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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 25658 | . 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 15724 | . 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 2837 | 1 ⊢ ∫𝐴𝐵 d𝑥 ∈ V |
| Colors of variables: wff setvar class |
| Syntax hints: ∧ wa 395 ∈ wcel 2108 Vcvv 3480 ⦋csb 3899 ifcif 4525 class class class wbr 5143 ↦ cmpt 5225 ‘cfv 6561 (class class class)co 7431 ℝcr 11154 0cc0 11155 ici 11157 · cmul 11160 ≤ cle 11296 / cdiv 11920 3c3 12322 ...cfz 13547 ↑cexp 14102 ℜcre 15136 Σcsu 15722 ∫2citg2 25651 ∫citg 25653 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-nul 5306 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-sn 4627 df-pr 4629 df-uni 4908 df-iota 6514 df-sum 15723 df-itg 25658 |
| This theorem is referenced by: ditgex 25887 ftc1lem1 26076 itgulm 26451 dmarea 27000 dfarea 27003 areaval 27007 ftc1anc 37708 itgsinexp 45970 wallispilem1 46080 wallispilem2 46081 |
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