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Theorem iooinsup 11459
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.
StepHypRef Expression
1 inrab 3436 . . 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 10000 . . . 4 |- ( ( A e. RR* /\ B e. RR* ) -> ( A (,) B ) = { z e. RR* | ( A < z /\ z < B ) } )
3 iooval 10000 . . . 4 |- ( ( C e. RR* /\ D e. RR* ) -> ( C (,) D ) = { z e. RR* | ( C < z /\ z < D ) } )
42, 3ineqan12d 3367 . . 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 11440 . . . . . . . 8 |- ( ( A e. RR* /\ C e. RR* /\ z e. RR* ) -> ( sup ( { A , C } , RR* , < ) < z <-> ( A < z /\ C < z ) ) )
65ad4ant124 1218 . . . . . . 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 11452 . . . . . . . . . 10 |- ( ( z e. RR* /\ B e. RR* /\ D e. RR* ) -> ( z < inf ( { B , D } , RR* , < ) <-> ( z < B /\ z < D ) ) )
873expb 1206 . . . . . . . . 9 |- ( ( z e. RR* /\ ( B e. RR* /\ D e. RR* ) ) -> ( z < inf ( { B , D } , RR* , < ) <-> ( z < B /\ z < D ) ) )
98ancoms 268 . . . . . . . 8 |- ( ( ( B e. RR* /\ D e. RR* ) /\ z e. RR* ) -> ( z < inf ( { B , D } , RR* , < ) <-> ( z < B /\ z < D ) ) )
109adantll 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 ) ) )
116, 10anbi12d 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 ) ) )
1311, 12bitr4di 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 ) ) ) )
1413rabbidva 2751 . . . 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 ) ) } )
1514an4s 588 . . 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 ) ) } )
161, 4, 153eqtr4a 2255 . 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 11434 . . . 4 |- ( ( A e. RR* /\ C e. RR* ) -> sup ( { A , C } , RR* , < ) e. RR* )
18 xrmincl 11448 . . . 4 |- ( ( B e. RR* /\ D e. RR* ) -> inf ( { B , D } , RR* , < ) e. RR* )
19 iooval 10000 . . . 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* , < ) ) } )
2017, 18, 19syl2an 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* , < ) ) } )
2120an4s 588 . 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* , < ) ) } )
2216, 21eqtr4d 2232 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* , < ) ) )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2167   {crab 2479   i^i cin 3156   {cpr 3624   class class class wbr 4034  (class class class)co 5925   supcsup 7057  infcinf 7058   RR*cxr 8077   < clt 8078   (,)cioo 9980
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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-mulrcl 7995  ax-addcom 7996  ax-mulcom 7997  ax-addass 7998  ax-mulass 7999  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-1rid 8003  ax-0id 8004  ax-rnegex 8005  ax-precex 8006  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-apti 8011  ax-pre-ltadd 8012  ax-pre-mulgt0 8013  ax-pre-mulext 8014  ax-arch 8015  ax-caucvg 8016
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 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-isom 5268  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-frec 6458  df-sup 7059  df-inf 7060  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-reap 8619  df-ap 8626  df-div 8717  df-inn 9008  df-2 9066  df-3 9067  df-4 9068  df-n0 9267  df-z 9344  df-uz 9619  df-rp 9746  df-xneg 9864  df-ioo 9984  df-seqfrec 10557  df-exp 10648  df-cj 11024  df-re 11025  df-im 11026  df-rsqrt 11180  df-abs 11181
This theorem is referenced by:  qtopbasss  14841  tgioo  14874
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