Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  esumeq1 Structured version   Visualization version   GIF version

Theorem esumeq1 30070
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 2621 . 2 (𝐴 = 𝐵𝐶 = 𝐶)
31, 2esumeq12d 30069 1 (𝐴 = 𝐵 → Σ*𝑘𝐴𝐶 = Σ*𝑘𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1481  Σ*cesum 30063
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-iota 5839  df-fv 5884  df-ov 6638  df-esum 30064
This theorem is referenced by:  esumrnmpt  30088  esumpad  30091  esumpad2  30092  esumpr  30102  esumpr2  30103  esumfzf  30105  esumpmono  30115  esumcvg  30122  esumcvg2  30123  esum2dlem  30128  measvun  30246  ddemeas  30273  oms0  30333  omssubadd  30336  carsgsigalem  30351  carsgclctunlem1  30353  carsgclctunlem2  30355  carsgclctun  30357  pmeasmono  30360  pmeasadd  30361
  Copyright terms: Public domain W3C validator