iooinsup - Intuitionistic Logic Explorer

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Theorem iooinsup 11320

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 3422	. . 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 9940	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐴(,)𝐵) = {𝑧 ∈ ℝ^ ∣ (𝐴 < 𝑧 ∧ 𝑧 < 𝐵)})
3		iooval 9940	. . . 4 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐷 ∈ ℝ^) → (𝐶(,)𝐷) = {𝑧 ∈ ℝ^ ∣ (𝐶 < 𝑧 ∧ 𝑧 < 𝐷)})
4	2, 3	ineqan12d 3353	. . 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 11301	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝑧 ∈ ℝ^) → (sup({𝐴, 𝐶}, ℝ^, < ) < 𝑧 ↔ (𝐴 < 𝑧 ∧ 𝐶 < 𝑧)))
6	5	ad4ant124 1218	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝐵 ∈ ℝ^ ∧ 𝐷 ∈ ℝ^)) ∧ 𝑧 ∈ ℝ^) → (sup({𝐴, 𝐶}, ℝ^*, < ) < 𝑧 ↔ (𝐴 < 𝑧 ∧ 𝐶 < 𝑧)))
7		xrltmininf 11313	. . . . . . . . . 10 ⊢ ((𝑧 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐷 ∈ ℝ^) → (𝑧 < inf({𝐵, 𝐷}, ℝ^, < ) ↔ (𝑧 < 𝐵 ∧ 𝑧 < 𝐷)))
8	7	3expb 1206	. . . . . . . . 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 476	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝐵 ∈ ℝ^ ∧ 𝐷 ∈ ℝ^)) ∧ 𝑧 ∈ ℝ^) → (𝑧 < inf({𝐵, 𝐷}, ℝ^*, < ) ↔ (𝑧 < 𝐵 ∧ 𝑧 < 𝐷)))
11	6, 10	anbi12d 473	. . . . . 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 586	. . . . . 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 2740	. . . 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 588	. . 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 2248	. 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 11295	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → sup({𝐴, 𝐶}, ℝ^, < ) ∈ ℝ^*)
18		xrmincl 11309	. . . 4 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐷 ∈ ℝ^) → inf({𝐵, 𝐷}, ℝ^, < ) ∈ ℝ^*)
19		iooval 9940	. . . 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 588	. 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 2225	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 104 ↔ wb 105 = wceq 1364 ∈ wcel 2160 {crab 2472 ∩ cin 3143 {cpr 3608 class class class wbr 4018 (class class class)co 5897 supcsup 7012 infcinf 7013 ℝ^*cxr 8022 < clt 8023 (,)cioo 9920

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-nul 4144 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-iinf 4605 ax-cnex 7933 ax-resscn 7934 ax-1cn 7935 ax-1re 7936 ax-icn 7937 ax-addcl 7938 ax-addrcl 7939 ax-mulcl 7940 ax-mulrcl 7941 ax-addcom 7942 ax-mulcom 7943 ax-addass 7944 ax-mulass 7945 ax-distr 7946 ax-i2m1 7947 ax-0lt1 7948 ax-1rid 7949 ax-0id 7950 ax-rnegex 7951 ax-precex 7952 ax-cnre 7953 ax-pre-ltirr 7954 ax-pre-ltwlin 7955 ax-pre-lttrn 7956 ax-pre-apti 7957 ax-pre-ltadd 7958 ax-pre-mulgt0 7959 ax-pre-mulext 7960 ax-arch 7961 ax-caucvg 7962

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-if 3550 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-id 4311 df-po 4314 df-iso 4315 df-iord 4384 df-on 4386 df-ilim 4387 df-suc 4389 df-iom 4608 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-f1 5240 df-fo 5241 df-f1o 5242 df-fv 5243 df-isom 5244 df-riota 5852 df-ov 5900 df-oprab 5901 df-mpo 5902 df-1st 6166 df-2nd 6167 df-recs 6331 df-frec 6417 df-sup 7014 df-inf 7015 df-pnf 8025 df-mnf 8026 df-xr 8027 df-ltxr 8028 df-le 8029 df-sub 8161 df-neg 8162 df-reap 8563 df-ap 8570 df-div 8661 df-inn 8951 df-2 9009 df-3 9010 df-4 9011 df-n0 9208 df-z 9285 df-uz 9560 df-rp 9686 df-xneg 9804 df-ioo 9924 df-seqfrec 10479 df-exp 10554 df-cj 10886 df-re 10887 df-im 10888 df-rsqrt 11042 df-abs 11043

This theorem is referenced by: qtopbasss 14498 tgioo 14523

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