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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 23917 | . 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 14895 | . 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 2856 | 1 ⊢ ∫𝐴𝐵 d𝑥 ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 387 ∈ wcel 2048 Vcvv 3409 ⦋csb 3782 ifcif 4344 class class class wbr 4923 ↦ cmpt 5002 ‘cfv 6182 (class class class)co 6970 ℝcr 10326 0cc0 10327 ici 10329 · cmul 10332 ≤ cle 10467 / cdiv 11090 3c3 11489 ...cfz 12701 ↑cexp 13237 ℜcre 14307 Σcsu 14893 ∫2citg2 23910 ∫citg 23912 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-ext 2745 ax-nul 5061 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ral 3087 df-rex 3088 df-v 3411 df-sbc 3678 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-nul 4174 df-sn 4436 df-pr 4438 df-uni 4707 df-iota 6146 df-sum 14894 df-itg 23917 |
This theorem is referenced by: ditgex 24143 ftc1lem1 24325 itgulm 24689 dmarea 25227 dfarea 25230 areaval 25234 ftc1anc 34364 itgsinexp 41616 wallispilem1 41727 wallispilem2 41728 |
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