Theorem cbvesum 34202

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 5186	. . . 4 ⊢ (𝑗 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐶)
7	6	oveq2i 7371	. . 3 ⊢ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑗 ∈ 𝐴 ↦ 𝐵)) = ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐶))
8	7	unieqi 4863	. 2 ⊢ ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑗 ∈ 𝐴 ↦ 𝐵)) = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐶))
9		df-esum 34188	. 2 ⊢ Σ*𝑗 ∈ 𝐴𝐵 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑗 ∈ 𝐴 ↦ 𝐵))
10		df-esum 34188	. 2 ⊢ Σ*𝑘 ∈ 𝐴𝐶 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐶))
11	8, 9, 10	3eqtr4i 2770	1 ⊢ Σ*𝑗 ∈ 𝐴𝐵 = Σ*𝑘 ∈ 𝐴𝐶

Syntax hints:   → wi 4   = wceq 1542  Ⅎwnfc 2884  ∪ cuni 4851   ↦ cmpt 5167  (class class class)co 7360  0cc0 11029  +∞cpnf 11167  [,]cicc 13292   ↾_s cress 17191  ℝ^*_𝑠cxrs 17455   tsums ctsu 24101  Σ^*cesum 34187

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-iota 6448  df-fv 6500  df-ov 7363  df-esum 34188

This theorem is referenced by:  esumfzf  34229  carsggect  34478