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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > esumeq12dvaf | Structured version Visualization version GIF version |
Description: Equality deduction for extended sum. (Contributed by Thierry Arnoux, 26-Mar-2017.) |
Ref | Expression |
---|---|
esumeq12dvaf.1 | ⊢ Ⅎ𝑘𝜑 |
esumeq12dvaf.2 | ⊢ (𝜑 → 𝐴 = 𝐵) |
esumeq12dvaf.3 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 = 𝐷) |
Ref | Expression |
---|---|
esumeq12dvaf | ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐶 = Σ*𝑘 ∈ 𝐵𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | esumeq12dvaf.1 | . . . . . 6 ⊢ Ⅎ𝑘𝜑 | |
2 | esumeq12dvaf.2 | . . . . . 6 ⊢ (𝜑 → 𝐴 = 𝐵) | |
3 | 1, 2 | alrimi 2201 | . . . . 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 3245 | . . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐶 = 𝐷) |
7 | mpteq12f 5232 | . . . . 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 7429 | . . 3 ⊢ (𝜑 → ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐶)) = ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐵 ↦ 𝐷))) |
10 | 9 | unieqd 4917 | . 2 ⊢ (𝜑 → ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐶)) = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐵 ↦ 𝐷))) |
11 | df-esum 33700 | . 2 ⊢ Σ*𝑘 ∈ 𝐴𝐶 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐶)) | |
12 | df-esum 33700 | . 2 ⊢ Σ*𝑘 ∈ 𝐵𝐷 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐵 ↦ 𝐷)) | |
13 | 10, 11, 12 | 3eqtr4g 2790 | 1 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐶 = Σ*𝑘 ∈ 𝐵𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∀wal 1531 = wceq 1533 Ⅎwnf 1777 ∈ wcel 2098 ∀wral 3051 ∪ cuni 4904 ↦ cmpt 5227 (class class class)co 7413 0cc0 11133 +∞cpnf 11270 [,]cicc 13354 ↾s cress 17203 ℝ*𝑠cxrs 17476 tsums ctsu 24043 Σ*cesum 33699 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-12 2166 ax-ext 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ral 3052 df-rab 3420 df-v 3465 df-dif 3944 df-un 3946 df-ss 3958 df-nul 4320 df-if 4526 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-br 5145 df-opab 5207 df-mpt 5228 df-iota 6495 df-fv 6551 df-ov 7416 df-esum 33700 |
This theorem is referenced by: esumeq12dva 33704 esumeq1d 33707 esumeq2d 33709 esumpinfval 33745 measvunilem0 33885 |
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