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Theorem gsumval 14452
Description: Expand out the substitutions in df-gsum 13405. (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 5539 . . . . 5  |-  ( Base `  G )  e.  _V
101, 9eqeltri 2353 . . . 4  |-  B  e. 
_V
1110a1i 10 . . 3  |-  ( ph  ->  B  e.  _V )
12 fex2 5401 . . 3  |-  ( ( F : A --> B  /\  A  e.  X  /\  B  e.  _V )  ->  F  e.  _V )
137, 8, 11, 12syl3anc 1182 . 2  |-  ( ph  ->  F  e.  _V )
14 fdm 5393 . . 3  |-  ( F : A --> B  ->  dom  F  =  A )
157, 14syl 15 . 2  |-  ( ph  ->  dom  F  =  A )
161, 2, 3, 4, 5, 6, 13, 15gsumvalx 14451 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 358   E.wex 1528    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   {crab 2547   _Vcvv 2788    \ cdif 3149    C_ wss 3152   ifcif 3565   `'ccnv 4688   dom cdm 4689   ran crn 4690   "cima 4692    o. ccom 4693   iotacio 5217   -->wf 5251   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5858   1c1 8738   ZZ>=cuz 10230   ...cfz 10782    seq cseq 11046   #chash 11337   Basecbs 13148   +g cplusg 13208   0gc0g 13400    gsumg cgsu 13401
This theorem is referenced by:  gsumress  14454  gsumval1  14456  gsumval2a  14459  gsumval3a  15189
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-recs 6388  df-rdg 6423  df-seq 11047  df-gsum 13405
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