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Theorem esumeq2 26630
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 2451 . . . . 5  |-  A  =  A
2 mpteq12 4472 . . . . 5  |-  ( ( A  =  A  /\  A. k  e.  A  B  =  C )  ->  (
k  e.  A  |->  B )  =  ( k  e.  A  |->  C ) )
31, 2mpan 670 . . . 4  |-  ( A. k  e.  A  B  =  C  ->  ( k  e.  A  |->  B )  =  ( k  e.  A  |->  C ) )
43oveq2d 6209 . . 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 4202 . 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 26622 . 2  |- Σ* k  e.  A B  =  U. (
( RR*ss  ( 0 [,] +oo ) ) tsums  ( k  e.  A  |->  B ) )
7 df-esum 26622 . 2  |- Σ* k  e.  A C  =  U. (
( RR*ss  ( 0 [,] +oo ) ) tsums  ( k  e.  A  |->  C ) )
85, 6, 73eqtr4g 2517 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 1370   A.wral 2795   U.cuni 4192    |-> cmpt 4451  (class class class)co 6193   0cc0 9386   +oocpnf 9519   [,]cicc 11407   ↾s cress 14286   RR*scxrs 14549   tsums ctsu 19821  Σ*cesum 26621
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-opab 4452  df-mpt 4453  df-iota 5482  df-fv 5527  df-ov 6196  df-esum 26622
This theorem is referenced by:  ddemeas  26789
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