Theorem esumeq2 34668

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 2760	. . . . 5 ⊢ 𝐴 = 𝐴
2		mpteq12 5192	. . . . 5 ⊢ ((𝐴 = 𝐴 ∧ ∀𝑘 ∈ 𝐴 𝐵 = 𝐶) → (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐶))
3	1, 2	mpan 703	. . . 4 ⊢ (∀𝑘 ∈ 𝐴 𝐵 = 𝐶 → (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐶))
4	3	oveq2d 7436	. . 3 ⊢ (∀𝑘 ∈ 𝐴 𝐵 = 𝐶 → ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) = ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐶)))
5	4	unieqd 4879	. 2 ⊢ (∀𝑘 ∈ 𝐴 𝐵 = 𝐶 → ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐶)))
6		df-esum 34660	. 2 ⊢ Σ*𝑘 ∈ 𝐴𝐵 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵))
7		df-esum 34660	. 2 ⊢ Σ*𝑘 ∈ 𝐴𝐶 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐶))
8	5, 6, 7	3eqtr4g 2820	1 ⊢ (∀𝑘 ∈ 𝐴 𝐵 = 𝐶 → Σ*𝑘 ∈ 𝐴𝐵 = Σ*𝑘 ∈ 𝐴𝐶)

This proof depends on syntax axioms:   → wi 4   = wceq 1569  ∀wral 3076  ∪ cuni 4866   ↦ cmpt 5185  (class class class)co 7420  0cc0 11200  +∞cpnf 11340  [,]cicc 13479   ↾_s cress 17408  ℝ^*_𝑠cxrs 17672   tsums ctsu 24445  Σ^*cesum 34659

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-12 2212  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-iota 6494  df-fv 6546  df-ov 7423  df-esum 34660

This theorem is used by: (None)