Theorem iooinsup 12021

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 3505	. . 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 10289	. . . 4 |- ( ( A e. RR* /\ B e. RR* ) -> ( A (,) B ) = { z e. RR* | ( A < z /\ z < B ) } )
3		iooval 10289	. . . 4 |- ( ( C e. RR* /\ D e. RR* ) -> ( C (,) D ) = { z e. RR* | ( C < z /\ z < D ) } )
4	2, 3	ineqan12d 3434	. . 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 12002	. . . . . . . 8 |- ( ( A e. RR* /\ C e. RR* /\ z e. RR* ) -> ( sup ( { A , C } , RR* , < ) < z <-> ( A < z /\ C < z ) ) )
6	5	ad4ant124 1247	. . . . . . 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 12014	. . . . . . . . . 10 |- ( ( z e. RR* /\ B e. RR* /\ D e. RR* ) -> ( z < inf ( { B , D } , RR* , < ) <-> ( z < B /\ z < D ) ) )
8	7	3expb 1235	. . . . . . . . 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 480	. . . . . . 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 477	. . . . . 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 592	. . . . . 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 2809	. . . 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 596	. . 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 2297	. 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 11996	. . . 4 |- ( ( A e. RR* /\ C e. RR* ) -> sup ( { A , C } , RR* , < ) e. RR* )
18		xrmincl 12010	. . . 4 |- ( ( B e. RR* /\ D e. RR* ) -> inf ( { B , D } , RR* , < ) e. RR* )
19		iooval 10289	. . . 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 596	. 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 2274	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 1402   e. wcel 2209   {crab 2532   i^i cin 3219   {cpr 3706   class class class wbr 4125  (class class class)co 6075   supcsup 7312  infcinf 7313   RR*cxr 8349   < clt 8350   (,)cioo 10269

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286  ax-pre-mulext 8287  ax-arch 8288  ax-caucvg 8289

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-isom 5381  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-frec 6652  df-sup 7314  df-inf 7315  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-reap 8893  df-ap 8900  df-div 8993  df-inn 9284  df-2 9342  df-3 9343  df-4 9344  df-n0 9543  df-z 9624  df-uz 9901  df-rp 10034  df-xneg 10153  df-ioo 10273  df-seqfrec 10863  df-exp 10954  df-cj 11585  df-re 11586  df-im 11587  df-rsqrt 11742  df-abs 11743

This theorem is referenced by:  qtopbasss  15545  tgioo  15578