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Theorem gsumval3a 15514
Description: Value of the group sum operation over an index set with finite support. (Contributed by Mario Carneiro, 7-Dec-2014.)
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  =/=  (/) )
gsumval3a.w  |-  W  =  ( `' F "
( _V  \  {  .0.  } ) )
gsumval3a.i  |-  ( ph  ->  -.  A  e.  ran  ... )
Assertion
Ref Expression
gsumval3a  |-  ( 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 gsumval3a
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 2438 . . 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 14774 . . . . . . 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 3467 . . . . 5  |-  ( ph  ->  ( _V  \  {
z  e.  B  |  A. y  e.  B  ( ( z  .+  y )  =  y  /\  ( y  .+  z )  =  y ) } )  =  ( _V  \  {  .0.  } ) )
98imaeq2d 5205 . . . 4  |-  ( ph  ->  ( `' F "
( _V  \  {
z  e.  B  |  A. y  e.  B  ( ( z  .+  y )  =  y  /\  ( y  .+  z )  =  y ) } ) )  =  ( `' F " ( _V  \  {  .0.  } ) ) )
10 gsumval3a.w . . . 4  |-  W  =  ( `' F "
( _V  \  {  .0.  } ) )
119, 10syl6reqr 2489 . . 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 14777 . 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 3378 . . . . . 6  |-  ( ph  ->  ( ran  F  C_  { z  e.  B  |  A. y  e.  B  ( ( z  .+  y )  =  y  /\  ( y  .+  z )  =  y ) }  <->  ran  F  C_  {  .0.  } ) )
17 ffn 5593 . . . . . . . . . . . 12  |-  ( F : A --> B  ->  F  Fn  A )
1813, 17syl 16 . . . . . . . . . . 11  |-  ( ph  ->  F  Fn  A )
1918adantr 453 . . . . . . . . . 10  |-  ( (
ph  /\  ran  F  C_  {  .0.  } )  ->  F  Fn  A )
20 simpr 449 . . . . . . . . . 10  |-  ( (
ph  /\  ran  F  C_  {  .0.  } )  ->  ran  F  C_  {  .0.  } )
21 df-f 5460 . . . . . . . . . 10  |-  ( F : A --> {  .0.  }  <-> 
( F  Fn  A  /\  ran  F  C_  {  .0.  } ) )
2219, 20, 21sylanbrc 647 . . . . . . . . 9  |-  ( (
ph  /\  ran  F  C_  {  .0.  } )  ->  F : A --> {  .0.  } )
23 disjdif 3702 . . . . . . . . 9  |-  ( {  .0.  }  i^i  ( _V  \  {  .0.  }
) )  =  (/)
24 fimacnvdisj 5623 . . . . . . . . 9  |-  ( ( F : A --> {  .0.  }  /\  ( {  .0.  }  i^i  ( _V  \  {  .0.  } ) )  =  (/) )  ->  ( `' F " ( _V 
\  {  .0.  }
) )  =  (/) )
2522, 23, 24sylancl 645 . . . . . . . 8  |-  ( (
ph  /\  ran  F  C_  {  .0.  } )  -> 
( `' F "
( _V  \  {  .0.  } ) )  =  (/) )
2610, 25syl5eq 2482 . . . . . . 7  |-  ( (
ph  /\  ran  F  C_  {  .0.  } )  ->  W  =  (/) )
2726ex 425 . . . . . 6  |-  ( ph  ->  ( ran  F  C_  {  .0.  }  ->  W  =  (/) ) )
2816, 27sylbid 208 . . . . 5  |-  ( ph  ->  ( ran  F  C_  { z  e.  B  |  A. y  e.  B  ( ( z  .+  y )  =  y  /\  ( y  .+  z )  =  y ) }  ->  W  =  (/) ) )
2928necon3ad 2639 . . . 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 3748 . . 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 gsumval3a.i . . 3  |-  ( ph  ->  -.  A  e.  ran  ... )
34 iffalse 3748 . . 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 2474 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 set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 360   E.wex 1551    = wceq 1653    e. wcel 1726    =/= wne 2601   A.wral 2707   E.wrex 2708   {crab 2711   _Vcvv 2958    \ cdif 3319    i^i cin 3321    C_ wss 3322   (/)c0 3630   ifcif 3741   {csn 3816   `'ccnv 4879   ran crn 4881   "cima 4883    o. ccom 4884   iotacio 5418    Fn wfn 5451   -->wf 5452   -1-1-onto->wf1o 5455   ` cfv 5456  (class class class)co 6083   Fincfn 7111   1c1 8993   ZZ>=cuz 10490   ...cfz 11045    seq cseq 11325   #chash 11620   Basecbs 13471   +g cplusg 13531   0gc0g 13725    gsumg cgsu 13726   Mndcmnd 14686  Cntzccntz 15116
This theorem is referenced by:  gsumval3  15516
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-riota 6551  df-recs 6635  df-rdg 6670  df-seq 11326  df-0g 13729  df-gsum 13730  df-mnd 14692
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