iooinsup - Intuitionistic Logic Explorer

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Theorem iooinsup 12062

Description: Intersection of two open intervals of extended reals. (Contributed by NM, 7-Feb-2007.) (Revised by Jim Kingdon, 22-May-2023.)

**Assertion**
Ref	Expression
iooinsup	⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐷 ∈ ℝ^)) → ((𝐴(,)𝐵) ∩ (𝐶(,)𝐷)) = (sup({𝐴, 𝐶}, ℝ^, < )(,)inf({𝐵, 𝐷}, ℝ^*, < )))

Proof of Theorem iooinsup Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		inrab 3505	. . 3 ⊢ ({𝑧 ∈ ℝ^* ∣ (𝐴 < 𝑧 ∧ 𝑧 < 𝐵)} ∩ {𝑧 ∈ ℝ^* ∣ (𝐶 < 𝑧 ∧ 𝑧 < 𝐷)}) = {𝑧 ∈ ℝ^* ∣ ((𝐴 < 𝑧 ∧ 𝑧 < 𝐵) ∧ (𝐶 < 𝑧 ∧ 𝑧 < 𝐷))}
2		iooval 10321	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐴(,)𝐵) = {𝑧 ∈ ℝ^ ∣ (𝐴 < 𝑧 ∧ 𝑧 < 𝐵)})
3		iooval 10321	. . . 4 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐷 ∈ ℝ^) → (𝐶(,)𝐷) = {𝑧 ∈ ℝ^ ∣ (𝐶 < 𝑧 ∧ 𝑧 < 𝐷)})
4	2, 3	ineqan12d 3434	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐷 ∈ ℝ^)) → ((𝐴(,)𝐵) ∩ (𝐶(,)𝐷)) = ({𝑧 ∈ ℝ^ ∣ (𝐴 < 𝑧 ∧ 𝑧 < 𝐵)} ∩ {𝑧 ∈ ℝ^* ∣ (𝐶 < 𝑧 ∧ 𝑧 < 𝐷)}))
5		xrmaxltsup 12043	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝑧 ∈ ℝ^) → (sup({𝐴, 𝐶}, ℝ^, < ) < 𝑧 ↔ (𝐴 < 𝑧 ∧ 𝐶 < 𝑧)))
6	5	ad4ant124 1247	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝐵 ∈ ℝ^ ∧ 𝐷 ∈ ℝ^)) ∧ 𝑧 ∈ ℝ^) → (sup({𝐴, 𝐶}, ℝ^*, < ) < 𝑧 ↔ (𝐴 < 𝑧 ∧ 𝐶 < 𝑧)))
7		xrltmininf 12055	. . . . . . . . . 10 ⊢ ((𝑧 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐷 ∈ ℝ^) → (𝑧 < inf({𝐵, 𝐷}, ℝ^, < ) ↔ (𝑧 < 𝐵 ∧ 𝑧 < 𝐷)))
8	7	3expb 1235	. . . . . . . . 9 ⊢ ((𝑧 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 𝐷 ∈ ℝ^)) → (𝑧 < inf({𝐵, 𝐷}, ℝ^, < ) ↔ (𝑧 < 𝐵 ∧ 𝑧 < 𝐷)))
9	8	ancoms 268	. . . . . . . 8 ⊢ (((𝐵 ∈ ℝ^* ∧ 𝐷 ∈ ℝ^) ∧ 𝑧 ∈ ℝ^) → (𝑧 < inf({𝐵, 𝐷}, ℝ^*, < ) ↔ (𝑧 < 𝐵 ∧ 𝑧 < 𝐷)))
10	9	adantll 480	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝐵 ∈ ℝ^ ∧ 𝐷 ∈ ℝ^)) ∧ 𝑧 ∈ ℝ^) → (𝑧 < inf({𝐵, 𝐷}, ℝ^*, < ) ↔ (𝑧 < 𝐵 ∧ 𝑧 < 𝐷)))
