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Theorem cbvesum 28271
Description: Change bound variable in an extended sum. (Contributed by Thierry Arnoux, 19-Jun-2017.)
Hypotheses
Ref Expression
cbvesum.1  |-  ( j  =  k  ->  B  =  C )
cbvesum.2  |-  F/_ k A
cbvesum.3  |-  F/_ j A
cbvesum.4  |-  F/_ k B
cbvesum.5  |-  F/_ j C
Assertion
Ref Expression
cbvesum  |- Σ* j  e.  A B  = Σ* k  e.  A C
Distinct variable group:    j, k
Allowed substitution hints:    A( j, k)    B( j, k)    C( j, k)

Proof of Theorem cbvesum
StepHypRef Expression
1 cbvesum.3 . . . . 5  |-  F/_ j A
2 cbvesum.2 . . . . 5  |-  F/_ k A
3 cbvesum.4 . . . . 5  |-  F/_ k B
4 cbvesum.5 . . . . 5  |-  F/_ j C
5 cbvesum.1 . . . . 5  |-  ( j  =  k  ->  B  =  C )
61, 2, 3, 4, 5cbvmptf 27715 . . . 4  |-  ( j  e.  A  |->  B )  =  ( k  e.  A  |->  C )
76oveq2i 6281 . . 3  |-  ( (
RR*ss  ( 0 [,] +oo ) ) tsums  ( j  e.  A  |->  B ) )  =  ( (
RR*ss  ( 0 [,] +oo ) ) tsums  ( k  e.  A  |->  C ) )
87unieqi 4244 . 2  |-  U. (
( RR*ss  ( 0 [,] +oo ) ) tsums  ( j  e.  A  |->  B ) )  =  U. (
( RR*ss  ( 0 [,] +oo ) ) tsums  ( k  e.  A  |->  C ) )
9 df-esum 28257 . 2  |- Σ* j  e.  A B  =  U. (
( RR*ss  ( 0 [,] +oo ) ) tsums  ( j  e.  A  |->  B ) )
10 df-esum 28257 . 2  |- Σ* k  e.  A C  =  U. (
( RR*ss  ( 0 [,] +oo ) ) tsums  ( k  e.  A  |->  C ) )
118, 9, 103eqtr4i 2493 1  |- Σ* j  e.  A B  = Σ* k  e.  A C
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1398   F/_wnfc 2602   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-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:  cbvesumv  28272  esumfzf  28298  carsggect  28526
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