Theorem esumeq2 34049

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 5177	. . . . 5 ⊢ ((𝐴 = 𝐴 ∧ ∀𝑘 ∈ 𝐴 𝐵 = 𝐶) → (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐶))
3	1, 2	mpan 690	. . . 4 ⊢ (∀𝑘 ∈ 𝐴 𝐵 = 𝐶 → (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐶))
4	3	oveq2d 7362	. . 3 ⊢ (∀𝑘 ∈ 𝐴 𝐵 = 𝐶 → ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) = ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐶)))
5	4	unieqd 4869	. 2 ⊢ (∀𝑘 ∈ 𝐴 𝐵 = 𝐶 → ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐶)))
6		df-esum 34041	. 2 ⊢ Σ*𝑘 ∈ 𝐴𝐵 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵))
7		df-esum 34041	. 2 ⊢ Σ*𝑘 ∈ 𝐴𝐶 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐶))
8	5, 6, 7	3eqtr4g 2791	1 ⊢ (∀𝑘 ∈ 𝐴 𝐵 = 𝐶 → Σ*𝑘 ∈ 𝐴𝐵 = Σ*𝑘 ∈ 𝐴𝐶)

Syntax hints:   → wi 4   = wceq 1541  ∀wral 3047  ∪ cuni 4856   ↦ cmpt 5170  (class class class)co 7346  0cc0 11006  +∞cpnf 11143  [,]cicc 13248   ↾_s cress 17141  ℝ^*_𝑠cxrs 17404   tsums ctsu 24041  Σ^*cesum 34040

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-12 2180  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-mpt 5171  df-iota 6437  df-fv 6489  df-ov 7349  df-esum 34041

This theorem is referenced by: (None)