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Mirrors > Home > MPE Home > Th. List > ditgcl | Structured version Visualization version GIF version |
Description: Closure of 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 |
---|---|
ditgcl | ⊢ (𝜑 → ⨜[𝐴 → 𝐵]𝐶 d𝑥 ∈ ℂ) |
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 13437 | . . . . 5 ⊢ ((𝑋 ∈ ℝ ∧ 𝑌 ∈ ℝ) → (𝐴 ∈ (𝑋[,]𝑌) ↔ (𝐴 ∈ ℝ ∧ 𝑋 ≤ 𝐴 ∧ 𝐴 ≤ 𝑌))) | |
5 | 2, 3, 4 | syl2anc 582 | . . . 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 1139 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) |
8 | ditgcl.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ (𝑋[,]𝑌)) | |
9 | elicc2 13437 | . . . . 5 ⊢ ((𝑋 ∈ ℝ ∧ 𝑌 ∈ ℝ) → (𝐵 ∈ (𝑋[,]𝑌) ↔ (𝐵 ∈ ℝ ∧ 𝑋 ≤ 𝐵 ∧ 𝐵 ≤ 𝑌))) | |
10 | 2, 3, 9 | syl2anc 582 | . . . 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 1139 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℝ) |
13 | simpr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → 𝐴 ≤ 𝐵) | |
14 | 13 | ditgpos 25873 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ⨜[𝐴 → 𝐵]𝐶 d𝑥 = ∫(𝐴(,)𝐵)𝐶 d𝑥) |
15 | 2 | rexrd 11305 | . . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ ℝ*) |
16 | 6 | simp2d 1140 | . . . . . . . . 9 ⊢ (𝜑 → 𝑋 ≤ 𝐴) |
17 | iooss1 13407 | . . . . . . . . 9 ⊢ ((𝑋 ∈ ℝ* ∧ 𝑋 ≤ 𝐴) → (𝐴(,)𝐵) ⊆ (𝑋(,)𝐵)) | |
18 | 15, 16, 17 | syl2anc 582 | . . . . . . . 8 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (𝑋(,)𝐵)) |
19 | 3 | rexrd 11305 | . . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ ℝ*) |
20 | 11 | simp3d 1141 | . . . . . . . . 9 ⊢ (𝜑 → 𝐵 ≤ 𝑌) |
21 | iooss2 13408 | . . . . . . . . 9 ⊢ ((𝑌 ∈ ℝ* ∧ 𝐵 ≤ 𝑌) → (𝑋(,)𝐵) ⊆ (𝑋(,)𝑌)) | |
22 | 19, 20, 21 | syl2anc 582 | . . . . . . . 8 ⊢ (𝜑 → (𝑋(,)𝐵) ⊆ (𝑋(,)𝑌)) |
23 | 18, 22 | sstrd 3989 | . . . . . . 7 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (𝑋(,)𝑌)) |
24 | 23 | sselda 3978 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ (𝑋(,)𝑌)) |
25 | ditgcl.c | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → 𝐶 ∈ 𝑉) | |
26 | 24, 25 | syldan 589 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝐶 ∈ 𝑉) |
27 | ioombl 25582 | . . . . . . 7 ⊢ (𝐴(,)𝐵) ∈ dom vol | |
28 | 27 | a1i 11 | . . . . . 6 ⊢ (𝜑 → (𝐴(,)𝐵) ∈ dom vol) |
29 | ditgcl.i | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐶) ∈ 𝐿1) | |
30 | 23, 28, 25, 29 | iblss 25822 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ 𝐶) ∈ 𝐿1) |
31 | 26, 30 | itgcl 25801 | . . . 4 ⊢ (𝜑 → ∫(𝐴(,)𝐵)𝐶 d𝑥 ∈ ℂ) |
32 | 31 | adantr 479 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ∫(𝐴(,)𝐵)𝐶 d𝑥 ∈ ℂ) |
33 | 14, 32 | eqeltrd 2826 | . 2 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ⨜[𝐴 → 𝐵]𝐶 d𝑥 ∈ ℂ) |
34 | simpr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → 𝐵 ≤ 𝐴) | |
35 | 12 | adantr 479 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → 𝐵 ∈ ℝ) |
36 | 7 | adantr 479 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → 𝐴 ∈ ℝ) |
37 | 34, 35, 36 | ditgneg 25874 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → ⨜[𝐴 → 𝐵]𝐶 d𝑥 = -∫(𝐵(,)𝐴)𝐶 d𝑥) |
38 | 11 | simp2d 1140 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ≤ 𝐵) |
39 | iooss1 13407 | . . . . . . . . . 10 ⊢ ((𝑋 ∈ ℝ* ∧ 𝑋 ≤ 𝐵) → (𝐵(,)𝐴) ⊆ (𝑋(,)𝐴)) | |
