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Theorem minabs 11742
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 11736 . . 3 |- ( ( A e. RR /\ B e. RR ) -> inf ( { A , B } , RR , < ) = -u sup ( { -u A , -u B } , RR , < ) )
2 renegcl 8403 . . . . 5 |- ( A e. RR -> -u A e. RR )
3 renegcl 8403 . . . . 5 |- ( B e. RR -> -u B e. RR )
4 maxabs 11715 . . . . 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 289 . . . 4 |- ( ( A e. RR /\ B e. RR ) -> sup ( { -u A , -u B } , RR , < ) = ( ( ( -u A + -u B ) + ( abs ` ( -u A - -u B ) ) ) / 2 ) )
65negeqd 8337 . . 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 2262 . 2 |- ( ( A e. RR /\ B e. RR ) -> inf ( { A , B } , RR , < ) = -u ( ( ( -u A + -u B ) + ( abs ` ( -u A - -u B ) ) ) / 2 ) )
8 simpl 109 . . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> A e. RR )
98recnd 8171 . . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> A e. CC )
109negcld 8440 . . . . 5 |- ( ( A e. RR /\ B e. RR ) -> -u A e. CC )
11 simpr 110 . . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> B e. RR )
1211recnd 8171 . . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> B e. CC )
1312negcld 8440 . . . . 5 |- ( ( A e. RR /\ B e. RR ) -> -u B e. CC )
1410, 13addcld 8162 . . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( -u A + -u B ) e. CC )
1510, 13subcld 8453 . . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( -u A - -u B ) e. CC )
1615abscld 11687 . . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( abs ` ( -u A - -u B ) ) e. RR )
1716recnd 8171 . . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( abs ` ( -u A - -u B ) ) e. CC )
1814, 17addcld 8162 . . 3 |- ( ( A e. RR /\ B e. RR ) -> ( ( -u A + -u B ) + ( abs ` ( -u A - -u B ) ) ) e. CC )
19 2cnd 9179 . . 3 |- ( ( A e. RR /\ B e. RR ) -> 2 e. CC )
20 2ap0 9199 . . . 4 |- 2 # 0
2120a1i 9 . . 3 |- ( ( A e. RR /\ B e. RR ) -> 2 # 0 )
2218, 19, 21divnegapd 8946 . 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 8467 . . . 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 8466 . . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> -u ( -u A + -u B ) = ( -u -u A + -u -u B ) )
259negnegd 8444 . . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> -u -u A = A )
2612negnegd 8444 . . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> -u -u B = B )
2725, 26oveq12d 6018 . . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( -u -u A + -u -u B ) = ( A + B ) )
2824, 27eqtrd 2262 . . . . 5 |- ( ( A e. RR /\ B e. RR ) -> -u ( -u A + -u B ) = ( A + B ) )
299, 12neg2subd 8470 . . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( -u A - -u B ) = ( B - A ) )
3029fveq2d 5630 . . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( abs ` ( -u A - -u B ) ) = ( abs ` ( B - A ) ) )
319, 12abssubd 11699 . . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( abs ` ( A - B ) ) = ( abs ` ( B - A ) ) )
3230, 31eqtr4d 2265 . . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( abs ` ( -u A - -u B ) ) = ( abs ` ( A - B ) ) )
3328, 32oveq12d 6018 . . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( -u ( -u A + -u B ) - ( abs ` ( -u A - -u B ) ) ) = ( ( A + B ) - ( abs ` ( A - B ) ) ) )
3423, 33eqtrd 2262 . . 3 |- ( ( A e. RR /\ B e. RR ) -> -u ( ( -u A + -u B ) + ( abs ` ( -u A - -u B ) ) ) = ( ( A + B ) - ( abs ` ( A - B ) ) ) )
3534oveq1d 6015 . 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 2266 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 104   = wceq 1395   e. wcel 2200   {cpr 3667   class class class wbr 4082   ` cfv 5317  (class class class)co 6000   supcsup 7145  infcinf 7146   RRcr 7994   0cc0 7995   + caddc 7998   < clt 8177   - cmin 8313   -ucneg 8314   # cap 8724   / cdiv 8815   2c2 9157   abscabs 11503
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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-mulrcl 8094  ax-addcom 8095  ax-mulcom 8096  ax-addass 8097  ax-mulass 8098  ax-distr 8099  ax-i2m1 8100  ax-0lt1 8101  ax-1rid 8102  ax-0id 8103  ax-rnegex 8104  ax-precex 8105  ax-cnre 8106  ax-pre-ltirr 8107  ax-pre-ltwlin 8108  ax-pre-lttrn 8109  ax-pre-apti 8110  ax-pre-ltadd 8111  ax-pre-mulgt0 8112  ax-pre-mulext 8113  ax-arch 8114  ax-caucvg 8115
This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4383  df-po 4386  df-iso 4387  df-iord 4456  df-on 4458  df-ilim 4459  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-isom 5326  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-recs 6449  df-frec 6535  df-sup 7147  df-inf 7148  df-pnf 8179  df-mnf 8180  df-xr 8181  df-ltxr 8182  df-le 8183  df-sub 8315  df-neg 8316  df-reap 8718  df-ap 8725  df-div 8816  df-inn 9107  df-2 9165  df-3 9166  df-4 9167  df-n0 9366  df-z 9443  df-uz 9719  df-rp 9846  df-seqfrec 10665  df-exp 10756  df-cj 11348  df-re 11349  df-im 11350  df-rsqrt 11504  df-abs 11505
This theorem is referenced by:  bdtri  11746  mincncf  15284
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