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Theorem cbvsum 12489
Description: Change bound variable in a sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 23-Aug-2014.)
Hypotheses
Ref Expression
cbvsum.1  |-  ( j  =  k  ->  B  =  C )
cbvsum.2  |-  F/_ k A
cbvsum.3  |-  F/_ j A
cbvsum.4  |-  F/_ k B
cbvsum.5  |-  F/_ j C
Assertion
Ref Expression
cbvsum  |-  sum_ j  e.  A  B  =  sum_ k  e.  A  C
Distinct variable group:    j, k
Allowed substitution hints:    A( j, k)    B( j, k)    C( j, k)

Proof of Theorem cbvsum
Dummy variables  f  m  n  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cbvsum.2 . . . . . . . . . . . . 13  |-  F/_ k A
21nfcri 2566 . . . . . . . . . . . 12  |-  F/ k  j  e.  A
3 cbvsum.4 . . . . . . . . . . . 12  |-  F/_ k B
4 nfcv 2572 . . . . . . . . . . . 12  |-  F/_ k
0
52, 3, 4nfif 3763 . . . . . . . . . . 11  |-  F/_ k if ( j  e.  A ,  B ,  0 )
6 cbvsum.3 . . . . . . . . . . . . 13  |-  F/_ j A
76nfcri 2566 . . . . . . . . . . . 12  |-  F/ j  k  e.  A
8 cbvsum.5 . . . . . . . . . . . 12  |-  F/_ j C
9 nfcv 2572 . . . . . . . . . . . 12  |-  F/_ j
0
107, 8, 9nfif 3763 . . . . . . . . . . 11  |-  F/_ j if ( k  e.  A ,  C ,  0 )
11 eleq1 2496 . . . . . . . . . . . 12  |-  ( j  =  k  ->  (
j  e.  A  <->  k  e.  A ) )
12 cbvsum.1 . . . . . . . . . . . 12  |-  ( j  =  k  ->  B  =  C )
13 eqidd 2437 . . . . . . . . . . . 12  |-  ( j  =  k  ->  0  =  0 )
1411, 12, 13ifbieq12d 3761 . . . . . . . . . . 11  |-  ( j  =  k  ->  if ( j  e.  A ,  B ,  0 )  =  if ( k  e.  A ,  C ,  0 ) )
155, 10, 14cbvmpt 4299 . . . . . . . . . 10  |-  ( j  e.  ZZ  |->  if ( j  e.  A ,  B ,  0 ) )  =  ( k  e.  ZZ  |->  if ( k  e.  A ,  C ,  0 ) )
1615a1i 11 . . . . . . . . 9  |-  (  T. 
->  ( j  e.  ZZ  |->  if ( j  e.  A ,  B ,  0 ) )  =  ( k  e.  ZZ  |->  if ( k  e.  A ,  C ,  0 ) ) )
1716seqeq3d 11331 . . . . . . . 8  |-  (  T. 
->  seq  m (  +  ,  ( j  e.  ZZ  |->  if ( j  e.  A ,  B ,  0 ) ) )  =  seq  m
(  +  ,  ( k  e.  ZZ  |->  if ( k  e.  A ,  C ,  0 ) ) ) )
1817breq1d 4222 . . . . . . 7  |-  (  T. 
->  (  seq  m (  +  ,  ( j  e.  ZZ  |->  if ( j  e.  A ,  B ,  0 ) ) )  ~~>  x  <->  seq  m (  +  ,  ( k  e.  ZZ  |->  if ( k  e.  A ,  C ,  0 ) ) )  ~~>  x ) )
1918anbi2d 685 . . . . . 6  |-  (  T. 
->  ( ( A  C_  ( ZZ>= `  m )  /\  seq  m (  +  ,  ( j  e.  ZZ  |->  if ( j  e.  A ,  B ,  0 ) ) )  ~~>  x )  <->  ( A  C_  ( ZZ>= `  m )  /\  seq  m (  +  ,  ( k  e.  ZZ  |->  if ( k  e.  A ,  C ,  0 ) ) )  ~~>  x ) ) )
2019rexbidv 2726 . . . . 5  |-  (  T. 
->  ( E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m )  /\  seq  m (  +  ,  ( j  e.  ZZ  |->  if ( j  e.  A ,  B ,  0 ) ) )  ~~>  x )  <->  E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m )  /\  seq  m (  +  ,  ( k  e.  ZZ  |->  if ( k  e.  A ,  C ,  0 ) ) )  ~~>  x ) ) )
213, 8, 12cbvcsb 3255 . . . . . . . . . . . . 13  |-  [_ (
f `  n )  /  j ]_ B  =  [_ ( f `  n )  /  k ]_ C
2221a1i 11 . . . . . . . . . . . 12  |-  (  T. 
->  [_ ( f `  n )  /  j ]_ B  =  [_ (
f `  n )  /  k ]_ C
)
2322mpteq2dv 4296 . . . . . . . . . . 11  |-  (  T. 
->  ( n  e.  NN  |->  [_ ( f `  n
)  /  j ]_ B )  =  ( n  e.  NN  |->  [_ ( f `  n
)  /  k ]_ C ) )
2423seqeq3d 11331 . . . . . . . . . 10  |-  (  T. 
->  seq  1 (  +  ,  ( n  e.  NN  |->  [_ ( f `  n )  /  j ]_ B ) )  =  seq  1 (  +  ,  ( n  e.  NN  |->  [_ ( f `  n )  /  k ]_ C ) ) )
2524fveq1d 5730 . . . . . . . . 9  |-  (  T. 
