Theorem cbvesum 34006

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 5275	. . . 4 ⊢ (𝑗 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐶)
7	6	oveq2i 7459	. . 3 ⊢ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑗 ∈ 𝐴 ↦ 𝐵)) = ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐶))
8	7	unieqi 4943	. 2 ⊢ ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑗 ∈ 𝐴 ↦ 𝐵)) = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐶))
9		df-esum 33992	. 2 ⊢ Σ*𝑗 ∈ 𝐴𝐵 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑗 ∈ 𝐴 ↦ 𝐵))
10		df-esum 33992	. 2 ⊢ Σ*𝑘 ∈ 𝐴𝐶 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐶))
11	8, 9, 10	3eqtr4i 2778	1 ⊢ Σ*𝑗 ∈ 𝐴𝐵 = Σ*𝑘 ∈ 𝐴𝐶

Syntax hints:   → wi 4   = wceq 1537  Ⅎwnfc 2893  ∪ cuni 4931   ↦ cmpt 5249  (class class class)co 7448  0cc0 11184  +∞cpnf 11321  [,]cicc 13410   ↾_s cress 17287  ℝ^*_𝑠cxrs 17560   tsums ctsu 24155  Σ^*cesum 33991

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-iota 6525  df-fv 6581  df-ov 7451  df-esum 33992

This theorem is referenced by:  esumfzf  34033  carsggect  34283