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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 529 | . . . 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 10179 | . . . 4 ⊢ (𝜑 → (𝐴 = 𝐵 ↔ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴))) |
6 | 2, 5 | bitr4d 271 | . . 3 ⊢ (𝜑 → (𝐵 ≤ 𝐴 ↔ 𝐴 = 𝐵)) |
7 | ditg0 23617 | . . . . 5 ⊢ ⨜[𝐵 → 𝐵]𝐶 d𝑥 = 0 | |
8 | neg0 10327 | . . . . 5 ⊢ -0 = 0 | |
9 | 7, 8 | eqtr4i 2647 | . . . 4 ⊢ ⨜[𝐵 → 𝐵]𝐶 d𝑥 = -0 |
10 | ditgeq2 23613 | . . . 4 ⊢ (𝐴 = 𝐵 → ⨜[𝐵 → 𝐴]𝐶 d𝑥 = ⨜[𝐵 → 𝐵]𝐶 d𝑥) | |
11 | oveq1 6657 | . . . . . . . 8 ⊢ (𝐴 = 𝐵 → (𝐴(,)𝐵) = (𝐵(,)𝐵)) | |
12 | iooid 12203 | . . . . . . . 8 ⊢ (𝐵(,)𝐵) = ∅ | |
13 | 11, 12 | syl6eq 2672 | . . . . . . 7 ⊢ (𝐴 = 𝐵 → (𝐴(,)𝐵) = ∅) |
14 | itgeq1 23539 | . . . . . . 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 23546 | . . . . . 6 ⊢ ∫∅𝐶 d𝑥 = 0 | |
17 | 15, 16 | syl6eq 2672 | . . . . 5 ⊢ (𝐴 = 𝐵 → ∫(𝐴(,)𝐵)𝐶 d𝑥 = 0) |
18 | 17 | negeqd 10275 | . . . 4 ⊢ (𝐴 = 𝐵 → -∫(𝐴(,)𝐵)𝐶 d𝑥 = -0) |
19 | 9, 10, 18 | 3eqtr4a 2682 | . . 3 ⊢ (𝐴 = 𝐵 → ⨜[𝐵 → 𝐴]𝐶 d𝑥 = -∫(𝐴(,)𝐵)𝐶 d𝑥) |
20 | 6, 19 | syl6bi 243 | . 2 ⊢ (𝜑 → (𝐵 ≤ 𝐴 → ⨜[𝐵 → 𝐴]𝐶 d𝑥 = -∫(𝐴(,)𝐵)𝐶 d𝑥)) |
21 | df-ditg 23611 | . . 3 ⊢ ⨜[𝐵 → 𝐴]𝐶 d𝑥 = if(𝐵 ≤ 𝐴, ∫(𝐵(,)𝐴)𝐶 d𝑥, -∫(𝐴(,)𝐵)𝐶 d𝑥) | |
22 | iffalse 4095 | . . 3 ⊢ (¬ 𝐵 ≤ 𝐴 → if(𝐵 ≤ 𝐴, ∫(𝐵(,)𝐴)𝐶 d𝑥, -∫(𝐴(,)𝐵)𝐶 d𝑥) = -∫(𝐴(,)𝐵)𝐶 d𝑥) | |
23 | 21, 22 | syl5eq 2668 | . 2 ⊢ (¬ 𝐵 ≤ 𝐴 → ⨜[𝐵 → 𝐴]𝐶 d𝑥 = -∫(𝐴(,)𝐵)𝐶 d𝑥) |
24 | 20, 23 | pm2.61d1 171 | 1 ⊢ (𝜑 → ⨜[𝐵 → 𝐴]𝐶 d𝑥 = -∫(𝐴(,)𝐵)𝐶 d𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∅c0 3915 ifcif 4086 class class class wbr 4653 (class class class)co 6650 ℝcr 9935 0cc0 9936 ≤ 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-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-disj 4621 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-ofr 6898 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-q 11789 df-rp 11833 df-xadd 11947 df-ioo 12179 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-fl 12593 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-sum 14417 df-xmet 19739 df-met 19740 df-ovol 23233 df-vol 23234 df-mbf 23388 df-itg1 23389 df-itg2 23390 df-itg 23392 df-0p 23437 df-ditg 23611 |
This theorem is referenced by: ditgcl 23622 ditgswap 23623 |
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