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Theorem gsumval 14775
Description: Expand out the substitutions in df-gsum 13728. (Contributed by Mario Carneiro, 7-Dec-2014.)
Hypotheses
Ref Expression
gsumval.b  |-  B  =  ( Base `  G
)
gsumval.z  |-  .0.  =  ( 0g `  G )
gsumval.p  |-  .+  =  ( +g  `  G )
gsumval.o  |-  O  =  { s  e.  B  |  A. t  e.  B  ( ( s  .+  t )  =  t  /\  ( t  .+  s )  =  t ) }
gsumval.w  |-  ( ph  ->  W  =  ( `' F " ( _V 
\  O ) ) )
gsumval.g  |-  ( ph  ->  G  e.  V )
gsumval.a  |-  ( ph  ->  A  e.  X )
gsumval.f  |-  ( ph  ->  F : A --> B )
Assertion
Ref Expression
gsumval  |-  ( ph  ->  ( G  gsumg  F )  =  if ( ran  F  C_  O ,  .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 ) ) ) ) ) ) )
Distinct variable groups:    t, s, x, B    f, m, n, x, ph    f, F, m, n, x    f, G, m, n, x    .+ , s,
t, x    f, O, m, n, x
Allowed substitution hints:    ph( t, s)    A( x, t, f, m, n, s)    B( f, m, n)    .+ ( f, m, n)    F( t, s)    G( t, s)    O( t, s)    V( x, t, f, m, n, s)    W( x, t, f, m, n, s)    X( x, t, f, m, n, s)    .0. ( x, t, f, m, n, s)

Proof of Theorem gsumval
StepHypRef Expression
1 gsumval.b . 2  |-  B  =  ( Base `  G
)
2 gsumval.z . 2  |-  .0.  =  ( 0g `  G )
3 gsumval.p . 2  |-  .+  =  ( +g  `  G )
4 gsumval.o . 2  |-  O  =  { s  e.  B  |  A. t  e.  B  ( ( s  .+  t )  =  t  /\  ( t  .+  s )  =  t ) }
5 gsumval.w . 2  |-  ( ph  ->  W  =  ( `' F " ( _V 
\  O ) ) )
6 gsumval.g . 2  |-  ( ph  ->  G  e.  V )
7 gsumval.f . . 3  |-  ( ph  ->  F : A --> B )
8 gsumval.a . . 3  |-  ( ph  ->  A  e.  X )
9 fvex 5742 . . . . 5  |-  ( Base `  G )  e.  _V
101, 9eqeltri 2506 . . . 4  |-  B  e. 
_V
1110a1i 11 . . 3  |-  ( ph  ->  B  e.  _V )
12 fex2 5603 . . 3  |-  ( ( F : A --> B  /\  A  e.  X  /\  B  e.  _V )  ->  F  e.  _V )
137, 8, 11, 12syl3anc 1184 . 2  |-  ( ph  ->  F  e.  _V )
14 fdm 5595 . . 3  |-  ( F : A --> B  ->  dom  F  =  A )
157, 14syl 16 . 2  |-  ( ph  ->  dom  F  =  A )
161, 2, 3, 4, 5, 6, 13, 15gsumvalx 14774 1  |-  ( ph  ->  ( G  gsumg  F )  =  if ( ran  F  C_  O ,  .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 ) ) ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   E.wex 1550    = wceq 1652    e. wcel 1725   A.wral 2705   E.wrex 2706   {crab 2709   _Vcvv 2956    \ cdif 3317    C_ wss 3320   ifcif 3739   `'ccnv 4877   dom cdm 4878   ran crn 4879   "cima 4881    o. ccom 4882   iotacio 5416   -->wf 5450   -1-1-onto->wf1o 5453   ` cfv 5454  (class class class)co 6081   1c1 8991   ZZ>=cuz 10488   ...cfz 11043    seq cseq 11323   #chash 11618   Basecbs 13469   +g cplusg 13529   0gc0g 13723    gsumg cgsu 13724
This theorem is referenced by:  gsumress  14777  gsumval1  14779  gsumval2a  14782  gsumval3a  15512
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-recs 6633  df-rdg 6668  df-seq 11324  df-gsum 13728
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