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Mirrors > Home > MPE Home > Th. List > itgex | Structured version Visualization version GIF version |
Description: An integral is a set. (Contributed by Mario Carneiro, 28-Jun-2014.) |
Ref | Expression |
---|---|
itgex | ⊢ ∫𝐴𝐵 d𝑥 ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-itg 25472 | . 2 ⊢ ∫𝐴𝐵 d𝑥 = Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0)))) | |
2 | sumex 15641 | . 2 ⊢ Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0)))) ∈ V | |
3 | 1, 2 | eqeltri 2828 | 1 ⊢ ∫𝐴𝐵 d𝑥 ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 ∈ wcel 2105 Vcvv 3473 ⦋csb 3893 ifcif 4528 class class class wbr 5148 ↦ cmpt 5231 ‘cfv 6543 (class class class)co 7412 ℝcr 11115 0cc0 11116 ici 11118 · cmul 11121 ≤ cle 11256 / cdiv 11878 3c3 12275 ...cfz 13491 ↑cexp 14034 ℜcre 15051 Σcsu 15639 ∫2citg2 25465 ∫citg 25467 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2702 ax-nul 5306 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1543 df-fal 1553 df-ex 1781 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-ne 2940 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-sn 4629 df-pr 4631 df-uni 4909 df-iota 6495 df-sum 15640 df-itg 25472 |
This theorem is referenced by: ditgex 25701 ftc1lem1 25890 itgulm 26259 dmarea 26803 dfarea 26806 areaval 26810 ftc1anc 37033 itgsinexp 45130 wallispilem1 45240 wallispilem2 45241 |
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