Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 23417 | . 2 ⊢ ⨜[𝐴 → 𝐵]𝐶 d𝑥 = if(𝐴 ≤ 𝐵, ∫(𝐴(,)𝐵)𝐶 d𝑥, -∫(𝐵(,)𝐴)𝐶 d𝑥) | |
2 | itgex 23343 | . . 3 ⊢ ∫(𝐴(,)𝐵)𝐶 d𝑥 ∈ V | |
3 | negex 10158 | . . 3 ⊢ -∫(𝐵(,)𝐴)𝐶 d𝑥 ∈ V | |
4 | 2, 3 | ifex 4106 | . 2 ⊢ if(𝐴 ≤ 𝐵, ∫(𝐴(,)𝐵)𝐶 d𝑥, -∫(𝐵(,)𝐴)𝐶 d𝑥) ∈ V |
5 | 1, 4 | eqeltri 2684 | 1 ⊢ ⨜[𝐴 → 𝐵]𝐶 d𝑥 ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1977 Vcvv 3173 ifcif 4036 class class class wbr 4583 (class class class)co 6549 ≤ cle 9954 -cneg 10146 (,)cioo 12046 ∫citg 23193 ⨜cdit 23416 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-nul 4717 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-uni 4373 df-iota 5768 df-fv 5812 df-ov 6552 df-neg 10148 df-sum 14265 df-itg 23198 df-ditg 23417 |
This theorem is referenced by: itgsubstlem 23615 |
Copyright terms: Public domain | W3C validator |