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Mirrors > Home > MPE Home > Th. List > ditgswap | Structured version Visualization version GIF version |
Description: Reverse a directed integral. (Contributed by Mario Carneiro, 13-Aug-2014.) |
Ref | Expression |
---|---|
ditgcl.x | ⊢ (𝜑 → 𝑋 ∈ ℝ) |
ditgcl.y | ⊢ (𝜑 → 𝑌 ∈ ℝ) |
ditgcl.a | ⊢ (𝜑 → 𝐴 ∈ (𝑋[,]𝑌)) |
ditgcl.b | ⊢ (𝜑 → 𝐵 ∈ (𝑋[,]𝑌)) |
ditgcl.c | ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → 𝐶 ∈ 𝑉) |
ditgcl.i | ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐶) ∈ 𝐿1) |
Ref | Expression |
---|---|
ditgswap | ⊢ (𝜑 → ⨜[𝐵 → 𝐴]𝐶 d𝑥 = -⨜[𝐴 → 𝐵]𝐶 d𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ditgcl.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ (𝑋[,]𝑌)) | |
2 | ditgcl.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ ℝ) | |
3 | ditgcl.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ ℝ) | |
4 | elicc2 13224 | . . . . 5 ⊢ ((𝑋 ∈ ℝ ∧ 𝑌 ∈ ℝ) → (𝐴 ∈ (𝑋[,]𝑌) ↔ (𝐴 ∈ ℝ ∧ 𝑋 ≤ 𝐴 ∧ 𝐴 ≤ 𝑌))) | |
5 | 2, 3, 4 | syl2anc 584 | . . . 4 ⊢ (𝜑 → (𝐴 ∈ (𝑋[,]𝑌) ↔ (𝐴 ∈ ℝ ∧ 𝑋 ≤ 𝐴 ∧ 𝐴 ≤ 𝑌))) |
6 | 1, 5 | mpbid 231 | . . 3 ⊢ (𝜑 → (𝐴 ∈ ℝ ∧ 𝑋 ≤ 𝐴 ∧ 𝐴 ≤ 𝑌)) |
7 | 6 | simp1d 1141 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) |
8 | ditgcl.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ (𝑋[,]𝑌)) | |
9 | elicc2 13224 | . . . . 5 ⊢ ((𝑋 ∈ ℝ ∧ 𝑌 ∈ ℝ) → (𝐵 ∈ (𝑋[,]𝑌) ↔ (𝐵 ∈ ℝ ∧ 𝑋 ≤ 𝐵 ∧ 𝐵 ≤ 𝑌))) | |
10 | 2, 3, 9 | syl2anc 584 | . . . 4 ⊢ (𝜑 → (𝐵 ∈ (𝑋[,]𝑌) ↔ (𝐵 ∈ ℝ ∧ 𝑋 ≤ 𝐵 ∧ 𝐵 ≤ 𝑌))) |
11 | 8, 10 | mpbid 231 | . . 3 ⊢ (𝜑 → (𝐵 ∈ ℝ ∧ 𝑋 ≤ 𝐵 ∧ 𝐵 ≤ 𝑌)) |
12 | 11 | simp1d 1141 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℝ) |
13 | simpr 485 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → 𝐴 ≤ 𝐵) | |
14 | 7 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ ℝ) |
15 | 12 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ ℝ) |
16 | 13, 14, 15 | ditgneg 25104 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ⨜[𝐵 → 𝐴]𝐶 d𝑥 = -∫(𝐴(,)𝐵)𝐶 d𝑥) |
17 | 13 | ditgpos 25103 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ⨜[𝐴 → 𝐵]𝐶 d𝑥 = ∫(𝐴(,)𝐵)𝐶 d𝑥) |
18 | 17 | negeqd 11295 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → -⨜[𝐴 → 𝐵]𝐶 d𝑥 = -∫(𝐴(,)𝐵)𝐶 d𝑥) |
19 | 16, 18 | eqtr4d 2780 | . 2 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ⨜[𝐵 → 𝐴]𝐶 d𝑥 = -⨜[𝐴 → 𝐵]𝐶 d𝑥) |
20 | 2 | rexrd 11105 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ ℝ*) |
21 | 11 | simp2d 1142 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ≤ 𝐵) |
22 | iooss1 13194 | . . . . . . . . . 10 ⊢ ((𝑋 ∈ ℝ* ∧ 𝑋 ≤ 𝐵) → (𝐵(,)𝐴) ⊆ (𝑋(,)𝐴)) | |
