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Theorem esumeq2 31352
Description: Equality theorem for extended sum. (Contributed by Thierry Arnoux, 24-Dec-2016.)
Assertion
Ref Expression
esumeq2 (∀𝑘𝐴 𝐵 = 𝐶 → Σ*𝑘𝐴𝐵 = Σ*𝑘𝐴𝐶)
Distinct variable group:   𝐴,𝑘
Allowed substitution hints:   𝐵(𝑘)   𝐶(𝑘)

Proof of Theorem esumeq2
StepHypRef Expression
1 eqid 2824 . . . . 5 𝐴 = 𝐴
2 mpteq12 5139 . . . . 5 ((𝐴 = 𝐴 ∧ ∀𝑘𝐴 𝐵 = 𝐶) → (𝑘𝐴𝐵) = (𝑘𝐴𝐶))
31, 2mpan 689 . . . 4 (∀𝑘𝐴 𝐵 = 𝐶 → (𝑘𝐴𝐵) = (𝑘𝐴𝐶))
43oveq2d 7165 . . 3 (∀𝑘𝐴 𝐵 = 𝐶 → ((ℝ*𝑠s (0[,]+∞)) tsums (𝑘𝐴𝐵)) = ((ℝ*𝑠s (0[,]+∞)) tsums (𝑘𝐴𝐶)))
54unieqd 4838 . 2 (∀𝑘𝐴 𝐵 = 𝐶 ((ℝ*𝑠s (0[,]+∞)) tsums (𝑘𝐴𝐵)) = ((ℝ*𝑠s (0[,]+∞)) tsums (𝑘𝐴𝐶)))
6 df-esum 31344 . 2 Σ*𝑘𝐴𝐵 = ((ℝ*𝑠s (0[,]+∞)) tsums (𝑘𝐴𝐵))
7 df-esum 31344 . 2 Σ*𝑘𝐴𝐶 = ((ℝ*𝑠s (0[,]+∞)) tsums (𝑘𝐴𝐶))
85, 6, 73eqtr4g 2884 1 (∀𝑘𝐴 𝐵 = 𝐶 → Σ*𝑘𝐴𝐵 = Σ*𝑘𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wral 3133   cuni 4824  cmpt 5132  (class class class)co 7149  0cc0 10535  +∞cpnf 10670  [,]cicc 12738  s cress 16484  *𝑠cxrs 16773   tsums ctsu 22734  Σ*cesum 31343
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-12 2179  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-ral 3138  df-v 3482  df-un 3924  df-in 3926  df-ss 3936  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-iota 6302  df-fv 6351  df-ov 7152  df-esum 31344
This theorem is referenced by: (None)
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