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Mirrors > Home > MPE Home > Th. List > Mathboxes > esumex | Structured version Visualization version GIF version |
Description: An extended sum is a set by definition. (Contributed by Thierry Arnoux, 5-Sep-2017.) |
Ref | Expression |
---|---|
esumex | ⊢ Σ*𝑘 ∈ 𝐴𝐵 ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-esum 31186 | . 2 ⊢ Σ*𝑘 ∈ 𝐴𝐵 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) | |
2 | ovex 7178 | . . 3 ⊢ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) ∈ V | |
3 | 2 | uniex 7454 | . 2 ⊢ ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) ∈ V |
4 | 1, 3 | eqeltri 2906 | 1 ⊢ Σ*𝑘 ∈ 𝐴𝐵 ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2105 Vcvv 3492 ∪ cuni 4830 ↦ cmpt 5137 (class class class)co 7145 0cc0 10525 +∞cpnf 10660 [,]cicc 12729 ↾s cress 16472 ℝ*𝑠cxrs 16761 tsums ctsu 22661 Σ*cesum 31185 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-sn 4558 df-pr 4560 df-uni 4831 df-iota 6307 df-fv 6356 df-ov 7148 df-esum 31186 |
This theorem is referenced by: esumcvg 31244 esumgect 31248 omssubaddlem 31456 omssubadd 31457 |
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