esumeq12dvaf - Mathbox for Thierry Arnoux

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Theorem esumeq12dvaf 34227

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 2227	. . . . 5 ⊢ (𝜑 → ∀𝑘 𝐴 = 𝐵)
4		esumeq12dvaf.3	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 = 𝐷)
5	4	ex 414	. . . . . 6 ⊢ (𝜑 → (𝑘 ∈ 𝐴 → 𝐶 = 𝐷))
6	1, 5	ralrimi 3239	. . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐶 = 𝐷)
7		mpteq12f 5160	. . . . 5 ⊢ ((∀𝑘 𝐴 = 𝐵 ∧ ∀𝑘 ∈ 𝐴 𝐶 = 𝐷) → (𝑘 ∈ 𝐴 ↦ 𝐶) = (𝑘 ∈ 𝐵 ↦ 𝐷))
8	3, 6, 7	syl2anc 591	. . . 4 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐶) = (𝑘 ∈ 𝐵 ↦ 𝐷))
9	8	oveq2d 7376	. . 3 ⊢ (𝜑 → ((ℝ^_𝑠 ↾_s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐶)) = ((ℝ^_𝑠 ↾_s (0[,]+∞)) tsums (𝑘 ∈ 𝐵 ↦ 𝐷)))
10	9	unieqd 4854	. 2 ⊢ (𝜑 → ∪ ((ℝ^_𝑠 ↾_s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐶)) = ∪ ((ℝ^_𝑠 ↾_s (0[,]+∞)) tsums (𝑘 ∈ 𝐵 ↦ 𝐷)))
11		df-esum 34224	. 2 ⊢ Σ^𝑘 ∈ 𝐴𝐶 = ∪ ((ℝ^_𝑠 ↾_s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐶))
12		df-esum 34224	. 2 ⊢ Σ^𝑘 ∈ 𝐵𝐷 = ∪ ((ℝ^_𝑠 ↾_s (0[,]+∞)) tsums (𝑘 ∈ 𝐵 ↦ 𝐷))
13	10, 11, 12	3eqtr4g 2801	1 ⊢ (𝜑 → Σ^𝑘 ∈ 𝐴𝐶 = Σ^𝑘 ∈ 𝐵𝐷)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 397 ∀wal 1546 = wceq 1548 Ⅎwnf 1791 ∈ wcel 2121 ∀wral 3055 ∪ cuni 4841 ↦ cmpt 5156 (class class class)co 7360 0cc0 11033 +∞cpnf 11171 [,]cicc 13296 ↾_s cress 17195 ℝ^*_𝑠cxrs 17459 tsums ctsu 24113 Σ^*cesum 34223

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-10 2154 ax-12 2191 ax-ext 2713

This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3an 1095 df-tru 1551 df-fal 1561 df-ex 1788 df-nf 1792 df-sb 2075 df-clab 2720 df-cleq 2733 df-clel 2816 df-ral 3056 df-rab 3394 df-v 3435 df-dif 3888 df-un 3890 df-ss 3902 df-nul 4265 df-if 4458 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4842 df-br 5076 df-opab 5138 df-mpt 5157 df-iota 6445 df-fv 6497 df-ov 7363 df-esum 34224

This theorem is referenced by: esumeq12dva 34228 esumeq1d 34231 esumeq2d 34233 esumpinfval 34269 measvunilem0 34409

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