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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 23437 | . 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 14462 | . 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 2726 | 1 ⊢ ∫𝐴𝐵 d𝑥 ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 ∈ wcel 2030 Vcvv 3231 ⦋csb 3566 ifcif 4119 class class class wbr 4685 ↦ cmpt 4762 ‘cfv 5926 (class class class)co 6690 ℝcr 9973 0cc0 9974 ici 9976 · cmul 9979 ≤ cle 10113 / cdiv 10722 3c3 11109 ...cfz 12364 ↑cexp 12900 ℜcre 13881 Σcsu 14460 ∫2citg2 23430 ∫citg 23432 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-nul 4822 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ral 2946 df-rex 2947 df-v 3233 df-sbc 3469 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-sn 4211 df-pr 4213 df-uni 4469 df-iota 5889 df-sum 14461 df-itg 23437 |
This theorem is referenced by: ditgex 23661 ftc1lem1 23843 itgulm 24207 dmarea 24729 dfarea 24732 areaval 24736 ftc1anc 33623 itgsinexp 40488 wallispilem1 40600 wallispilem2 40601 |
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