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Theorem itgioo 25737

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 13325	. . . 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 13321	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
6	3, 4, 5	syl2anc 584	. . 3 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
7	3	rexrd 11154	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
8	4	rexrd 11154	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
9		icc0 13285	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴[,]𝐵) = ∅ ↔ 𝐵 < 𝐴))
10	7, 8, 9	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → ((𝐴[,]𝐵) = ∅ ↔ 𝐵 < 𝐴))
11	10	biimpar 477	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → (𝐴[,]𝐵) = ∅)
12	11	difeq1d 4073	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → ((𝐴[,]𝐵) ∖ (𝐴(,)𝐵)) = (∅ ∖ (𝐴(,)𝐵)))
13		0dif 4353	. . . . . . 7 ⊢ (∅ ∖ (𝐴(,)𝐵)) = ∅
14		0ss 4348	. . . . . . 7 ⊢ ∅ ⊆ {𝐴, 𝐵}
15	13, 14	eqsstri 3979	. . . . . 6 ⊢ (∅ ∖ (𝐴(,)𝐵)) ⊆ {𝐴, 𝐵}
16	12, 15	eqsstrdi 3977	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → ((𝐴[,]𝐵) ∖ (𝐴(,)𝐵)) ⊆ {𝐴, 𝐵})
17		uncom 4106	. . . . . . . . 9 ⊢ ({𝐴, 𝐵} ∪ (𝐴(,)𝐵)) = ((𝐴(,)𝐵) ∪ {𝐴, 𝐵})
18	7	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ ℝ^*)
19	8	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ ℝ^*)
20		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → 𝐴 ≤ 𝐵)
21		prunioo 13373	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → ((𝐴(,)𝐵) ∪ {𝐴, 𝐵}) = (𝐴[,]𝐵))
22	18, 19, 20, 21	syl3anc 1373	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ((𝐴(,)𝐵) ∪ {𝐴, 𝐵}) = (𝐴[,]𝐵))
23	17, 22	eqtr2id 2778	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → (𝐴[,]𝐵) = ({𝐴, 𝐵} ∪ (𝐴(,)𝐵)))
24	23	difeq1d 4073	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ((𝐴[,]𝐵) ∖ (𝐴(,)𝐵)) = (({𝐴, 𝐵} ∪ (𝐴(,)𝐵)) ∖ (𝐴(,)𝐵)))
25		difun2 4429	. . . . . . 7 ⊢ (({𝐴, 𝐵} ∪ (𝐴(,)𝐵)) ∖ (𝐴(,)𝐵)) = ({𝐴, 𝐵} ∖ (𝐴(,)𝐵))
26	24, 25	eqtrdi 2781	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ((𝐴[,]𝐵) ∖ (𝐴(,)𝐵)) = ({𝐴, 𝐵} ∖ (𝐴(,)𝐵)))
27		difss 4084	. . . . . 6 ⊢ ({𝐴, 𝐵} ∖ (𝐴(,)𝐵)) ⊆ {𝐴, 𝐵}
28	26, 27	eqsstrdi 3977	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ((𝐴[,]𝐵) ∖ (𝐴(,)𝐵)) ⊆ {𝐴, 𝐵})
29	16, 28, 4, 3	ltlecasei 11213	. . . 4 ⊢ (𝜑 → ((𝐴[,]𝐵) ∖ (𝐴(,)𝐵)) ⊆ {𝐴, 𝐵})
30	3, 4	prssd 4772	. . . 4 ⊢ (𝜑 → {𝐴, 𝐵} ⊆ ℝ)
31		prfi 9203	. . . . 5 ⊢ {𝐴, 𝐵} ∈ Fin
32		ovolfi 25415	. . . . 5 ⊢ (({𝐴, 𝐵} ∈ Fin ∧ {𝐴, 𝐵} ⊆ ℝ) → (vol*‘{𝐴, 𝐵}) = 0)
33	31, 30, 32	sylancr 587	. . . 4 ⊢ (𝜑 → (vol*‘{𝐴, 𝐵}) = 0)
34		ovolssnul 25408	. . . 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 1373	. . 3 ⊢ (𝜑 → (vol*‘((𝐴[,]𝐵) ∖ (𝐴(,)𝐵))) = 0)
36		itgioo.3	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐶 ∈ ℂ)
37	2, 6, 35, 36	itgss3 25736	. 2 ⊢ (𝜑 → (((𝑥 ∈ (𝐴(,)𝐵) ↦ 𝐶) ∈ 𝐿¹ ↔ (𝑥 ∈ (𝐴[,]𝐵) ↦ 𝐶) ∈ 𝐿¹) ∧ ∫(𝐴(,)𝐵)𝐶 d𝑥 = ∫(𝐴[,]𝐵)𝐶 d𝑥))
38	37	simprd 495	1 ⊢ (𝜑 → ∫(𝐴(,)𝐵)𝐶 d𝑥 = ∫(𝐴[,]𝐵)𝐶 d𝑥)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1541 ∈ wcel 2110 ∖ cdif 3897 ∪ cun 3898 ⊆ wss 3900 ∅c0 4281 {cpr 4576 class class class wbr 5089 ↦ cmpt 5170 ‘cfv 6477 (class class class)co 7341 Fincfn 8864 ℂcc 10996 ℝcr 10997 0cc0 10998 ℝ^*cxr 11137 < clt 11138 ≤ cle 11139 (,)cioo 13237 [,]cicc 13240 vol*covol 25383 𝐿¹cibl 25538 ∫citg 25539

