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Theorem esumeq2 34037
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 2737 . . . . 5 𝐴 = 𝐴
2 mpteq12 5234 . . . . 5 ((𝐴 = 𝐴 ∧ ∀𝑘𝐴 𝐵 = 𝐶) → (𝑘𝐴𝐵) = (𝑘𝐴𝐶))
31, 2mpan 690 . . . 4 (∀𝑘𝐴 𝐵 = 𝐶 → (𝑘𝐴𝐵) = (𝑘𝐴𝐶))
43oveq2d 7447 . . 3 (∀𝑘𝐴 𝐵 = 𝐶 → ((ℝ*𝑠s (0[,]+∞)) tsums (𝑘𝐴𝐵)) = ((ℝ*𝑠s (0[,]+∞)) tsums (𝑘𝐴𝐶)))
54unieqd 4920 . 2 (∀𝑘𝐴 𝐵 = 𝐶 ((ℝ*𝑠s (0[,]+∞)) tsums (𝑘𝐴𝐵)) = ((ℝ*𝑠s (0[,]+∞)) tsums (𝑘𝐴𝐶)))
6 df-esum 34029 . 2 Σ*𝑘𝐴𝐵 = ((ℝ*𝑠s (0[,]+∞)) tsums (𝑘𝐴𝐵))
7 df-esum 34029 . 2 Σ*𝑘𝐴𝐶 = ((ℝ*𝑠s (0[,]+∞)) tsums (𝑘𝐴𝐶))
85, 6, 73eqtr4g 2802 1 (∀𝑘𝐴 𝐵 = 𝐶 → Σ*𝑘𝐴𝐵 = Σ*𝑘𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wral 3061   cuni 4907  cmpt 5225  (class class class)co 7431  0cc0 11155  +∞cpnf 11292  [,]cicc 13390  s cress 17274  *𝑠cxrs 17545   tsums ctsu 24134  Σ*cesum 34028
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-iota 6514  df-fv 6569  df-ov 7434  df-esum 34029
This theorem is referenced by: (None)
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