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Mirrors > Home > MPE Home > Th. List > Mathboxes > esumeq2 | Structured version Visualization version GIF version |
Description: Equality theorem for extended sum. (Contributed by Thierry Arnoux, 24-Dec-2016.) |
Ref | Expression |
---|---|
esumeq2 | ⊢ (∀𝑘 ∈ 𝐴 𝐵 = 𝐶 → Σ*𝑘 ∈ 𝐴𝐵 = Σ*𝑘 ∈ 𝐴𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2735 | . . . . 5 ⊢ 𝐴 = 𝐴 | |
2 | mpteq12 5240 | . . . . 5 ⊢ ((𝐴 = 𝐴 ∧ ∀𝑘 ∈ 𝐴 𝐵 = 𝐶) → (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐶)) | |
3 | 1, 2 | mpan 690 | . . . 4 ⊢ (∀𝑘 ∈ 𝐴 𝐵 = 𝐶 → (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐶)) |
4 | 3 | oveq2d 7447 | . . 3 ⊢ (∀𝑘 ∈ 𝐴 𝐵 = 𝐶 → ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) = ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐶))) |
5 | 4 | unieqd 4925 | . 2 ⊢ (∀𝑘 ∈ 𝐴 𝐵 = 𝐶 → ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐶))) |
6 | df-esum 34009 | . 2 ⊢ Σ*𝑘 ∈ 𝐴𝐵 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) | |
7 | df-esum 34009 | . 2 ⊢ Σ*𝑘 ∈ 𝐴𝐶 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐶)) | |
8 | 5, 6, 7 | 3eqtr4g 2800 | 1 ⊢ (∀𝑘 ∈ 𝐴 𝐵 = 𝐶 → Σ*𝑘 ∈ 𝐴𝐵 = Σ*𝑘 ∈ 𝐴𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∀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: (None) |
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