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Mathbox for Thierry Arnoux |
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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 30430 | . 2 ⊢ Σ*𝑘 ∈ 𝐴𝐵 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) | |
2 | ovex 6823 | . . 3 ⊢ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) ∈ V | |
3 | 2 | uniex 7100 | . 2 ⊢ ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) ∈ V |
4 | 1, 3 | eqeltri 2846 | 1 ⊢ Σ*𝑘 ∈ 𝐴𝐵 ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2145 Vcvv 3351 ∪ cuni 4574 ↦ cmpt 4863 (class class class)co 6793 0cc0 10138 +∞cpnf 10273 [,]cicc 12383 ↾s cress 16065 ℝ*𝑠cxrs 16368 tsums ctsu 22149 Σ*cesum 30429 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-un 7096 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ral 3066 df-rex 3067 df-v 3353 df-sbc 3588 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-sn 4317 df-pr 4319 df-uni 4575 df-iota 5994 df-fv 6039 df-ov 6796 df-esum 30430 |
This theorem is referenced by: esumcvg 30488 esumgect 30492 omssubaddlem 30701 omssubadd 30702 |
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