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Theorem esumeq2 33332
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 2730 . . . . 5 𝐴 = 𝐴
2 mpteq12 5239 . . . . 5 ((𝐴 = 𝐴 ∧ ∀𝑘𝐴 𝐵 = 𝐶) → (𝑘𝐴𝐵) = (𝑘𝐴𝐶))
31, 2mpan 686 . . . 4 (∀𝑘𝐴 𝐵 = 𝐶 → (𝑘𝐴𝐵) = (𝑘𝐴𝐶))
43oveq2d 7427 . . 3 (∀𝑘𝐴 𝐵 = 𝐶 → ((ℝ*𝑠s (0[,]+∞)) tsums (𝑘𝐴𝐵)) = ((ℝ*𝑠s (0[,]+∞)) tsums (𝑘𝐴𝐶)))
54unieqd 4921 . 2 (∀𝑘𝐴 𝐵 = 𝐶 ((ℝ*𝑠s (0[,]+∞)) tsums (𝑘𝐴𝐵)) = ((ℝ*𝑠s (0[,]+∞)) tsums (𝑘𝐴𝐶)))
6 df-esum 33324 . 2 Σ*𝑘𝐴𝐵 = ((ℝ*𝑠s (0[,]+∞)) tsums (𝑘𝐴𝐵))
7 df-esum 33324 . 2 Σ*𝑘𝐴𝐶 = ((ℝ*𝑠s (0[,]+∞)) tsums (𝑘𝐴𝐶))
85, 6, 73eqtr4g 2795 1 (∀𝑘𝐴 𝐵 = 𝐶 → Σ*𝑘𝐴𝐵 = Σ*𝑘𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wral 3059   cuni 4907  cmpt 5230  (class class class)co 7411  0cc0 11112  +∞cpnf 11249  [,]cicc 13331  s cress 17177  *𝑠cxrs 17450   tsums ctsu 23850  Σ*cesum 33323
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-12 2169  ax-ext 2701
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ral 3060  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-iota 6494  df-fv 6550  df-ov 7414  df-esum 33324
This theorem is referenced by: (None)
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