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Theorem sumeq1 10105
Description: Equality theorem for a sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jun-2019.)
Assertion
Ref Expression
sumeq1  |-  ( A  =  B  ->  sum_ k  e.  A  C  =  sum_ k  e.  B  C
)

Proof of Theorem sumeq1
Dummy variables  f  m  n  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sseq1 2994 . . . . . 6  |-  ( A  =  B  ->  ( A  C_  ( ZZ>= `  m
)  <->  B  C_  ( ZZ>= `  m ) ) )
2 simpl 106 . . . . . . . . . . 11  |-  ( ( A  =  B  /\  n  e.  ZZ )  ->  A  =  B )
32eleq2d 2123 . . . . . . . . . 10  |-  ( ( A  =  B  /\  n  e.  ZZ )  ->  ( n  e.  A  <->  n  e.  B ) )
43ifbid 3377 . . . . . . . . 9  |-  ( ( A  =  B  /\  n  e.  ZZ )  ->  if ( n  e.  A ,  [_ n  /  k ]_ C ,  0 )  =  if ( n  e.  B ,  [_ n  /  k ]_ C ,  0 ) )
54mpteq2dva 3875 . . . . . . . 8  |-  ( A  =  B  ->  (
n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  k ]_ C ,  0 ) )  =  ( n  e.  ZZ  |->  if ( n  e.  B ,  [_ n  /  k ]_ C ,  0 ) ) )
6 iseqeq3 9380 . . . . . . . 8  |-  ( ( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  k ]_ C ,  0 ) )  =  ( n  e.  ZZ  |->  if ( n  e.  B ,  [_ n  /  k ]_ C ,  0 ) )  ->  seq m
(  +  ,  ( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  k ]_ C ,  0 ) ) ,  CC )  =  seq m (  +  ,  ( n  e.  ZZ  |->  if ( n  e.  B ,  [_ n  /  k ]_ C ,  0 ) ) ,  CC ) )
75, 6syl 14 . . . . . . 7  |-  ( A  =  B  ->  seq m (  +  , 
( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  k ]_ C ,  0 ) ) ,  CC )  =  seq m (  +  ,  ( n  e.  ZZ  |->  if ( n  e.  B ,  [_ n  /  k ]_ C ,  0 ) ) ,  CC ) )
87breq1d 3802 . . . . . 6  |-  ( A  =  B  ->  (  seq m (  +  , 
( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  k ]_ C ,  0 ) ) ,  CC )  ~~>  x  <->  seq m (  +  ,  ( n  e.  ZZ  |->  if ( n  e.  B ,  [_ n  /  k ]_ C ,  0 ) ) ,  CC )  ~~>  x ) )
91, 8anbi12d 450 . . . . 5  |-  ( A  =  B  ->  (
( A  C_  ( ZZ>=
`  m )  /\  seq m (  +  , 
( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  k ]_ C ,  0 ) ) ,  CC )  ~~>  x )  <->  ( B  C_  ( ZZ>= `  m )  /\  seq m (  +  ,  ( n  e.  ZZ  |->  if ( n  e.  B ,  [_ n  /  k ]_ C ,  0 ) ) ,  CC )  ~~>  x ) ) )
109rexbidv 2344 . . . 4  |-  ( A  =  B  ->  ( E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m )  /\  seq m (  +  , 
( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  k ]_ C ,  0 ) ) ,  CC )  ~~>  x )  <->  E. m  e.  ZZ  ( B  C_  ( ZZ>= `  m )  /\  seq m (  +  ,  ( n  e.  ZZ  |->  if ( n  e.  B ,  [_ n  /  k ]_ C ,  0 ) ) ,  CC )  ~~>  x ) ) )
11 f1oeq3 5147 . . . . . . 7  |-  ( A  =  B  ->  (
f : ( 1 ... m ) -1-1-onto-> A  <->  f :
( 1 ... m
)
-1-1-onto-> B ) )
1211anbi1d 446 . . . . . 6  |-  ( A  =  B  ->  (
( f : ( 1 ... m ) -1-1-onto-> A  /\  x  =  (  seq 1 (  +  ,  ( n  e.  NN  |->  [_ ( f `  n )  /  k ]_ C ) ,  CC ) `  m )
)  <->  ( f : ( 1 ... m
)
-1-1-onto-> B  /\  x  =  (  seq 1 (  +  ,  ( n  e.  NN  |->  [_ ( f `  n )  /  k ]_ C ) ,  CC ) `  m )
) ) )
1312exbidv 1722 . . . . 5  |-  ( A  =  B  ->  ( E. f ( f : ( 1 ... m
)
-1-1-onto-> A  /\  x  =  (  seq 1 (  +  ,  ( n  e.  NN  |->  [_ ( f `  n )  /  k ]_ C ) ,  CC ) `  m )
)  <->  E. f ( f : ( 1 ... m ) -1-1-onto-> B  /\  x  =  (  seq 1 (  +  ,  ( n  e.  NN  |->  [_ (
