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Theorem cbvesum 31305
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 5137 . . . 4 (𝑗𝐴𝐵) = (𝑘𝐴𝐶)
76oveq2i 7140 . . 3 ((ℝ*𝑠s (0[,]+∞)) tsums (𝑗𝐴𝐵)) = ((ℝ*𝑠s (0[,]+∞)) tsums (𝑘𝐴𝐶))
87unieqi 4823 . 2 ((ℝ*𝑠s (0[,]+∞)) tsums (𝑗𝐴𝐵)) = ((ℝ*𝑠s (0[,]+∞)) tsums (𝑘𝐴𝐶))
9 df-esum 31291 . 2 Σ*𝑗𝐴𝐵 = ((ℝ*𝑠s (0[,]+∞)) tsums (𝑗𝐴𝐵))
10 df-esum 31291 . 2 Σ*𝑘𝐴𝐶 = ((ℝ*𝑠s (0[,]+∞)) tsums (𝑘𝐴𝐶))
118, 9, 103eqtr4i 2853 1 Σ*𝑗𝐴𝐵 = Σ*𝑘𝐴𝐶
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wnfc 2957   cuni 4810  cmpt 5118  (class class class)co 7129  0cc0 10511  +∞cpnf 10646  [,]cicc 12716  s cress 16459  *𝑠cxrs 16748   tsums ctsu 22706  Σ*cesum 31290
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-rab 3134  df-v 3472  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4811  df-br 5039  df-opab 5101  df-mpt 5119  df-iota 6286  df-fv 6335  df-ov 7132  df-esum 31291
This theorem is referenced by:  cbvesumv  31306  esumfzf  31332  carsggect  31580
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