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Theorem itgioo 25777

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 13353	. . . 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 13349	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
6	3, 4, 5	syl2anc 585	. . 3 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
7	3	rexrd 11186	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
8	4	rexrd 11186	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
9		icc0 13313	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴[,]𝐵) = ∅ ↔ 𝐵 < 𝐴))
10	7, 8, 9	syl2anc 585	. . . . . . . 8 ⊢ (𝜑 → ((𝐴[,]𝐵) = ∅ ↔ 𝐵 < 𝐴))
11	10	biimpar 477	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → (𝐴[,]𝐵) = ∅)
12	11	difeq1d 4078	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → ((𝐴[,]𝐵) ∖ (𝐴(,)𝐵)) = (∅ ∖ (𝐴(,)𝐵)))
13		0dif 4358	. . . . . . 7 ⊢ (∅ ∖ (𝐴(,)𝐵)) = ∅
14		0ss 4353	. . . . . . 7 ⊢ ∅ ⊆ {𝐴, 𝐵}
15	13, 14	eqsstri 3981	. . . . . 6 ⊢ (∅ ∖ (𝐴(,)𝐵)) ⊆ {𝐴, 𝐵}
16	12, 15	eqsstrdi 3979	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → ((𝐴[,]𝐵) ∖ (𝐴(,)𝐵)) ⊆ {𝐴, 𝐵})
17		uncom 4111	. . . . . . . . 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 13401	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → ((𝐴(,)𝐵) ∪ {𝐴, 𝐵}) = (𝐴[,]𝐵))
22	18, 19, 20, 21	syl3anc 1374	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ((𝐴(,)𝐵) ∪ {𝐴, 𝐵}) = (𝐴[,]𝐵))
23	17, 22	eqtr2id 2785	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → (𝐴[,]𝐵) = ({𝐴, 𝐵} ∪ (𝐴(,)𝐵)))
24	23	difeq1d 4078	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ((𝐴[,]𝐵) ∖ (𝐴(,)𝐵)) = (({𝐴, 𝐵} ∪ (𝐴(,)𝐵)) ∖ (𝐴(,)𝐵)))
25		difun2 4434	. . . . . . 7 ⊢ (({𝐴, 𝐵} ∪ (𝐴(,)𝐵)) ∖ (𝐴(,)𝐵)) = ({𝐴, 𝐵} ∖ (𝐴(,)𝐵))
26	24, 25	eqtrdi 2788	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ((𝐴[,]𝐵) ∖ (𝐴(,)𝐵)) = ({𝐴, 𝐵} ∖ (𝐴(,)𝐵)))
27		difss 4089	. . . . . 6 ⊢ ({𝐴, 𝐵} ∖ (𝐴(,)𝐵)) ⊆ {𝐴, 𝐵}
28	26, 27	eqsstrdi 3979	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ((𝐴[,]𝐵) ∖ (𝐴(,)𝐵)) ⊆ {𝐴, 𝐵})
29	16, 28, 4, 3	ltlecasei 11245	. . . 4 ⊢ (𝜑 → ((𝐴[,]𝐵) ∖ (𝐴(,)𝐵)) ⊆ {𝐴, 𝐵})
30	3, 4	prssd 4779	. . . 4 ⊢ (𝜑 → {𝐴, 𝐵} ⊆ ℝ)
31		prfi 9228	. . . . 5 ⊢ {𝐴, 𝐵} ∈ Fin
32		ovolfi 25455	. . . . 5 ⊢ (({𝐴, 𝐵} ∈ Fin ∧ {𝐴, 𝐵} ⊆ ℝ) → (vol*‘{𝐴, 𝐵}) = 0)
33	31, 30, 32	sylancr 588	. . . 4 ⊢ (𝜑 → (vol*‘{𝐴, 𝐵}) = 0)
34		ovolssnul 25448	. . . 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 1374	. . 3 ⊢ (𝜑 → (vol*‘((𝐴[,]𝐵) ∖ (𝐴(,)𝐵))) = 0)
36		itgioo.3	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐶 ∈ ℂ)
37	2, 6, 35, 36	itgss3 25776	. 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 1542 ∈ wcel 2114 ∖ cdif 3899 ∪ cun 3900 ⊆ wss 3902 ∅c0 4286 {cpr 4583 class class class wbr 5099 ↦ cmpt 5180 ‘cfv 6493 (class class class)co 7360 Fincfn 8887 ℂcc 11028 ℝcr 11029 0cc0 11030 ℝ^*cxr 11169 < clt 11170 ≤ cle 11171 (,)cioo 13265 [,]cicc 13268 vol*covol 25423 𝐿¹cibl 25578 ∫citg 25579

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5225 ax-sep 5242 ax-nul 5252 ax-pow 5311 ax-pr 5378 ax-un 7682 ax-inf2 9554 ax-cnex 11086 ax-resscn 11087 ax-1cn 11088 ax-icn 11089 ax-addcl 11090 ax-addrcl 11091 ax-mulcl 11092 ax-mulrcl 11093 ax-mulcom 11094 ax-addass 11095 ax-mulass 11096 ax-distr 11097 ax-i2m1 11098 ax-1ne0 11099 ax-1rid 11100 ax-rnegex 11101 ax-rrecex 11102 ax-cnre 11103 ax-pre-lttri 11104 ax-pre-lttrn 11105 ax-pre-ltadd 11106 ax-pre-mulgt0 11107 ax-pre-sup 11108 ax-addf 11109

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3351 df-reu 3352 df-rab 3401 df-v 3443 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-symdif 4206 df-nul 4287 df-if 4481 df-pw 4557 df-sn 4582 df-pr 4584 df-op 4588 df-uni 4865 df-int 4904 df-iun 4949 df-disj 5067 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-se 5579 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-of 7624 df-ofr 7625 df-om 7811 df-1st 7935 df-2nd 7936 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-2o 8400 df-er 8637 df-map 8769 df-pm 8770 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-fi 9318 df-sup 9349 df-inf 9350 df-oi 9419 df-dju 9817 df-card 9855 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12150 df-2 12212 df-3 12213 df-4 12214 df-n0 12406 df-z 12493 df-uz 12756 df-q 12866 df-rp 12910 df-xneg 13030 df-xadd 13031 df-xmul 13032 df-ioo 13269 df-ico 13271 df-icc 13272 df-fz 13428 df-fzo 13575 df-fl 13716 df-mod 13794 df-seq 13929 df-exp 13989 df-hash 14258 df-cj 15026 df-re 15027 df-im 15028 df-sqrt 15162 df-abs 15163 df-clim 15415 df-sum 15614 df-rest 17346 df-topgen 17367 df-psmet 21305 df-xmet 21306 df-met 21307 df-bl 21308 df-mopn 21309 df-top 22842 df-topon 22859 df-bases 22894 df-cmp 23335 df-ovol 25425 df-vol 25426 df-mbf 25580 df-itg1 25581 df-itg2 25582 df-ibl 25583 df-itg 25584

This theorem is referenced by: itgpowd 26017 lcmineqlem10 42329 lcmineqlem12 42331 itgioocnicc 46257 itgiccshift 46260 itgperiod 46261 fourierdlem73 46459 fourierdlem81 46467 fourierdlem82 46468 fourierdlem111 46497

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