ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  xrminltinf Unicode version
Theorem xrminltinf 11791
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 10036 . . . 4 |- ( B e. RR* -> -e B e. RR* )
213ad2ant2 1043 . . 3 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> -e B e. RR* )
3 xnegcl 10036 . . . 4 |- ( C e. RR* -> -e C e. RR* )
433ad2ant3 1044 . . 3 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> -e C e. RR* )
5 xnegcl 10036 . . . 4 |- ( A e. RR* -> -e A e. RR* )
653ad2ant1 1042 . . 3 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> -e A e. RR* )
7 xrltmaxsup 11776 . . 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 1271 . 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 11784 . . . . 5 |- ( ( B e. RR* /\ C e. RR* ) -> inf ( { B , C } , RR* , < ) = -e sup ( { -e B , -e C } , RR* , < ) )
1093adant1 1039 . . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> inf ( { B , C } , RR* , < ) = -e sup ( { -e B , -e C } , RR* , < ) )
11 xnegneg 10037 . . . . . 6 |- ( A e. RR* -> -e -e A = A )
1211eqcomd 2235 . . . . 5 |- ( A e. RR* -> A = -e -e A )
13123ad2ant1 1042 . . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> A = -e -e A )
1410, 13breq12d 4096 . . 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 11771 . . . . 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 10040 . . . 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 1022 . . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> B e. RR* )
21 simp1 1021 . . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> A e. RR* )
22 xltneg 10040 . . . 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 1023 . . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> C e. RR* )
25 xltneg 10040 . . . 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 798 . 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 713   /\ w3a 1002   = wceq 1395   e. wcel 2200   {cpr 3667   class class class wbr 4083   supcsup 7157  infcinf 7158   RR*cxr 8188   < clt 8189   -ecxne 9973
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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8098  ax-resscn 8099  ax-1cn 8100  ax-1re 8101  ax-icn 8102  ax-addcl 8103  ax-addrcl 8104  ax-mulcl 8105  ax-mulrcl 8106  ax-addcom 8107  ax-mulcom 8108  ax-addass 8109  ax-mulass 8110  ax-distr 8111  ax-i2m1 8112  ax-0lt1 8113  ax-1rid 8114  ax-0id 8115  ax-rnegex 8116  ax-precex 8117  ax-cnre 8118  ax-pre-ltirr 8119  ax-pre-ltwlin 8120  ax-pre-lttrn 8121  ax-pre-apti 8122  ax-pre-ltadd 8123  ax-pre-mulgt0 8124  ax-pre-mulext 8125  ax-arch 8126  ax-caucvg 8127
This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-isom 5327  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-recs 6457  df-frec 6543  df-sup 7159  df-inf 7160  df-pnf 8191  df-mnf 8192  df-xr 8193  df-ltxr 8194  df-le 8195  df-sub 8327  df-neg 8328  df-reap 8730  df-ap 8737  df-div 8828  df-inn 9119  df-2 9177  df-3 9178  df-4 9179  df-n0 9378  df-z 9455  df-uz 9731  df-rp 9858  df-xneg 9976  df-seqfrec 10678  df-exp 10769  df-cj 11361  df-re 11362  df-im 11363  df-rsqrt 11517  df-abs 11518
This theorem is referenced by:  bdbl  15185
  Copyright terms: Public domain W3C validator