voliooico - Mathbox for Glauco Siliprandi

	Mathbox for Glauco Siliprandi		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > voliooico		Structured version Visualization version GIF version

Theorem voliooico 42271

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 4472	. . . . . 6 ⊢ (𝐴 < 𝐵 → if(𝐴 < 𝐵, (𝐵 − 𝐴), 0) = (𝐵 − 𝐴))
2	1	adantl 484	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝐴 < 𝐵) → if(𝐴 < 𝐵, (𝐵 − 𝐴), 0) = (𝐵 − 𝐴))
3		voliooico.2	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ ℝ)
4	3	recnd 10663	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ ℂ)
5	4	subidd 10979	. . . . . . . 8 ⊢ (𝜑 → (𝐵 − 𝐵) = 0)
6	5	eqcomd 2827	. . . . . . 7 ⊢ (𝜑 → 0 = (𝐵 − 𝐵))
7	6	ad2antrr 724	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ ¬ 𝐴 < 𝐵) → 0 = (𝐵 − 𝐵))
8		iffalse 4475	. . . . . . 7 ⊢ (¬ 𝐴 < 𝐵 → if(𝐴 < 𝐵, (𝐵 − 𝐴), 0) = 0)
9	8	adantl 484	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ ¬ 𝐴 < 𝐵) → if(𝐴 < 𝐵, (𝐵 − 𝐴), 0) = 0)
10		simpll 765	. . . . . . 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 487	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → 𝐴 ≤ 𝐵)
15	14	adantr 483	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ ¬ 𝐴 < 𝐵) → 𝐴 ≤ 𝐵)
16		simpr 487	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ ¬ 𝐴 < 𝐵) → ¬ 𝐴 < 𝐵)
17	12, 13, 15, 16	lenlteq 41625	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ ¬ 𝐴 < 𝐵) → 𝐴 = 𝐵)
18		oveq2 7158	. . . . . . . 8 ⊢ (𝐴 = 𝐵 → (𝐵 − 𝐴) = (𝐵 − 𝐵))
19	18	adantl 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝐵 − 𝐴) = (𝐵 − 𝐵))
20	10, 17, 19	syl2anc 586	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ ¬ 𝐴 < 𝐵) → (𝐵 − 𝐴) = (𝐵 − 𝐵))
21	7, 9, 20	3eqtr4d 2866	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ ¬ 𝐴 < 𝐵) → if(𝐴 < 𝐵, (𝐵 − 𝐴), 0) = (𝐵 − 𝐴))
22	2, 21	pm2.61dan 811	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → if(𝐴 < 𝐵, (𝐵 − 𝐴), 0) = (𝐵 − 𝐴))
23	22	eqcomd 2827	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → (𝐵 − 𝐴) = if(𝐴 < 𝐵, (𝐵 − 𝐴), 0))
24	11	adantr 483	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ ℝ)
25	3	adantr 483	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ ℝ)
26		volioo 24164	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → (vol‘(𝐴(,)𝐵)) = (𝐵 − 𝐴))
27	24, 25, 14, 26	syl3anc 1367	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → (vol‘(𝐴(,)𝐵)) = (𝐵 − 𝐴))
28		volico 42262	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (vol‘(𝐴[,)𝐵)) = if(𝐴 < 𝐵, (𝐵 − 𝐴), 0))
29	11, 3, 28	syl2anc 586	. . . 4 ⊢ (𝜑 → (vol‘(𝐴[,)𝐵)) = if(𝐴 < 𝐵, (𝐵 − 𝐴), 0))
30	29	adantr 483	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → (vol‘(𝐴[,)𝐵)) = if(𝐴 < 𝐵, (𝐵 − 𝐴), 0))
31	23, 27, 30	3eqtr4d 2866	. 2 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → (vol‘(𝐴(,)𝐵)) = (vol‘(𝐴[,)𝐵)))
32		simpl 485	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐴 ≤ 𝐵) → 𝜑)
33		simpr 487	. . . 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 10781	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐴 ≤ 𝐵) → (𝐵 < 𝐴 ↔ ¬ 𝐴 ≤ 𝐵))
37	33, 36	mpbird 259	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐴 ≤ 𝐵) → 𝐵 < 𝐴)
38	3	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → 𝐵 ∈ ℝ)
39	11	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → 𝐴 ∈ ℝ)
40		simpr 487	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → 𝐵 < 𝐴)
41	38, 39, 40	ltled 10782	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → 𝐵 ≤ 𝐴)
42	39	rexrd 10685	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → 𝐴 ∈ ℝ^*)
43	38	rexrd 10685	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → 𝐵 ∈ ℝ^*)
44		ioo0 12757	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴(,)𝐵) = ∅ ↔ 𝐵 ≤ 𝐴))
45	42, 43, 44	syl2anc 586	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → ((𝐴(,)𝐵) = ∅ ↔ 𝐵 ≤ 𝐴))
46	41, 45	mpbird 259	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → (𝐴(,)𝐵) = ∅)
47		ico0 12778	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴[,)𝐵) = ∅ ↔ 𝐵 ≤ 𝐴))
48	42, 43, 47	syl2anc 586	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → ((𝐴[,)𝐵) = ∅ ↔ 𝐵 ≤ 𝐴))
49	41, 48	mpbird 259	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → (𝐴[,)𝐵) = ∅)
50	46, 49	eqtr4d 2859	. . . 4 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → (𝐴(,)𝐵) = (𝐴[,)𝐵))
51	50	fveq2d 6668	. . 3 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → (vol‘(𝐴(,)𝐵)) = (vol‘(𝐴[,)𝐵)))
52	32, 37, 51	syl2anc 586	. 2 ⊢ ((𝜑 ∧ ¬ 𝐴 ≤ 𝐵) → (vol‘(𝐴(,)𝐵)) = (vol‘(𝐴[,)𝐵)))
53	31, 52	pm2.61dan 811	1 ⊢ (𝜑 → (vol‘(𝐴(,)𝐵)) = (vol‘(𝐴[,)𝐵)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∅c0 4290 ifcif 4466 class class class wbr 5058 ‘cfv 6349 (class class class)co 7150 ℝcr 10530 0cc0 10531 ℝ^*cxr 10668 < clt 10669 ≤ cle 10670 − cmin 10864 (,)cioo 12732 [,)cico 12734 volcvol 24058

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-inf2 9098 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-se 5509 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-oadd 8100 df-er 8283 df-map 8402 df-pm 8403 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-fi 8869 df-sup 8900 df-inf 8901 df-oi 8968 df-dju 9324 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-n0 11892 df-z 11976 df-uz 12238 df-q 12343 df-rp 12384 df-xneg 12501 df-xadd 12502 df-xmul 12503 df-ioo 12736 df-ico 12738 df-icc 12739 df-fz 12887 df-fzo 13028 df-fl 13156 df-seq 13364 df-exp 13424 df-hash 13685 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-clim 14839 df-rlim 14840 df-sum 15037 df-rest 16690 df-topgen 16711 df-psmet 20531 df-xmet 20532 df-met 20533 df-bl 20534 df-mopn 20535 df-top 21496 df-topon 21513 df-bases 21548 df-cmp 21989 df-ovol 24059 df-vol 24060

This theorem is referenced by: voliooicof 42275 vonn0ioo2 42966

W3C validator