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Mirrors > Home > MPE Home > Th. List > ditgneg | Structured version Visualization version GIF version |
Description: Value of the directed integral in the backward direction. (Contributed by Mario Carneiro, 13-Aug-2014.) |
Ref | Expression |
---|---|
ditgpos.1 | ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
ditgneg.2 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
ditgneg.3 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
Ref | Expression |
---|---|
ditgneg | ⊢ (𝜑 → ⨜[𝐵 → 𝐴]𝐶 d𝑥 = -∫(𝐴(,)𝐵)𝐶 d𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ditgpos.1 | . . . . 5 ⊢ (𝜑 → 𝐴 ≤ 𝐵) | |
2 | 1 | biantrurd 535 | . . . 4 ⊢ (𝜑 → (𝐵 ≤ 𝐴 ↔ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴))) |
3 | ditgneg.2 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
4 | ditgneg.3 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
5 | 3, 4 | letri3d 10776 | . . . 4 ⊢ (𝜑 → (𝐴 = 𝐵 ↔ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴))) |
6 | 2, 5 | bitr4d 284 | . . 3 ⊢ (𝜑 → (𝐵 ≤ 𝐴 ↔ 𝐴 = 𝐵)) |
7 | ditg0 24445 | . . . . 5 ⊢ ⨜[𝐵 → 𝐵]𝐶 d𝑥 = 0 | |
8 | neg0 10926 | . . . . 5 ⊢ -0 = 0 | |
9 | 7, 8 | eqtr4i 2847 | . . . 4 ⊢ ⨜[𝐵 → 𝐵]𝐶 d𝑥 = -0 |
10 | ditgeq2 24441 | . . . 4 ⊢ (𝐴 = 𝐵 → ⨜[𝐵 → 𝐴]𝐶 d𝑥 = ⨜[𝐵 → 𝐵]𝐶 d𝑥) | |
11 | oveq1 7157 | . . . . . . . 8 ⊢ (𝐴 = 𝐵 → (𝐴(,)𝐵) = (𝐵(,)𝐵)) | |
12 | iooid 12760 | . . . . . . . 8 ⊢ (𝐵(,)𝐵) = ∅ | |
13 | 11, 12 | syl6eq 2872 | . . . . . . 7 ⊢ (𝐴 = 𝐵 → (𝐴(,)𝐵) = ∅) |
14 | itgeq1 24367 | . . . . . . 7 ⊢ ((𝐴(,)𝐵) = ∅ → ∫(𝐴(,)𝐵)𝐶 d𝑥 = ∫∅𝐶 d𝑥) | |
15 | 13, 14 | syl 17 | . . . . . 6 ⊢ (𝐴 = 𝐵 → ∫(𝐴(,)𝐵)𝐶 d𝑥 = ∫∅𝐶 d𝑥) |
16 | itg0 24374 | . . . . . 6 ⊢ ∫∅𝐶 d𝑥 = 0 | |
17 | 15, 16 | syl6eq 2872 | . . . . 5 ⊢ (𝐴 = 𝐵 → ∫(𝐴(,)𝐵)𝐶 d𝑥 = 0) |
18 | 17 | negeqd 10874 | . . . 4 ⊢ (𝐴 = 𝐵 → -∫(𝐴(,)𝐵)𝐶 d𝑥 = -0) |
19 | 9, 10, 18 | 3eqtr4a 2882 | . . 3 ⊢ (𝐴 = 𝐵 → ⨜[𝐵 → 𝐴]𝐶 d𝑥 = -∫(𝐴(,)𝐵)𝐶 d𝑥) |
20 | 6, 19 | syl6bi 255 | . 2 ⊢ (𝜑 → (𝐵 ≤ 𝐴 → ⨜[𝐵 → 𝐴]𝐶 d𝑥 = -∫(𝐴(,)𝐵)𝐶 d𝑥)) |
21 | df-ditg 24439 | . . 3 ⊢ ⨜[𝐵 → 𝐴]𝐶 d𝑥 = if(𝐵 ≤ 𝐴, ∫(𝐵(,)𝐴)𝐶 d𝑥, -∫(𝐴(,)𝐵)𝐶 d𝑥) | |
22 | iffalse 4475 | . . 3 ⊢ (¬ 𝐵 ≤ 𝐴 → if(𝐵 ≤ 𝐴, ∫(𝐵(,)𝐴)𝐶 d𝑥, -∫(𝐴(,)𝐵)𝐶 d𝑥) = -∫(𝐴(,)𝐵)𝐶 d𝑥) | |
23 | 21, 22 | syl5eq 2868 | . 2 ⊢ (¬ 𝐵 ≤ 𝐴 → ⨜[𝐵 → 𝐴]𝐶 d𝑥 = -∫(𝐴(,)𝐵)𝐶 d𝑥) |
24 | 20, 23 | pm2.61d1 182 | 1 ⊢ (𝜑 → ⨜[𝐵 → 𝐴]𝐶 d𝑥 = -∫(𝐴(,)𝐵)𝐶 d𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∅c0 4290 ifcif 4466 class class class wbr 5058 (class class class)co 7150 ℝcr 10530 0cc0 10531 ≤ cle 10670 -cneg 10865 (,)cioo 12732 ∫citg 24213 ⨜cdit 24438 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-inf2 9098 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 ax-addf 10610 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-disj 5024 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-se 5509 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-ofr 7404 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-oadd 8100 df-er 8283 df-map 8402 df-pm 8403 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-sup 8900 df-inf 8901 df-oi 8968 df-dju 9324 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-n0 11892 df-z 11976 df-uz 12238 df-q 12343 df-rp 12384 df-xadd 12502 df-ioo 12736 df-ico 12738 df-icc 12739 df-fz 12887 df-fzo 13028 df-fl 13156 df-seq 13364 df-exp 13424 df-hash 13685 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-clim 14839 df-sum 15037 df-xmet 20532 df-met 20533 df-ovol 24059 df-vol 24060 df-mbf 24214 df-itg1 24215 df-itg2 24216 df-itg 24218 df-0p 24265 df-ditg 24439 |
This theorem is referenced by: ditgcl 24450 ditgswap 24451 |
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