->  (  seq  1
(  +  ,  ( n  e.  NN  |->  [_ ( f `  n
)  /  j ]_ B ) ) `  m )  =  (  seq  1 (  +  ,  ( n  e.  NN  |->  [_ ( f `  n )  /  k ]_ C ) ) `  m ) )
2625eqeq2d 2447 . . . . . . . 8  |-  (  T. 
->  ( x  =  (  seq  1 (  +  ,  ( n  e.  NN  |->  [_ ( f `  n )  /  j ]_ B ) ) `  m )  <->  x  =  (  seq  1 (  +  ,  ( n  e.  NN  |->  [_ ( f `  n )  /  k ]_ C ) ) `  m ) ) )
2726anbi2d 685 . . . . . . 7  |-  (  T. 
->  ( ( f : ( 1 ... m
)
-1-1-onto-> A  /\  x  =  (  seq  1 (  +  ,  ( n  e.  NN  |->  [_ ( f `  n )  /  j ]_ B ) ) `  m ) )  <->  ( f : ( 1 ... m ) -1-1-onto-> A  /\  x  =  (  seq  1 (  +  ,  ( n  e.  NN  |->  [_ (
f `  n )  /  k ]_ C
) ) `  m
) ) ) )
2827exbidv 1636 . . . . . 6  |-  (  T. 
->  ( E. f ( f : ( 1 ... m ) -1-1-onto-> A  /\  x  =  (  seq  1 (  +  , 
( n  e.  NN  |->  [_ ( f `  n
)  /  j ]_ B ) ) `  m ) )  <->  E. f
( f : ( 1 ... m ) -1-1-onto-> A  /\  x  =  (  seq  1 (  +  ,  ( n  e.  NN  |->  [_ ( f `  n )  /  k ]_ C ) ) `  m ) ) ) )
2928rexbidv 2726 . . . . 5  |-  (  T. 
->  ( E. m  e.  NN  E. f ( f : ( 1 ... m ) -1-1-onto-> A  /\  x  =  (  seq  1 (  +  , 
( n  e.  NN  |->  [_ ( f `  n
)  /  j ]_ B ) ) `  m ) )  <->  E. m  e.  NN  E. f ( f : ( 1 ... m ) -1-1-onto-> A  /\  x  =  (  seq  1 (  +  , 
( n  e.  NN  |->  [_ ( f `  n
)  /  k ]_ C ) ) `  m ) ) ) )
3020, 29orbi12d 691 . . . 4  |-  (  T. 
->  ( ( E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m )  /\  seq  m (  +  ,  ( j  e.  ZZ  |->  if ( j  e.  A ,  B ,  0 ) ) )  ~~>  x )  \/ 
E. m  e.  NN  E. f ( f : ( 1 ... m
)
-1-1-onto-> A  /\  x  =  (  seq  1 (  +  ,  ( n  e.  NN  |->  [_ ( f `  n )  /  j ]_ B ) ) `  m ) ) )  <-> 
( 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 ) ) ) ) )
3130iotabidv 5439 . . 3  |-  (  T. 
->  ( iota x ( E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m )  /\  seq  m (  +  , 
( j  e.  ZZ  |->  if ( j  e.  A ,  B ,  0 ) ) )  ~~>  x )  \/  E. m  e.  NN  E. f ( f : ( 1 ... m ) -1-1-onto-> A  /\  x  =  (  seq  1 (  +  , 
( n  e.  NN  |->  [_ ( f `  n
)  /  j ]_ B ) ) `  m ) ) ) )  =  ( 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 ) ) ) ) )
3231trud 1332 . 2  |-  ( iota
x ( E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m )  /\  seq  m (  +  ,  ( j  e.  ZZ  |->  if ( j  e.  A ,  B ,  0 ) ) )  ~~>  x )  \/ 
E. m  e.  NN  E. f ( f : ( 1 ... m
)
-1-1-onto-> A  /\  x  =  (  seq  1 (  +  ,  ( n  e.  NN  |->  [_ ( f `  n )  /  j ]_ B ) ) `  m ) ) ) )  =  ( 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 ) ) ) )
33 df-sum 12480 . 2  |-  sum_ j  e.  A  B  =  ( iota x ( E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m
)  /\  seq  m (  +  ,  ( j  e.  ZZ  |->  if ( j  e.  A ,  B ,  0 ) ) )  ~~>  x )  \/  E. m  e.  NN  E. f ( f : ( 1 ... m ) -1-1-onto-> A  /\  x  =  (  seq  1 (  +  , 
( n  e.  NN  |->  [_ ( f `  n
)  /  j ]_ B ) ) `  m ) ) ) )
34 df-sum 12480 . 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 ) ) ) )
3532, 33, 343eqtr4i 2466 1  |-  sum_ j  e.  A  B  =  sum_ k  e.  A  C
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    /\ wa 359    T. wtru 1325   E.wex 1550    = wceq 1652    e. wcel 1725   F/_wnfc 2559   E.wrex 2706   [_csb 3251    C_ wss 3320   ifcif 3739   class class class wbr 4212    e. cmpt 4266   iotacio 5416   -1-1-onto->wf1o 5453   ` cfv 5454  (class class class)co 6081   0cc0 8990   1c1 8991    + caddc 8993   NNcn 10000   ZZcz 10282   ZZ>=cuz 10488   ...cfz 11043    seq cseq 11323    ~~> cli 12278   sum_csu 12479
This theorem is referenced by:  cbvsumv  12490  cbvsumi  12491  esumpfinvalf  24466
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-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
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-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  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-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-cnv 4886  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-recs 6633  df-rdg 6668  df-seq 11324  df-sum 12480
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