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Theorem ltmininf 10342
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 940 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  B  e.  RR )
21renegcld 7628 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  -u B  e.  RR )
3 simp3 941 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  C  e.  RR )
43renegcld 7628 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  -u C  e.  RR )
5 simp1 939 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  A  e.  RR )
65renegcld 7628 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  -u A  e.  RR )
7 maxltsup 10330 . . 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 1170 . 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 10338 . . . . 5  |-  ( ( B  e.  RR  /\  C  e.  RR )  -> inf ( { B ,  C } ,  RR ,  <  )  =  -u sup ( { -u B ,  -u C } ,  RR ,  <  ) )
109breq2d 3818 . . . 4  |-  ( ( B  e.  RR  /\  C  e.  RR )  ->  ( A  < inf ( { B ,  C } ,  RR ,  <  )  <->  A  <  -u sup ( {
-u B ,  -u C } ,  RR ,  <  ) ) )
11103adant1 957 . . 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 10322 . . . . 5  |-  ( (
-u B  e.  RR  /\  -u C  e.  RR )  ->  sup ( { -u B ,  -u C } ,  RR ,  <  )  e.  RR )
132, 4, 12syl2anc 403 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  sup ( { -u B ,  -u C } ,  RR ,  <  )  e.  RR )
14 ltnegcon2 7712 . . . 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 403 . . 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 186 . 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 7767 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <  B  <->  -u B  <  -u A ) )
185, 3ltnegd 7767 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <  C  <->  -u C  <  -u A ) )
1917, 18anbi12d 457 . 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 218 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 102    <-> wb 103    /\ w3a 920    e. wcel 1434   {cpr 3418   class class class wbr 3806   supcsup 6491  infcinf 6492   RRcr 7119    < clt 7292   -ucneg 7424
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3914  ax-sep 3917  ax-nul 3925  ax-pow 3969  ax-pr 3993  ax-un 4217  ax-setind 4309  ax-iinf 4358  ax-cnex 7206  ax-resscn 7207  ax-1cn 7208  ax-1re 7209  ax-icn 7210  ax-addcl 7211  ax-addrcl 7212  ax-mulcl 7213  ax-mulrcl 7214  ax-addcom 7215  ax-mulcom 7216  ax-addass 7217  ax-mulass 7218  ax-distr 7219  ax-i2m1 7220  ax-0lt1 7221  ax-1rid 7222  ax-0id 7223  ax-rnegex 7224  ax-precex 7225  ax-cnre 7226  ax-pre-ltirr 7227  ax-pre-ltwlin 7228  ax-pre-lttrn 7229  ax-pre-apti 7230  ax-pre-ltadd 7231  ax-pre-mulgt0 7232  ax-pre-mulext 7233  ax-arch 7234  ax-caucvg 7235
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rmo 2361  df-rab 2362  df-v 2613  df-sbc 2826  df-csb 2919  df-dif 2985  df-un 2987  df-in 2989  df-ss 2996  df-nul 3269  df-if 3370  df-pw 3403  df-sn 3423  df-pr 3424  df-op 3426  df-uni 3623  df-int 3658  df-iun 3701  df-br 3807  df-opab 3861  df-mpt 3862  df-tr 3897  df-id 4077  df-po 4080  df-iso 4081  df-iord 4150  df-on 4152  df-ilim 4153  df-suc 4155  df-iom 4361  df-xp 4398  df-rel 4399  df-cnv 4400  df-co 4401  df-dm 4402  df-rn 4403  df-res 4404  df-ima 4405  df-iota 4918  df-fun 4955  df-fn 4956  df-f 4957  df-f1 4958  df-fo 4959  df-f1o 4960  df-fv 4961  df-isom 4962  df-riota 5521  df-ov 5568  df-oprab 5569  df-mpt2 5570  df-1st 5820  df-2nd 5821  df-recs 5976  df-frec 6062  df-sup 6493  df-inf 6494  df-pnf 7294  df-mnf 7295  df-xr 7296  df-ltxr 7297  df-le 7298  df-sub 7425  df-neg 7426  df-reap 7819  df-ap 7826  df-div 7905  df-inn 8184  df-2 8242  df-3 8243  df-4 8244  df-n0 8433  df-z 8510  df-uz 8778  df-rp 8893  df-iseq 9599  df-iexp 9650  df-cj 9955  df-re 9956  df-im 9957  df-rsqrt 10110  df-abs 10111
This theorem is referenced by: (None)
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