iooinsup - Intuitionistic Logic Explorer

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Theorem iooinsup 11240

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 3399	. . 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 9865	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐴(,)𝐵) = {𝑧 ∈ ℝ^ ∣ (𝐴 < 𝑧 ∧ 𝑧 < 𝐵)})
3		iooval 9865	. . . 4 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐷 ∈ ℝ^) → (𝐶(,)𝐷) = {𝑧 ∈ ℝ^ ∣ (𝐶 < 𝑧 ∧ 𝑧 < 𝐷)})
4	2, 3	ineqan12d 3330	. . 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 11221	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝑧 ∈ ℝ^) → (sup({𝐴, 𝐶}, ℝ^, < ) < 𝑧 ↔ (𝐴 < 𝑧 ∧ 𝐶 < 𝑧)))
6	5	ad4ant124 1211	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝐵 ∈ ℝ^ ∧ 𝐷 ∈ ℝ^)) ∧ 𝑧 ∈ ℝ^) → (sup({𝐴, 𝐶}, ℝ^*, < ) < 𝑧 ↔ (𝐴 < 𝑧 ∧ 𝐶 < 𝑧)))
7		xrltmininf 11233	. . . . . . . . . 10 ⊢ ((𝑧 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐷 ∈ ℝ^) → (𝑧 < inf({𝐵, 𝐷}, ℝ^, < ) ↔ (𝑧 < 𝐵 ∧ 𝑧 < 𝐷)))
8	7	3expb 1199	. . . . . . . . 9 ⊢ ((𝑧 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 𝐷 ∈ ℝ^)) → (𝑧 < inf({𝐵, 𝐷}, ℝ^, < ) ↔ (𝑧 < 𝐵 ∧ 𝑧 < 𝐷)))
9	8	ancoms 266	. . . . . . . 8 ⊢ (((𝐵 ∈ ℝ^* ∧ 𝐷 ∈ ℝ^) ∧ 𝑧 ∈ ℝ^) → (𝑧 < inf({𝐵, 𝐷}, ℝ^*, < ) ↔ (𝑧 < 𝐵 ∧ 𝑧 < 𝐷)))
10	9	adantll 473	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝐵 ∈ ℝ^ ∧ 𝐷 ∈ ℝ^)) ∧ 𝑧 ∈ ℝ^) → (𝑧 < inf({𝐵, 𝐷}, ℝ^*, < ) ↔ (𝑧 < 𝐵 ∧ 𝑧 < 𝐷)))
11	6, 10	anbi12d 470	. . . . . 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 581	. . . . . 6 ⊢ (((𝐴 < 𝑧 ∧ 𝑧 < 𝐵) ∧ (𝐶 < 𝑧 ∧ 𝑧 < 𝐷)) ↔ ((𝐴 < 𝑧 ∧ 𝐶 < 𝑧) ∧ (𝑧 < 𝐵 ∧ 𝑧 < 𝐷)))
13	11, 12	bitr4di 197	. . . . 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 2718	. . . 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 583	. . 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 2229	. 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 11215	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → sup({𝐴, 𝐶}, ℝ^, < ) ∈ ℝ^*)
18		xrmincl 11229	. . . 4 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐷 ∈ ℝ^) → inf({𝐵, 𝐷}, ℝ^, < ) ∈ ℝ^*)
19		iooval 9865	. . . 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 287	. . 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 583	. 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 2206	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

Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1348 ∈ wcel 2141 {crab 2452 ∩ cin 3120 {cpr 3584 class class class wbr 3989 (class class class)co 5853 supcsup 6959 infcinf 6960 ℝ^*cxr 7953 < clt 7954 (,)cioo 9845

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-mulrcl 7873 ax-addcom 7874 ax-mulcom 7875 ax-addass 7876 ax-mulass 7877 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-1rid 7881 ax-0id 7882 ax-rnegex 7883 ax-precex 7884 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-apti 7889 ax-pre-ltadd 7890 ax-pre-mulgt0 7891 ax-pre-mulext 7892 ax-arch 7893 ax-caucvg 7894

This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3527 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-id 4278 df-po 4281 df-iso 4282 df-iord 4351 df-on 4353 df-ilim 4354 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-isom 5207 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-recs 6284 df-frec 6370 df-sup 6961 df-inf 6962 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-reap 8494 df-ap 8501 df-div 8590 df-inn 8879 df-2 8937 df-3 8938 df-4 8939 df-n0 9136 df-z 9213 df-uz 9488 df-rp 9611 df-xneg 9729 df-ioo 9849 df-seqfrec 10402 df-exp 10476 df-cj 10806 df-re 10807 df-im 10808 df-rsqrt 10962 df-abs 10963

This theorem is referenced by: qtopbasss 13315 tgioo 13340

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