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Theorem sumeq1f 12161
Description: Equality theorem for a sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jul-2013.)
Hypotheses
Ref Expression
sumeq1f.1  |-  F/_ k A
sumeq1f.2  |-  F/_ k B
Assertion
Ref Expression
sumeq1f  |-  ( A  =  B  ->  sum_ k  e.  A  C  =  sum_ k  e.  B  C
)

Proof of Theorem sumeq1f
Dummy variables  f  m  n  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sseq1 3199 . . . . . 6  |-  ( A  =  B  ->  ( A  C_  ( ZZ>= `  m
)  <->  B  C_  ( ZZ>= `  m ) ) )
2 sumeq1f.1 . . . . . . . . . 10  |-  F/_ k A
3 sumeq1f.2 . . . . . . . . . 10  |-  F/_ k B
42, 3nfeq 2426 . . . . . . . . 9  |-  F/ k  A  =  B
5 simpl 443 . . . . . . . . . . 11  |-  ( ( A  =  B  /\  k  e.  ZZ )  ->  A  =  B )
65eleq2d 2350 . . . . . . . . . 10  |-  ( ( A  =  B  /\  k  e.  ZZ )  ->  ( k  e.  A  <->  k  e.  B ) )
76ifbid 3583 . . . . . . . . 9  |-  ( ( A  =  B  /\  k  e.  ZZ )  ->  if ( k  e.  A ,  C , 
0 )  =  if ( k  e.  B ,  C ,  0 ) )
84, 7mpteq2da 4105 . . . . . . . 8  |-  ( A  =  B  ->  (
k  e.  ZZ  |->  if ( k  e.  A ,  C ,  0 ) )  =  ( k  e.  ZZ  |->  if ( k  e.  B ,  C ,  0 ) ) )
98seqeq3d 11054 . . . . . . 7  |-  ( A  =  B  ->  seq  m (  +  , 
( k  e.  ZZ  |->  if ( k  e.  A ,  C ,  0 ) ) )  =  seq  m (  +  , 
( k  e.  ZZ  |->  if ( k  e.  B ,  C ,  0 ) ) ) )
109breq1d 4033 . . . . . 6  |-  ( A  =  B  ->  (  seq  m (  +  , 
( k  e.  ZZ  |->  if ( k  e.  A ,  C ,  0 ) ) )  ~~>  x  <->  seq  m (  +  ,  ( k  e.  ZZ  |->  if ( k  e.  B ,  C ,  0 ) ) )  ~~>  x ) )
111, 10anbi12d 691 . . . . 5  |-  ( A  =  B  ->  (
( A  C_  ( ZZ>=
`  m )  /\  seq  m (  +  , 
( k  e.  ZZ  |->  if ( k  e.  A ,  C ,  0 ) ) )  ~~>  x )  <-> 
( B  C_  ( ZZ>=
`  m )  /\  seq  m (  +  , 
( k  e.  ZZ  |->  if ( k  e.  B ,  C ,  0 ) ) )  ~~>  x ) ) )
1211rexbidv 2564 . . . 4  |-  ( A  =  B  ->  ( E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m )  /\  seq  m (  +  , 
( k  e.  ZZ  |->  if ( k  e.  A ,  C ,  0 ) ) )  ~~>  x )  <->  E. m  e.  ZZ  ( B  C_  ( ZZ>= `  m )  /\  seq  m (  +  , 
( k  e.  ZZ  |->  if ( k  e.  B ,  C ,  0 ) ) )  ~~>  x ) ) )
13 f1oeq3 5465 . . . . . . 7  |-  ( A  =  B  ->  (
f : ( 1 ... m ) -1-1-onto-> A  <->  f :
( 1 ... m
)
-1-1-onto-> B ) )
1413anbi1d 685 . . . . . 6  |-  ( A  =  B  ->  (
( f : ( 1 ... m ) -1-1-onto-> A  /\  x  =  (  seq  1 (  +  ,  ( n  e.  NN  |->  [_ ( f `  n )  /  k ]_ C ) ) `  m ) )  <->  ( f : ( 1 ... m ) -1-1-onto-> B  /\  x  =  (  seq  1 (  +  ,  ( n  e.  NN  |->  [_ (
f `  n )  /  k ]_ C
) ) `  m
) ) ) )
1514exbidv 1612 . . . . 5  |-  ( A  =  B  ->  ( E. f ( f : ( 1 ... m
)
-1-1-onto-> A  /\  x  =  (  seq  1 (  +  ,  ( n  e.  NN  |->  [_ ( f `  n )  /  k ]_ C ) ) `  m ) )  <->  E. f
( f : ( 1 ... m ) -1-1-onto-> B  /\  x  =  (  seq  1 (  +  ,  ( n  e.  NN  |->  [_ ( f `  n )  /  k ]_ C ) ) `  m ) ) ) )
1615rexbidv 2564 . . . 4  |-  ( A  =  B  ->  ( E. m  e.  NN  E. f ( f : ( 1 ... m
)
