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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 533 | . . . 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 11255 | . . . 4 ⊢ (𝜑 → (𝐴 = 𝐵 ↔ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴))) |
6 | 2, 5 | bitr4d 281 | . . 3 ⊢ (𝜑 → (𝐵 ≤ 𝐴 ↔ 𝐴 = 𝐵)) |
7 | ditg0 25169 | . . . . 5 ⊢ ⨜[𝐵 → 𝐵]𝐶 d𝑥 = 0 | |
8 | neg0 11405 | . . . . 5 ⊢ -0 = 0 | |
9 | 7, 8 | eqtr4i 2768 | . . . 4 ⊢ ⨜[𝐵 → 𝐵]𝐶 d𝑥 = -0 |
10 | ditgeq2 25165 | . . . 4 ⊢ (𝐴 = 𝐵 → ⨜[𝐵 → 𝐴]𝐶 d𝑥 = ⨜[𝐵 → 𝐵]𝐶 d𝑥) | |
11 | oveq1 7358 | . . . . . . . 8 ⊢ (𝐴 = 𝐵 → (𝐴(,)𝐵) = (𝐵(,)𝐵)) | |
12 | iooid 13246 | . . . . . . . 8 ⊢ (𝐵(,)𝐵) = ∅ | |
13 | 11, 12 | eqtrdi 2793 | . . . . . . 7 ⊢ (𝐴 = 𝐵 → (𝐴(,)𝐵) = ∅) |
14 | itgeq1 25089 | . . . . . . 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 25096 | . . . . . 6 ⊢ ∫∅𝐶 d𝑥 = 0 | |
17 | 15, 16 | eqtrdi 2793 | . . . . 5 ⊢ (𝐴 = 𝐵 → ∫(𝐴(,)𝐵)𝐶 d𝑥 = 0) |
18 | 17 | negeqd 11353 | . . . 4 ⊢ (𝐴 = 𝐵 → -∫(𝐴(,)𝐵)𝐶 d𝑥 = -0) |
19 | 9, 10, 18 | 3eqtr4a 2803 | . . 3 ⊢ (𝐴 = 𝐵 → ⨜[𝐵 → 𝐴]𝐶 d𝑥 = -∫(𝐴(,)𝐵)𝐶 d𝑥) |
20 | 6, 19 | syl6bi 252 | . 2 ⊢ (𝜑 → (𝐵 ≤ 𝐴 → ⨜[𝐵 → 𝐴]𝐶 d𝑥 = -∫(𝐴(,)𝐵)𝐶 d𝑥)) |
21 | df-ditg 25163 | . . 3 ⊢ ⨜[𝐵 → 𝐴]𝐶 d𝑥 = if(𝐵 ≤ 𝐴, ∫(𝐵(,)𝐴)𝐶 d𝑥, -∫(𝐴(,)𝐵)𝐶 d𝑥) | |
22 | iffalse 4493 | . . 3 ⊢ (¬ 𝐵 ≤ 𝐴 → if(𝐵 ≤ 𝐴, ∫(𝐵(,)𝐴)𝐶 d𝑥, -∫(𝐴(,)𝐵)𝐶 d𝑥) = -∫(𝐴(,)𝐵)𝐶 d𝑥) | |
23 | 21, 22 | eqtrid 2789 | . 2 ⊢ (¬ 𝐵 ≤ 𝐴 → ⨜[𝐵 → 𝐴]𝐶 d𝑥 = -∫(𝐴(,)𝐵)𝐶 d𝑥) |
24 | 20, 23 | pm2.61d1 180 | 1 ⊢ (𝜑 → ⨜[𝐵 → 𝐴]𝐶 d𝑥 = -∫(𝐴(,)𝐵)𝐶 d𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∅c0 4280 ifcif 4484 class class class wbr 5103 (class class class)co 7351 ℝcr 11008 0cc0 11009 ≤ cle 11148 -cneg 11344 (,)cioo 13218 ∫citg 24934 ⨜cdit 25162 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-inf2 9535 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087 ax-addf 11088 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-disj 5069 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-se 5587 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-of 7609 df-ofr 7610 df-om 7795 df-1st 7913 df-2nd 7914 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-1o 8404 df-2o 8405 df-er 8606 df-map 8725 df-pm 8726 df-en 8842 df-dom 8843 df-sdom 8844 df-fin 8845 df-sup 9336 df-inf 9337 df-oi 9404 df-dju 9795 df-card 9833 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-sub 11345 df-neg 11346 df-div 11771 df-nn 12112 df-2 12174 df-3 12175 df-n0 12372 df-z 12458 df-uz 12722 df-q 12828 df-rp 12870 df-xadd 12988 df-ioo 13222 df-ico 13224 df-icc 13225 df-fz 13379 df-fzo 13522 df-fl 13651 df-seq 13861 df-exp 13922 df-hash 14185 df-cj 14944 df-re 14945 df-im 14946 df-sqrt 15080 df-abs 15081 df-clim 15330 df-sum 15531 df-xmet 20742 df-met 20743 df-ovol 24780 df-vol 24781 df-mbf 24935 df-itg1 24936 df-itg2 24937 df-itg 24939 df-0p 24986 df-ditg 25163 |
This theorem is referenced by: ditgcl 25174 ditgswap 25175 |
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