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Theorem minabs 11035
Description: The minimum of two real numbers in terms of absolute value. (Contributed by Jim Kingdon, 15-May-2023.)
Assertion
Ref Expression
minabs  |-  ( ( A  e.  RR  /\  B  e.  RR )  -> inf ( { A ,  B } ,  RR ,  <  )  =  ( ( ( A  +  B
)  -  ( abs `  ( A  -  B
) ) )  / 
2 ) )

Proof of Theorem minabs
StepHypRef Expression
1 minmax 11029 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  -> inf ( { A ,  B } ,  RR ,  <  )  =  -u sup ( { -u A ,  -u B } ,  RR ,  <  ) )
2 renegcl 8043 . . . . 5  |-  ( A  e.  RR  ->  -u A  e.  RR )
3 renegcl 8043 . . . . 5  |-  ( B  e.  RR  ->  -u B  e.  RR )
4 maxabs 11009 . . . . 5  |-  ( (
-u A  e.  RR  /\  -u B  e.  RR )  ->  sup ( { -u A ,  -u B } ,  RR ,  <  )  =  ( ( (
-u A  +  -u B )  +  ( abs `  ( -u A  -  -u B ) ) )  /  2
) )
52, 3, 4syl2an 287 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  sup ( { -u A ,  -u B } ,  RR ,  <  )  =  ( ( (
-u A  +  -u B )  +  ( abs `  ( -u A  -  -u B ) ) )  /  2
) )
65negeqd 7977 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  -> 
-u sup ( { -u A ,  -u B } ,  RR ,  <  )  =  -u ( ( (
-u A  +  -u B )  +  ( abs `  ( -u A  -  -u B ) ) )  /  2
) )
71, 6eqtrd 2173 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  -> inf ( { A ,  B } ,  RR ,  <  )  =  -u (
( ( -u A  +  -u B )  +  ( abs `  ( -u A  -  -u B
) ) )  / 
2 ) )
8 simpl 108 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  A  e.  RR )
98recnd 7814 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  A  e.  CC )
109negcld 8080 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  -> 
-u A  e.  CC )
11 simpr 109 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  B  e.  RR )
1211recnd 7814 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  B  e.  CC )
1312negcld 8080 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  -> 
-u B  e.  CC )
1410, 13addcld 7805 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( -u A  +  -u B )  e.  CC )
1510, 13subcld 8093 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( -u A  -  -u B )  e.  CC )
1615abscld 10981 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( abs `  ( -u A  -  -u B
) )  e.  RR )
1716recnd 7814 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( abs `  ( -u A  -  -u B
) )  e.  CC )
1814, 17addcld 7805 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( -u A  +  -u B )  +  ( abs `  ( -u A  -  -u B
) ) )  e.  CC )
19 2cnd 8813 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  2  e.  CC )
20 2ap0 8833 . . . 4  |-  2 #  0
2120a1i 9 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  2 #  0 )
2218, 19, 21divnegapd 8583 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  -> 
-u ( ( (
-u A  +  -u B )  +  ( abs `  ( -u A  -  -u B ) ) )  /  2
)  =  ( -u ( ( -u A  +  -u B )  +  ( abs `  ( -u A  -  -u B
) ) )  / 
2 ) )
2314, 17negdi2d 8107 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  -> 
-u ( ( -u A  +  -u B )  +  ( abs `  ( -u A  -  -u B
) ) )  =  ( -u ( -u A  +  -u B )  -  ( abs `  ( -u A  -  -u B
) ) ) )
2410, 13negdid 8106 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  -> 
-u ( -u A  +  -u B )  =  ( -u -u A  +  -u -u B ) )
259negnegd 8084 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  -> 
-u -u A  =  A )
2612negnegd 8084 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  -> 
-u -u B  =  B )
2725, 26oveq12d 5796 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( -u -u A  +  -u -u B )  =  ( A  +  B
) )
