esumeq12dvaf - Mathbox for Thierry Arnoux

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Theorem esumeq12dvaf 34663

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 2249	. . . . 5 ⊢ (𝜑 → ∀𝑘 𝐴 = 𝐵)
4		esumeq12dvaf.3	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 = 𝐷)
5	4	ex 418	. . . . . 6 ⊢ (𝜑 → (𝑘 ∈ 𝐴 → 𝐶 = 𝐷))
6	1, 5	ralrimi 3260	. . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐶 = 𝐷)
7		mpteq12f 5189	. . . . 5 ⊢ ((∀𝑘 𝐴 = 𝐵 ∧ ∀𝑘 ∈ 𝐴 𝐶 = 𝐷) → (𝑘 ∈ 𝐴 ↦ 𝐶) = (𝑘 ∈ 𝐵 ↦ 𝐷))
8	3, 6, 7	syl2anc 596	. . . 4 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐶) = (𝑘 ∈ 𝐵 ↦ 𝐷))
9	8	oveq2d 7436	. . 3 ⊢ (𝜑 → ((ℝ^_𝑠 ↾_s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐶)) = ((ℝ^_𝑠 ↾_s (0[,]+∞)) tsums (𝑘 ∈ 𝐵 ↦ 𝐷)))
10	9	unieqd 4879	. 2 ⊢ (𝜑 → ∪ ((ℝ^_𝑠 ↾_s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐶)) = ∪ ((ℝ^_𝑠 ↾_s (0[,]+∞)) tsums (𝑘 ∈ 𝐵 ↦ 𝐷)))
11		df-esum 34660	. 2 ⊢ Σ^𝑘 ∈ 𝐴𝐶 = ∪ ((ℝ^_𝑠 ↾_s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐶))
12		df-esum 34660	. 2 ⊢ Σ^𝑘 ∈ 𝐵𝐷 = ∪ ((ℝ^_𝑠 ↾_s (0[,]+∞)) tsums (𝑘 ∈ 𝐵 ↦ 𝐷))
13	10, 11, 12	3eqtr4g 2820	1 ⊢ (𝜑 → Σ^𝑘 ∈ 𝐴𝐶 = Σ^𝑘 ∈ 𝐵𝐷)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∀wal 1567 = wceq 1569 Ⅎwnf 1815 ∈ wcel 2144 ∀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: esumeq12dva 34664 esumeq1d 34667 esumeq2d 34669 esumpinfval 34705 measvunilem0 34846

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