iooinsup - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > iooinsup		GIF version

Theorem iooinsup 11276

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 3407	. . 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 9902	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐴(,)𝐵) = {𝑧 ∈ ℝ^ ∣ (𝐴 < 𝑧 ∧ 𝑧 < 𝐵)})
3		iooval 9902	. . . 4 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐷 ∈ ℝ^) → (𝐶(,)𝐷) = {𝑧 ∈ ℝ^ ∣ (𝐶 < 𝑧 ∧ 𝑧 < 𝐷)})
4	2, 3	ineqan12d 3338	. . 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 11257	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝑧 ∈ ℝ^) → (sup({𝐴, 𝐶}, ℝ^, < ) < 𝑧 ↔ (𝐴 < 𝑧 ∧ 𝐶 < 𝑧)))
6	5	ad4ant124 1216	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝐵 ∈ ℝ^ ∧ 𝐷 ∈ ℝ^)) ∧ 𝑧 ∈ ℝ^) → (sup({𝐴, 𝐶}, ℝ^*, < ) < 𝑧 ↔ (𝐴 < 𝑧 ∧ 𝐶 < 𝑧)))
7		xrltmininf 11269	. . . . . . . . . 10 ⊢ ((𝑧 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐷 ∈ ℝ^) → (𝑧 < inf({𝐵, 𝐷}, ℝ^, < ) ↔ (𝑧 < 𝐵 ∧ 𝑧 < 𝐷)))
8	7	3expb 1204	. . . . . . . . 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 2725	. . . 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 2236	. 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 11251	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → sup({𝐴, 𝐶}, ℝ^, < ) ∈ ℝ^*)
18		xrmincl 11265	. . . 4 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐷 ∈ ℝ^) → inf({𝐵, 𝐷}, ℝ^, < ) ∈ ℝ^*)
19		iooval 9902	. . . 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 2213	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 1353 ∈ wcel 2148 {crab 2459 ∩ cin 3128 {cpr 3593 class class class wbr 4001 (class class class)co 5870 supcsup 6976 infcinf 6977 ℝ^*cxr 7985 < clt 7986 (,)cioo 9882

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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4116 ax-sep 4119 ax-nul 4127 ax-pow 4172 ax-pr 4207 ax-un 4431 ax-setind 4534 ax-iinf 4585 ax-cnex 7897 ax-resscn 7898 ax-1cn 7899 ax-1re 7900 ax-icn 7901 ax-addcl 7902 ax-addrcl 7903 ax-mulcl 7904 ax-mulrcl 7905 ax-addcom 7906 ax-mulcom 7907 ax-addass 7908 ax-mulass 7909 ax-distr 7910 ax-i2m1 7911 ax-0lt1 7912 ax-1rid 7913 ax-0id 7914 ax-rnegex 7915 ax-precex 7916 ax-cnre 7917 ax-pre-ltirr 7918 ax-pre-ltwlin 7919 ax-pre-lttrn 7920 ax-pre-apti 7921 ax-pre-ltadd 7922 ax-pre-mulgt0 7923 ax-pre-mulext 7924 ax-arch 7925 ax-caucvg 7926

This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-if 3535 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3809 df-int 3844 df-iun 3887 df-br 4002 df-opab 4063 df-mpt 4064 df-tr 4100 df-id 4291 df-po 4294 df-iso 4295 df-iord 4364 df-on 4366 df-ilim 4367 df-suc 4369 df-iom 4588 df-xp 4630 df-rel 4631 df-cnv 4632 df-co 4633 df-dm 4634 df-rn 4635 df-res 4636 df-ima 4637 df-iota 5175 df-fun 5215 df-fn 5216 df-f 5217 df-f1 5218 df-fo 5219 df-f1o 5220 df-fv 5221 df-isom 5222 df-riota 5826 df-ov 5873 df-oprab 5874 df-mpo 5875 df-1st 6136 df-2nd 6137 df-recs 6301 df-frec 6387 df-sup 6978 df-inf 6979 df-pnf 7988 df-mnf 7989 df-xr 7990 df-ltxr 7991 df-le 7992 df-sub 8124 df-neg 8125 df-reap 8526 df-ap 8533 df-div 8624 df-inn 8914 df-2 8972 df-3 8973 df-4 8974 df-n0 9171 df-z 9248 df-uz 9523 df-rp 9648 df-xneg 9766 df-ioo 9886 df-seqfrec 10439 df-exp 10513 df-cj 10842 df-re 10843 df-im 10844 df-rsqrt 10998 df-abs 10999

This theorem is referenced by: qtopbasss 13803 tgioo 13828

W3C validator