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Theorem ltmininf 11621
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 1001 . . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> B e. RR )
21renegcld 8472 . . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> -u B e. RR )
3 simp3 1002 . . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> C e. RR )
43renegcld 8472 . . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> -u C e. RR )
5 simp1 1000 . . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> A e. RR )
65renegcld 8472 . . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> -u A e. RR )
7 maxltsup 11604 . . 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 1250 . 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 11616 . . . . 5 |- ( ( B e. RR /\ C e. RR ) -> inf ( { B , C } , RR , < ) = -u sup ( { -u B , -u C } , RR , < ) )
109breq2d 4063 . . . 4 |- ( ( B e. RR /\ C e. RR ) -> ( A < inf ( { B , C } , RR , < ) <-> A < -u sup ( { -u B , -u C } , RR , < ) ) )
11103adant1 1018 . . 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 11596 . . . . 5 |- ( ( -u B e. RR /\ -u C e. RR ) -> sup ( { -u B , -u C } , RR , < ) e. RR )
132, 4, 12syl2anc 411 . . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> sup ( { -u B , -u C } , RR , < ) e. RR )
14 ltnegcon2 8557 . . . 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 411 . . 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 188 . 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 8616 . . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < B <-> -u B < -u A ) )
185, 3ltnegd 8616 . . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < C <-> -u C < -u A ) )
1917, 18anbi12d 473 . 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 220 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 981   e. wcel 2177   {cpr 3639   class class class wbr 4051   supcsup 7099  infcinf 7100   RRcr 7944   < clt 8127   -ucneg 8264
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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4167  ax-sep 4170  ax-nul 4178  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-setind 4593  ax-iinf 4644  ax-cnex 8036  ax-resscn 8037  ax-1cn 8038  ax-1re 8039  ax-icn 8040  ax-addcl 8041  ax-addrcl 8042  ax-mulcl 8043  ax-mulrcl 8044  ax-addcom 8045  ax-mulcom 8046  ax-addass 8047  ax-mulass 8048  ax-distr 8049  ax-i2m1 8050  ax-0lt1 8051  ax-1rid 8052  ax-0id 8053  ax-rnegex 8054  ax-precex 8055  ax-cnre 8056  ax-pre-ltirr 8057  ax-pre-ltwlin 8058  ax-pre-lttrn 8059  ax-pre-apti 8060  ax-pre-ltadd 8061  ax-pre-mulgt0 8062  ax-pre-mulext 8063  ax-arch 8064  ax-caucvg 8065
This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-if 3576  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-int 3892  df-iun 3935  df-br 4052  df-opab 4114  df-mpt 4115  df-tr 4151  df-id 4348  df-po 4351  df-iso 4352  df-iord 4421  df-on 4423  df-ilim 4424  df-suc 4426  df-iom 4647  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696  df-iota 5241  df-fun 5282  df-fn 5283  df-f 5284  df-f1 5285  df-fo 5286  df-f1o 5287  df-fv 5288  df-isom 5289  df-riota 5912  df-ov 5960  df-oprab 5961  df-mpo 5962  df-1st 6239  df-2nd 6240  df-recs 6404  df-frec 6490  df-sup 7101  df-inf 7102  df-pnf 8129  df-mnf 8130  df-xr 8131  df-ltxr 8132  df-le 8133  df-sub 8265  df-neg 8266  df-reap 8668  df-ap 8675  df-div 8766  df-inn 9057  df-2 9115  df-3 9116  df-4 9117  df-n0 9316  df-z 9393  df-uz 9669  df-rp 9796  df-seqfrec 10615  df-exp 10706  df-cj 11228  df-re 11229  df-im 11230  df-rsqrt 11384  df-abs 11385
This theorem is referenced by:  rpmincl  11624  mul0inf  11627  reccn2ap  11699  addcncntoplem  15108  mulcncflem  15154  suplociccreex  15171  dveflem  15273
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