f `  n )  /  k ]_ C
) ,  CC ) `
 m ) ) ) )
1413rexbidv 2344 . . . 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 ) ,  CC ) `  m )
)  <->  E. m  e.  NN  E. f ( f : ( 1 ... m
)
-1-1-onto-> B  /\  x  =  (  seq 1 (  +  ,  ( n  e.  NN  |->  [_ ( f `  n )  /  k ]_ C ) ,  CC ) `  m )
) ) )
1510, 14orbi12d 717 . . 3  |-  ( A  =  B  ->  (
( E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m )  /\  seq m (  +  ,  ( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  k ]_ C ,  0 ) ) ,  CC )  ~~>  x )  \/  E. m  e.  NN  E. f ( f : ( 1 ... m ) -1-1-onto-> A  /\  x  =  (  seq 1 (  +  , 
( n  e.  NN  |->  [_ ( f `  n
)  /  k ]_ C ) ,  CC ) `  m )
) )  <->  ( E. m  e.  ZZ  ( B  C_  ( ZZ>= `  m
)  /\  seq m
(  +  ,  ( n  e.  ZZ  |->  if ( n  e.  B ,  [_ n  /  k ]_ C ,  0 ) ) ,  CC )  ~~>  x )  \/  E. m  e.  NN  E. f
( f : ( 1 ... m ) -1-1-onto-> B  /\  x  =  (  seq 1 (  +  ,  ( n  e.  NN  |->  [_ ( f `  n )  /  k ]_ C ) ,  CC ) `  m )
) ) ) )
1615iotabidv 4916 . 2  |-  ( A  =  B  ->  ( iota x ( E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m )  /\  seq m (  +  ,  ( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  k ]_ C ,  0 ) ) ,  CC )  ~~>  x )  \/  E. m  e.  NN  E. f ( f : ( 1 ... m ) -1-1-onto-> A  /\  x  =  (  seq 1 (  +  , 
( n  e.  NN  |->  [_ ( f `  n
)  /  k ]_ C ) ,  CC ) `  m )
) ) )  =  ( iota x ( E. m  e.  ZZ  ( B  C_  ( ZZ>= `  m )  /\  seq m (  +  , 
( n  e.  ZZ  |->  if ( n  e.  B ,  [_ n  /  k ]_ C ,  0 ) ) ,  CC )  ~~>  x )  \/  E. m  e.  NN  E. f
( f : ( 1 ... m ) -1-1-onto-> B  /\  x  =  (  seq 1 (  +  ,  ( n  e.  NN  |->  [_ ( f `  n )  /  k ]_ C ) ,  CC ) `  m )
) ) ) )
17 df-sum 10104 . 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 ) ) ,  CC )  ~~>  x )  \/  E. m  e.  NN  E. f
( f : ( 1 ... m ) -1-1-onto-> A  /\  x  =  (  seq 1 (  +  ,  ( n  e.  NN  |->  [_ ( f `  n )  /  k ]_ C ) ,  CC ) `  m )
) ) )
18 df-sum 10104 . 2  |-  sum_ k  e.  B  C  =  ( iota x ( E. m  e.  ZZ  ( B  C_  ( ZZ>= `  m
)  /\  seq m
(  +  ,  ( n  e.  ZZ  |->  if ( n  e.  B ,  [_ n  /  k ]_ C ,  0 ) ) ,  CC )  ~~>  x )  \/  E. m  e.  NN  E. f
( f : ( 1 ... m ) -1-1-onto-> B  /\  x  =  (  seq 1 (  +  ,  ( n  e.  NN  |->  [_ ( f `  n )  /  k ]_ C ) ,  CC ) `  m )
) ) )
1916, 17, 183eqtr4g 2113 1  |-  ( A  =  B  ->  sum_ k  e.  A  C  =  sum_ k  e.  B  C
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    \/ wo 639    = wceq 1259   E.wex 1397    e. wcel 1409   E.wrex 2324   [_csb 2880    C_ wss 2945   ifcif 3359   class class class wbr 3792    |-> cmpt 3846   iotacio 4893   -1-1-onto->wf1o 4929   ` cfv 4930  (class class class)co 5540   CCcc 6945   0cc0 6947   1c1 6948    + caddc 6950   NNcn 7990   ZZcz 8302   ZZ>=cuz 8569   ...cfz 8976    seqcseq 9375    ~~> cli 10030   sum_csu 10103
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-if 3360  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-mpt 3848  df-cnv 4381  df-dm 4383  df-rn 4384  df-res 4385  df-iota 4895  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-recs 5951  df-frec 6009  df-iseq 9376  df-sum 10104
This theorem is referenced by: (None)
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