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Theorem gsumval3a 15024
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
StepHypRef Expression
1 gsumval3.b . . 3  |-  B  =  ( Base `  G
)
2 gsumval3.0 . . 3  |-  .0.  =  ( 0g `  G )
3 gsumval3.p . . 3  |-  .+  =  ( +g  `  G )
4 eqid 2253 . . 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 14284 . . . . . . 7  |-  ( G  e.  Mnd  ->  { z  e.  B  |  A. y  e.  B  (
( z  .+  y
)  =  y  /\  ( y  .+  z
)  =  y ) }  =  {  .0.  } )
75, 6syl 17 . . . . . 6  |-  ( ph  ->  { z  e.  B  |  A. y  e.  B  ( ( z  .+  y )  =  y  /\  ( y  .+  z )  =  y ) }  =  {  .0.  } )
87difeq2d 3211 . . . . 5  |-  ( ph  ->  ( _V  \  {
z  e.  B  |  A. y  e.  B  ( ( z  .+  y )  =  y  /\  ( y  .+  z )  =  y ) } )  =  ( _V  \  {  .0.  } ) )
98imaeq2d 4919 . . . 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 2304 . . 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 14287 . 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 3127 . . . . . 6  |-  ( ph  ->  ( ran  F  C_  { z  e.  B  |  A. y  e.  B  ( ( z  .+  y )  =  y  /\  ( y  .+  z )  =  y ) }  <->  ran  F  C_  {  .0.  } ) )
17 ffn 5246 . . . . . . . . . . . 12  |-  ( F : A --> B  ->  F  Fn  A )
1813, 17syl 17 . . . . . . . . . . 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 4604 . . . . . . . . . 10  |-  ( F : A --> {  .0.  }  <-> 
( F  Fn  A  /\  ran  F  C_  {  .0.  } ) )
2219, 20, 21sylanbrc 648 . . . . . . . . 9  |-  ( (
ph  /\  ran  F  C_  {  .0.  } )  ->  F : A --> {  .0.  } )
23 disjdif 3432 . . . . . . . . 9  |-  ( {  .0.  }  i^i  ( _V  \  {  .0.  }
) )  =  (/)
24 fimacnvdisj 5276 . . . . . . . . 9  |-  ( ( F : A --> {  .0.  }  /\  ( {  .0.  }  i^i  ( _V  \  {  .0.  } ) )  =  (/) )  ->  ( `' F " ( _V 
\  {  .0.  }
) )  =  (/) )
2522, 23, 24sylancl 646 . . . . . . . 8  |-  ( (
ph  /\  ran  F  C_  {  .0.  } )  -> 
( `' F "
( _V  \  {  .0.  } ) )  =  (/) )
2610, 25syl5eq 2297 . . . . . . 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 2448 . . . 4  |-  ( ph  ->  ( W  =/=  (/)  ->  -.  ran  F  C_  { z  e.  B  |  A. y  e.  B  (
( z  .+  y
)  =  y  /\  ( y  .+  z
)  =  y ) } ) )
3015, 29mpd 16 . . 3  |-  ( ph  ->  -.  ran  F  C_  { z  e.  B  |  A. y  e.  B  ( ( z  .+  y )  =  y  /\  ( y  .+  z )  =  y ) } )
31 iffalse 3477 . . 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 17 . 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 3477 . . 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 17 . 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 2289 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 5    -> wi 6    /\ wa 360   E.wex 1537    = wceq 1619    e. wcel 1621    =/= wne 2412   A.wral 2509   E.wrex 2510   {crab 2512   _Vcvv 2727    \ cdif 3075    i^i cin 3077    C_ wss 3078   (/)c0 3362   ifcif 3470   {csn 3544   `'ccnv 4579   ran crn 4581   "cima 4583    o. ccom 4584    Fn wfn 4587   -->wf 4588   -1-1-onto->wf1o 4591   ` cfv 4592  (class class class)co 5710   iotacio 6141   Fincfn 6749   1c1 8618   ZZ>=cuz 10109   ...cfz 10660    seq cseq 10924   #chash 11215   Basecbs 13022   +g cplusg 13082   0gc0g 13274    gsumg cgsu 13275   Mndcmnd 14196  Cntzccntz 14626
This theorem is referenced by:  gsumval3  15026
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-iota 6143  df-riota 6190  df-recs 6274  df-rdg 6309  df-seq 10925  df-0g 13278  df-gsum 13279  df-mnd 14202
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