esumeq12dvaf - Mathbox for Thierry Arnoux

	Mathbox for Thierry Arnoux		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > esumeq12dvaf		Structured version Visualization version GIF version

Theorem esumeq12dvaf 34137

Description: Equality deduction for extended sum. (Contributed by Thierry Arnoux, 26-Mar-2017.)

**Hypotheses**
Ref	Expression
esumeq12dvaf.1	⊢ Ⅎ𝑘𝜑
esumeq12dvaf.2	⊢ (𝜑 → 𝐴 = 𝐵)
esumeq12dvaf.3	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 = 𝐷)

**Assertion**
Ref	Expression
esumeq12dvaf	⊢ (𝜑 → Σ^𝑘 ∈ 𝐴𝐶 = Σ^𝑘 ∈ 𝐵𝐷)

**Proof of Theorem esumeq12dvaf**
Step	Hyp	Ref	Expression
1		esumeq12dvaf.1	. . . . . 6 ⊢ Ⅎ𝑘𝜑
2		esumeq12dvaf.2	. . . . . 6 ⊢ (𝜑 → 𝐴 = 𝐵)
3	1, 2	alrimi 2218	. . . . 5 ⊢ (𝜑 → ∀𝑘 𝐴 = 𝐵)
4		esumeq12dvaf.3	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 = 𝐷)
5	4	ex 412	. . . . . 6 ⊢ (𝜑 → (𝑘 ∈ 𝐴 → 𝐶 = 𝐷))
6	1, 5	ralrimi 3232	. . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐶 = 𝐷)
7		mpteq12f 5181	. . . . 5 ⊢ ((∀𝑘 𝐴 = 𝐵 ∧ ∀𝑘 ∈ 𝐴 𝐶 = 𝐷) → (𝑘 ∈ 𝐴 ↦ 𝐶) = (𝑘 ∈ 𝐵 ↦ 𝐷))
8	3, 6, 7	syl2anc 584	. . . 4 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐶) = (𝑘 ∈ 𝐵 ↦ 𝐷))
9	8	oveq2d 7372	. . 3 ⊢ (𝜑 → ((ℝ^_𝑠 ↾_s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐶)) = ((ℝ^_𝑠 ↾_s (0[,]+∞)) tsums (𝑘 ∈ 𝐵 ↦ 𝐷)))
10	9	unieqd 4874	. 2 ⊢ (𝜑 → ∪ ((ℝ^_𝑠 ↾_s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐶)) = ∪ ((ℝ^_𝑠 ↾_s (0[,]+∞)) tsums (𝑘 ∈ 𝐵 ↦ 𝐷)))
11		df-esum 34134	. 2 ⊢ Σ^𝑘 ∈ 𝐴𝐶 = ∪ ((ℝ^_𝑠 ↾_s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐶))
12		df-esum 34134	. 2 ⊢ Σ^𝑘 ∈ 𝐵𝐷 = ∪ ((ℝ^_𝑠 ↾_s (0[,]+∞)) tsums (𝑘 ∈ 𝐵 ↦ 𝐷))
13	10, 11, 12	3eqtr4g 2794	1 ⊢ (𝜑 → Σ^𝑘 ∈ 𝐴𝐶 = Σ^𝑘 ∈ 𝐵𝐷)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∀wal 1539 = wceq 1541 Ⅎwnf 1784 ∈ wcel 2113 ∀wral 3049 ∪ cuni 4861 ↦ cmpt 5177 (class class class)co 7356 0cc0 11024 +∞cpnf 11161 [,]cicc 13262 ↾_s cress 17155 ℝ^*_𝑠cxrs 17419 tsums ctsu 24068 Σ^*cesum 34133

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 2115 ax-9 2123 ax-10 2146 ax-12 2182 ax-ext 2706

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 2713 df-cleq 2726 df-clel 2809 df-ral 3050 df-rab 3398 df-v 3440 df-dif 3902 df-un 3904 df-ss 3916 df-nul 4284 df-if 4478 df-sn 4579 df-pr 4581 df-op 4585 df-uni 4862 df-br 5097 df-opab 5159 df-mpt 5178 df-iota 6446 df-fv 6498 df-ov 7359 df-esum 34134

This theorem is referenced by: esumeq12dva 34138 esumeq1d 34141 esumeq2d 34143 esumpinfval 34179 measvunilem0 34319

W3C validator