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Theorem xrminltinf 11299
Description: Two ways of saying an extended real is greater than the minimum of two others. (Contributed by Jim Kingdon, 19-May-2023.)
Assertion
Ref Expression
xrminltinf |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> (inf ( { B , C } , RR* , < ) < A <-> ( B < A \/ C < A ) ) )
Proof of Theorem xrminltinf
StepHypRef Expression
1 xnegcl 9851 . . . 4 |- ( B e. RR* -> -e B e. RR* )
213ad2ant2 1021 . . 3 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> -e B e. RR* )
3 xnegcl 9851 . . . 4 |- ( C e. RR* -> -e C e. RR* )
433ad2ant3 1022 . . 3 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> -e C e. RR* )
5 xnegcl 9851 . . . 4 |- ( A e. RR* -> -e A e. RR* )
653ad2ant1 1020 . . 3 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> -e A e. RR* )
7 xrltmaxsup 11284 . . 3 |- ( ( -e B e. RR* /\ -e C e. RR* /\ -e A e. RR* ) -> ( -e A < sup ( { -e B , -e C } , RR* , < ) <-> ( -e A < -e B \/ -e A < -e C ) ) )
82, 4, 6, 7syl3anc 1249 . 2 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( -e A < sup ( { -e B , -e C } , RR* , < ) <-> ( -e A < -e B \/ -e A < -e C ) ) )
9 xrminmax 11292 . . . . 5 |- ( ( B e. RR* /\ C e. RR* ) -> inf ( { B , C } , RR* , < ) = -e sup ( { -e B , -e C } , RR* , < ) )
1093adant1 1017 . . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> inf ( { B , C } , RR* , < ) = -e sup ( { -e B , -e C } , RR* , < ) )
11 xnegneg 9852 . . . . . 6 |- ( A e. RR* -> -e -e A = A )
1211eqcomd 2195 . . . . 5 |- ( A e. RR* -> A = -e -e A )
13123ad2ant1 1020 . . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> A = -e -e A )
1410, 13breq12d 4031 . . 3 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> (inf ( { B , C } , RR* , < ) < A <-> -e sup ( { -e B , -e C } , RR* , < ) < -e -e A ) )
15 xrmaxcl 11279 . . . . 5 |- ( ( -e B e. RR* /\ -e C e. RR* ) -> sup ( { -e B , -e C } , RR* , < ) e. RR* )
162, 4, 15syl2anc 411 . . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> sup ( { -e B , -e C } , RR* , < ) e. RR* )
17 xltneg 9855 . . . 4 |- ( ( -e A e. RR* /\ sup ( { -e B , -e C } , RR* , < ) e. RR* ) -> ( -e A < sup ( { -e B , -e C } , RR* , < ) <-> -e sup ( { -e B , -e C } , RR* , < ) < -e -e A ) )
186, 16, 17syl2anc 411 . . 3 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( -e A < sup ( { -e B , -e C } , RR* , < ) <-> -e sup ( { -e B , -e C } , RR* , < ) < -e -e A ) )
1914, 18bitr4d 191 . 2 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> (inf ( { B , C } , RR* , < ) < A <-> -e A < sup ( { -e B , -e C } , RR* , < ) ) )
20 simp2 1000 . . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> B e. RR* )
21 simp1 999 . . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> A e. RR* )
22 xltneg 9855 . . . 4 |- ( ( B e. RR* /\ A e. RR* ) -> ( B < A <-> -e A < -e B ) )
2320, 21, 22syl2anc 411 . . 3 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( B < A <-> -e A < -e B ) )
24 simp3 1001 . . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> C e. RR* )
25 xltneg 9855 . . . 4 |- ( ( C e. RR* /\ A e. RR* ) -> ( C < A <-> -e A < -e C ) )
2624, 21, 25syl2anc 411 . . 3 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( C < A <-> -e A < -e C ) )
2723, 26orbi12d 794 . 2 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( ( B < A \/ C < A ) <-> ( -e A < -e B \/ -e A < -e C ) ) )
288, 19, 273bitr4d 220 1 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> (inf ( { B , C } , RR* , < ) < A <-> ( B < A \/ C < A ) ) )
Colors of variables: wff set class
Syntax hints:   -> wi 4   <-> wb 105   \/ wo 709   /\ 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:  bdbl  14406
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