iocinif - Mathbox for Thierry Arnoux

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Theorem iocinif 32758

Description: Relate intersection of two open-below, closed-above intervals with the same upper bound with a conditional construct. (Contributed by Thierry Arnoux, 7-Aug-2017.)

**Assertion**
Ref	Expression
iocinif	⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → ((𝐴(,]𝐶) ∩ (𝐵(,]𝐶)) = if(𝐴 < 𝐵, (𝐵(,]𝐶), (𝐴(,]𝐶)))

**Proof of Theorem iocinif**
Step	Hyp	Ref	Expression
1		exmid 894	. . 3 ⊢ (𝐴 < 𝐵 ∨ ¬ 𝐴 < 𝐵)
2		xrltle 13165	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 < 𝐵 → 𝐴 ≤ 𝐵))
3	2	imp 406	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → 𝐴 ≤ 𝐵)
4	3	3adantl3 1169	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → 𝐴 ≤ 𝐵)
5		iocinioc2 32756	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝐴 ≤ 𝐵) → ((𝐴(,]𝐶) ∩ (𝐵(,]𝐶)) = (𝐵(,]𝐶))
6	4, 5	syldan 591	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → ((𝐴(,]𝐶) ∩ (𝐵(,]𝐶)) = (𝐵(,]𝐶))
7	6	ex 412	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐴 < 𝐵 → ((𝐴(,]𝐶) ∩ (𝐵(,]𝐶)) = (𝐵(,]𝐶)))
8	7	ancld 550	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐴 < 𝐵 → (𝐴 < 𝐵 ∧ ((𝐴(,]𝐶) ∩ (𝐵(,]𝐶)) = (𝐵(,]𝐶))))
9		simpl2 1193	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ ¬ 𝐴 < 𝐵) → 𝐵 ∈ ℝ^)
10		simpl1 1192	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ ¬ 𝐴 < 𝐵) → 𝐴 ∈ ℝ^)
11		simpr 484	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ ¬ 𝐴 < 𝐵) → ¬ 𝐴 < 𝐵)
12		xrlenlt 11300	. . . . . . . . 9 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*) → (𝐵 ≤ 𝐴 ↔ ¬ 𝐴 < 𝐵))
13	12	biimpar 477	. . . . . . . 8 ⊢ (((𝐵 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*) ∧ ¬ 𝐴 < 𝐵) → 𝐵 ≤ 𝐴)
14	9, 10, 11, 13	syl21anc 837	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ ¬ 𝐴 < 𝐵) → 𝐵 ≤ 𝐴)
15		3ancoma 1097	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ↔ (𝐴 ∈ ℝ^ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*))
16		incom 4184	. . . . . . . . 9 ⊢ ((𝐵(,]𝐶) ∩ (𝐴(,]𝐶)) = ((𝐴(,]𝐶) ∩ (𝐵(,]𝐶))
17		iocinioc2 32756	. . . . . . . . 9 ⊢ (((𝐵 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝐵 ≤ 𝐴) → ((𝐵(,]𝐶) ∩ (𝐴(,]𝐶)) = (𝐴(,]𝐶))
18	16, 17	eqtr3id 2784	. . . . . . . 8 ⊢ (((𝐵 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝐵 ≤ 𝐴) → ((𝐴(,]𝐶) ∩ (𝐵(,]𝐶)) = (𝐴(,]𝐶))
19	15, 18	sylanbr 582	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝐵 ≤ 𝐴) → ((𝐴(,]𝐶) ∩ (𝐵(,]𝐶)) = (𝐴(,]𝐶))
20	14, 19	syldan 591	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ ¬ 𝐴 < 𝐵) → ((𝐴(,]𝐶) ∩ (𝐵(,]𝐶)) = (𝐴(,]𝐶))
21	20	ex 412	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (¬ 𝐴 < 𝐵 → ((𝐴(,]𝐶) ∩ (𝐵(,]𝐶)) = (𝐴(,]𝐶)))
22	21	ancld 550	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (¬ 𝐴 < 𝐵 → (¬ 𝐴 < 𝐵 ∧ ((𝐴(,]𝐶) ∩ (𝐵(,]𝐶)) = (𝐴(,]𝐶))))
23	8, 22	orim12d 966	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → ((𝐴 < 𝐵 ∨ ¬ 𝐴 < 𝐵) → ((𝐴 < 𝐵 ∧ ((𝐴(,]𝐶) ∩ (𝐵(,]𝐶)) = (𝐵(,]𝐶)) ∨ (¬ 𝐴 < 𝐵 ∧ ((𝐴(,]𝐶) ∩ (𝐵(,]𝐶)) = (𝐴(,]𝐶)))))
24	1, 23	mpi 20	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → ((𝐴 < 𝐵 ∧ ((𝐴(,]𝐶) ∩ (𝐵(,]𝐶)) = (𝐵(,]𝐶)) ∨ (¬ 𝐴 < 𝐵 ∧ ((𝐴(,]𝐶) ∩ (𝐵(,]𝐶)) = (𝐴(,]𝐶))))
25		eqif 4542	. 2 ⊢ (((𝐴(,]𝐶) ∩ (𝐵(,]𝐶)) = if(𝐴 < 𝐵, (𝐵(,]𝐶), (𝐴(,]𝐶)) ↔ ((𝐴 < 𝐵 ∧ ((𝐴(,]𝐶) ∩ (𝐵(,]𝐶)) = (𝐵(,]𝐶)) ∨ (¬ 𝐴 < 𝐵 ∧ ((𝐴(,]𝐶) ∩ (𝐵(,]𝐶)) = (𝐴(,]𝐶))))
26	24, 25	sylibr 234	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → ((𝐴(,]𝐶) ∩ (𝐵(,]𝐶)) = if(𝐴 < 𝐵, (𝐵(,]𝐶), (𝐴(,]𝐶)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 = wceq 1540 ∈ wcel 2108 ∩ cin 3925 ifcif 4500 class class class wbr 5119 (class class class)co 7405 ℝ^*cxr 11268 < clt 11269 ≤ cle 11270 (,]cioc 13363

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-cnex 11185 ax-resscn 11186 ax-pre-lttri 11203 ax-pre-lttrn 11204

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-op 4608 df-uni 4884 df-br 5120 df-opab 5182 df-mpt 5202 df-id 5548 df-po 5561 df-so 5562 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-ov 7408 df-oprab 7409 df-mpo 7410 df-er 8719 df-en 8960 df-dom 8961 df-sdom 8962 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275 df-ioc 13367

This theorem is referenced by: pnfneige0 33982

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