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 23198 | . 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 14266 | . 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 2684 | 1 ⊢ ∫𝐴𝐵 d𝑥 ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 ∈ wcel 1977 Vcvv 3173 ⦋csb 3499 ifcif 4036 class class class wbr 4583 ↦ cmpt 4643 ‘cfv 5804 (class class class)co 6549 ℝcr 9814 0cc0 9815 ici 9817 · cmul 9820 ≤ cle 9954 / cdiv 10563 3c3 10948 ...cfz 12197 ↑cexp 12722 ℜcre 13685 Σcsu 14264 ∫2citg2 23191 ∫citg 23193 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-nul 4717 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-sn 4126 df-pr 4128 df-uni 4373 df-iota 5768 df-sum 14265 df-itg 23198 |
This theorem is referenced by: ditgex 23422 ftc1lem1 23602 itgulm 23966 dmarea 24484 dfarea 24487 areaval 24491 ftc1anc 32663 itgsinexp 38846 wallispilem1 38958 wallispilem2 38959 |
Copyright terms: Public domain | W3C validator |