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Theorem esumeq1 32673
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 2738 . 2 (𝐴 = 𝐵𝐶 = 𝐶)
31, 2esumeq12d 32672 1 (𝐴 = 𝐵 → Σ*𝑘𝐴𝐶 = Σ*𝑘𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  Σ*cesum 32666
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-iota 6453  df-fv 6509  df-ov 7365  df-esum 32667
This theorem is referenced by:  esumrnmpt  32691  esumpad  32694  esumpad2  32695  esumpr  32705  esumpr2  32706  esumfzf  32708  esumpmono  32718  esumcvg  32725  esumcvg2  32726  esum2dlem  32731  measvun  32848  ddemeas  32875  oms0  32937  omssubadd  32940  carsgsigalem  32955  carsgclctunlem1  32957  carsgclctunlem2  32959  carsgclctun  32961  pmeasmono  32964  pmeasadd  32965
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