Mathbox for Thierry Arnoux < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  esumeq2d Structured version   Visualization version   GIF version

Theorem esumeq2d 29426
 Description: Equality deduction for extended sum. (Contributed by Thierry Arnoux, 21-Sep-2016.)
Hypotheses
Ref Expression
esumeq2d.0 𝑘𝜑
esumeq2d.1 (𝜑 → ∀𝑘𝐴 𝐵 = 𝐶)
Assertion
Ref Expression
esumeq2d (𝜑 → Σ*𝑘𝐴𝐵 = Σ*𝑘𝐴𝐶)

Proof of Theorem esumeq2d
StepHypRef Expression
1 esumeq2d.0 . 2 𝑘𝜑
2 eqidd 2611 . 2 (𝜑𝐴 = 𝐴)
3 esumeq2d.1 . . 3 (𝜑 → ∀𝑘𝐴 𝐵 = 𝐶)
43r19.21bi 2916 . 2 ((𝜑𝑘𝐴) → 𝐵 = 𝐶)
51, 2, 4esumeq12dvaf 29420 1 (𝜑 → Σ*𝑘𝐴𝐵 = Σ*𝑘𝐴𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475  Ⅎwnf 1699  ∀wral 2896  Σ*cesum 29416 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-iota 5768  df-fv 5812  df-ov 6552  df-esum 29417 This theorem is referenced by:  esumeq2dv  29427  esumpad  29444  esumlef  29451  esumrnmpt2  29457  voliune  29619  omssubadd  29689  carsggect  29707  omsmeas  29712  dstrvprob  29860
 Copyright terms: Public domain W3C validator