23 | 20, 21, 22 | syl2anc 584 | . . . . . . . . 9 ⊢ (𝜑 → (𝐵(,)𝐴) ⊆ (𝑋(,)𝐴)) |
24 | 3 | rexrd 11105 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ∈ ℝ*) |
25 | 6 | simp3d 1143 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ≤ 𝑌) |
26 | iooss2 13195 | . . . . . . . . . 10 ⊢ ((𝑌 ∈ ℝ* ∧ 𝐴 ≤ 𝑌) → (𝑋(,)𝐴) ⊆ (𝑋(,)𝑌)) | |
27 | 24, 25, 26 | syl2anc 584 | . . . . . . . . 9 ⊢ (𝜑 → (𝑋(,)𝐴) ⊆ (𝑋(,)𝑌)) |
28 | 23, 27 | sstrd 3941 | . . . . . . . 8 ⊢ (𝜑 → (𝐵(,)𝐴) ⊆ (𝑋(,)𝑌)) |
29 | 28 | sselda 3931 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,)𝐴)) → 𝑥 ∈ (𝑋(,)𝑌)) |
30 | ditgcl.i | . . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐶) ∈ 𝐿1) | |
31 | iblmbf 25015 | . . . . . . . . 9 ⊢ ((𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐶) ∈ 𝐿1 → (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐶) ∈ MblFn) | |
32 | 30, 31 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐶) ∈ MblFn) |
33 | ditgcl.c | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → 𝐶 ∈ 𝑉) | |
34 | 32, 33 | mbfmptcl 24883 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → 𝐶 ∈ ℂ) |
35 | 29, 34 | syldan 591 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,)𝐴)) → 𝐶 ∈ ℂ) |
36 | ioombl 24812 | . . . . . . . 8 ⊢ (𝐵(,)𝐴) ∈ dom vol | |
37 | 36 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → (𝐵(,)𝐴) ∈ dom vol) |
38 | 28, 37, 33, 30 | iblss 25052 | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (𝐵(,)𝐴) ↦ 𝐶) ∈ 𝐿1) |
39 | 35, 38 | itgcl 25031 | . . . . 5 ⊢ (𝜑 → ∫(𝐵(,)𝐴)𝐶 d𝑥 ∈ ℂ) |
40 | 39 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → ∫(𝐵(,)𝐴)𝐶 d𝑥 ∈ ℂ) |
41 | 40 | negnegd 11403 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → --∫(𝐵(,)𝐴)𝐶 d𝑥 = ∫(𝐵(,)𝐴)𝐶 d𝑥) |
42 | simpr 485 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → 𝐵 ≤ 𝐴) | |
43 | 12 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → 𝐵 ∈ ℝ) |
44 | 7 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → 𝐴 ∈ ℝ) |
45 | 42, 43, 44 | ditgneg 25104 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → ⨜[𝐴 → 𝐵]𝐶 d𝑥 = -∫(𝐵(,)𝐴)𝐶 d𝑥) |
46 | 45 | negeqd 11295 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → -⨜[𝐴 → 𝐵]𝐶 d𝑥 = --∫(𝐵(,)𝐴)𝐶 d𝑥) |
47 | 42 | ditgpos 25103 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → ⨜[𝐵 → 𝐴]𝐶 d𝑥 = ∫(𝐵(,)𝐴)𝐶 d𝑥) |
48 | 41, 46, 47 | 3eqtr4rd 2788 | . 2 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → ⨜[𝐵 → 𝐴]𝐶 d𝑥 = -⨜[𝐴 → 𝐵]𝐶 d𝑥) |
