Theorem cbvesumv 34045

Description: Change bound variable in an extended sum. (Contributed by Thierry Arnoux, 19-Jun-2017.)

Hypothesis

Ref	Expression
cbvesum.1	⊢ (𝑗 = 𝑘 → 𝐵 = 𝐶)

Assertion

Ref	Expression
cbvesumv	⊢ Σ*𝑗 ∈ 𝐴𝐵 = Σ*𝑘 ∈ 𝐴𝐶

Distinct variable groups:   𝑗,𝑘,𝐴   𝐵,𝑘   𝐶,𝑗

Allowed substitution hints:   𝐵(𝑗)   𝐶(𝑘)

Proof of Theorem cbvesumv

Step	Hyp	Ref	Expression
1		cbvesum.1	. . . . 5 ⊢ (𝑗 = 𝑘 → 𝐵 = 𝐶)
2	1	cbvmptv 5254	. . . 4 ⊢ (𝑗 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐶)
3	2	oveq2i 7443	. . 3 ⊢ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑗 ∈ 𝐴 ↦ 𝐵)) = ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐶))
4	3	unieqi 4918	. 2 ⊢ ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑗 ∈ 𝐴 ↦ 𝐵)) = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐶))
5		df-esum 34030	. 2 ⊢ Σ*𝑗 ∈ 𝐴𝐵 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑗 ∈ 𝐴 ↦ 𝐵))
6		df-esum 34030	. 2 ⊢ Σ*𝑘 ∈ 𝐴𝐶 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐶))
7	4, 5, 6	3eqtr4i 2774	1 ⊢ Σ*𝑗 ∈ 𝐴𝐵 = Σ*𝑘 ∈ 𝐴𝐶

Syntax hints:   → wi 4   = wceq 1539  ∪ cuni 4906   ↦ cmpt 5224  (class class class)co 7432  0cc0 11156  +∞cpnf 11293  [,]cicc 13391   ↾_s cress 17275  ℝ^*_𝑠cxrs 17546   tsums ctsu 24135  Σ^*cesum 34029

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-iota 6513  df-fv 6568  df-ov 7435  df-esum 34030

This theorem is referenced by:  esumcvg2  34089  omssubadd  34303  totprob  34430