-1-1-onto-> A  /\  x  =  (  seq  1 (  +  ,  ( n  e.  NN  |->  [_ ( f `  n )  /  k ]_ C ) ) `  m ) )  <->  E. m  e.  NN  E. f ( f : ( 1 ... m ) -1-1-onto-> B  /\  x  =  (  seq  1 (  +  , 
( n  e.  NN  |->  [_ ( f `  n
)  /  k ]_ C ) ) `  m ) ) ) )
1712, 16orbi12d 690 . . 3  |-  ( A  =  B  ->  (
( E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m )  /\  seq  m (  +  ,  ( k  e.  ZZ  |->  if ( k  e.  A ,  C ,  0 ) ) )  ~~>  x )  \/ 
E. m  e.  NN  E. f ( f : ( 1 ... m
)
-1-1-onto-> A  /\  x  =  (  seq  1 (  +  ,  ( n  e.  NN  |->  [_ ( f `  n )  /  k ]_ C ) ) `  m ) ) )  <-> 
( E. m  e.  ZZ  ( B  C_  ( ZZ>= `  m )  /\  seq  m (  +  ,  ( k  e.  ZZ  |->  if ( k  e.  B ,  C ,  0 ) ) )  ~~>  x )  \/ 
E. m  e.  NN  E. f ( f : ( 1 ... m
)
-1-1-onto-> B  /\  x  =  (  seq  1 (  +  ,  ( n  e.  NN  |->  [_ ( f `  n )  /  k ]_ C ) ) `  m ) ) ) ) )
1817iotabidv 5240 . 2  |-  ( A  =  B  ->  ( iota x ( E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m )  /\  seq  m (  +  ,  ( k  e.  ZZ  |->  if ( k  e.  A ,  C ,  0 ) ) )  ~~>  x )  \/ 
E. m  e.  NN  E. f ( f : ( 1 ... m
)
-1-1-onto-> A  /\  x  =  (  seq  1 (  +  ,  ( n  e.  NN  |->  [_ ( f `  n )  /  k ]_ C ) ) `  m ) ) ) )  =  ( iota
x ( E. m  e.  ZZ  ( B  C_  ( ZZ>= `  m )  /\  seq  m (  +  ,  ( k  e.  ZZ  |->  if ( k  e.  B ,  C ,  0 ) ) )  ~~>  x )  \/ 
E. m  e.  NN  E. f ( f : ( 1 ... m
)
-1-1-onto-> B  /\  x  =  (  seq  1 (  +  ,  ( n  e.  NN  |->  [_ ( f `  n )  /  k ]_ C ) ) `  m ) ) ) ) )
19 df-sum 12159 . 2  |-  sum_ k  e.  A  C  =  ( iota x ( E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m
)  /\  seq  m (  +  ,  ( k  e.  ZZ  |->  if ( k  e.  A ,  C ,  0 ) ) )  ~~>  x )  \/  E. m  e.  NN  E. f ( f : ( 1 ... m ) -1-1-onto-> A  /\  x  =  (  seq  1 (  +  , 
( n  e.  NN  |->  [_ ( f `  n
)  /  k ]_ C ) ) `  m ) ) ) )
20 df-sum 12159 . 2  |-  sum_ k  e.  B  C  =  ( iota x ( E. m  e.  ZZ  ( B  C_  ( ZZ>= `  m
)  /\  seq  m (  +  ,  ( k  e.  ZZ  |->  if ( k  e.  B ,  C ,  0 ) ) )  ~~>  x )  \/  E. m  e.  NN  E. f ( f : ( 1 ... m ) -1-1-onto-> B  /\  x  =  (  seq  1 (  +  , 
( n  e.  NN  |->  [_ ( f `  n
)  /  k ]_ C ) ) `  m ) ) ) )
2118, 19, 203eqtr4g 2340 1  |-  ( A  =  B  ->  sum_ k  e.  A  C  =  sum_ k  e.  B  C
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   F/_wnfc 2406   E.wrex 2544   [_csb 3081    C_ wss 3152   ifcif 3565   class class class wbr 4023    e. cmpt 4077   iotacio 5217   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5858   0cc0 8737   1c1 8738    + caddc 8740   NNcn 9746   ZZcz 10024   ZZ>=cuz 10230   ...cfz 10782    seq cseq 11046    ~~> cli 11958   sum_csu 12158
This theorem is referenced by:  sumeq1  12162
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-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
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-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  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-sum 12159
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