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Theorem sumeq1f 12402
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 3305 . . . . . 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 2523 . . . . . . . . 9  |-  F/ k  A  =  B
5 simpl 444 . . . . . . . . . . 11  |-  ( ( A  =  B  /\  k  e.  ZZ )  ->  A  =  B )
65eleq2d 2447 . . . . . . . . . 10  |-  ( ( A  =  B  /\  k  e.  ZZ )  ->  ( k  e.  A  <->  k  e.  B ) )
76ifbid 3693 . . . . . . . . 9  |-  ( ( A  =  B  /\  k  e.  ZZ )  ->  if ( k  e.  A ,  C , 
0 )  =  if ( k  e.  B ,  C ,  0 ) )
84, 7mpteq2da 4228 . . . . . . . 8  |-  ( A  =  B  ->  (
k  e.  ZZ  |->  if ( k  e.  A ,  C ,  0 ) )  =  ( k  e.  ZZ  |->  if ( k  e.  B ,  C ,  0 ) ) )
98seqeq3d 11251 . . . . . . 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 4156 . . . . . 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 692 . . . . 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 2663 . . . 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 5600 . . . . . . 7  |-  ( A  =  B  ->  (
f : ( 1 ... m ) -1-1-onto-> A  <->  f :
( 1 ... m
)
-1-1-onto-> B ) )
1413anbi1d 686 . . . . . 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 1633 . . . . 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 2663 . . . 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 691 . . 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 5372 . 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 12400 . 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 12400 . 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 2437 1  |-  ( A  =  B  ->  sum_ k  e.  A  C  =  sum_ k  e.  B  C
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1717   F/_wnfc 2503   E.wrex 2643   [_csb 3187    C_ wss 3256   ifcif 3675   class class class wbr 4146    e. cmpt 4200   iotacio 5349   -1-1-onto->wf1o 5386   ` cfv 5387  (class class class)co 6013   0cc0 8916   1c1 8917    + caddc 8919   NNcn 9925   ZZcz 10207   ZZ>=cuz 10413   ...cfz 10968    seq cseq 11243    ~~> cli 12198   sum_csu 12399
This theorem is referenced by:  sumeq1  12403
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-mpt 4202  df-cnv 4819  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-recs 6562  df-rdg 6597  df-seq 11244  df-sum 12400
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