Theorem cbvesum 34044

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 5250	. . . 4 ⊢ (𝑗 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐶)
7	6	oveq2i 7443	. . 3 ⊢ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑗 ∈ 𝐴 ↦ 𝐵)) = ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐶))
8	7	unieqi 4918	. 2 ⊢ ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑗 ∈ 𝐴 ↦ 𝐵)) = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐶))
9		df-esum 34030	. 2 ⊢ Σ*𝑗 ∈ 𝐴𝐵 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑗 ∈ 𝐴 ↦ 𝐵))
10		df-esum 34030	. 2 ⊢ Σ*𝑘 ∈ 𝐴𝐶 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐶))
11	8, 9, 10	3eqtr4i 2774	1 ⊢ Σ*𝑗 ∈ 𝐴𝐵 = Σ*𝑘 ∈ 𝐴𝐶

Syntax hints:   → wi 4   = wceq 1539  Ⅎwnfc 2889  ∪ 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-10 2140  ax-11 2156  ax-12 2176  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-nf 1783  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  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:  esumfzf  34071  carsggect  34321