esumeq12dvaf - Mathbox for Thierry Arnoux

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Theorem esumeq12dvaf 33327

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 2204	. . . . 5 ⊢ (𝜑 → ∀𝑘 𝐴 = 𝐵)
4		esumeq12dvaf.3	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 = 𝐷)
5	4	ex 411	. . . . . 6 ⊢ (𝜑 → (𝑘 ∈ 𝐴 → 𝐶 = 𝐷))
6	1, 5	ralrimi 3252	. . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐶 = 𝐷)
7		mpteq12f 5235	. . . . 5 ⊢ ((∀𝑘 𝐴 = 𝐵 ∧ ∀𝑘 ∈ 𝐴 𝐶 = 𝐷) → (𝑘 ∈ 𝐴 ↦ 𝐶) = (𝑘 ∈ 𝐵 ↦ 𝐷))
8	3, 6, 7	syl2anc 582	. . . 4 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐶) = (𝑘 ∈ 𝐵 ↦ 𝐷))
9	8	oveq2d 7427	. . 3 ⊢ (𝜑 → ((ℝ^_𝑠 ↾_s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐶)) = ((ℝ^_𝑠 ↾_s (0[,]+∞)) tsums (𝑘 ∈ 𝐵 ↦ 𝐷)))
10	9	unieqd 4921	. 2 ⊢ (𝜑 → ∪ ((ℝ^_𝑠 ↾_s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐶)) = ∪ ((ℝ^_𝑠 ↾_s (0[,]+∞)) tsums (𝑘 ∈ 𝐵 ↦ 𝐷)))
11		df-esum 33324	. 2 ⊢ Σ^𝑘 ∈ 𝐴𝐶 = ∪ ((ℝ^_𝑠 ↾_s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐶))
12		df-esum 33324	. 2 ⊢ Σ^𝑘 ∈ 𝐵𝐷 = ∪ ((ℝ^_𝑠 ↾_s (0[,]+∞)) tsums (𝑘 ∈ 𝐵 ↦ 𝐷))
13	10, 11, 12	3eqtr4g 2795	1 ⊢ (𝜑 → Σ^𝑘 ∈ 𝐴𝐶 = Σ^𝑘 ∈ 𝐵𝐷)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 ∀wal 1537 = wceq 1539 Ⅎwnf 1783 ∈ wcel 2104 ∀wral 3059 ∪ cuni 4907 ↦ cmpt 5230 (class class class)co 7411 0cc0 11112 +∞cpnf 11249 [,]cicc 13331 ↾_s cress 17177 ℝ^*_𝑠cxrs 17450 tsums ctsu 23850 Σ^*cesum 33323

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-12 2169 ax-ext 2701

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-clab 2708 df-cleq 2722 df-clel 2808 df-ral 3060 df-rab 3431 df-v 3474 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-iota 6494 df-fv 6550 df-ov 7414 df-esum 33324

This theorem is referenced by: esumeq12dva 33328 esumeq1d 33331 esumeq2d 33333 esumpinfval 33369 measvunilem0 33509

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