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Theorem gsumval3a 15205
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 2296 . . 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 14465 . . . . . . 7  |-  ( G  e.  Mnd  ->  { z  e.  B  |  A. y  e.  B  (
( z  .+  y
)  =  y  /\  ( y  .+  z
)  =  y ) }  =  {  .0.  } )
75, 6syl 15 . . . . . 6  |-  ( ph  ->  { z  e.  B  |  A. y  e.  B  ( ( z  .+  y )  =  y  /\  ( y  .+  z )  =  y ) }  =  {  .0.  } )
87difeq2d 3307 . . . . 5  |-  ( ph  ->  ( _V  \  {
z  e.  B  |  A. y  e.  B  ( ( z  .+  y )  =  y  /\  ( y  .+  z )  =  y ) } )  =  ( _V  \  {  .0.  } ) )
98imaeq2d 5028 . . . 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 2347 . . 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 14468 . 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 3219 . . . . . 6  |-  ( ph  ->  ( ran  F  C_  { z  e.  B  |  A. y  e.  B  ( ( z  .+  y )  =  y  /\  ( y  .+  z )  =  y ) }  <->  ran  F  C_  {  .0.  } ) )
17 ffn 5405 . . . . . . . . . . . 12  |-  ( F : A --> B  ->  F  Fn  A )
1813, 17syl 15 . . . . . . . . . . 11  |-  ( ph  ->  F  Fn  A )
1918adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  ran  F  C_  {  .0.  } )  ->  F  Fn  A )
20 simpr 447 . . . . . . . . . 10  |-  ( (
ph  /\  ran  F  C_  {  .0.  } )  ->  ran  F  C_  {  .0.  } )
21 df-f 5275 . . . . . . . . . 10  |-  ( F : A --> {  .0.  }  <-> 
( F  Fn  A  /\  ran  F  C_  {  .0.  } ) )
2219, 20, 21sylanbrc 645 . . . . . . . . 9  |-  ( (
ph  /\  ran  F  C_  {  .0.  } )  ->  F : A --> {  .0.  } )
23 disjdif 3539 . . . . . . . . 9  |-  ( {  .0.  }  i^i  ( _V  \  {  .0.  }
) )  =  (/)
24 fimacnvdisj 5435 . . . . . . . . 9  |-  ( ( F : A --> {  .0.  }  /\  ( {  .0.  }  i^i  ( _V  \  {  .0.  } ) )  =  (/) )  ->  ( `' F " ( _V 
\  {  .0.  }
) )  =  (/) )
2522, 23, 24sylancl 643 . . . . . . . 8  |-  ( (
ph  /\  ran  F  C_  {  .0.  } )  -> 
( `' F "
( _V  \  {  .0.  } ) )  =  (/) )
2610, 25syl5eq 2340 . . . . . . 7  |-  ( (
ph  /\  ran  F  C_  {  .0.  } )  ->  W  =  (/) )
2726ex 423 . . . . . 6  |-  ( ph  ->  ( ran  F  C_  {  .0.  }  ->  W  =  (/) ) )
2816, 27sylbid 206 . . . . 5  |-  ( ph  ->  ( ran  F  C_  { z  e.  B  |  A. y  e.  B  ( ( z  .+  y )  =  y  /\  ( y  .+  z )  =  y ) }  ->  W  =  (/) ) )
2928necon3ad 2495 . . . 4  |-  ( ph  ->  ( W  =/=  (/)  ->  -.  ran  F  C_  { z  e.  B  |  A. y  e.  B  (
( z  .+  y
)  =  y  /\  ( y  .+  z
)  =  y ) } ) )
3015, 29mpd 14 . . 3  |-  ( ph  ->  -.  ran  F  C_  { z  e.  B  |  A. y  e.  B  ( ( z  .+  y )  =  y  /\  ( y  .+  z )  =  y ) } )
31 iffalse 3585 . . 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 15 . 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 3585 . . 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 15 . 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 2332 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 358   E.wex 1531    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557   {crab 2560   _Vcvv 2801    \ cdif 3162    i^i cin 3164    C_ wss 3165   (/)c0 3468   ifcif 3578   {csn 3653   `'ccnv 4704   ran crn 4706   "cima 4708    o. ccom 4709   iotacio 5233    Fn wfn 5266   -->wf 5267   -1-1-onto->wf1o 5270   ` cfv 5271  (class class class)co 5874   Fincfn 6879   1c1 8754   ZZ>=cuz 10246   ...cfz 10798    seq cseq 11062   #chash 11353   Basecbs 13164   +g cplusg 13224   0gc0g 13416    gsumg cgsu 13417   Mndcmnd 14377  Cntzccntz 14807
This theorem is referenced by:  gsumval3  15207
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-recs 6404  df-rdg 6439  df-seq 11063  df-0g 13420  df-gsum 13421  df-mnd 14383
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