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Theorem esumeq1 26626
Description: Equality theorem for an extended sum. (Contributed by Thierry Arnoux, 18-Feb-2017.)
Assertion
Ref Expression
esumeq1  |-  ( A  =  B  -> Σ* k  e.  A C  = Σ* k  e.  B C )
Distinct variable groups:    A, k    B, k
Allowed substitution hint:    C( k)

Proof of Theorem esumeq1
StepHypRef Expression
1 id 22 . 2  |-  ( A  =  B  ->  A  =  B )
2 eqidd 2452 . 2  |-  ( A  =  B  ->  C  =  C )
31, 2esumeq12d 26625 1  |-  ( A  =  B  -> Σ* k  e.  A C  = Σ* k  e.  B C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370  Σ*cesum 26619
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 3072  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-br 4393  df-opab 4451  df-mpt 4452  df-iota 5481  df-fv 5526  df-ov 6195  df-esum 26620
This theorem is referenced by:  esumpr  26652  esumpr2  26653  esumfzf  26654  esumpmono  26664  esumcvg  26671  esumcvg2  26672  measvun  26759  ddemeas  26788  oms0  26846
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