Theorem xrltmininf 11297

Description: Two ways of saying an extended real is less than the minimum of two others. (Contributed by NM, 7-Feb-2007.) (Revised by Jim Kingdon, 3-May-2023.)

Assertion

Ref	Expression
xrltmininf	|- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( A < inf ( { B , C } , RR* , < ) <-> ( A < B /\ A < C ) ) )

Proof of Theorem xrltmininf

Step	Hyp	Ref	Expression
1		xrminmax 11292	. . . 4 |- ( ( B e. RR* /\ C e. RR* ) -> inf ( { B , C } , RR* , < ) = -e sup ( { -e B , -e C } , RR* , < ) )
2	1	3adant1 1017	. . 3 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> inf ( { B , C } , RR* , < ) = -e sup ( { -e B , -e C } , RR* , < ) )
3	2	breq2d 4030	. 2 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( A < inf ( { B , C } , RR* , < ) <-> A < -e sup ( { -e B , -e C } , RR* , < ) ) )
4		simp2 1000	. . . . 5 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> B e. RR* )
5	4	xnegcld 9874	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> -e B e. RR* )
6		simp3 1001	. . . . 5 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> C e. RR* )
7	6	xnegcld 9874	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> -e C e. RR* )
8		simp1 999	. . . . 5 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> A e. RR* )
9	8	xnegcld 9874	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> -e A e. RR* )
10		xrmaxltsup 11285	. . . 4 |- ( ( -e B e. RR* /\ -e C e. RR* /\ -e A e. RR* ) -> ( sup ( { -e B , -e C } , RR* , < ) < -e A <-> ( -e B < -e A /\ -e C < -e A ) ) )
11	5, 7, 9, 10	syl3anc 1249	. . 3 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( sup ( { -e B , -e C } , RR* , < ) < -e A <-> ( -e B < -e A /\ -e C < -e A ) ) )
12		xrmaxcl 11279	. . . . . . 7 |- ( ( -e B e. RR* /\ -e C e. RR* ) -> sup ( { -e B , -e C } , RR* , < ) e. RR* )
13	5, 7, 12	syl2anc 411	. . . . . 6 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> sup ( { -e B , -e C } , RR* , < ) e. RR* )
14	13	xnegcld 9874	. . . . 5 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> -e sup ( { -e B , -e C } , RR* , < ) e. RR* )
15		xltneg 9855	. . . . 5 |- ( ( A e. RR* /\ -e sup ( { -e B , -e C } , RR* , < ) e. RR* ) -> ( A < -e sup ( { -e B , -e C } , RR* , < ) <-> -e -e sup ( { -e B , -e C } , RR* , < ) < -e A ) )
16	8, 14, 15	syl2anc 411	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( A < -e sup ( { -e B , -e C } , RR* , < ) <-> -e -e sup ( { -e B , -e C } , RR* , < ) < -e A ) )
17		xnegneg 9852	. . . . . 6 |- ( sup ( { -e B , -e C } , RR* , < ) e. RR* -> -e -e sup ( { -e B , -e C } , RR* , < ) = sup ( { -e B , -e C } , RR* , < ) )
18	13, 17	syl 14	. . . . 5 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> -e -e sup ( { -e B , -e C } , RR* , < ) = sup ( { -e B , -e C } , RR* , < ) )
19	18	breq1d 4028	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( -e -e sup ( { -e B , -e C } , RR* , < ) < -e A <-> sup ( { -e B , -e C } , RR* , < ) < -e A ) )
20	16, 19	bitrd 188	. . 3 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( A < -e sup ( { -e B , -e C } , RR* , < ) <-> sup ( { -e B , -e C } , RR* , < ) < -e A ) )
21		xltneg 9855	. . . . 5 |- ( ( A e. RR* /\ B e. RR* ) -> ( A < B <-> -e B < -e A ) )
22	21	3adant3 1019	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( A < B <-> -e B < -e A ) )
23		xltneg 9855	. . . . 5 |- ( ( A e. RR* /\ C e. RR* ) -> ( A < C <-> -e C < -e A ) )
24	23	3adant2 1018	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( A < C <-> -e C < -e A ) )
25	22, 24	anbi12d 473	. . 3 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( ( A < B /\ A < C ) <-> ( -e B < -e A /\ -e C < -e A ) ) )
26	11, 20, 25	3bitr4d 220	. 2 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( A < -e sup ( { -e B , -e C } , RR* , < ) <-> ( A < B /\ A < C ) ) )
27	3, 26	bitrd 188	1 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( A < inf ( { B , C } , RR* , < ) <-> ( A < B /\ A < C ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 980   = wceq 1364   e. wcel 2160   {cpr 3608   class class class wbr 4018   supcsup 7000  infcinf 7001   RR*cxr 8010   < clt 8011   -ecxne 9788

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-iinf 4602  ax-cnex 7921  ax-resscn 7922  ax-1cn 7923  ax-1re 7924  ax-icn 7925  ax-addcl 7926  ax-addrcl 7927  ax-mulcl 7928  ax-mulrcl 7929  ax-addcom 7930  ax-mulcom 7931  ax-addass 7932  ax-mulass 7933  ax-distr 7934  ax-i2m1 7935  ax-0lt1 7936  ax-1rid 7937  ax-0id 7938  ax-rnegex 7939  ax-precex 7940  ax-cnre 7941  ax-pre-ltirr 7942  ax-pre-ltwlin 7943  ax-pre-lttrn 7944  ax-pre-apti 7945  ax-pre-ltadd 7946  ax-pre-mulgt0 7947  ax-pre-mulext 7948  ax-arch 7949  ax-caucvg 7950

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 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4308  df-po 4311  df-iso 4312  df-iord 4381  df-on 4383  df-ilim 4384  df-suc 4386  df-iom 4605  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-f1 5236  df-fo 5237  df-f1o 5238  df-fv 5239  df-isom 5240  df-riota 5847  df-ov 5894  df-oprab 5895  df-mpo 5896  df-1st 6159  df-2nd 6160  df-recs 6324  df-frec 6410  df-sup 7002  df-inf 7003  df-pnf 8013  df-mnf 8014  df-xr 8015  df-ltxr 8016  df-le 8017  df-sub 8149  df-neg 8150  df-reap 8551  df-ap 8558  df-div 8649  df-inn 8939  df-2 8997  df-3 8998  df-4 8999  df-n0 9196  df-z 9273  df-uz 9548  df-rp 9673  df-xneg 9791  df-seqfrec 10465  df-exp 10539  df-cj 10870  df-re 10871  df-im 10872  df-rsqrt 11026  df-abs 11027

This theorem is referenced by:  xrminrpcl  11301  iooinsup  11304  blininf  14327  bdxmet  14404  bdmopn  14407