voliooico - Mathbox for Glauco Siliprandi

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Theorem voliooico 45948

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 4537	. . . . . 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 11287	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ ℂ)
5	4	subidd 11606	. . . . . . . 8 ⊢ (𝜑 → (𝐵 − 𝐵) = 0)
6	5	eqcomd 2741	. . . . . . 7 ⊢ (𝜑 → 0 = (𝐵 − 𝐵))
7	6	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ ¬ 𝐴 < 𝐵) → 0 = (𝐵 − 𝐵))
8		iffalse 4540	. . . . . . 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 767	. . . . . . 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 45314	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ ¬ 𝐴 < 𝐵) → 𝐴 = 𝐵)
18		oveq2 7439	. . . . . . . 8 ⊢ (𝐴 = 𝐵 → (𝐵 − 𝐴) = (𝐵 − 𝐵))
19	18	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝐵 − 𝐴) = (𝐵 − 𝐵))
20	10, 17, 19	syl2anc 584	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ ¬ 𝐴 < 𝐵) → (𝐵 − 𝐴) = (𝐵 − 𝐵))
21	7, 9, 20	3eqtr4d 2785	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ ¬ 𝐴 < 𝐵) → if(𝐴 < 𝐵, (𝐵 − 𝐴), 0) = (𝐵 − 𝐴))
22	2, 21	pm2.61dan 813	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → if(𝐴 < 𝐵, (𝐵 − 𝐴), 0) = (𝐵 − 𝐴))
23	22	eqcomd 2741	. . 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 25618	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → (vol‘(𝐴(,)𝐵)) = (𝐵 − 𝐴))
27	24, 25, 14, 26	syl3anc 1370	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → (vol‘(𝐴(,)𝐵)) = (𝐵 − 𝐴))
28		volico 45939	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (vol‘(𝐴[,)𝐵)) = if(𝐴 < 𝐵, (𝐵 − 𝐴), 0))
29	11, 3, 28	syl2anc 584	. . . 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 2785	. 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 11406	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐴 ≤ 𝐵) → (𝐵 < 𝐴 ↔ ¬ 𝐴 ≤ 𝐵))
37	33, 36	mpbird 257	. . 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 11407	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → 𝐵 ≤ 𝐴)
42	39	rexrd 11309	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → 𝐴 ∈ ℝ^*)
43	38	rexrd 11309	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → 𝐵 ∈ ℝ^*)
44		ioo0 13409	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴(,)𝐵) = ∅ ↔ 𝐵 ≤ 𝐴))
45	42, 43, 44	syl2anc 584	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → ((𝐴(,)𝐵) = ∅ ↔ 𝐵 ≤ 𝐴))
46	41, 45	mpbird 257	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → (𝐴(,)𝐵) = ∅)
47		ico0 13430	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴[,)𝐵) = ∅ ↔ 𝐵 ≤ 𝐴))
48	42, 43, 47	syl2anc 584	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → ((𝐴[,)𝐵) = ∅ ↔ 𝐵 ≤ 𝐴))
49	41, 48	mpbird 257	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → (𝐴[,)𝐵) = ∅)
50	46, 49	eqtr4d 2778	. . . 4 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → (𝐴(,)𝐵) = (𝐴[,)𝐵))
51	50	fveq2d 6911	. . 3 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → (vol‘(𝐴(,)𝐵)) = (vol‘(𝐴[,)𝐵)))
52	32, 37, 51	syl2anc 584	. 2 ⊢ ((𝜑 ∧ ¬ 𝐴 ≤ 𝐵) → (vol‘(𝐴(,)𝐵)) = (vol‘(𝐴[,)𝐵)))
53	31, 52	pm2.61dan 813	1 ⊢ (𝜑 → (vol‘(𝐴(,)𝐵)) = (vol‘(𝐴[,)𝐵)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∅c0 4339 ifcif 4531 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 ℝcr 11152 0cc0 11153 ℝ^*cxr 11292 < clt 11293 ≤ cle 11294 − cmin 11490 (,)cioo 13384 [,)cico 13386 volcvol 25512

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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-pm 8868 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fi 9449 df-sup 9480 df-inf 9481 df-oi 9548 df-dju 9939 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12612 df-uz 12877 df-q 12989 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-ioo 13388 df-ico 13390 df-icc 13391 df-fz 13545 df-fzo 13692 df-fl 13829 df-seq 14040 df-exp 14100 df-hash 14367 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-clim 15521 df-rlim 15522 df-sum 15720 df-rest 17469 df-topgen 17490 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-top 22916 df-topon 22933 df-bases 22969 df-cmp 23411 df-ovol 25513 df-vol 25514

This theorem is referenced by: voliooicof 45952 vonn0ioo2 46646

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