Theorem iooinsup 11788

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 e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) ) -> ( ( A (,) B ) i^i ( C (,) D ) ) = ( sup ( { A , C } , RR* , < ) (,)inf ( { B , D } , RR* , < ) ) )

Proof of Theorem iooinsup

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		inrab 3476	. . 3 |- ( { z e. RR* | ( A < z /\ z < B ) } i^i { z e. RR* | ( C < z /\ z < D ) } ) = { z e. RR* | ( ( A < z /\ z < B ) /\ ( C < z /\ z < D ) ) }
2		iooval 10104	. . . 4 |- ( ( A e. RR* /\ B e. RR* ) -> ( A (,) B ) = { z e. RR* | ( A < z /\ z < B ) } )
3		iooval 10104	. . . 4 |- ( ( C e. RR* /\ D e. RR* ) -> ( C (,) D ) = { z e. RR* | ( C < z /\ z < D ) } )
4	2, 3	ineqan12d 3407	. . 3 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) ) -> ( ( A (,) B ) i^i ( C (,) D ) ) = ( { z e. RR* | ( A < z /\ z < B ) } i^i { z e. RR* | ( C < z /\ z < D ) } ) )
5		xrmaxltsup 11769	. . . . . . . 8 |- ( ( A e. RR* /\ C e. RR* /\ z e. RR* ) -> ( sup ( { A , C } , RR* , < ) < z <-> ( A < z /\ C < z ) ) )
6	5	ad4ant124 1240	. . . . . . 7 |- ( ( ( ( A e. RR* /\ C e. RR* ) /\ ( B e. RR* /\ D e. RR* ) ) /\ z e. RR* ) -> ( sup ( { A , C } , RR* , < ) < z <-> ( A < z /\ C < z ) ) )
7		xrltmininf 11781	. . . . . . . . . 10 |- ( ( z e. RR* /\ B e. RR* /\ D e. RR* ) -> ( z < inf ( { B , D } , RR* , < ) <-> ( z < B /\ z < D ) ) )
8	7	3expb 1228	. . . . . . . . 9 |- ( ( z e. RR* /\ ( B e. RR* /\ D e. RR* ) ) -> ( z < inf ( { B , D } , RR* , < ) <-> ( z < B /\ z < D ) ) )
9	8	ancoms 268	. . . . . . . 8 |- ( ( ( B e. RR* /\ D e. RR* ) /\ z e. RR* ) -> ( z < inf ( { B , D } , RR* , < ) <-> ( z < B /\ z < D ) ) )
10	9	adantll 476	. . . . . . 7 |- ( ( ( ( A e. RR* /\ C e. RR* ) /\ ( B e. RR* /\ D e. RR* ) ) /\ z e. RR* ) -> ( z < inf ( { B , D } , RR* , < ) <-> ( z < B /\ z < D ) ) )
11	6, 10	anbi12d 473	. . . . . 6 |- ( ( ( ( A e. RR* /\ C e. RR* ) /\ ( B e. RR* /\ D e. RR* ) ) /\ z e. RR* ) -> ( ( sup ( { A , C } , RR* , < ) < z /\ z < inf ( { B , D } , RR* , < ) ) <-> ( ( 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 e. RR* /\ C e. RR* ) /\ ( B e. RR* /\ D e. RR* ) ) /\ z e. RR* ) -> ( ( sup ( { A , C } , RR* , < ) < z /\ z < inf ( { B , D } , RR* , < ) ) <-> ( ( A < z /\ z < B ) /\ ( C < z /\ z < D ) ) ) )
14	13	rabbidva 2787	. . . 4 |- ( ( ( A e. RR* /\ C e. RR* ) /\ ( B e. RR* /\ D e. RR* ) ) -> { z e. RR* | ( sup ( { A , C } , RR* , < ) < z /\ z < inf ( { B , D } , RR* , < ) ) } = { z e. RR* | ( ( A < z /\ z < B ) /\ ( C < z /\ z < D ) ) } )
15	14	an4s 590	. . 3 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) ) -> { z e. RR* | ( sup ( { A , C } , RR* , < ) < z /\ z < inf ( { B , D } , RR* , < ) ) } = { z e. RR* | ( ( A < z /\ z < B ) /\ ( C < z /\ z < D ) ) } )
16	1, 4, 15	3eqtr4a 2288	. 2 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) ) -> ( ( A (,) B ) i^i ( C (,) D ) ) = { z e. RR* | ( sup ( { A , C } , RR* , < ) < z /\ z < inf ( { B , D } , RR* , < ) ) } )
17		xrmaxcl 11763	. . . 4 |- ( ( A e. RR* /\ C e. RR* ) -> sup ( { A , C } , RR* , < ) e. RR* )
18		xrmincl 11777	. . . 4 |- ( ( B e. RR* /\ D e. RR* ) -> inf ( { B , D } , RR* , < ) e. RR* )
19		iooval 10104	. . . 4 |- ( ( sup ( { A , C } , RR* , < ) e. RR* /\ inf ( { B , D } , RR* , < ) e. RR* ) -> ( sup ( { A , C } , RR* , < ) (,)inf ( { B , D } , RR* , < ) ) = { z e. RR* | ( sup ( { A , C } , RR* , < ) < z /\ z < inf ( { B , D } , RR* , < ) ) } )
20	17, 18, 19	syl2an 289	. . 3 |- ( ( ( A e. RR* /\ C e. RR* ) /\ ( B e. RR* /\ D e. RR* ) ) -> ( sup ( { A , C } , RR* , < ) (,)inf ( { B , D } , RR* , < ) ) = { z e. RR* | ( sup ( { A , C } , RR* , < ) < z /\ z < inf ( { B , D } , RR* , < ) ) } )
21	20	an4s 590	. 2 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) ) -> ( sup ( { A , C } , RR* , < ) (,)inf ( { B , D } , RR* , < ) ) = { z e. RR* | ( sup ( { A , C } , RR* , < ) < z /\ z < inf ( { B , D } , RR* , < ) ) } )
22	16, 21	eqtr4d 2265	1 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) ) -> ( ( A (,) B ) i^i ( C (,) D ) ) = ( sup ( { A , C } , RR* , < ) (,)inf ( { B , D } , RR* , < ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   {crab 2512   i^i cin 3196   {cpr 3667   class class class wbr 4083  (class class class)co 6001   supcsup 7149  infcinf 7150   RR*cxr 8180   < clt 8181   (,)cioo 10084

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-mulrcl 8098  ax-addcom 8099  ax-mulcom 8100  ax-addass 8101  ax-mulass 8102  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-1rid 8106  ax-0id 8107  ax-rnegex 8108  ax-precex 8109  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-apti 8114  ax-pre-ltadd 8115  ax-pre-mulgt0 8116  ax-pre-mulext 8117  ax-arch 8118  ax-caucvg 8119

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-isom 5327  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-recs 6451  df-frec 6537  df-sup 7151  df-inf 7152  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-reap 8722  df-ap 8729  df-div 8820  df-inn 9111  df-2 9169  df-3 9170  df-4 9171  df-n0 9370  df-z 9447  df-uz 9723  df-rp 9850  df-xneg 9968  df-ioo 10088  df-seqfrec 10670  df-exp 10761  df-cj 11353  df-re 11354  df-im 11355  df-rsqrt 11509  df-abs 11510

This theorem is referenced by:  qtopbasss  15195  tgioo  15228