ditgcl - Metamath Proof Explorer

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Theorem ditgcl 25815

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 13327	. . . . 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 232	. . 3 ⊢ (𝜑 → (𝐴 ∈ ℝ ∧ 𝑋 ≤ 𝐴 ∧ 𝐴 ≤ 𝑌))
7	6	simp1d 1142	. 2 ⊢ (𝜑 → 𝐴 ∈ ℝ)
8		ditgcl.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ (𝑋[,]𝑌))
9		elicc2 13327	. . . . 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 232	. . 3 ⊢ (𝜑 → (𝐵 ∈ ℝ ∧ 𝑋 ≤ 𝐵 ∧ 𝐵 ≤ 𝑌))
12	11	simp1d 1142	. 2 ⊢ (𝜑 → 𝐵 ∈ ℝ)
13		simpr 484	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → 𝐴 ≤ 𝐵)
14	13	ditgpos 25813	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ⨜[𝐴 → 𝐵]𝐶 d𝑥 = ∫(𝐴(,)𝐵)𝐶 d𝑥)
15	2	rexrd 11182	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ ℝ^*)
16	6	simp2d 1143	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ≤ 𝐴)
17		iooss1 13296	. . . . . . . . 9 ⊢ ((𝑋 ∈ ℝ^* ∧ 𝑋 ≤ 𝐴) → (𝐴(,)𝐵) ⊆ (𝑋(,)𝐵))
18	15, 16, 17	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (𝑋(,)𝐵))
19	3	rexrd 11182	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ ℝ^*)
20	11	simp3d 1144	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ≤ 𝑌)
21		iooss2 13297	. . . . . . . . 9 ⊢ ((𝑌 ∈ ℝ^* ∧ 𝐵 ≤ 𝑌) → (𝑋(,)𝐵) ⊆ (𝑋(,)𝑌))
22	19, 20, 21	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → (𝑋(,)𝐵) ⊆ (𝑋(,)𝑌))
23	18, 22	sstrd 3944	. . . . . . 7 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (𝑋(,)𝑌))
24	23	sselda 3933	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ (𝑋(,)𝑌))
25		ditgcl.c	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → 𝐶 ∈ 𝑉)
26	24, 25	syldan 591	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝐶 ∈ 𝑉)
27		ioombl 25522	. . . . . . 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 25762	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ 𝐶) ∈ 𝐿¹)
31	26, 30	itgcl 25741	. . . 4 ⊢ (𝜑 → ∫(𝐴(,)𝐵)𝐶 d𝑥 ∈ ℂ)
32	31	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ∫(𝐴(,)𝐵)𝐶 d𝑥 ∈ ℂ)
33	14, 32	eqeltrd 2836	. 2 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ⨜[𝐴 → 𝐵]𝐶 d𝑥 ∈ ℂ)
34		simpr 484	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → 𝐵 ≤ 𝐴)
35	12	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → 𝐵 ∈ ℝ)
36	7	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → 𝐴 ∈ ℝ)
37	34, 35, 36	ditgneg 25814	. . 3 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → ⨜[𝐴 → 𝐵]𝐶 d𝑥 = -∫(𝐵(,)𝐴)𝐶 d𝑥)
38	11	simp2d 1143	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ≤ 𝐵)
39		iooss1 13296	. . . . . . . . . 10 ⊢ ((𝑋 ∈ ℝ^* ∧ 𝑋 ≤ 𝐵) → (𝐵(,)𝐴) ⊆ (𝑋(,)𝐴))
40	15, 38, 39	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → (𝐵(,)𝐴) ⊆ (𝑋(,)𝐴))
41	6	simp3d 1144	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ≤ 𝑌)
42		iooss2 13297	. . . . . . . . . 10 ⊢ ((𝑌 ∈ ℝ^* ∧ 𝐴 ≤ 𝑌) → (𝑋(,)𝐴) ⊆ (𝑋(,)𝑌))
43	19, 41, 42	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → (𝑋(,)𝐴) ⊆ (𝑋(,)𝑌))
44	40, 43	sstrd 3944	. . . . . . . 8 ⊢ (𝜑 → (𝐵(,)𝐴) ⊆ (𝑋(,)𝑌))
45	44	sselda 3933	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,)𝐴)) → 𝑥 ∈ (𝑋(,)𝑌))
46	45, 25	syldan 591	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,)𝐴)) → 𝐶 ∈ 𝑉)
47		ioombl 25522	. . . . . . . 8 ⊢ (𝐵(,)𝐴) ∈ dom vol
48	47	a1i 11	. . . . . . 7 ⊢ (𝜑 → (𝐵(,)𝐴) ∈ dom vol)
49	44, 48, 25, 29	iblss 25762	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (𝐵(,)𝐴) ↦ 𝐶) ∈ 𝐿¹)
50	46, 49	itgcl 25741	. . . . 5 ⊢ (𝜑 → ∫(𝐵(,)𝐴)𝐶 d𝑥 ∈ ℂ)
51	50	negcld 11479	. . . 4 ⊢ (𝜑 → -∫(𝐵(,)𝐴)𝐶 d𝑥 ∈ ℂ)
52	51	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → -∫(𝐵(,)𝐴)𝐶 d𝑥 ∈ ℂ)
53	37, 52	eqeltrd 2836	. 2 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → ⨜[𝐴 → 𝐵]𝐶 d𝑥 ∈ ℂ)
54	7, 12, 33, 53	lecasei 11239	1 ⊢ (𝜑 → ⨜[𝐴 → 𝐵]𝐶 d𝑥 ∈ ℂ)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 ∈ wcel 2113 ⊆ wss 3901 class class class wbr 5098 ↦ cmpt 5179 dom cdm 5624 (class class class)co 7358 ℂcc 11024 ℝcr 11025 ℝ^*cxr 11165 ≤ cle 11167 -cneg 11365 (,)cioo 13261 [,]cicc 13264 volcvol 25420 𝐿¹cibl 25574 ∫citg 25575 ⨜cdit 25803

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 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-inf2 9550 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103 ax-pre-sup 11104 ax-addf 11105

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-disj 5066 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-ofr 7623 df-om 7809 df-1st 7933 df-2nd 7934 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-2o 8398 df-er 8635 df-map 8765 df-pm 8766 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-sup 9345 df-inf 9346 df-oi 9415 df-dju 9813 df-card 9851 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-div 11795 df-nn 12146 df-2 12208 df-3 12209 df-4 12210 df-n0 12402 df-z 12489 df-uz 12752 df-q 12862 df-rp 12906 df-xadd 13027 df-ioo 13265 df-ico 13267 df-icc 13268 df-fz 13424 df-fzo 13571 df-fl 13712 df-mod 13790 df-seq 13925 df-exp 13985 df-hash 14254 df-cj 15022 df-re 15023 df-im 15024 df-sqrt 15158 df-abs 15159 df-clim 15411 df-rlim 15412 df-sum 15610 df-xmet 21302 df-met 21303 df-ovol 25421 df-vol 25422 df-mbf 25576 df-itg1 25577 df-itg2 25578 df-ibl 25579 df-itg 25580 df-0p 25627 df-ditg 25804

This theorem is referenced by: ditgsplit 25818 itgsubstlem 26011

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