11	6, 10	anbi12d 477	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝐵 ∈ ℝ^ ∧ 𝐷 ∈ ℝ^)) ∧ 𝑧 ∈ ℝ^) → ((sup({𝐴, 𝐶}, ℝ^, < ) < 𝑧 ∧ 𝑧 < inf({𝐵, 𝐷}, ℝ^, < )) ↔ ((𝐴 < 𝑧 ∧ 𝐶 < 𝑧) ∧ (𝑧 < 𝐵 ∧ 𝑧 < 𝐷))))
12		an4 592	. . . . . 6 ⊢ (((𝐴 < 𝑧 ∧ 𝑧 < 𝐵) ∧ (𝐶 < 𝑧 ∧ 𝑧 < 𝐷)) ↔ ((𝐴 < 𝑧 ∧ 𝐶 < 𝑧) ∧ (𝑧 < 𝐵 ∧ 𝑧 < 𝐷)))
13	11, 12	bitr4di 198	. . . . 5 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝐵 ∈ ℝ^ ∧ 𝐷 ∈ ℝ^)) ∧ 𝑧 ∈ ℝ^) → ((sup({𝐴, 𝐶}, ℝ^, < ) < 𝑧 ∧ 𝑧 < inf({𝐵, 𝐷}, ℝ^, < )) ↔ ((𝐴 < 𝑧 ∧ 𝑧 < 𝐵) ∧ (𝐶 < 𝑧 ∧ 𝑧 < 𝐷))))
14	13	rabbidva 2809	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝐵 ∈ ℝ^ ∧ 𝐷 ∈ ℝ^)) → {𝑧 ∈ ℝ^ ∣ (sup({𝐴, 𝐶}, ℝ^, < ) < 𝑧 ∧ 𝑧 < inf({𝐵, 𝐷}, ℝ^, < ))} = {𝑧 ∈ ℝ^* ∣ ((𝐴 < 𝑧 ∧ 𝑧 < 𝐵) ∧ (𝐶 < 𝑧 ∧ 𝑧 < 𝐷))})
15	14	an4s 596	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐷 ∈ ℝ^)) → {𝑧 ∈ ℝ^ ∣ (sup({𝐴, 𝐶}, ℝ^, < ) < 𝑧 ∧ 𝑧 < inf({𝐵, 𝐷}, ℝ^, < ))} = {𝑧 ∈ ℝ^* ∣ ((𝐴 < 𝑧 ∧ 𝑧 < 𝐵) ∧ (𝐶 < 𝑧 ∧ 𝑧 < 𝐷))})
16	1, 4, 15	3eqtr4a 2297	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐷 ∈ ℝ^)) → ((𝐴(,)𝐵) ∩ (𝐶(,)𝐷)) = {𝑧 ∈ ℝ^ ∣ (sup({𝐴, 𝐶}, ℝ^, < ) < 𝑧 ∧ 𝑧 < inf({𝐵, 𝐷}, ℝ^, < ))})
17		xrmaxcl 12037	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → sup({𝐴, 𝐶}, ℝ^, < ) ∈ ℝ^*)
18		xrmincl 12051	. . . 4 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐷 ∈ ℝ^) → inf({𝐵, 𝐷}, ℝ^, < ) ∈ ℝ^*)
19		iooval 10321	. . . 4 ⊢ ((sup({𝐴, 𝐶}, ℝ^, < ) ∈ ℝ^ ∧ inf({𝐵, 𝐷}, ℝ^, < ) ∈ ℝ^) → (sup({𝐴, 𝐶}, ℝ^, < )(,)inf({𝐵, 𝐷}, ℝ^, < )) = {𝑧 ∈ ℝ^* ∣ (sup({𝐴, 𝐶}, ℝ^, < ) < 𝑧 ∧ 𝑧 < inf({𝐵, 𝐷}, ℝ^, < ))})
20	17, 18, 19	syl2an 289	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝐵 ∈ ℝ^ ∧ 𝐷 ∈ ℝ^)) → (sup({𝐴, 𝐶}, ℝ^, < )(,)inf({𝐵, 𝐷}, ℝ^, < )) = {𝑧 ∈ ℝ^ ∣ (sup({𝐴, 𝐶}, ℝ^, < ) < 𝑧 ∧ 𝑧 < inf({𝐵, 𝐷}, ℝ^, < ))})
21	20	an4s 596	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐷 ∈ ℝ^)) → (sup({𝐴, 𝐶}, ℝ^, < )(,)inf({𝐵, 𝐷}, ℝ^, < )) = {𝑧 ∈ ℝ^ ∣ (sup({𝐴, 𝐶}, ℝ^, < ) < 𝑧 ∧ 𝑧 < inf({𝐵, 𝐷}, ℝ^, < ))})
22	16, 21	eqtr4d 2274	1 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐷 ∈ ℝ^)) → ((𝐴(,)𝐵) ∩ (𝐶(,)𝐷)) = (sup({𝐴, 𝐶}, ℝ^, < )(,)inf({𝐵, 𝐷}, ℝ^*, < )))

