Theorem cbvesum 32010

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

Hypotheses

Ref	Expression
cbvesum.1	⊢ (𝑗 = 𝑘 → 𝐵 = 𝐶)
cbvesum.2	⊢ Ⅎ𝑘𝐴
cbvesum.3	⊢ Ⅎ𝑗𝐴
cbvesum.4	⊢ Ⅎ𝑘𝐵
cbvesum.5	⊢ Ⅎ𝑗𝐶

Assertion

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

Distinct variable group:   𝑗,𝑘

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

Proof of Theorem cbvesum

Step	Hyp	Ref	Expression
1		cbvesum.3	. . . . 5 ⊢ Ⅎ𝑗𝐴
2		cbvesum.2	. . . . 5 ⊢ Ⅎ𝑘𝐴
3		cbvesum.4	. . . . 5 ⊢ Ⅎ𝑘𝐵
4		cbvesum.5	. . . . 5 ⊢ Ⅎ𝑗𝐶
5		cbvesum.1	. . . . 5 ⊢ (𝑗 = 𝑘 → 𝐵 = 𝐶)
6	1, 2, 3, 4, 5	cbvmptf 5183	. . . 4 ⊢ (𝑗 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐶)
7	6	oveq2i 7286	. . 3 ⊢ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑗 ∈ 𝐴 ↦ 𝐵)) = ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐶))
8	7	unieqi 4852	. 2 ⊢ ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑗 ∈ 𝐴 ↦ 𝐵)) = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐶))
9		df-esum 31996	. 2 ⊢ Σ*𝑗 ∈ 𝐴𝐵 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑗 ∈ 𝐴 ↦ 𝐵))
10		df-esum 31996	. 2 ⊢ Σ*𝑘 ∈ 𝐴𝐶 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐶))
11	8, 9, 10	3eqtr4i 2776	1 ⊢ Σ*𝑗 ∈ 𝐴𝐵 = Σ*𝑘 ∈ 𝐴𝐶

Syntax hints:   → wi 4   = wceq 1539  Ⅎwnfc 2887  ∪ cuni 4839   ↦ cmpt 5157  (class class class)co 7275  0cc0 10871  +∞cpnf 11006  [,]cicc 13082   ↾_s cress 16941  ℝ^*_𝑠cxrs 17211   tsums ctsu 23277  Σ^*cesum 31995

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-iota 6391  df-fv 6441  df-ov 7278  df-esum 31996

This theorem is referenced by:  cbvesumv  32011  esumfzf  32037  carsggect  32285