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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 12795 | . . . . 5 ⊢ ((𝑋 ∈ ℝ ∧ 𝑌 ∈ ℝ) → (𝐴 ∈ (𝑋[,]𝑌) ↔ (𝐴 ∈ ℝ ∧ 𝑋 ≤ 𝐴 ∧ 𝐴 ≤ 𝑌))) | |
5 | 2, 3, 4 | syl2anc 586 | . . . 4 ⊢ (𝜑 → (𝐴 ∈ (𝑋[,]𝑌) ↔ (𝐴 ∈ ℝ ∧ 𝑋 ≤ 𝐴 ∧ 𝐴 ≤ 𝑌))) |
6 | 1, 5 | mpbid 234 | . . 3 ⊢ (𝜑 → (𝐴 ∈ ℝ ∧ 𝑋 ≤ 𝐴 ∧ 𝐴 ≤ 𝑌)) |
7 | 6 | simp1d 1138 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) |
8 | ditgcl.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ (𝑋[,]𝑌)) | |
9 | elicc2 12795 | . . . . 5 ⊢ ((𝑋 ∈ ℝ ∧ 𝑌 ∈ ℝ) → (𝐵 ∈ (𝑋[,]𝑌) ↔ (𝐵 ∈ ℝ ∧ 𝑋 ≤ 𝐵 ∧ 𝐵 ≤ 𝑌))) | |
10 | 2, 3, 9 | syl2anc 586 | . . . 4 ⊢ (𝜑 → (𝐵 ∈ (𝑋[,]𝑌) ↔ (𝐵 ∈ ℝ ∧ 𝑋 ≤ 𝐵 ∧ 𝐵 ≤ 𝑌))) |
11 | 8, 10 | mpbid 234 | . . 3 ⊢ (𝜑 → (𝐵 ∈ ℝ ∧ 𝑋 ≤ 𝐵 ∧ 𝐵 ≤ 𝑌)) |
12 | 11 | simp1d 1138 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℝ) |
13 | simpr 487 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → 𝐴 ≤ 𝐵) | |
14 | 13 | ditgpos 24448 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ⨜[𝐴 → 𝐵]𝐶 d𝑥 = ∫(𝐴(,)𝐵)𝐶 d𝑥) |
15 | 2 | rexrd 10685 | . . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ ℝ*) |
16 | 6 | simp2d 1139 | . . . . . . . . 9 ⊢ (𝜑 → 𝑋 ≤ 𝐴) |
17 | iooss1 12767 | . . . . . . . . 9 ⊢ ((𝑋 ∈ ℝ* ∧ 𝑋 ≤ 𝐴) → (𝐴(,)𝐵) ⊆ (𝑋(,)𝐵)) | |
18 | 15, 16, 17 | syl2anc 586 | . . . . . . . 8 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (𝑋(,)𝐵)) |
19 | 3 | rexrd 10685 | . . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ ℝ*) |
20 | 11 | simp3d 1140 | . . . . . . . . 9 ⊢ (𝜑 → 𝐵 ≤ 𝑌) |
21 | iooss2 12768 | . . . . . . . . 9 ⊢ ((𝑌 ∈ ℝ* ∧ 𝐵 ≤ 𝑌) → (𝑋(,)𝐵) ⊆ (𝑋(,)𝑌)) | |
22 | 19, 20, 21 | syl2anc 586 | . . . . . . . 8 ⊢ (𝜑 → (𝑋(,)𝐵) ⊆ (𝑋(,)𝑌)) |
23 | 18, 22 | sstrd 3977 | . . . . . . 7 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (𝑋(,)𝑌)) |
24 | 23 | sselda 3967 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ (𝑋(,)𝑌)) |
25 | ditgcl.c | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → 𝐶 ∈ 𝑉) | |
26 | 24, 25 | syldan 593 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝐶 ∈ 𝑉) |
27 | ioombl 24160 | . . . . . . 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 24399 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ 𝐶) ∈ 𝐿1) |
31 | 26, 30 | itgcl 24378 | . . . 4 ⊢ (𝜑 → ∫(𝐴(,)𝐵)𝐶 d𝑥 ∈ ℂ) |
32 | 31 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ∫(𝐴(,)𝐵)𝐶 d𝑥 ∈ ℂ) |
33 | 14, 32 | eqeltrd 2913 | . 2 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ⨜[𝐴 → 𝐵]𝐶 d𝑥 ∈ ℂ) |
34 | simpr 487 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → 𝐵 ≤ 𝐴) | |
35 | 12 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → 𝐵 ∈ ℝ) |
36 | 7 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → 𝐴 ∈ ℝ) |
37 | 34, 35, 36 | ditgneg 24449 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → ⨜[𝐴 → 𝐵]𝐶 d𝑥 = -∫(𝐵(,)𝐴)𝐶 d𝑥) |
38 | 11 | simp2d 1139 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ≤ 𝐵) |
