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Theorem cbvesum 30089
 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 4746 . . . 4 (𝑗𝐴𝐵) = (𝑘𝐴𝐶)
76oveq2i 6658 . . 3 ((ℝ*𝑠s (0[,]+∞)) tsums (𝑗𝐴𝐵)) = ((ℝ*𝑠s (0[,]+∞)) tsums (𝑘𝐴𝐶))
87unieqi 4443 . 2 ((ℝ*𝑠s (0[,]+∞)) tsums (𝑗𝐴𝐵)) = ((ℝ*𝑠s (0[,]+∞)) tsums (𝑘𝐴𝐶))
9 df-esum 30075 . 2 Σ*𝑗𝐴𝐵 = ((ℝ*𝑠s (0[,]+∞)) tsums (𝑗𝐴𝐵))
10 df-esum 30075 . 2 Σ*𝑘𝐴𝐶 = ((ℝ*𝑠s (0[,]+∞)) tsums (𝑘𝐴𝐶))
118, 9, 103eqtr4i 2653 1 Σ*𝑗𝐴𝐵 = Σ*𝑘𝐴𝐶
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1482  Ⅎwnfc 2750  ∪ cuni 4434   ↦ cmpt 4727  (class class class)co 6647  0cc0 9933  +∞cpnf 10068  [,]cicc 12175   ↾s cress 15852  ℝ*𝑠cxrs 16154   tsums ctsu 21923  Σ*cesum 30074 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-rex 2917  df-rab 2920  df-v 3200  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-opab 4711  df-mpt 4728  df-iota 5849  df-fv 5894  df-ov 6650  df-esum 30075 This theorem is referenced by:  cbvesumv  30090  esumfzf  30116  carsggect  30365
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