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Theorem ltmininf 11162
Description: Two ways of saying a number is less than the minimum of two others. (Contributed by Jim Kingdon, 10-Feb-2022.)
Assertion
Ref Expression
ltmininf |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < inf ( { B , C } , RR , < ) <-> ( A < B /\ A < C ) ) )
Proof of Theorem ltmininf
StepHypRef Expression
1 simp2 987 . . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> B e. RR )
21renegcld 8269 . . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> -u B e. RR )
3 simp3 988 . . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> C e. RR )
43renegcld 8269 . . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> -u C e. RR )
5 simp1 986 . . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> A e. RR )
65renegcld 8269 . . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> -u A e. RR )
7 maxltsup 11146 . . 3 |- ( ( -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 ) ) )
82, 4, 6, 7syl3anc 1227 . 2 |- ( ( 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 ) ) )
9 minmax 11157 . . . . 5 |- ( ( B e. RR /\ C e. RR ) -> inf ( { B , C } , RR , < ) = -u sup ( { -u B , -u C } , RR , < ) )
109breq2d 3988 . . . 4 |- ( ( B e. RR /\ C e. RR ) -> ( A < inf ( { B , C } , RR , < ) <-> A < -u sup ( { -u B , -u C } , RR , < ) ) )
11103adant1 1004 . . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < inf ( { B , C } , RR , < ) <-> A < -u sup ( { -u B , -u C } , RR , < ) ) )
12 maxcl 11138 . . . . 5 |- ( ( -u B e. RR /\ -u C e. RR ) -> sup ( { -u B , -u C } , RR , < ) e. RR )
132, 4, 12syl2anc 409 . . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> sup ( { -u B , -u C } , RR , < ) e. RR )
14 ltnegcon2 8353 . . . 4 |- ( ( A e. RR /\ sup ( { -u B , -u C } , RR , < ) e. RR ) -> ( A < -u sup ( { -u B , -u C } , RR , < ) <-> sup ( { -u B , -u C } , RR , < ) < -u A ) )
155, 13, 14syl2anc 409 . . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < -u sup ( { -u B , -u C } , RR , < ) <-> sup ( { -u B , -u C } , RR , < ) < -u A ) )
1611, 15bitrd 187 . 2 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < inf ( { B , C } , RR , < ) <-> sup ( { -u B , -u C } , RR , < ) < -u A ) )
175, 1ltnegd 8412 . . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < B <-> -u B < -u A ) )
185, 3ltnegd 8412 . . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < C <-> -u C < -u A ) )
1917, 18anbi12d 465 . 2 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A < B /\ A < C ) <-> ( -u B < -u A /\ -u C < -u A ) ) )
208, 16, 193bitr4d 219 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 103   <-> wb 104   /\ w3a 967   e. wcel 2135   {cpr 3571   class class class wbr 3976   supcsup 6938  infcinf 6939   RRcr 7743   < clt 7924   -ucneg 8061
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-coll 4091  ax-sep 4094  ax-nul 4102  ax-pow 4147  ax-pr 4181  ax-un 4405  ax-setind 4508  ax-iinf 4559  ax-cnex 7835  ax-resscn 7836  ax-1cn 7837  ax-1re 7838  ax-icn 7839  ax-addcl 7840  ax-addrcl 7841  ax-mulcl 7842  ax-mulrcl 7843  ax-addcom 7844  ax-mulcom 7845  ax-addass 7846  ax-mulass 7847  ax-distr 7848  ax-i2m1 7849  ax-0lt1 7850  ax-1rid 7851  ax-0id 7852  ax-rnegex 7853  ax-precex 7854  ax-cnre 7855  ax-pre-ltirr 7856  ax-pre-ltwlin 7857  ax-pre-lttrn 7858  ax-pre-apti 7859  ax-pre-ltadd 7860  ax-pre-mulgt0 7861  ax-pre-mulext 7862  ax-arch 7863  ax-caucvg 7864
This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 968  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-nel 2430  df-ral 2447  df-rex 2448  df-reu 2449  df-rmo 2450  df-rab 2451  df-v 2723  df-sbc 2947  df-csb 3041  df-dif 3113  df-un 3115  df-in 3117  df-ss 3124  df-nul 3405  df-if 3516  df-pw 3555  df-sn 3576  df-pr 3577  df-op 3579  df-uni 3784  df-int 3819  df-iun 3862  df-br 3977  df-opab 4038  df-mpt 4039  df-tr 4075  df-id 4265  df-po 4268  df-iso 4269  df-iord 4338  df-on 4340  df-ilim 4341  df-suc 4343  df-iom 4562  df-xp 4604  df-rel 4605  df-cnv 4606  df-co 4607  df-dm 4608  df-rn 4609  df-res 4610  df-ima 4611  df-iota 5147  df-fun 5184  df-fn 5185  df-f 5186  df-f1 5187  df-fo 5188  df-f1o 5189  df-fv 5190  df-isom 5191  df-riota 5792  df-ov 5839  df-oprab 5840  df-mpo 5841  df-1st 6100  df-2nd 6101  df-recs 6264  df-frec 6350  df-sup 6940  df-inf 6941  df-pnf 7926  df-mnf 7927  df-xr 7928  df-ltxr 7929  df-le 7930  df-sub 8062  df-neg 8063  df-reap 8464  df-ap 8471  df-div 8560  df-inn 8849  df-2 8907  df-3 8908  df-4 8909  df-n0 9106  df-z 9183  df-uz 9458  df-rp 9581  df-seqfrec 10371  df-exp 10445  df-cj 10770  df-re 10771  df-im 10772  df-rsqrt 10926  df-abs 10927
This theorem is referenced by:  rpmincl  11165  mul0inf  11168  reccn2ap  11240  addcncntoplem  13092  mulcncflem  13131  suplociccreex  13143  dveflem  13228
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