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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 13336 | . . . . 5 ⊢ ((𝑋 ∈ ℝ ∧ 𝑌 ∈ ℝ) → (𝐴 ∈ (𝑋[,]𝑌) ↔ (𝐴 ∈ ℝ ∧ 𝑋 ≤ 𝐴 ∧ 𝐴 ≤ 𝑌))) | |
5 | 2, 3, 4 | syl2anc 585 | . . . 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 1143 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) |
8 | ditgcl.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ (𝑋[,]𝑌)) | |
9 | elicc2 13336 | . . . . 5 ⊢ ((𝑋 ∈ ℝ ∧ 𝑌 ∈ ℝ) → (𝐵 ∈ (𝑋[,]𝑌) ↔ (𝐵 ∈ ℝ ∧ 𝑋 ≤ 𝐵 ∧ 𝐵 ≤ 𝑌))) | |
10 | 2, 3, 9 | syl2anc 585 | . . . 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 1143 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℝ) |
13 | simpr 486 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → 𝐴 ≤ 𝐵) | |
14 | 13 | ditgpos 25236 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ⨜[𝐴 → 𝐵]𝐶 d𝑥 = ∫(𝐴(,)𝐵)𝐶 d𝑥) |
15 | 2 | rexrd 11212 | . . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ ℝ*) |
16 | 6 | simp2d 1144 | . . . . . . . . 9 ⊢ (𝜑 → 𝑋 ≤ 𝐴) |
17 | iooss1 13306 | . . . . . . . . 9 ⊢ ((𝑋 ∈ ℝ* ∧ 𝑋 ≤ 𝐴) → (𝐴(,)𝐵) ⊆ (𝑋(,)𝐵)) | |
18 | 15, 16, 17 | syl2anc 585 | . . . . . . . 8 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (𝑋(,)𝐵)) |
19 | 3 | rexrd 11212 | . . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ ℝ*) |
20 | 11 | simp3d 1145 | . . . . . . . . 9 ⊢ (𝜑 → 𝐵 ≤ 𝑌) |
21 | iooss2 13307 | . . . . . . . . 9 ⊢ ((𝑌 ∈ ℝ* ∧ 𝐵 ≤ 𝑌) → (𝑋(,)𝐵) ⊆ (𝑋(,)𝑌)) | |
22 | 19, 20, 21 | syl2anc 585 | . . . . . . . 8 ⊢ (𝜑 → (𝑋(,)𝐵) ⊆ (𝑋(,)𝑌)) |
23 | 18, 22 | sstrd 3959 | . . . . . . 7 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (𝑋(,)𝑌)) |
24 | 23 | sselda 3949 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ (𝑋(,)𝑌)) |
25 | ditgcl.c | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → 𝐶 ∈ 𝑉) | |
26 | 24, 25 | syldan 592 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝐶 ∈ 𝑉) |
27 | ioombl 24945 | . . . . . . 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 25185 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ 𝐶) ∈ 𝐿1) |
31 | 26, 30 | itgcl 25164 | . . . 4 ⊢ (𝜑 → ∫(𝐴(,)𝐵)𝐶 d𝑥 ∈ ℂ) |
32 | 31 | adantr 482 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ∫(𝐴(,)𝐵)𝐶 d𝑥 ∈ ℂ) |
33 | 14, 32 | eqeltrd 2838 | . 2 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ⨜[𝐴 → 𝐵]𝐶 d𝑥 ∈ ℂ) |
34 | simpr 486 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → 𝐵 ≤ 𝐴) | |
35 | 12 | adantr 482 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → 𝐵 ∈ ℝ) |
36 | 7 | adantr 482 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → 𝐴 ∈ ℝ) |
37 | 34, 35, 36 | ditgneg 25237 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → ⨜[𝐴 → 𝐵]𝐶 d𝑥 = -∫(𝐵(,)𝐴)𝐶 d𝑥) |
38 | 11 | simp2d 1144 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ≤ 𝐵) |
39 | iooss1 13306 | . . . . . . . . . 10 ⊢ ((𝑋 ∈ ℝ* ∧ 𝑋 ≤ 𝐵) → (𝐵(,)𝐴) ⊆ (𝑋(,)𝐴)) | |
