esumeq12dvaf - Mathbox for Thierry Arnoux

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Theorem esumeq12dvaf 34012

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 2211	. . . . 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 3255	. . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐶 = 𝐷)
7		mpteq12f 5236	. . . . 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 7447	. . 3 ⊢ (𝜑 → ((ℝ^_𝑠 ↾_s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐶)) = ((ℝ^_𝑠 ↾_s (0[,]+∞)) tsums (𝑘 ∈ 𝐵 ↦ 𝐷)))
10	9	unieqd 4925	. 2 ⊢ (𝜑 → ∪ ((ℝ^_𝑠 ↾_s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐶)) = ∪ ((ℝ^_𝑠 ↾_s (0[,]+∞)) tsums (𝑘 ∈ 𝐵 ↦ 𝐷)))
11		df-esum 34009	. 2 ⊢ Σ^𝑘 ∈ 𝐴𝐶 = ∪ ((ℝ^_𝑠 ↾_s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐶))
12		df-esum 34009	. 2 ⊢ Σ^𝑘 ∈ 𝐵𝐷 = ∪ ((ℝ^_𝑠 ↾_s (0[,]+∞)) tsums (𝑘 ∈ 𝐵 ↦ 𝐷))
13	10, 11, 12	3eqtr4g 2800	1 ⊢ (𝜑 → Σ^𝑘 ∈ 𝐴𝐶 = Σ^𝑘 ∈ 𝐵𝐷)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∀wal 1535 = wceq 1537 Ⅎwnf 1780 ∈ wcel 2106 ∀wral 3059 ∪ cuni 4912 ↦ cmpt 5231 (class class class)co 7431 0cc0 11153 +∞cpnf 11290 [,]cicc 13387 ↾_s cress 17274 ℝ^*_𝑠cxrs 17547 tsums ctsu 24150 Σ^*cesum 34008

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-12 2175 ax-ext 2706

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-iota 6516 df-fv 6571 df-ov 7434 df-esum 34009

This theorem is referenced by: esumeq12dva 34013 esumeq1d 34016 esumeq2d 34018 esumpinfval 34054 measvunilem0 34194

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