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 30431
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 2771 . 2 ((𝜑𝑘𝐴) → 𝐶 = 𝐶)
41, 2, 3esumeq12dvaf 30427 1 (𝜑 → Σ*𝑘𝐴𝐶 = Σ*𝑘𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1630  wnf 1855  wcel 2144  Σ*cesum 30423
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  df-iota 5994  df-fv 6039  df-ov 6795  df-esum 30424
This theorem is referenced by:  esummono  30450  esumrnmpt2  30464  esumfzf  30465  hasheuni  30481  esum2dlem  30488  measvuni  30611  ddemeas  30633  omssubadd  30696  carsggect  30714
  Copyright terms: Public domain W3C validator