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Theorem esumeq1 29423
Description: Equality theorem for an extended sum. (Contributed by Thierry Arnoux, 18-Feb-2017.)
Assertion
Ref Expression
esumeq1 (𝐴 = 𝐵 → Σ*𝑘𝐴𝐶 = Σ*𝑘𝐵𝐶)
Distinct variable groups:   𝐴,𝑘   𝐵,𝑘
Allowed substitution hint:   𝐶(𝑘)

Proof of Theorem esumeq1
StepHypRef Expression
1 id 22 . 2 (𝐴 = 𝐵𝐴 = 𝐵)
2 eqidd 2611 . 2 (𝐴 = 𝐵𝐶 = 𝐶)
31, 2esumeq12d 29422 1 (𝐴 = 𝐵 → Σ*𝑘𝐴𝐶 = Σ*𝑘𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1475  Σ*cesum 29416
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-iota 5768  df-fv 5812  df-ov 6552  df-esum 29417
This theorem is referenced by:  esumrnmpt  29441  esumpad  29444  esumpad2  29445  esumpr  29455  esumpr2  29456  esumfzf  29458  esumpmono  29468  esumcvg  29475  esumcvg2  29476  esum2dlem  29481  measvun  29599  ddemeas  29626  oms0  29686  omssubadd  29689  carsgsigalem  29704  carsgclctunlem1  29706  carsgclctunlem2  29708  carsgclctun  29710  pmeasmono  29713  pmeasadd  29714
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