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Theorem esumeq2 29921
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 2621 . . . . 5 𝐴 = 𝐴
2 mpteq12 4706 . . . . 5 ((𝐴 = 𝐴 ∧ ∀𝑘𝐴 𝐵 = 𝐶) → (𝑘𝐴𝐵) = (𝑘𝐴𝐶))
31, 2mpan 705 . . . 4 (∀𝑘𝐴 𝐵 = 𝐶 → (𝑘𝐴𝐵) = (𝑘𝐴𝐶))
43oveq2d 6631 . . 3 (∀𝑘𝐴 𝐵 = 𝐶 → ((ℝ*𝑠s (0[,]+∞)) tsums (𝑘𝐴𝐵)) = ((ℝ*𝑠s (0[,]+∞)) tsums (𝑘𝐴𝐶)))
54unieqd 4419 . 2 (∀𝑘𝐴 𝐵 = 𝐶 ((ℝ*𝑠s (0[,]+∞)) tsums (𝑘𝐴𝐵)) = ((ℝ*𝑠s (0[,]+∞)) tsums (𝑘𝐴𝐶)))
6 df-esum 29913 . 2 Σ*𝑘𝐴𝐵 = ((ℝ*𝑠s (0[,]+∞)) tsums (𝑘𝐴𝐵))
7 df-esum 29913 . 2 Σ*𝑘𝐴𝐶 = ((ℝ*𝑠s (0[,]+∞)) tsums (𝑘𝐴𝐶))
85, 6, 73eqtr4g 2680 1 (∀𝑘𝐴 𝐵 = 𝐶 → Σ*𝑘𝐴𝐵 = Σ*𝑘𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wral 2908   cuni 4409  cmpt 4683  (class class class)co 6615  0cc0 9896  +∞cpnf 10031  [,]cicc 12136  s cress 15801  *𝑠cxrs 16100   tsums ctsu 21869  Σ*cesum 29912
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-iota 5820  df-fv 5865  df-ov 6618  df-esum 29913
This theorem is referenced by: (None)
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