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Theorem esumeq2 31194
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 2818 . . . . 5 𝐴 = 𝐴
2 mpteq12 5144 . . . . 5 ((𝐴 = 𝐴 ∧ ∀𝑘𝐴 𝐵 = 𝐶) → (𝑘𝐴𝐵) = (𝑘𝐴𝐶))
31, 2mpan 686 . . . 4 (∀𝑘𝐴 𝐵 = 𝐶 → (𝑘𝐴𝐵) = (𝑘𝐴𝐶))
43oveq2d 7161 . . 3 (∀𝑘𝐴 𝐵 = 𝐶 → ((ℝ*𝑠s (0[,]+∞)) tsums (𝑘𝐴𝐵)) = ((ℝ*𝑠s (0[,]+∞)) tsums (𝑘𝐴𝐶)))
54unieqd 4840 . 2 (∀𝑘𝐴 𝐵 = 𝐶 ((ℝ*𝑠s (0[,]+∞)) tsums (𝑘𝐴𝐵)) = ((ℝ*𝑠s (0[,]+∞)) tsums (𝑘𝐴𝐶)))
6 df-esum 31186 . 2 Σ*𝑘𝐴𝐵 = ((ℝ*𝑠s (0[,]+∞)) tsums (𝑘𝐴𝐵))
7 df-esum 31186 . 2 Σ*𝑘𝐴𝐶 = ((ℝ*𝑠s (0[,]+∞)) tsums (𝑘𝐴𝐶))
85, 6, 73eqtr4g 2878 1 (∀𝑘𝐴 𝐵 = 𝐶 → Σ*𝑘𝐴𝐵 = Σ*𝑘𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wral 3135   cuni 4830  cmpt 5137  (class class class)co 7145  0cc0 10525  +∞cpnf 10660  [,]cicc 12729  s cress 16472  *𝑠cxrs 16761   tsums ctsu 22661  Σ*cesum 31185
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-iota 6307  df-fv 6356  df-ov 7148  df-esum 31186
This theorem is referenced by: (None)
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