40 | 15, 38, 39 | syl2anc 585 | . . . . . . . . 9 ⊢ (𝜑 → (𝐵(,)𝐴) ⊆ (𝑋(,)𝐴)) |
41 | 6 | simp3d 1145 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ≤ 𝑌) |
42 | iooss2 13307 | . . . . . . . . . 10 ⊢ ((𝑌 ∈ ℝ* ∧ 𝐴 ≤ 𝑌) → (𝑋(,)𝐴) ⊆ (𝑋(,)𝑌)) | |
43 | 19, 41, 42 | syl2anc 585 | . . . . . . . . 9 ⊢ (𝜑 → (𝑋(,)𝐴) ⊆ (𝑋(,)𝑌)) |
44 | 40, 43 | sstrd 3959 | . . . . . . . 8 ⊢ (𝜑 → (𝐵(,)𝐴) ⊆ (𝑋(,)𝑌)) |
45 | 44 | sselda 3949 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,)𝐴)) → 𝑥 ∈ (𝑋(,)𝑌)) |
46 | 45, 25 | syldan 592 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,)𝐴)) → 𝐶 ∈ 𝑉) |
47 | ioombl 24945 | . . . . . . . 8 ⊢ (𝐵(,)𝐴) ∈ dom vol | |
48 | 47 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → (𝐵(,)𝐴) ∈ dom vol) |
49 | 44, 48, 25, 29 | iblss 25185 | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (𝐵(,)𝐴) ↦ 𝐶) ∈ 𝐿1) |
50 | 46, 49 | itgcl 25164 | . . . . 5 ⊢ (𝜑 → ∫(𝐵(,)𝐴)𝐶 d𝑥 ∈ ℂ) |
51 | 50 | negcld 11506 | . . . 4 ⊢ (𝜑 → -∫(𝐵(,)𝐴)𝐶 d𝑥 ∈ ℂ) |
52 | 51 | adantr 482 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → -∫(𝐵(,)𝐴)𝐶 d𝑥 ∈ ℂ) |
53 | 37, 52 | eqeltrd 2838 | . 2 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → ⨜[𝐴 → 𝐵]𝐶 d𝑥 ∈ ℂ) |
54 | 7, 12, 33, 53 | lecasei 11268 | 1 ⊢ (𝜑 → ⨜[𝐴 → 𝐵]𝐶 d𝑥 ∈ ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 ∈ wcel 2107 ⊆ wss 3915 class class class wbr 5110 ↦ cmpt 5193 dom cdm 5638 (class class class)co 7362 ℂcc 11056 ℝcr 11057 ℝ*cxr 11195 ≤ cle 11197 -cneg 11393 (,)cioo 13271 [,]cicc 13274 volcvol 24843 𝐿1cibl 24997 ∫citg 24998 ⨜cdit 25226 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-inf2 9584 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 ax-pre-sup 11136 ax-addf 11137 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-disj 5076 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-se 5594 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-isom 6510 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-of 7622 df-ofr 7623 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-2o 8418 df-er 8655 df-map 8774 df-pm 8775 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-sup 9385 df-inf 9386 df-oi 9453 df-dju 9844 df-card 9882 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-div 11820 df-nn 12161 df-2 12223 df-3 12224 df-4 12225 df-n0 12421 df-z 12507 df-uz 12771 df-q 12881 df-rp 12923 df-xadd 13041 df-ioo 13275 df-ico 13277 df-icc 13278 df-fz 13432 df-fzo 13575 df-fl 13704 df-mod 13782 df-seq 13914 df-exp 13975 df-hash 14238 df-cj 14991 df-re 14992 df-im 14993 df-sqrt 15127 df-abs 15128 df-clim 15377 df-rlim 15378 df-sum 15578 df-xmet 20805 df-met 20806 df-ovol 24844 df-vol 24845 df-mbf 24999 df-itg1 25000 df-itg2 25001 df-ibl 25002 df-itg 25003 df-0p 25050 df-ditg 25227 |
This theorem is referenced by: ditgsplit 25241 itgsubstlem 25428 |
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