voliooico - Mathbox for Glauco Siliprandi

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Theorem voliooico 43423

Description: An open interval and a left-closed, right-open interval with the same real bounds, have the same Lebesgue measure. (Contributed by Glauco Siliprandi, 3-Mar-2021.)

**Hypotheses**
Ref	Expression
voliooico.1	⊢ (𝜑 → 𝐴 ∈ ℝ)
voliooico.2	⊢ (𝜑 → 𝐵 ∈ ℝ)

**Assertion**
Ref	Expression
voliooico	⊢ (𝜑 → (vol‘(𝐴(,)𝐵)) = (vol‘(𝐴[,)𝐵)))

**Proof of Theorem voliooico**
Step	Hyp	Ref	Expression
1		iftrue 4462	. . . . . 6 ⊢ (𝐴 < 𝐵 → if(𝐴 < 𝐵, (𝐵 − 𝐴), 0) = (𝐵 − 𝐴))
2	1	adantl 481	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝐴 < 𝐵) → if(𝐴 < 𝐵, (𝐵 − 𝐴), 0) = (𝐵 − 𝐴))
3		voliooico.2	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ ℝ)
4	3	recnd 10934	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ ℂ)
5	4	subidd 11250	. . . . . . . 8 ⊢ (𝜑 → (𝐵 − 𝐵) = 0)
6	5	eqcomd 2744	. . . . . . 7 ⊢ (𝜑 → 0 = (𝐵 − 𝐵))
7	6	ad2antrr 722	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ ¬ 𝐴 < 𝐵) → 0 = (𝐵 − 𝐵))
8		iffalse 4465	. . . . . . 7 ⊢ (¬ 𝐴 < 𝐵 → if(𝐴 < 𝐵, (𝐵 − 𝐴), 0) = 0)
9	8	adantl 481	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ ¬ 𝐴 < 𝐵) → if(𝐴 < 𝐵, (𝐵 − 𝐴), 0) = 0)
10		simpll 763	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ ¬ 𝐴 < 𝐵) → 𝜑)
11		voliooico.1	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℝ)
12	10, 11	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ ¬ 𝐴 < 𝐵) → 𝐴 ∈ ℝ)
13	10, 3	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ ¬ 𝐴 < 𝐵) → 𝐵 ∈ ℝ)
14		simpr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → 𝐴 ≤ 𝐵)
15	14	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ ¬ 𝐴 < 𝐵) → 𝐴 ≤ 𝐵)
16		simpr 484	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ ¬ 𝐴 < 𝐵) → ¬ 𝐴 < 𝐵)
17	12, 13, 15, 16	lenlteq 42793	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ ¬ 𝐴 < 𝐵) → 𝐴 = 𝐵)
18		oveq2 7263	. . . . . . . 8 ⊢ (𝐴 = 𝐵 → (𝐵 − 𝐴) = (𝐵 − 𝐵))
19	18	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝐵 − 𝐴) = (𝐵 − 𝐵))
20	10, 17, 19	syl2anc 583	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ ¬ 𝐴 < 𝐵) → (𝐵 − 𝐴) = (𝐵 − 𝐵))
21	7, 9, 20	3eqtr4d 2788	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ ¬ 𝐴 < 𝐵) → if(𝐴 < 𝐵, (𝐵 − 𝐴), 0) = (𝐵 − 𝐴))
22	2, 21	pm2.61dan 809	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → if(𝐴 < 𝐵, (𝐵 − 𝐴), 0) = (𝐵 − 𝐴))
23	22	eqcomd 2744	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → (𝐵 − 𝐴) = if(𝐴 < 𝐵, (𝐵 − 𝐴), 0))
24	11	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ ℝ)
25	3	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ ℝ)
26		volioo 24638	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → (vol‘(𝐴(,)𝐵)) = (𝐵 − 𝐴))
27	24, 25, 14, 26	syl3anc 1369	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → (vol‘(𝐴(,)𝐵)) = (𝐵 − 𝐴))
28		volico 43414	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (vol‘(𝐴[,)𝐵)) = if(𝐴 < 𝐵, (𝐵 − 𝐴), 0))
29	11, 3, 28	syl2anc 583	. . . 4 ⊢ (𝜑 → (vol‘(𝐴[,)𝐵)) = if(𝐴 < 𝐵, (𝐵 − 𝐴), 0))
30	29	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → (vol‘(𝐴[,)𝐵)) = if(𝐴 < 𝐵, (𝐵 − 𝐴), 0))
31	23, 27, 30	3eqtr4d 2788	. 2 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → (vol‘(𝐴(,)𝐵)) = (vol‘(𝐴[,)𝐵)))
