Theorem esumeq2 32758

Description: Equality theorem for extended sum. (Contributed by Thierry Arnoux, 24-Dec-2016.)

Assertion

Ref	Expression
esumeq2	⊢ (∀𝑘 ∈ 𝐴 𝐵 = 𝐶 → Σ*𝑘 ∈ 𝐴𝐵 = Σ*𝑘 ∈ 𝐴𝐶)

Distinct variable group:   𝐴,𝑘

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

Proof of Theorem esumeq2

Step	Hyp	Ref	Expression
1		eqid 2731	. . . . 5 ⊢ 𝐴 = 𝐴
2		mpteq12 5217	. . . . 5 ⊢ ((𝐴 = 𝐴 ∧ ∀𝑘 ∈ 𝐴 𝐵 = 𝐶) → (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐶))
3	1, 2	mpan 688	. . . 4 ⊢ (∀𝑘 ∈ 𝐴 𝐵 = 𝐶 → (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐶))
4	3	oveq2d 7393	. . 3 ⊢ (∀𝑘 ∈ 𝐴 𝐵 = 𝐶 → ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) = ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐶)))
5	4	unieqd 4899	. 2 ⊢ (∀𝑘 ∈ 𝐴 𝐵 = 𝐶 → ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐶)))
6		df-esum 32750	. 2 ⊢ Σ*𝑘 ∈ 𝐴𝐵 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵))
7		df-esum 32750	. 2 ⊢ Σ*𝑘 ∈ 𝐴𝐶 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐶))
8	5, 6, 7	3eqtr4g 2796	1 ⊢ (∀𝑘 ∈ 𝐴 𝐵 = 𝐶 → Σ*𝑘 ∈ 𝐴𝐵 = Σ*𝑘 ∈ 𝐴𝐶)

Syntax hints:   → wi 4   = wceq 1541  ∀wral 3060  ∪ cuni 4885   ↦ cmpt 5208  (class class class)co 7377  0cc0 11075  +∞cpnf 11210  [,]cicc 13292   ↾_s cress 17138  ℝ^*_𝑠cxrs 17411   tsums ctsu 23529  Σ^*cesum 32749

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rab 3419  df-v 3461  df-dif 3931  df-un 3933  df-in 3935  df-ss 3945  df-nul 4303  df-if 4507  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4886  df-br 5126  df-opab 5188  df-mpt 5209  df-iota 6468  df-fv 6524  df-ov 7380  df-esum 32750

This theorem is referenced by: (None)