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Theorem xrminltinf 10880
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 9456 . . . 4 |- ( B e. RR* -> -e B e. RR* )
213ad2ant2 971 . . 3 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> -e B e. RR* )
3 xnegcl 9456 . . . 4 |- ( C e. RR* -> -e C e. RR* )
433ad2ant3 972 . . 3 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> -e C e. RR* )
5 xnegcl 9456 . . . 4 |- ( A e. RR* -> -e A e. RR* )
653ad2ant1 970 . . 3 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> -e A e. RR* )
7 xrltmaxsup 10865 . . 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 1184 . 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 10873 . . . . 5 |- ( ( B e. RR* /\ C e. RR* ) -> inf ( { B , C } , RR* , < ) = -e sup ( { -e B , -e C } , RR* , < ) )
1093adant1 967 . . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> inf ( { B , C } , RR* , < ) = -e sup ( { -e B , -e C } , RR* , < ) )
11 xnegneg 9457 . . . . . 6 |- ( A e. RR* -> -e -e A = A )
1211eqcomd 2105 . . . . 5 |- ( A e. RR* -> A = -e -e A )
13123ad2ant1 970 . . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> A = -e -e A )
1410, 13breq12d 3888 . . 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 10860 . . . . 5 |- ( ( -e B e. RR* /\ -e C e. RR* ) -> sup ( { -e B , -e C } , RR* , < ) e. RR* )
162, 4, 15syl2anc 406 . . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> sup ( { -e B , -e C } , RR* , < ) e. RR* )
17 xltneg 9460 . . . 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 406 . . 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 190 . 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 950 . . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> B e. RR* )
21 simp1 949 . . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> A e. RR* )
22 xltneg 9460 . . . 4 |- ( ( B e. RR* /\ A e. RR* ) -> ( B < A <-> -e A < -e B ) )
2320, 21, 22syl2anc 406 . . 3 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( B < A <-> -e A < -e B ) )
24 simp3 951 . . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> C e. RR* )
25 xltneg 9460 . . . 4 |- ( ( C e. RR* /\ A e. RR* ) -> ( C < A <-> -e A < -e C ) )
2624, 21, 25syl2anc 406 . . 3 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( C < A <-> -e A < -e C ) )
2723, 26orbi12d 748 . 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 219 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 104   \/ wo 670   /\ w3a 930   = wceq 1299   e. wcel 1448   {cpr 3475   class class class wbr 3875   supcsup 6784  infcinf 6785   RR*cxr 7671   < clt 7672   -ecxne 9397
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-iinf 4440  ax-cnex 7586  ax-resscn 7587  ax-1cn 7588  ax-1re 7589  ax-icn 7590  ax-addcl 7591  ax-addrcl 7592  ax-mulcl 7593  ax-mulrcl 7594  ax-addcom 7595  ax-mulcom 7596  ax-addass 7597  ax-mulass 7598  ax-distr 7599  ax-i2m1 7600  ax-0lt1 7601  ax-1rid 7602  ax-0id 7603  ax-rnegex 7604  ax-precex 7605  ax-cnre 7606  ax-pre-ltirr 7607  ax-pre-ltwlin 7608  ax-pre-lttrn 7609  ax-pre-apti 7610  ax-pre-ltadd 7611  ax-pre-mulgt0 7612  ax-pre-mulext 7613  ax-arch 7614  ax-caucvg 7615
This theorem depends on definitions:  df-bi 116  df-dc 787  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-nel 2363  df-ral 2380  df-rex 2381  df-reu 2382  df-rmo 2383  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-if 3422  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-tr 3967  df-id 4153  df-po 4156  df-iso 4157  df-iord 4226  df-on 4228  df-ilim 4229  df-suc 4231  df-iom 4443  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-isom 5068  df-riota 5662  df-ov 5709  df-oprab 5710  df-mpo 5711  df-1st 5969  df-2nd 5970  df-recs 6132  df-frec 6218  df-sup 6786  df-inf 6787  df-pnf 7674  df-mnf 7675  df-xr 7676  df-ltxr 7677  df-le 7678  df-sub 7806  df-neg 7807  df-reap 8203  df-ap 8210  df-div 8294  df-inn 8579  df-2 8637  df-3 8638  df-4 8639  df-n0 8830  df-z 8907  df-uz 9177  df-rp 9292  df-xneg 9400  df-seqfrec 10060  df-exp 10134  df-cj 10455  df-re 10456  df-im 10457  df-rsqrt 10610  df-abs 10611
This theorem is referenced by:  bdbl  12431
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