Colors of variables: wff set class

This proof depends on syntax axioms: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1402 ∈ wcel 2209 {crab 2532 ∩ cin 3219 {cpr 3710 class class class wbr 4130 (class class class)co 6085 supcsup 7323 infcinf 7324 ℝ^*cxr 8360 < clt 8361 (,)cioo 10301

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4246 ax-sep 4249 ax-nul 4259 ax-pow 4311 ax-pr 4346 ax-un 4578 ax-setind 4684 ax-iinf 4735 ax-cnex 8271 ax-resscn 8272 ax-1cn 8273 ax-1re 8274 ax-icn 8275 ax-addcl 8276 ax-addrcl 8277 ax-mulcl 8278 ax-mulrcl 8279 ax-addcom 8280 ax-mulcom 8281 ax-addass 8282 ax-mulass 8283 ax-distr 8284 ax-i2m1 8285 ax-0lt1 8286 ax-1rid 8287 ax-0id 8288 ax-rnegex 8289 ax-precex 8290 ax-cnre 8291 ax-pre-ltirr 8292 ax-pre-ltwlin 8293 ax-pre-lttrn 8294 ax-pre-apti 8295 ax-pre-ltadd 8296 ax-pre-mulgt0 8297 ax-pre-mulext 8298 ax-arch 8299 ax-caucvg 8300

This proof depends on definitions: df-bi 117 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rmo 2536 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3639 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-iun 4014 df-br 4131 df-opab 4193 df-mpt 4194 df-tr 4230 df-id 4438 df-po 4441 df-iso 4442 df-iord 4511 df-on 4513 df-ilim 4514 df-suc 4516 df-iom 4738 df-xp 4780 df-rel 4781 df-cnv 4782 df-co 4783 df-dm 4784 df-rn 4785 df-res 4786 df-ima 4787 df-iota 5337 df-fun 5379 df-fn 5380 df-f 5381 df-f1 5382 df-fo 5383 df-f1o 5384 df-fv 5385 df-isom 5386 df-riota 6038 df-ov 6088 df-oprab 6089 df-mpo 6090 df-1st 6374 df-2nd 6375 df-recs 6576 df-frec 6662 df-sup 7325 df-inf 7326 df-pnf 8363 df-mnf 8364 df-xr 8365 df-ltxr 8366 df-le 8367 df-sub 8501 df-neg 8502 df-reap 8906 df-ap 8913 df-div 9006 df-inn 9308 df-2 9366 df-3 9367 df-4 9368 df-n0 9569 df-z 9650 df-uz 9932 df-rp 10066 df-xneg 10185 df-ioo 10305 df-seqfrec 10900 df-exp 10991 df-cj 11623 df-re 11624 df-im 11625 df-rsqrt 11780 df-abs 11781

This theorem is used by: qtopbasss 15713 tgioo 15746

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