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

Theorem esumeq1 30426
 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 2761 . 2 (𝐴 = 𝐵𝐶 = 𝐶)
31, 2esumeq12d 30425 1 (𝐴 = 𝐵 → Σ*𝑘𝐴𝐶 = Σ*𝑘𝐵𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1632  Σ*cesum 30419 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-iota 6012  df-fv 6057  df-ov 6817  df-esum 30420 This theorem is referenced by:  esumrnmpt  30444  esumpad  30447  esumpad2  30448  esumpr  30458  esumpr2  30459  esumfzf  30461  esumpmono  30471  esumcvg  30478  esumcvg2  30479  esum2dlem  30484  measvun  30602  ddemeas  30629  oms0  30689  omssubadd  30692  carsgsigalem  30707  carsgclctunlem1  30709  carsgclctunlem2  30711  carsgclctun  30713  pmeasmono  30716  pmeasadd  30717
 Copyright terms: Public domain W3C validator