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Theorem cbvesum 30620
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 4941 . . . 4 (𝑗𝐴𝐵) = (𝑘𝐴𝐶)
76oveq2i 6889 . . 3 ((ℝ*𝑠s (0[,]+∞)) tsums (𝑗𝐴𝐵)) = ((ℝ*𝑠s (0[,]+∞)) tsums (𝑘𝐴𝐶))
87unieqi 4637 . 2 ((ℝ*𝑠s (0[,]+∞)) tsums (𝑗𝐴𝐵)) = ((ℝ*𝑠s (0[,]+∞)) tsums (𝑘𝐴𝐶))
9 df-esum 30606 . 2 Σ*𝑗𝐴𝐵 = ((ℝ*𝑠s (0[,]+∞)) tsums (𝑗𝐴𝐵))
10 df-esum 30606 . 2 Σ*𝑘𝐴𝐶 = ((ℝ*𝑠s (0[,]+∞)) tsums (𝑘𝐴𝐶))
118, 9, 103eqtr4i 2831 1 Σ*𝑗𝐴𝐵 = Σ*𝑘𝐴𝐶
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1653  wnfc 2928   cuni 4628  cmpt 4922  (class class class)co 6878  0cc0 10224  +∞cpnf 10360  [,]cicc 12427  s cress 16185  *𝑠cxrs 16475   tsums ctsu 22257  Σ*cesum 30605
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-rex 3095  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-mpt 4923  df-iota 6064  df-fv 6109  df-ov 6881  df-esum 30606
This theorem is referenced by:  cbvesumv  30621  esumfzf  30647  carsggect  30896
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