ditgcl - Metamath Proof Explorer

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Theorem ditgcl 24456

Description: Closure of a directed integral. (Contributed by Mario Carneiro, 13-Aug-2014.)

**Hypotheses**
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)

**Assertion**
Ref	Expression
ditgcl	⊢ (𝜑 → ⨜[𝐴 → 𝐵]𝐶 d𝑥 ∈ ℂ)

Distinct variable groups: 𝑥,𝐴 𝑥,𝐵 𝜑,𝑥 𝑥,𝑉 𝑥,𝑋 𝑥,𝑌 Allowed substitution hint: 𝐶(𝑥)

**Proof of Theorem ditgcl**
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 12802	. . . . 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 12802	. . . . 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 24454	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ⨜[𝐴 → 𝐵]𝐶 d𝑥 = ∫(𝐴(,)𝐵)𝐶 d𝑥)
15	2	rexrd 10691	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ ℝ^*)
16	6	simp2d 1139	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ≤ 𝐴)
17		iooss1 12774	. . . . . . . . 9 ⊢ ((𝑋 ∈ ℝ^* ∧ 𝑋 ≤ 𝐴) → (𝐴(,)𝐵) ⊆ (𝑋(,)𝐵))
18	15, 16, 17	syl2anc 586	. . . . . . . 8 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (𝑋(,)𝐵))
19	3	rexrd 10691	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ ℝ^*)
20	11	simp3d 1140	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ≤ 𝑌)
21		iooss2 12775	. . . . . . . . 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 24166	. . . . . . 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 24405	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ 𝐶) ∈ 𝐿¹)
31	26, 30	itgcl 24384	. . . 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 24455	. . 3 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → ⨜[𝐴 → 𝐵]𝐶 d𝑥 = -∫(𝐵(,)𝐴)𝐶 d𝑥)
38	11	simp2d 1139	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ≤ 𝐵)
39		iooss1 12774	. . . . . . . . . 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 12775	. . . . . . . . . 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 24166	. . . . . . . 8 ⊢ (𝐵(,)𝐴) ∈ dom vol
48	47	a1i 11	. . . . . . 7 ⊢ (𝜑 → (𝐵(,)𝐴) ∈ dom vol)
49	44, 48, 25, 29	iblss 24405	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (𝐵(,)𝐴) ↦ 𝐶) ∈ 𝐿¹)
50	46, 49	itgcl 24384	. . . . 5 ⊢ (𝜑 → ∫(𝐵(,)𝐴)𝐶 d𝑥 ∈ ℂ)
51	50	negcld 10984	. . . 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 10746	1 ⊢ (𝜑 → ⨜[𝐴 → 𝐵]𝐶 d𝑥 ∈ ℂ)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 ∈ wcel 2114 ⊆ wss 3936 class class class wbr 5066 ↦ cmpt 5146 dom cdm 5555 (class class class)co 7156 ℂcc 10535 ℝcr 10536 ℝ^*cxr 10674 ≤ cle 10676 -cneg 10871 (,)cioo 12739 [,]cicc 12742 volcvol 24064 𝐿¹cibl 24218 ∫citg 24219 ⨜cdit 24444

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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 ax-addf 10616

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3496 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 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-disj 5032 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-ofr 7410 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-er 8289 df-map 8408 df-pm 8409 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-sup 8906 df-inf 8907 df-oi 8974 df-dju 9330 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-n0 11899 df-z 11983 df-uz 12245 df-q 12350 df-rp 12391 df-xadd 12509 df-ioo 12743 df-ico 12745 df-icc 12746 df-fz 12894 df-fzo 13035 df-fl 13163 df-mod 13239 df-seq 13371 df-exp 13431 df-hash 13692 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-clim 14845 df-rlim 14846 df-sum 15043 df-xmet 20538 df-met 20539 df-ovol 24065 df-vol 24066 df-mbf 24220 df-itg1 24221 df-itg2 24222 df-ibl 24223 df-itg 24224 df-0p 24271 df-ditg 24445

This theorem is referenced by: ditgsplit 24459 itgsubstlem 24645

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