40 | 15, 38, 39 | syl2anc 582 | . . . . . . . . 9 ⊢ (𝜑 → (𝐵(,)𝐴) ⊆ (𝑋(,)𝐴)) |
41 | 6 | simp3d 1141 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ≤ 𝑌) |
42 | iooss2 13408 | . . . . . . . . . 10 ⊢ ((𝑌 ∈ ℝ* ∧ 𝐴 ≤ 𝑌) → (𝑋(,)𝐴) ⊆ (𝑋(,)𝑌)) | |
43 | 19, 41, 42 | syl2anc 582 | . . . . . . . . 9 ⊢ (𝜑 → (𝑋(,)𝐴) ⊆ (𝑋(,)𝑌)) |
44 | 40, 43 | sstrd 3989 | . . . . . . . 8 ⊢ (𝜑 → (𝐵(,)𝐴) ⊆ (𝑋(,)𝑌)) |
45 | 44 | sselda 3978 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,)𝐴)) → 𝑥 ∈ (𝑋(,)𝑌)) |
46 | 45, 25 | syldan 589 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,)𝐴)) → 𝐶 ∈ 𝑉) |
47 | ioombl 25582 | . . . . . . . 8 ⊢ (𝐵(,)𝐴) ∈ dom vol | |
48 | 47 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → (𝐵(,)𝐴) ∈ dom vol) |
49 | 44, 48, 25, 29 | iblss 25822 | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (𝐵(,)𝐴) ↦ 𝐶) ∈ 𝐿1) |
50 | 46, 49 | itgcl 25801 | . . . . 5 ⊢ (𝜑 → ∫(𝐵(,)𝐴)𝐶 d𝑥 ∈ ℂ) |
51 | 50 | negcld 11599 | . . . 4 ⊢ (𝜑 → -∫(𝐵(,)𝐴)𝐶 d𝑥 ∈ ℂ) |
52 | 51 | adantr 479 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → -∫(𝐵(,)𝐴)𝐶 d𝑥 ∈ ℂ) |
53 | 37, 52 | eqeltrd 2826 | . 2 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → ⨜[𝐴 → 𝐵]𝐶 d𝑥 ∈ ℂ) |
54 | 7, 12, 33, 53 | lecasei 11361 | 1 ⊢ (𝜑 → ⨜[𝐴 → 𝐵]𝐶 d𝑥 ∈ ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 ∈ wcel 2099 ⊆ wss 3946 class class class wbr 5145 ↦ cmpt 5228 dom cdm 5674 (class class class)co 7416 ℂcc 11147 ℝcr 11148 ℝ*cxr 11288 ≤ cle 11290 -cneg 11486 (,)cioo 13372 [,]cicc 13375 volcvol 25480 𝐿1cibl 25634 ∫citg 25635 ⨜cdit 25863 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5282 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 ax-inf2 9677 ax-cnex 11205 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226 ax-pre-sup 11227 ax-addf 11228 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-int 4947 df-iun 4995 df-disj 5111 df-br 5146 df-opab 5208 df-mpt 5229 df-tr 5263 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-se 5630 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6304 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-isom 6555 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7682 df-ofr 7683 df-om 7869 df-1st 7995 df-2nd 7996 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-1o 8488 df-2o 8489 df-er 8726 df-map 8849 df-pm 8850 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-sup 9478 df-inf 9479 df-oi 9546 df-dju 9937 df-card 9975 df-pnf 11291 df-mnf 11292 df-xr 11293 df-ltxr 11294 df-le 11295 df-sub 11487 df-neg 11488 df-div 11913 df-nn 12259 df-2 12321 df-3 12322 df-4 12323 df-n0 12519 df-z 12605 df-uz 12869 df-q 12979 df-rp 13023 df-xadd 13141 df-ioo 13376 df-ico 13378 df-icc 13379 df-fz 13533 df-fzo 13676 df-fl 13806 df-mod 13884 df-seq 14016 df-exp 14076 df-hash 14343 df-cj 15099 df-re 15100 df-im 15101 df-sqrt 15235 df-abs 15236 df-clim 15485 df-rlim 15486 df-sum 15686 df-xmet 21332 df-met 21333 df-ovol 25481 df-vol 25482 df-mbf 25636 df-itg1 25637 df-itg2 25638 df-ibl 25639 df-itg 25640 df-0p 25687 df-ditg 25864 |
This theorem is referenced by: ditgsplit 25878 itgsubstlem 26071 |
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