Theorem iooinsup 11917

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 3481	. . 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 10204	. . . 4 |- ( ( A e. RR* /\ B e. RR* ) -> ( A (,) B ) = { z e. RR* | ( A < z /\ z < B ) } )
3		iooval 10204	. . . 4 |- ( ( C e. RR* /\ D e. RR* ) -> ( C (,) D ) = { z e. RR* | ( C < z /\ z < D ) } )
4	2, 3	ineqan12d 3412	. . 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 11898	. . . . . . . 8 |- ( ( A e. RR* /\ C e. RR* /\ z e. RR* ) -> ( sup ( { A , C } , RR* , < ) < z <-> ( A < z /\ C < z ) ) )
6	5	ad4ant124 1243	. . . . . . 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 11910	. . . . . . . . . 10 |- ( ( z e. RR* /\ B e. RR* /\ D e. RR* ) -> ( z < inf ( { B , D } , RR* , < ) <-> ( z < B /\ z < D ) ) )
8	7	3expb 1231	. . . . . . . . 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 2791	. . . 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 2290	. 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 11892	. . . 4 |- ( ( A e. RR* /\ C e. RR* ) -> sup ( { A , C } , RR* , < ) e. RR* )
18		xrmincl 11906	. . . 4 |- ( ( B e. RR* /\ D e. RR* ) -> inf ( { B , D } , RR* , < ) e. RR* )
19		iooval 10204	. . . 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 2267	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 1398   e. wcel 2202   {crab 2515   i^i cin 3200   {cpr 3674   class class class wbr 4093  (class class class)co 6028   supcsup 7241  infcinf 7242   RR*cxr 8272   < clt 8273   (,)cioo 10184

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-mulrcl 8191  ax-addcom 8192  ax-mulcom 8193  ax-addass 8194  ax-mulass 8195  ax-distr 8196  ax-i2m1 8197  ax-0lt1 8198  ax-1rid 8199  ax-0id 8200  ax-rnegex 8201  ax-precex 8202  ax-cnre 8203  ax-pre-ltirr 8204  ax-pre-ltwlin 8205  ax-pre-lttrn 8206  ax-pre-apti 8207  ax-pre-ltadd 8208  ax-pre-mulgt0 8209  ax-pre-mulext 8210  ax-arch 8211  ax-caucvg 8212

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-isom 5342  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-frec 6600  df-sup 7243  df-inf 7244  df-pnf 8275  df-mnf 8276  df-xr 8277  df-ltxr 8278  df-le 8279  df-sub 8411  df-neg 8412  df-reap 8814  df-ap 8821  df-div 8912  df-inn 9203  df-2 9261  df-3 9262  df-4 9263  df-n0 9462  df-z 9541  df-uz 9817  df-rp 9950  df-xneg 10068  df-ioo 10188  df-seqfrec 10773  df-exp 10864  df-cj 11482  df-re 11483  df-im 11484  df-rsqrt 11638  df-abs 11639

This theorem is referenced by:  qtopbasss  15332  tgioo  15365