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Theorem itgioo 26048

Description: Equality of integrals on open and closed intervals. (Contributed by Mario Carneiro, 2-Sep-2014.)

**Hypotheses**
Ref	Expression
itgioo.1	⊢ (𝜑 → 𝐴 ∈ ℝ)
itgioo.2	⊢ (𝜑 → 𝐵 ∈ ℝ)
itgioo.3	⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐶 ∈ ℂ)

**Assertion**
Ref	Expression
itgioo	⊢ (𝜑 → ∫(𝐴(,)𝐵)𝐶 d𝑥 = ∫(𝐴[,]𝐵)𝐶 d𝑥)

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

**Proof of Theorem itgioo**
Step	Hyp	Ref	Expression
1		ioossicc 13488	. . . 4 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
2	1	a1i 11	. . 3 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵))
3		itgioo.1	. . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ)
4		itgioo.2	. . . 4 ⊢ (𝜑 → 𝐵 ∈ ℝ)
5		iccssre 13484	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
6	3, 4, 5	syl2anc 596	. . 3 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
7	3	rexrd 11286	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
8	4	rexrd 11286	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
9		icc0 13448	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴[,]𝐵) = ∅ ↔ 𝐵 < 𝐴))
10	7, 8, 9	syl2anc 596	. . . . . . . 8 ⊢ (𝜑 → ((𝐴[,]𝐵) = ∅ ↔ 𝐵 < 𝐴))
11	10	biimpar 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → (𝐴[,]𝐵) = ∅)
12	11	difeq1d 4076	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → ((𝐴[,]𝐵) ∖ (𝐴(,)𝐵)) = (∅ ∖ (𝐴(,)𝐵)))
13		0dif 4359	. . . . . . 7 ⊢ (∅ ∖ (𝐴(,)𝐵)) = ∅
14		0ss 4353	. . . . . . 7 ⊢ ∅ ⊆ {𝐴, 𝐵}
15	13, 14	eqsstri 3980	. . . . . 6 ⊢ (∅ ∖ (𝐴(,)𝐵)) ⊆ {𝐴, 𝐵}
16	12, 15	eqsstrdi 3978	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → ((𝐴[,]𝐵) ∖ (𝐴(,)𝐵)) ⊆ {𝐴, 𝐵})
17		uncom 4108	. . . . . . . . 9 ⊢ ({𝐴, 𝐵} ∪ (𝐴(,)𝐵)) = ((𝐴(,)𝐵) ∪ {𝐴, 𝐵})
18	7	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ ℝ^*)
19	8	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ ℝ^*)
20		simpr 490	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → 𝐴 ≤ 𝐵)
21		prunioo 13536	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → ((𝐴(,)𝐵) ∪ {𝐴, 𝐵}) = (𝐴[,]𝐵))
22	18, 19, 20, 21	syl3anc 1398	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ((𝐴(,)𝐵) ∪ {𝐴, 𝐵}) = (𝐴[,]𝐵))
23	17, 22	eqtr2id 2810	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → (𝐴[,]𝐵) = ({𝐴, 𝐵} ∪ (𝐴(,)𝐵)))
24	23	difeq1d 4076	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ((𝐴[,]𝐵) ∖ (𝐴(,)𝐵)) = (({𝐴, 𝐵} ∪ (𝐴(,)𝐵)) ∖ (𝐴(,)𝐵)))
25		difun2 4440	. . . . . . 7 ⊢ (({𝐴, 𝐵} ∪ (𝐴(,)𝐵)) ∖ (𝐴(,)𝐵)) = ({𝐴, 𝐵} ∖ (𝐴(,)𝐵))
26	24, 25	eqtrdi 2813	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ((𝐴[,]𝐵) ∖ (𝐴(,)𝐵)) = ({𝐴, 𝐵} ∖ (𝐴(,)𝐵)))
27		difss 4086	. . . . . 6 ⊢ ({𝐴, 𝐵} ∖ (𝐴(,)𝐵)) ⊆ {𝐴, 𝐵}
28	26, 27	eqsstrdi 3978	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ((𝐴[,]𝐵) ∖ (𝐴(,)𝐵)) ⊆ {𝐴, 𝐵})
29	16, 28, 4, 3	ltlecasei 11345	. . . 4 ⊢ (𝜑 → ((𝐴[,]𝐵) ∖ (𝐴(,)𝐵)) ⊆ {𝐴, 𝐵})
30	3, 4	prssd 4786	. . . 4 ⊢ (𝜑 → {𝐴, 𝐵} ⊆ ℝ)
31		prfi 9296	. . . . 5 ⊢ {𝐴, 𝐵} ∈ Fin
32		ovolfi 25726	. . . . 5 ⊢ (({𝐴, 𝐵} ∈ Fin ∧ {𝐴, 𝐵} ⊆ ℝ) → (vol*‘{𝐴, 𝐵}) = 0)
33	31, 30, 32	sylancr 599	. . . 4 ⊢ (𝜑 → (vol*‘{𝐴, 𝐵}) = 0)
34		ovolssnul 25719	. . . 4 ⊢ ((((𝐴[,]𝐵) ∖ (𝐴(,)𝐵)) ⊆ {𝐴, 𝐵} ∧ {𝐴, 𝐵} ⊆ ℝ ∧ (vol‘{𝐴, 𝐵}) = 0) → (vol‘((𝐴[,]𝐵) ∖ (𝐴(,)𝐵))) = 0)
35	29, 30, 33, 34	syl3anc 1398	. . 3 ⊢ (𝜑 → (vol*‘((𝐴[,]𝐵) ∖ (𝐴(,)𝐵))) = 0)
36		itgioo.3	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐶 ∈ ℂ)
37	2, 6, 35, 36	itgss3 26047	. 2 ⊢ (𝜑 → (((𝑥 ∈ (𝐴(,)𝐵) ↦ 𝐶) ∈ 𝐿¹ ↔ (𝑥 ∈ (𝐴[,]𝐵) ↦ 𝐶) ∈ 𝐿¹) ∧ ∫(𝐴(,)𝐵)𝐶 d𝑥 = ∫(𝐴[,]𝐵)𝐶 d𝑥))
38	37	simprd 501	1 ⊢ (𝜑 → ∫(𝐴(,)𝐵)𝐶 d𝑥 = ∫(𝐴[,]𝐵)𝐶 d𝑥)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ∖ cdif 3899 ∪ cun 3900 ⊆ wss 3902 ∅c0 4282 {cpr 4589 class class class wbr 5107 ↦ cmpt 5190 ‘cfv 6537 (class class class)co 7416 Fincfn 8955 ℂcc 11125 ℝcr 11126 0cc0 11127 ℝ^*cxr 11269 < clt 11270 ≤ cle 11271 (,)cioo 13400 [,]cicc 13403 vol*covol 25694 𝐿¹cibl 25849 ∫citg 25850