32		simpl 482	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐴 ≤ 𝐵) → 𝜑)
33		simpr 484	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐴 ≤ 𝐵) → ¬ 𝐴 ≤ 𝐵)
34	32, 3	syl 17	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐴 ≤ 𝐵) → 𝐵 ∈ ℝ)
35	32, 11	syl 17	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐴 ≤ 𝐵) → 𝐴 ∈ ℝ)
36	34, 35	ltnled 11052	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐴 ≤ 𝐵) → (𝐵 < 𝐴 ↔ ¬ 𝐴 ≤ 𝐵))
37	33, 36	mpbird 256	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐴 ≤ 𝐵) → 𝐵 < 𝐴)
38	3	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → 𝐵 ∈ ℝ)
39	11	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → 𝐴 ∈ ℝ)
40		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → 𝐵 < 𝐴)
41	38, 39, 40	ltled 11053	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → 𝐵 ≤ 𝐴)
42	39	rexrd 10956	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → 𝐴 ∈ ℝ^*)
43	38	rexrd 10956	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → 𝐵 ∈ ℝ^*)
44		ioo0 13033	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴(,)𝐵) = ∅ ↔ 𝐵 ≤ 𝐴))
45	42, 43, 44	syl2anc 583	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → ((𝐴(,)𝐵) = ∅ ↔ 𝐵 ≤ 𝐴))
46	41, 45	mpbird 256	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → (𝐴(,)𝐵) = ∅)
47		ico0 13054	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴[,)𝐵) = ∅ ↔ 𝐵 ≤ 𝐴))
48	42, 43, 47	syl2anc 583	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → ((𝐴[,)𝐵) = ∅ ↔ 𝐵 ≤ 𝐴))
49	41, 48	mpbird 256	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → (𝐴[,)𝐵) = ∅)
50	46, 49	eqtr4d 2781	. . . 4 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → (𝐴(,)𝐵) = (𝐴[,)𝐵))
51	50	fveq2d 6760	. . 3 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → (vol‘(𝐴(,)𝐵)) = (vol‘(𝐴[,)𝐵)))
52	32, 37, 51	syl2anc 583	. 2 ⊢ ((𝜑 ∧ ¬ 𝐴 ≤ 𝐵) → (vol‘(𝐴(,)𝐵)) = (vol‘(𝐴[,)𝐵)))
53	31, 52	pm2.61dan 809	1 ⊢ (𝜑 → (vol‘(𝐴(,)𝐵)) = (vol‘(𝐴[,)𝐵)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1539 ∈ wcel 2108 ∅c0 4253 ifcif 4456 class class class wbr 5070 ‘cfv 6418 (class class class)co 7255 ℝcr 10801 0cc0 10802 ℝ^*cxr 10939 < clt 10940 ≤ cle 10941 − cmin 11135 (,)cioo 13008 [,)cico 13010 volcvol 24532

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-inf2 9329 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-2o 8268 df-er 8456 df-map 8575 df-pm 8576 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fi 9100 df-sup 9131 df-inf 9132 df-oi 9199 df-dju 9590 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-n0 12164 df-z 12250 df-uz 12512 df-q 12618 df-rp 12660 df-xneg 12777 df-xadd 12778 df-xmul 12779 df-ioo 13012 df-ico 13014 df-icc 13015 df-fz 13169 df-fzo 13312 df-fl 13440 df-seq 13650 df-exp 13711 df-hash 13973 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-clim 15125 df-rlim 15126 df-sum 15326 df-rest 17050 df-topgen 17071 df-psmet 20502 df-xmet 20503 df-met 20504 df-bl 20505 df-mopn 20506 df-top 21951 df-topon 21968 df-bases 22004 df-cmp 22446 df-ovol 24533 df-vol 24534

This theorem is referenced by: voliooicof 43427 vonn0ioo2 44118

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