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Mirrors > Home > MPE Home > Th. List > ditgex | Structured version Visualization version GIF version |
Description: A directed integral is a set. (Contributed by Mario Carneiro, 7-Sep-2014.) |
Ref | Expression |
---|---|
ditgex | ⊢ ⨜[𝐴 → 𝐵]𝐶 d𝑥 ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ditg 25364 | . 2 ⊢ ⨜[𝐴 → 𝐵]𝐶 d𝑥 = if(𝐴 ≤ 𝐵, ∫(𝐴(,)𝐵)𝐶 d𝑥, -∫(𝐵(,)𝐴)𝐶 d𝑥) | |
2 | itgex 25288 | . . 3 ⊢ ∫(𝐴(,)𝐵)𝐶 d𝑥 ∈ V | |
3 | negex 11458 | . . 3 ⊢ -∫(𝐵(,)𝐴)𝐶 d𝑥 ∈ V | |
4 | 2, 3 | ifex 4579 | . 2 ⊢ if(𝐴 ≤ 𝐵, ∫(𝐴(,)𝐵)𝐶 d𝑥, -∫(𝐵(,)𝐴)𝐶 d𝑥) ∈ V |
5 | 1, 4 | eqeltri 2830 | 1 ⊢ ⨜[𝐴 → 𝐵]𝐶 d𝑥 ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2107 Vcvv 3475 ifcif 4529 class class class wbr 5149 (class class class)co 7409 ≤ cle 11249 -cneg 11445 (,)cioo 13324 ∫citg 25135 ⨜cdit 25363 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-nul 5307 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2942 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-uni 4910 df-iota 6496 df-fv 6552 df-ov 7412 df-neg 11447 df-sum 15633 df-itg 25140 df-ditg 25364 |
This theorem is referenced by: itgsubstlem 25565 |
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