ... n )  /\  x  =  (  seq m
(  .+  ,  F
) `  n )
) ) ,  ( iota x E. f
( f : ( 1 ... ( # `  W ) ) -1-1-onto-> W  /\  x  =  (  seq 1 (  .+  , 
( F  o.  f
) ) `  ( # `
 W ) ) ) ) )  =  ( iota x E. f ( f : ( 1 ... ( # `
 W ) ) -1-1-onto-> W  /\  x  =  (  seq 1 (  .+  ,  ( F  o.  f ) ) `  ( # `  W ) ) ) ) )
3614, 32, 353eqtrd 2499 1  |-  ( ph  ->  ( G  gsumg  F )  =  ( iota x E. f
( f : ( 1 ... ( # `  W ) ) -1-1-onto-> W  /\  x  =  (  seq 1 (  .+  , 
( F  o.  f
) ) `  ( # `
 W ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1370   E.wex 1587    e. wcel 1758    =/= wne 2648   A.wral 2799   E.wrex 2800   {crab 2803   _Vcvv 3078    \ cdif 3434    i^i cin 3436    C_ wss 3437   (/)c0 3746   ifcif 3900   {csn 3986   `'ccnv 4948   ran crn 4950   "cima 4952    o. ccom 4953   iotacio 5488    Fn wfn 5522   -->wf 5523   -1-1-onto->wf1o 5526   ` cfv 5527  (class class class)co 6201   Fincfn 7421   1c1 9395   ZZ>=cuz 10973   ...cfz 11555    seqcseq 11924   #chash 12221   Basecbs 14293   +g cplusg 14358   0gc0g 14498    gsumg cgsu 14499   Mndcmnd 15529  Cntzccntz 15953
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-recs 6943  df-rdg 6977  df-seq 11925  df-0g 14500  df-gsum 14501  df-mnd 15535
This theorem is referenced by:  gsumval3OLD  16504
  Copyright terms: Public domain W3C validator