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Theorem cbvsum 13519
Description: Change bound variable in a sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jun-2019.)
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.4 . . . . . . . . . . . . 13  |-  F/_ k B
2 cbvsum.5 . . . . . . . . . . . . 13  |-  F/_ j C
3 cbvsum.1 . . . . . . . . . . . . 13  |-  ( j  =  k  ->  B  =  C )
41, 2, 3cbvcsb 3353 . . . . . . . . . . . 12  |-  [_ n  /  j ]_ B  =  [_ n  /  k ]_ C
54a1i 11 . . . . . . . . . . 11  |-  ( T. 
->  [_ n  /  j ]_ B  =  [_ n  /  k ]_ C
)
65ifeq1d 3875 . . . . . . . . . 10  |-  ( T. 
->  if ( n  e.  A ,  [_ n  /  j ]_ B ,  0 )  =  if ( n  e.  A ,  [_ n  /  k ]_ C ,  0 ) )
76mpteq2dv 4454 . . . . . . . . 9  |-  ( T. 
->  ( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  j ]_ B ,  0 ) )  =  ( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  k ]_ C ,  0 ) ) )
87seqeq3d 12018 . . . . . . . 8  |-  ( T. 
->  seq m (  +  ,  ( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  j ]_ B ,  0 ) ) )  =  seq m
(  +  ,  ( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  k ]_ C ,  0 ) ) ) )
98trud 1408 . . . . . . 7  |-  seq m
(  +  ,  ( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  j ]_ B ,  0 ) ) )  =  seq m (  +  , 
( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  k ]_ C ,  0 ) ) )
109breq1i 4374 . . . . . 6  |-  (  seq m (  +  , 
( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  j ]_ B ,  0 ) ) )  ~~>  x  <->  seq m
(  +  ,  ( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  k ]_ C ,  0 ) ) )  ~~>  x )
1110anbi2i 692 . . . . 5  |-  ( ( A  C_  ( ZZ>= `  m )  /\  seq m (  +  , 
( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  j ]_ B ,  0 ) ) )  ~~>  x )  <-> 
( A  C_  ( ZZ>=
`  m )  /\  seq m (  +  , 
( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  k ]_ C ,  0 ) ) )  ~~>  x ) )
1211rexbii 2884 . . . 4  |-  ( E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m
)  /\  seq m
(  +  ,  ( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  j ]_ B ,  0 ) ) )  ~~>  x )  <->  E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m )  /\  seq m (  +  , 
( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  k ]_ C ,  0 ) ) )  ~~>  x ) )
131, 2, 3cbvcsb 3353 . . . . . . . . . . . . 13  |-  [_ (
f `  n )  /  j ]_ B  =  [_ ( f `  n )  /  k ]_ C
1413a1i 11 . . . . . . . . . . . 12  |-  ( T. 
->  [_ ( f `  n )  /  j ]_ B  =  [_ (
f `  n )  /  k ]_ C
)
1514mpteq2dv 4454 . . . . . . . . . . 11  |-  ( T. 
->  ( n  e.  NN  |->  [_ ( f `  n
)  /  j ]_ B )  =  ( n  e.  NN  |->  [_ ( f `  n
)  /  k ]_ C ) )
1615seqeq3d 12018 . . . . . . . . . 10  |-  ( T. 
->  seq 1 (  +  ,  ( n  e.  NN  |->  [_ ( f `  n )  /  j ]_ B ) )  =  seq 1 (  +  ,  ( n  e.  NN  |->  [_ ( f `  n )  /  k ]_ C ) ) )
1716trud 1408 . . . . . . . . 9  |-  seq 1
(  +  ,  ( n  e.  NN  |->  [_ ( f `  n
)  /  j ]_ B ) )  =  seq 1 (  +  ,  ( n  e.  NN  |->  [_ ( f `  n )  /  k ]_ C ) )
1817fveq1i 5775 . . . . . . . 8  |-  (  seq 1 (  +  , 
( n  e.  NN  |->  [_ ( f `  n
)  /  j ]_ B ) ) `  m )  =  (  seq 1 (  +  ,  ( n  e.  NN  |->  [_ ( f `  n )  /  k ]_ C ) ) `  m )
1918eqeq2i 2400 . . . . . . 7  |-  ( x  =  (  seq 1
(  +  ,  ( n  e.  NN  |->  [_ ( f `  n
)  /  j ]_ B ) ) `  m )  <->  x  =  (  seq 1 (  +  ,  ( n  e.  NN  |->  [_ ( f `  n )  /  k ]_ C ) ) `  m ) )
2019anbi2i 692 . . . . . 6  |-  ( ( 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
) ) )
2120exbii 1675 . . . . 5  |-  ( 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 ) ) )
2221rexbii 2884 . . . 4  |-  ( 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 ) ) )
2312, 22orbi12i 519 . . 3  |-  ( ( E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m )  /\  seq m (  +  , 
( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  j ]_ 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 (  +  ,  ( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  k ]_ 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 ) ) ) )
2423iotabii 5482 . 2  |-  ( iota
x ( E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m )  /\  seq m (  +  ,  ( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  j ]_ 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 (  +  ,  ( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  k ]_ 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 ) ) ) )
25 df-sum 13511 . 2  |-  sum_ j  e.  A  B  =  ( iota x ( E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m
)  /\  seq m
(  +  ,  ( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  j ]_ 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 ) ) ) )
26 df-sum 13511 . 2  |-  sum_ k  e.  A  C  =  ( iota x ( E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m
)  /\  seq m
(  +  ,  ( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  k ]_ 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 ) ) ) )
2724, 25, 263eqtr4i 2421 1  |-  sum_ j  e.  A  B  =  sum_ k  e.  A  C
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 366    /\ wa 367    = wceq 1399   T. wtru 1400   E.wex 1620    e. wcel 1826   F/_wnfc 2530   E.wrex 2733   [_csb 3348    C_ wss 3389   ifcif 3857   class class class wbr 4367    |-> cmpt 4425   iotacio 5458   -1-1-onto->wf1o 5495   ` cfv 5496  (class class class)co 6196   0cc0 9403   1c1 9404    + caddc 9406   NNcn 10452   ZZcz 10781   ZZ>=cuz 11001   ...cfz 11593    seqcseq 12010    ~~> cli 13309   sum_csu 13510
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-br 4368  df-opab 4426  df-mpt 4427  df-cnv 4921  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fv 5504  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-recs 6960  df-rdg 6994  df-seq 12011  df-sum 13511
This theorem is referenced by:  cbvsumv  13520  cbvsumi  13521  esumpfinvalf  28224  fsumclf  31733  fsumsplitf  31734  fsummulc1f  31735  dvnmul  31906
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