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Theorem cbvesum 31200
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 5156 . . . 4 (𝑗𝐴𝐵) = (𝑘𝐴𝐶)
76oveq2i 7156 . . 3 ((ℝ*𝑠s (0[,]+∞)) tsums (𝑗𝐴𝐵)) = ((ℝ*𝑠s (0[,]+∞)) tsums (𝑘𝐴𝐶))
87unieqi 4839 . 2 ((ℝ*𝑠s (0[,]+∞)) tsums (𝑗𝐴𝐵)) = ((ℝ*𝑠s (0[,]+∞)) tsums (𝑘𝐴𝐶))
9 df-esum 31186 . 2 Σ*𝑗𝐴𝐵 = ((ℝ*𝑠s (0[,]+∞)) tsums (𝑗𝐴𝐵))
10 df-esum 31186 . 2 Σ*𝑘𝐴𝐶 = ((ℝ*𝑠s (0[,]+∞)) tsums (𝑘𝐴𝐶))
118, 9, 103eqtr4i 2851 1 Σ*𝑗𝐴𝐵 = Σ*𝑘𝐴𝐶
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wnfc 2958   cuni 4830  cmpt 5137  (class class class)co 7145  0cc0 10525  +∞cpnf 10660  [,]cicc 12729  s cress 16472  *𝑠cxrs 16761   tsums ctsu 22661  Σ*cesum 31185
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-iota 6307  df-fv 6356  df-ov 7148  df-esum 31186
This theorem is referenced by:  cbvesumv  31201  esumfzf  31227  carsggect  31475
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