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 2112 ax-9 2120 ax-10 2143 ax-11 2159 ax-12 2179 ax-ext 2702 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7663 ax-inf2 9526 ax-cnex 11054 ax-resscn 11055 ax-1cn 11056 ax-icn 11057 ax-addcl 11058 ax-addrcl 11059 ax-mulcl 11060 ax-mulrcl 11061 ax-mulcom 11062 ax-addass 11063 ax-mulass 11064 ax-distr 11065 ax-i2m1 11066 ax-1ne0 11067 ax-1rid 11068 ax-rnegex 11069 ax-rrecex 11070 ax-cnre 11071 ax-pre-lttri 11072 ax-pre-lttrn 11073 ax-pre-ltadd 11074 ax-pre-mulgt0 11075 ax-pre-sup 11076 ax-addf 11077

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 2067 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3344 df-reu 3345 df-rab 3394 df-v 3436 df-sbc 3740 df-csb 3849 df-dif 3903 df-un 3905 df-in 3907 df-ss 3917 df-pss 3920 df-symdif 4201 df-nul 4282 df-if 4474 df-pw 4550 df-sn 4575 df-pr 4577 df-op 4581 df-uni 4858 df-int 4896 df-iun 4941 df-disj 5057 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-se 5568 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6244 df-ord 6305 df-on 6306 df-lim 6307 df-suc 6308 df-iota 6433 df-fun 6479 df-fn 6480 df-f 6481 df-f1 6482 df-fo 6483 df-f1o 6484 df-fv 6485 df-isom 6486 df-riota 7298 df-ov 7344 df-oprab 7345 df-mpo 7346 df-of 7605 df-ofr 7606 df-om 7792 df-1st 7916 df-2nd 7917 df-frecs 8206 df-wrecs 8237 df-recs 8286 df-rdg 8324 df-1o 8380 df-2o 8381 df-er 8617 df-map 8747 df-pm 8748 df-en 8865 df-dom 8866 df-sdom 8867 df-fin 8868 df-fi 9290 df-sup 9321 df-inf 9322 df-oi 9391 df-dju 9786 df-card 9824 df-pnf 11140 df-mnf 11141 df-xr 11142 df-ltxr 11143 df-le 11144 df-sub 11338 df-neg 11339 df-div 11767 df-nn 12118 df-2 12180 df-3 12181 df-4 12182 df-n0 12374 df-z 12461 df-uz 12725 df-q 12839 df-rp 12883 df-xneg 13003 df-xadd 13004 df-xmul 13005 df-ioo 13241 df-ico 13243 df-icc 13244 df-fz 13400 df-fzo 13547 df-fl 13688 df-mod 13766 df-seq 13901 df-exp 13961 df-hash 14230 df-cj 14998 df-re 14999 df-im 15000 df-sqrt 15134 df-abs 15135 df-clim 15387 df-sum 15586 df-rest 17318 df-topgen 17339 df-psmet 21276 df-xmet 21277 df-met 21278 df-bl 21279 df-mopn 21280 df-top 22802 df-topon 22819 df-bases 22854 df-cmp 23295 df-ovol 25385 df-vol 25386 df-mbf 25540 df-itg1 25541 df-itg2 25542 df-ibl 25543 df-itg 25544

This theorem is referenced by: itgpowd 25977 lcmineqlem10 42050 lcmineqlem12 42052 itgioocnicc 45994 itgiccshift 45997 itgperiod 45998 fourierdlem73 46196 fourierdlem81 46204 fourierdlem82 46205 fourierdlem111 46234

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