![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 25672 | . 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 15721 | . 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 2835 | 1 ⊢ ∫𝐴𝐵 d𝑥 ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 ∈ wcel 2106 Vcvv 3478 ⦋csb 3908 ifcif 4531 class class class wbr 5148 ↦ cmpt 5231 ‘cfv 6563 (class class class)co 7431 ℝcr 11152 0cc0 11153 ici 11155 · cmul 11158 ≤ cle 11294 / cdiv 11918 3c3 12320 ...cfz 13544 ↑cexp 14099 ℜcre 15133 Σcsu 15719 ∫2citg2 25665 ∫citg 25667 |
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-ext 2706 ax-nul 5312 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-sn 4632 df-pr 4634 df-uni 4913 df-iota 6516 df-sum 15720 df-itg 25672 |
This theorem is referenced by: ditgex 25902 ftc1lem1 26091 itgulm 26466 dmarea 27015 dfarea 27018 areaval 27022 ftc1anc 37688 itgsinexp 45911 wallispilem1 46021 wallispilem2 46022 |
Copyright terms: Public domain | W3C validator |