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Theorem lemininf 11915
Description: Two ways of saying a number is less than or equal to the minimum of two others. (Contributed by NM, 3-Aug-2007.)
Assertion
Ref Expression
lemininf |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A <_ inf ( { B , C } , RR , < ) <-> ( A <_ B /\ A <_ C ) ) )
Proof of Theorem lemininf
StepHypRef Expression
1 simp2 1025 . . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> B e. RR )
2 simp3 1026 . . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> C e. RR )
3 minmax 11911 . . . 4 |- ( ( B e. RR /\ C e. RR ) -> inf ( { B , C } , RR , < ) = -u sup ( { -u B , -u C } , RR , < ) )
41, 2, 3syl2anc 411 . . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> inf ( { B , C } , RR , < ) = -u sup ( { -u B , -u C } , RR , < ) )
54breq2d 4120 . 2 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A <_ inf ( { B , C } , RR , < ) <-> A <_ -u sup ( { -u B , -u C } , RR , < ) ) )
61renegcld 8652 . . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> -u B e. RR )
72renegcld 8652 . . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> -u C e. RR )
8 maxcl 11891 . . . 4 |- ( ( -u B e. RR /\ -u C e. RR ) -> sup ( { -u B , -u C } , RR , < ) e. RR )
96, 7, 8syl2anc 411 . . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> sup ( { -u B , -u C } , RR , < ) e. RR )
10 simp1 1024 . . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> A e. RR )
11 lenegcon2 8740 . . 3 |- ( ( sup ( { -u B , -u C } , RR , < ) e. RR /\ A e. RR ) -> ( sup ( { -u B , -u C } , RR , < ) <_ -u A <-> A <_ -u sup ( { -u B , -u C } , RR , < ) ) )
129, 10, 11syl2anc 411 . 2 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( sup ( { -u B , -u C } , RR , < ) <_ -u A <-> A <_ -u sup ( { -u B , -u C } , RR , < ) ) )
1310renegcld 8652 . . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> -u A e. RR )
14 maxleastb 11895 . . . 4 |- ( ( -u B e. RR /\ -u C e. RR /\ -u A e. RR ) -> ( sup ( { -u B , -u C } , RR , < ) <_ -u A <-> ( -u B <_ -u A /\ -u C <_ -u A ) ) )
156, 7, 13, 14syl3anc 1274 . . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( sup ( { -u B , -u C } , RR , < ) <_ -u A <-> ( -u B <_ -u A /\ -u C <_ -u A ) ) )
1610, 1lenegd 8797 . . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A <_ B <-> -u B <_ -u A ) )
1710, 2lenegd 8797 . . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A <_ C <-> -u C <_ -u A ) )
1816, 17anbi12d 473 . . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A <_ B /\ A <_ C ) <-> ( -u B <_ -u A /\ -u C <_ -u A ) ) )
1915, 18bitr4d 191 . 2 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( sup ( { -u B , -u C } , RR , < ) <_ -u A <-> ( A <_ B /\ A <_ C ) ) )
205, 12, 193bitr2d 216 1 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A <_ inf ( { B , C } , RR , < ) <-> ( A <_ B /\ A <_ C ) ) )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2203   {cpr 3689   class class class wbr 4108   supcsup 7272  infcinf 7273   RRcr 8125   < clt 8307   <_ cle 8308   -ucneg 8444
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 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-iinf 4709  ax-cnex 8217  ax-resscn 8218  ax-1cn 8219  ax-1re 8220  ax-icn 8221  ax-addcl 8222  ax-addrcl 8223  ax-mulcl 8224  ax-mulrcl 8225  ax-addcom 8226  ax-mulcom 8227  ax-addass 8228  ax-mulass 8229  ax-distr 8230  ax-i2m1 8231  ax-0lt1 8232  ax-1rid 8233  ax-0id 8234  ax-rnegex 8235  ax-precex 8236  ax-cnre 8237  ax-pre-ltirr 8238  ax-pre-ltwlin 8239  ax-pre-lttrn 8240  ax-pre-apti 8241  ax-pre-ltadd 8242  ax-pre-mulgt0 8243  ax-pre-mulext 8244  ax-arch 8245  ax-caucvg 8246
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 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-if 3620  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-id 4413  df-po 4416  df-iso 4417  df-iord 4486  df-on 4488  df-ilim 4489  df-suc 4491  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-isom 5360  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-recs 6535  df-frec 6621  df-sup 7274  df-inf 7275  df-pnf 8309  df-mnf 8310  df-xr 8311  df-ltxr 8312  df-le 8313  df-sub 8445  df-neg 8446  df-reap 8848  df-ap 8855  df-div 8946  df-inn 9237  df-2 9295  df-3 9296  df-4 9297  df-n0 9496  df-z 9577  df-uz 9853  df-rp 9986  df-seqfrec 10809  df-exp 10900  df-cj 11523  df-re 11524  df-im 11525  df-rsqrt 11679  df-abs 11680
This theorem is referenced by:  mul0inf  11922  pc2dvds  13024
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