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

Theorem cbvesum 32681
Description: Change bound variable in an extended sum. (Contributed by Thierry Arnoux, 19-Jun-2017.)
Hypotheses
Ref Expression
cbvesum.1 (𝑗 = 𝑘𝐵 = 𝐶)
cbvesum.2 𝑘𝐴
cbvesum.3 𝑗𝐴
cbvesum.4 𝑘𝐵
cbvesum.5 𝑗𝐶
Assertion
Ref Expression
cbvesum Σ*𝑗𝐴𝐵 = Σ*𝑘𝐴𝐶
Distinct variable group:   𝑗,𝑘
Allowed substitution hints:   𝐴(𝑗,𝑘)   𝐵(𝑗,𝑘)   𝐶(𝑗,𝑘)

Proof of Theorem cbvesum
StepHypRef Expression
1 cbvesum.3 . . . . 5 𝑗𝐴
2 cbvesum.2 . . . . 5 𝑘𝐴
3 cbvesum.4 . . . . 5 𝑘𝐵
4 cbvesum.5 . . . . 5 𝑗𝐶
5 cbvesum.1 . . . . 5 (𝑗 = 𝑘𝐵 = 𝐶)
61, 2, 3, 4, 5cbvmptf 5219 . . . 4 (𝑗𝐴𝐵) = (𝑘𝐴𝐶)
76oveq2i 7373 . . 3 ((ℝ*𝑠s (0[,]+∞)) tsums (𝑗𝐴𝐵)) = ((ℝ*𝑠s (0[,]+∞)) tsums (𝑘𝐴𝐶))
87unieqi 4883 . 2 ((ℝ*𝑠s (0[,]+∞)) tsums (𝑗𝐴𝐵)) = ((ℝ*𝑠s (0[,]+∞)) tsums (𝑘𝐴𝐶))
9 df-esum 32667 . 2 Σ*𝑗𝐴𝐵 = ((ℝ*𝑠s (0[,]+∞)) tsums (𝑗𝐴𝐵))
10 df-esum 32667 . 2 Σ*𝑘𝐴𝐶 = ((ℝ*𝑠s (0[,]+∞)) tsums (𝑘𝐴𝐶))
118, 9, 103eqtr4i 2775 1 Σ*𝑗𝐴𝐵 = Σ*𝑘𝐴𝐶
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wnfc 2888   cuni 4870  cmpt 5193  (class class class)co 7362  0cc0 11058  +∞cpnf 11193  [,]cicc 13274  s cress 17119  *𝑠cxrs 17389   tsums ctsu 23493  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-11 2155  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-nfc 2890  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:  cbvesumv  32682  esumfzf  32708  carsggect  32958
  Copyright terms: Public domain W3C validator