Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  esumeq1d Structured version   Visualization version   GIF version

Theorem esumeq1d 32674
Description: Equality theorem for an extended sum. (Contributed by Thierry Arnoux, 19-Oct-2017.)
Hypotheses
Ref Expression
esumeq1d.0 𝑘𝜑
esumeq1d.1 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
esumeq1d (𝜑 → Σ*𝑘𝐴𝐶 = Σ*𝑘𝐵𝐶)

Proof of Theorem esumeq1d
StepHypRef Expression
1 esumeq1d.0 . 2 𝑘𝜑
2 esumeq1d.1 . 2 (𝜑𝐴 = 𝐵)
3 eqidd 2738 . 2 ((𝜑𝑘𝐴) → 𝐶 = 𝐶)
41, 2, 3esumeq12dvaf 32670 1 (𝜑 → Σ*𝑘𝐴𝐶 = Σ*𝑘𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wnf 1786  wcel 2107  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:  esummono  32693  esumrnmpt2  32707  esumfzf  32708  hasheuni  32724  esum2dlem  32731  measvuni  32853  ddemeas  32875  omssubadd  32940  carsggect  32958
  Copyright terms: Public domain W3C validator