2824, 27eqtrd 2173 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  -> 
-u ( -u A  +  -u B )  =  ( A  +  B
) )
299, 12neg2subd 8110 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( -u A  -  -u B )  =  ( B  -  A ) )
3029fveq2d 5429 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( abs `  ( -u A  -  -u B
) )  =  ( abs `  ( B  -  A ) ) )
319, 12abssubd 10993 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( abs `  ( A  -  B )
)  =  ( abs `  ( B  -  A
) ) )
3230, 31eqtr4d 2176 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( abs `  ( -u A  -  -u B
) )  =  ( abs `  ( A  -  B ) ) )
3328, 32oveq12d 5796 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( -u ( -u A  +  -u B )  -  ( abs `  ( -u A  -  -u B
) ) )  =  ( ( A  +  B )  -  ( abs `  ( A  -  B ) ) ) )
3423, 33eqtrd 2173 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  -> 
-u ( ( -u A  +  -u B )  +  ( abs `  ( -u A  -  -u B
) ) )  =  ( ( A  +  B )  -  ( abs `  ( A  -  B ) ) ) )
3534oveq1d 5793 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( -u ( (
-u A  +  -u B )  +  ( abs `  ( -u A  -  -u B ) ) )  /  2
)  =  ( ( ( A  +  B
)  -  ( abs `  ( A  -  B
) ) )  / 
2 ) )
367, 22, 353eqtrd 2177 1  |-  ( ( A  e.  RR  /\  B  e.  RR )  -> inf ( { A ,  B } ,  RR ,  <  )  =  ( ( ( A  +  B
)  -  ( abs `  ( A  -  B
) ) )  / 
2 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1332    e. wcel 1481   {cpr 3529   class class class wbr 3933   ` cfv 5127  (class class class)co 5778   supcsup 6873  infcinf 6874   RRcr 7639   0cc0 7640    + caddc 7643    < clt 7820    - cmin 7953   -ucneg 7954   # cap 8363    / cdiv 8452   2c2 8791   abscabs 10797
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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4047  ax-sep 4050  ax-nul 4058  ax-pow 4102  ax-pr 4135  ax-un 4359  ax-setind 4456  ax-iinf 4506  ax-cnex 7731  ax-resscn 7732  ax-1cn 7733  ax-1re 7734  ax-icn 7735  ax-addcl 7736  ax-addrcl 7737  ax-mulcl 7738  ax-mulrcl 7739  ax-addcom 7740  ax-mulcom 7741  ax-addass 7742  ax-mulass 7743  ax-distr 7744  ax-i2m1 7745  ax-0lt1 7746  ax-1rid 7747  ax-0id 7748  ax-rnegex 7749  ax-precex 7750  ax-cnre 7751  ax-pre-ltirr 7752  ax-pre-ltwlin 7753  ax-pre-lttrn 7754  ax-pre-apti 7755  ax-pre-ltadd 7756  ax-pre-mulgt0 7757  ax-pre-mulext 7758  ax-arch 7759  ax-caucvg 7760
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rmo 2425  df-rab 2426  df-v 2689  df-sbc 2911  df-csb 3005  df-dif 3074  df-un 3076  df-in 3078  df-ss 3085  df-nul 3365  df-if 3476  df-pw 3513  df-sn 3534  df-pr 3535  df-op 3537  df-uni 3741  df-int 3776  df-iun 3819  df-br 3934  df-opab 3994  df-mpt 3995  df-tr 4031  df-id 4219  df-po 4222  df-iso 4223  df-iord 4292  df-on 4294  df-ilim 4295  df-suc 4297  df-iom 4509  df-xp 4549  df-rel 4550  df-cnv 4551  df-co 4552  df-dm 4553  df-rn 4554  df-res 4555  df-ima 4556  df-iota 5092  df-fun 5129  df-fn 5130  df-f 5131  df-f1 5132  df-fo 5133  df-f1o 5134  df-fv 5135  df-isom 5136  df-riota 5734  df-ov 5781  df-oprab 5782  df-mpo 5783  df-1st 6042  df-2nd 6043  df-recs 6206  df-frec 6292  df-sup 6875  df-inf 6876  df-pnf 7822  df-mnf 7823  df-xr 7824  df-ltxr 7825  df-le 7826  df-sub 7955  df-neg 7956  df-reap 8357  df-ap 8364  df-div 8453  df-inn 8741  df-2 8799  df-3 8800  df-4 8801  df-n0 8998  df-z 9075  df-uz 9347  df-rp 9467  df-seqfrec 10246  df-exp 10320  df-cj 10642  df-re 10643  df-im 10644  df-rsqrt 10798  df-abs 10799
This theorem is referenced by:  bdtri  11039
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