39 | iooss1 12767 | . . . . . . . . . 10 ⊢ ((𝑋 ∈ ℝ* ∧ 𝑋 ≤ 𝐵) → (𝐵(,)𝐴) ⊆ (𝑋(,)𝐴)) | |
40 | 15, 38, 39 | syl2anc 586 | . . . . . . . . 9 ⊢ (𝜑 → (𝐵(,)𝐴) ⊆ (𝑋(,)𝐴)) |
41 | 6 | simp3d 1140 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ≤ 𝑌) |
42 | iooss2 12768 | . . . . . . . . . 10 ⊢ ((𝑌 ∈ ℝ* ∧ 𝐴 ≤ 𝑌) → (𝑋(,)𝐴) ⊆ (𝑋(,)𝑌)) | |
43 | 19, 41, 42 | syl2anc 586 | . . . . . . . . 9 ⊢ (𝜑 → (𝑋(,)𝐴) ⊆ (𝑋(,)𝑌)) |
44 | 40, 43 | sstrd 3977 | . . . . . . . 8 ⊢ (𝜑 → (𝐵(,)𝐴) ⊆ (𝑋(,)𝑌)) |
45 | 44 | sselda 3967 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,)𝐴)) → 𝑥 ∈ (𝑋(,)𝑌)) |
46 | 45, 25 | syldan 593 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,)𝐴)) → 𝐶 ∈ 𝑉) |
47 | ioombl 24160 | . . . . . . . 8 ⊢ (𝐵(,)𝐴) ∈ dom vol | |
48 | 47 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → (𝐵(,)𝐴) ∈ dom vol) |
49 | 44, 48, 25, 29 | iblss 24399 | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (𝐵(,)𝐴) ↦ 𝐶) ∈ 𝐿1) |
50 | 46, 49 | itgcl 24378 | . . . . 5 ⊢ (𝜑 → ∫(𝐵(,)𝐴)𝐶 d𝑥 ∈ ℂ) |
51 | 50 | negcld 10978 | . . . 4 ⊢ (𝜑 → -∫(𝐵(,)𝐴)𝐶 d𝑥 ∈ ℂ) |
52 | 51 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → -∫(𝐵(,)𝐴)𝐶 d𝑥 ∈ ℂ) |
53 | 37, 52 | eqeltrd 2913 | . 2 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → ⨜[𝐴 → 𝐵]𝐶 d𝑥 ∈ ℂ) |
54 | 7, 12, 33, 53 | lecasei 10740 | 1 ⊢ (𝜑 → ⨜[𝐴 → 𝐵]𝐶 d𝑥 ∈ ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 ∈ wcel 2110 ⊆ wss 3936 class class class wbr 5059 ↦ cmpt 5139 dom cdm 5550 (class class class)co 7150 ℂcc 10529 ℝcr 10530 ℝ*cxr 10668 ≤ cle 10670 -cneg 10865 (,)cioo 12732 [,]cicc 12735 volcvol 24058 𝐿1cibl 24212 ∫citg 24213 ⨜cdit 24438 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-inf2 9098 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 ax-addf 10610 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-disj 5025 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-se 5510 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-isom 6359 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-ofr 7404 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-oadd 8100 df-er 8283 df-map 8402 df-pm 8403 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-sup 8900 df-inf 8901 df-oi 8968 df-dju 9324 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-n0 11892 df-z 11976 df-uz 12238 df-q 12343 df-rp 12384 df-xadd 12502 df-ioo 12736 df-ico 12738 df-icc 12739 df-fz 12887 df-fzo 13028 df-fl 13156 df-mod 13232 df-seq 13364 df-exp 13424 df-hash 13685 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-clim 14839 df-rlim 14840 df-sum 15037 df-xmet 20532 df-met 20533 df-ovol 24059 df-vol 24060 df-mbf 24214 df-itg1 24215 df-itg2 24216 df-ibl 24217 df-itg 24218 df-0p 24265 df-ditg 24439 |
This theorem is referenced by: ditgsplit 24453 itgsubstlem 24639 |
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