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7739 ax-inf2 9623 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204 ax-pre-sup 11205 ax-addf 11206

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-symdif 4202 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4871 df-int 4911 df-iun 4956 df-disj 5075 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-se 5613 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7681 df-ofr 7682 df-om 7866 df-1st 7989 df-2nd 7990 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-1o 8458 df-2o 8459 df-er 8699 df-map 8831 df-pm 8832 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-fi 9384 df-sup 9415 df-inf 9416 df-oi 9485 df-dju 9909 df-card 9947 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11470 df-neg 11471 df-div 11899 df-nn 12261 df-2 12330 df-3 12331 df-4 12332 df-n0 12532 df-z 12619 df-uz 12891 df-q 13001 df-rp 13045 df-xneg 13165 df-xadd 13166 df-xmul 13167 df-ioo 13404 df-ico 13406 df-icc 13407 df-fz 13564 df-fzo 13712 df-fl 13855 df-mod 13933 df-seq 14068 df-exp 14128 df-hash 14397 df-cj 15188 df-re 15189 df-im 15190 df-sqrt 15324 df-abs 15325 df-clim 15577 df-sum 15776 df-rest 17511 df-topgen 17532 df-psmet 21581 df-xmet 21582 df-met 21583 df-bl 21584 df-mopn 21585 df-top 23123 df-topon 23140 df-bases 23175 df-cmp 23616 df-ovol 25696 df-vol 25697 df-mbf 25851 df-itg1 25852 df-itg2 25853 df-ibl 25854 df-itg 25855

This theorem is used by: itgpowd 26282 lcmineqlem10 42906 lcmineqlem12 42908 itgioocnicc 46807 itgiccshift 46810 itgperiod 46811 fourierdlem73 47009 fourierdlem81 47017 fourierdlem82 47018 fourierdlem111 47047

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