Theorem iooinsup 11857

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 3478	. . 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 10145	. . . 4 |- ( ( A e. RR* /\ B e. RR* ) -> ( A (,) B ) = { z e. RR* | ( A < z /\ z < B ) } )
3		iooval 10145	. . . 4 |- ( ( C e. RR* /\ D e. RR* ) -> ( C (,) D ) = { z e. RR* | ( C < z /\ z < D ) } )
4	2, 3	ineqan12d 3409	. . 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 11838	. . . . . . . 8 |- ( ( A e. RR* /\ C e. RR* /\ z e. RR* ) -> ( sup ( { A , C } , RR* , < ) < z <-> ( A < z /\ C < z ) ) )
6	5	ad4ant124 1242	. . . . . . 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 11850	. . . . . . . . . 10 |- ( ( z e. RR* /\ B e. RR* /\ D e. RR* ) -> ( z < inf ( { B , D } , RR* , < ) <-> ( z < B /\ z < D ) ) )
8	7	3expb 1230	. . . . . . . . 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 588	. . . . . 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 2789	. . . 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 592	. . 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 2289	. 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 11832	. . . 4 |- ( ( A e. RR* /\ C e. RR* ) -> sup ( { A , C } , RR* , < ) e. RR* )
18		xrmincl 11846	. . . 4 |- ( ( B e. RR* /\ D e. RR* ) -> inf ( { B , D } , RR* , < ) e. RR* )
19		iooval 10145	. . . 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 592	. 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 2266	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 1397   e. wcel 2201   {crab 2513   i^i cin 3198   {cpr 3669   class class class wbr 4087  (class class class)co 6020   supcsup 7183  infcinf 7184   RR*cxr 8215   < clt 8216   (,)cioo 10125

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2203  ax-14 2204  ax-ext 2212  ax-coll 4203  ax-sep 4206  ax-nul 4214  ax-pow 4263  ax-pr 4298  ax-un 4529  ax-setind 4634  ax-iinf 4685  ax-cnex 8125  ax-resscn 8126  ax-1cn 8127  ax-1re 8128  ax-icn 8129  ax-addcl 8130  ax-addrcl 8131  ax-mulcl 8132  ax-mulrcl 8133  ax-addcom 8134  ax-mulcom 8135  ax-addass 8136  ax-mulass 8137  ax-distr 8138  ax-i2m1 8139  ax-0lt1 8140  ax-1rid 8141  ax-0id 8142  ax-rnegex 8143  ax-precex 8144  ax-cnre 8145  ax-pre-ltirr 8146  ax-pre-ltwlin 8147  ax-pre-lttrn 8148  ax-pre-apti 8149  ax-pre-ltadd 8150  ax-pre-mulgt0 8151  ax-pre-mulext 8152  ax-arch 8153  ax-caucvg 8154

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ne 2402  df-nel 2497  df-ral 2514  df-rex 2515  df-reu 2516  df-rmo 2517  df-rab 2518  df-v 2803  df-sbc 3031  df-csb 3127  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-nul 3494  df-if 3605  df-pw 3653  df-sn 3674  df-pr 3675  df-op 3677  df-uni 3893  df-int 3928  df-iun 3971  df-br 4088  df-opab 4150  df-mpt 4151  df-tr 4187  df-id 4389  df-po 4392  df-iso 4393  df-iord 4462  df-on 4464  df-ilim 4465  df-suc 4467  df-iom 4688  df-xp 4730  df-rel 4731  df-cnv 4732  df-co 4733  df-dm 4734  df-rn 4735  df-res 4736  df-ima 4737  df-iota 5285  df-fun 5327  df-fn 5328  df-f 5329  df-f1 5330  df-fo 5331  df-f1o 5332  df-fv 5333  df-isom 5334  df-riota 5973  df-ov 6023  df-oprab 6024  df-mpo 6025  df-1st 6305  df-2nd 6306  df-recs 6473  df-frec 6559  df-sup 7185  df-inf 7186  df-pnf 8218  df-mnf 8219  df-xr 8220  df-ltxr 8221  df-le 8222  df-sub 8354  df-neg 8355  df-reap 8757  df-ap 8764  df-div 8855  df-inn 9146  df-2 9204  df-3 9205  df-4 9206  df-n0 9405  df-z 9482  df-uz 9758  df-rp 9891  df-xneg 10009  df-ioo 10129  df-seqfrec 10713  df-exp 10804  df-cj 11422  df-re 11423  df-im 11424  df-rsqrt 11578  df-abs 11579

This theorem is referenced by:  qtopbasss  15271  tgioo  15304