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Theorem esumeq2 28265
Description: Equality theorem for extended sum. (Contributed by Thierry Arnoux, 24-Dec-2016.)
Assertion
Ref Expression
esumeq2  |-  ( A. k  e.  A  B  =  C  -> Σ* k  e.  A B  = Σ* k  e.  A C )
Distinct variable group:    A, k
Allowed substitution hints:    B( k)    C( k)

Proof of Theorem esumeq2
StepHypRef Expression
1 eqid 2454 . . . . 5  |-  A  =  A
2 mpteq12 4518 . . . . 5  |-  ( ( A  =  A  /\  A. k  e.  A  B  =  C )  ->  (
k  e.  A  |->  B )  =  ( k  e.  A  |->  C ) )
31, 2mpan 668 . . . 4  |-  ( A. k  e.  A  B  =  C  ->  ( k  e.  A  |->  B )  =  ( k  e.  A  |->  C ) )
43oveq2d 6286 . . 3  |-  ( A. k  e.  A  B  =  C  ->  ( (
RR*ss  ( 0 [,] +oo ) ) tsums  ( k  e.  A  |->  B ) )  =  ( (
RR*ss  ( 0 [,] +oo ) ) tsums  ( k  e.  A  |->  C ) ) )
54unieqd 4245 . 2  |-  ( A. k  e.  A  B  =  C  ->  U. (
( RR*ss  ( 0 [,] +oo ) ) tsums  ( k  e.  A  |->  B ) )  =  U. (
( RR*ss  ( 0 [,] +oo ) ) tsums  ( k  e.  A  |->  C ) ) )
6 df-esum 28257 . 2  |- Σ* k  e.  A B  =  U. (
( RR*ss  ( 0 [,] +oo ) ) tsums  ( k  e.  A  |->  B ) )
7 df-esum 28257 . 2  |- Σ* k  e.  A C  =  U. (
( RR*ss  ( 0 [,] +oo ) ) tsums  ( k  e.  A  |->  C ) )
85, 6, 73eqtr4g 2520 1  |-  ( A. k  e.  A  B  =  C  -> Σ* k  e.  A B  = Σ* k  e.  A C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1398   A.wral 2804   U.cuni 4235    |-> cmpt 4497  (class class class)co 6270   0cc0 9481   +oocpnf 9614   [,]cicc 11535   ↾s cress 14717   RR*scxrs 14989   tsums ctsu 20790  Σ*cesum 28256
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-iota 5534  df-fv 5578  df-ov 6273  df-esum 28257
This theorem is referenced by: (None)
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