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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 23611 | . 2 ⊢ ⨜[𝐴 → 𝐵]𝐶 d𝑥 = if(𝐴 ≤ 𝐵, ∫(𝐴(,)𝐵)𝐶 d𝑥, -∫(𝐵(,)𝐴)𝐶 d𝑥) | |
2 | itgex 23537 | . . 3 ⊢ ∫(𝐴(,)𝐵)𝐶 d𝑥 ∈ V | |
3 | negex 10279 | . . 3 ⊢ -∫(𝐵(,)𝐴)𝐶 d𝑥 ∈ V | |
4 | 2, 3 | ifex 4156 | . 2 ⊢ if(𝐴 ≤ 𝐵, ∫(𝐴(,)𝐵)𝐶 d𝑥, -∫(𝐵(,)𝐴)𝐶 d𝑥) ∈ V |
5 | 1, 4 | eqeltri 2697 | 1 ⊢ ⨜[𝐴 → 𝐵]𝐶 d𝑥 ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1990 Vcvv 3200 ifcif 4086 class class class wbr 4653 (class class class)co 6650 ≤ cle 10075 -cneg 10267 (,)cioo 12175 ∫citg 23387 ⨜cdit 23610 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-nul 4789 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-uni 4437 df-iota 5851 df-fv 5896 df-ov 6653 df-neg 10269 df-sum 14417 df-itg 23392 df-ditg 23611 |
This theorem is referenced by: itgsubstlem 23811 |
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