49 | 7, 12, 19, 48 | lecasei 11161 | 1 ⊢ (𝜑 → ⨜[𝐵 → 𝐴]𝐶 d𝑥 = -⨜[𝐴 → 𝐵]𝐶 d𝑥) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 ⊆ wss 3897 class class class wbr 5087 ↦ cmpt 5170 dom cdm 5608 (class class class)co 7317 ℂcc 10949 ℝcr 10950 ℝ*cxr 11088 ≤ cle 11090 -cneg 11286 (,)cioo 13159 [,]cicc 13162 volcvol 24710 MblFncmbf 24861 𝐿1cibl 24864 ∫citg 24865 ⨜cdit 25093 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2708 ax-rep 5224 ax-sep 5238 ax-nul 5245 ax-pow 5303 ax-pr 5367 ax-un 7630 ax-inf2 9477 ax-cnex 11007 ax-resscn 11008 ax-1cn 11009 ax-icn 11010 ax-addcl 11011 ax-addrcl 11012 ax-mulcl 11013 ax-mulrcl 11014 ax-mulcom 11015 ax-addass 11016 ax-mulass 11017 ax-distr 11018 ax-i2m1 11019 ax-1ne0 11020 ax-1rid 11021 ax-rnegex 11022 ax-rrecex 11023 ax-cnre 11024 ax-pre-lttri 11025 ax-pre-lttrn 11026 ax-pre-ltadd 11027 ax-pre-mulgt0 11028 ax-pre-sup 11029 ax-addf 11030 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 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 3350 df-reu 3351 df-rab 3405 df-v 3443 df-sbc 3727 df-csb 3843 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3916 df-nul 4268 df-if 4472 df-pw 4547 df-sn 4572 df-pr 4574 df-op 4578 df-uni 4851 df-int 4893 df-iun 4939 df-disj 5053 df-br 5088 df-opab 5150 df-mpt 5171 df-tr 5205 df-id 5507 df-eprel 5513 df-po 5521 df-so 5522 df-fr 5563 df-se 5564 df-we 5565 df-xp 5614 df-rel 5615 df-cnv 5616 df-co 5617 df-dm 5618 df-rn 5619 df-res 5620 df-ima 5621 df-pred 6225 df-ord 6292 df-on 6293 df-lim 6294 df-suc 6295 df-iota 6418 df-fun 6468 df-fn 6469 df-f 6470 df-f1 6471 df-fo 6472 df-f1o 6473 df-fv 6474 df-isom 6475 df-riota 7274 df-ov 7320 df-oprab 7321 df-mpo 7322 df-of 7575 df-ofr 7576 df-om 7760 df-1st 7878 df-2nd 7879 df-frecs 8146 df-wrecs 8177 df-recs 8251 df-rdg 8290 df-1o 8346 df-2o 8347 df-er 8548 df-map 8667 df-pm 8668 df-en 8784 df-dom 8785 df-sdom 8786 df-fin 8787 df-sup 9278 df-inf 9279 df-oi 9346 df-dju 9737 df-card 9775 df-pnf 11091 df-mnf 11092 df-xr 11093 df-ltxr 11094 df-le 11095 df-sub 11287 df-neg 11288 df-div 11713 df-nn 12054 df-2 12116 df-3 12117 df-4 12118 df-n0 12314 df-z 12400 df-uz 12663 df-q 12769 df-rp 12811 df-xadd 12929 df-ioo 13163 df-ico 13165 df-icc 13166 df-fz 13320 df-fzo 13463 df-fl 13592 df-mod 13670 df-seq 13802 df-exp 13863 df-hash 14125 df-cj 14889 df-re 14890 df-im 14891 df-sqrt 15025 df-abs 15026 df-clim 15276 df-rlim 15277 df-sum 15477 df-xmet 20673 df-met 20674 df-ovol 24711 df-vol 24712 df-mbf 24866 df-itg1 24867 df-itg2 24868 df-ibl 24869 df-itg 24870 df-0p 24917 df-ditg 25094 |
This theorem is referenced by: ditgsplit 25108 ftc2ditg 25293 |
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