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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 24448 | . 2 ⊢ ⨜[𝐴 → 𝐵]𝐶 d𝑥 = if(𝐴 ≤ 𝐵, ∫(𝐴(,)𝐵)𝐶 d𝑥, -∫(𝐵(,)𝐴)𝐶 d𝑥) | |
2 | itgex 24374 | . . 3 ⊢ ∫(𝐴(,)𝐵)𝐶 d𝑥 ∈ V | |
3 | negex 10887 | . . 3 ⊢ -∫(𝐵(,)𝐴)𝐶 d𝑥 ∈ V | |
4 | 2, 3 | ifex 4518 | . 2 ⊢ if(𝐴 ≤ 𝐵, ∫(𝐴(,)𝐵)𝐶 d𝑥, -∫(𝐵(,)𝐴)𝐶 d𝑥) ∈ V |
5 | 1, 4 | eqeltri 2912 | 1 ⊢ ⨜[𝐴 → 𝐵]𝐶 d𝑥 ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2113 Vcvv 3497 ifcif 4470 class class class wbr 5069 (class class class)co 7159 ≤ cle 10679 -cneg 10874 (,)cioo 12741 ∫citg 24222 ⨜cdit 24447 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-nul 5213 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-uni 4842 df-iota 6317 df-fv 6366 df-ov 7162 df-neg 10876 df-sum 15046 df-itg 24227 df-ditg 24448 |
This theorem is referenced by: itgsubstlem 24648 |
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