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Theorem minabs 11713
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 11707 . . 3 |- ( ( A e. RR /\ B e. RR ) -> inf ( { A , B } , RR , < ) = -u sup ( { -u A , -u B } , RR , < ) )
2 renegcl 8375 . . . . 5 |- ( A e. RR -> -u A e. RR )
3 renegcl 8375 . . . . 5 |- ( B e. RR -> -u B e. RR )
4 maxabs 11686 . . . . 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 8309 . . 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 2242 . 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 8143 . . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> A e. CC )
109negcld 8412 . . . . 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 8143 . . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> B e. CC )
1312negcld 8412 . . . . 5 |- ( ( A e. RR /\ B e. RR ) -> -u B e. CC )
1410, 13addcld 8134 . . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( -u A + -u B ) e. CC )
1510, 13subcld 8425 . . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( -u A - -u B ) e. CC )
1615abscld 11658 . . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( abs ` ( -u A - -u B ) ) e. RR )
1716recnd 8143 . . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( abs ` ( -u A - -u B ) ) e. CC )
1814, 17addcld 8134 . . 3 |- ( ( A e. RR /\ B e. RR ) -> ( ( -u A + -u B ) + ( abs ` ( -u A - -u B ) ) ) e. CC )
19 2cnd 9151 . . 3 |- ( ( A e. RR /\ B e. RR ) -> 2 e. CC )
20 2ap0 9171 . . . 4 |- 2 # 0
2120a1i 9 . . 3 |- ( ( A e. RR /\ B e. RR ) -> 2 # 0 )
2218, 19, 21divnegapd 8918 . 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 8439 . . . 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 8438 . . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> -u ( -u A + -u B ) = ( -u -u A + -u -u B ) )
259negnegd 8416 . . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> -u -u A = A )
2612negnegd 8416 . . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> -u -u B = B )
2725, 26oveq12d 5992 . . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( -u -u A + -u -u B ) = ( A + B ) )
2824, 27eqtrd 2242 . . . . 5 |- ( ( A e. RR /\ B e. RR ) -> -u ( -u A + -u B ) = ( A + B ) )
299, 12neg2subd 8442 . . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( -u A - -u B ) = ( B - A ) )
3029fveq2d 5607 . . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( abs ` ( -u A - -u B ) ) = ( abs ` ( B - A ) ) )
319, 12abssubd 11670 . . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( abs ` ( A - B ) ) = ( abs ` ( B - A ) ) )
3230, 31eqtr4d 2245 . . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( abs ` ( -u A - -u B ) ) = ( abs ` ( A - B ) ) )
3328, 32oveq12d 5992 . . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( -u ( -u A + -u B ) - ( abs ` ( -u A - -u B ) ) ) = ( ( A + B ) - ( abs ` ( A - B ) ) ) )
3423, 33eqtrd 2242 . . 3 |- ( ( A e. RR /\ B e. RR ) -> -u ( ( -u A + -u B ) + ( abs ` ( -u A - -u B ) ) ) = ( ( A + B ) - ( abs ` ( A - B ) ) ) )
3534oveq1d 5989 . 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 2246 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 1375   e. wcel 2180   {cpr 3647   class class class wbr 4062   ` cfv 5294  (class class class)co 5974   supcsup 7117  infcinf 7118   RRcr 7966   0cc0 7967   + caddc 7970   < clt 8149   - cmin 8285   -ucneg 8286   # cap 8696   / cdiv 8787   2c2 9129   abscabs 11474
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 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-13 2182  ax-14 2183  ax-ext 2191  ax-coll 4178  ax-sep 4181  ax-nul 4189  ax-pow 4237  ax-pr 4272  ax-un 4501  ax-setind 4606  ax-iinf 4657  ax-cnex 8058  ax-resscn 8059  ax-1cn 8060  ax-1re 8061  ax-icn 8062  ax-addcl 8063  ax-addrcl 8064  ax-mulcl 8065  ax-mulrcl 8066  ax-addcom 8067  ax-mulcom 8068  ax-addass 8069  ax-mulass 8070  ax-distr 8071  ax-i2m1 8072  ax-0lt1 8073  ax-1rid 8074  ax-0id 8075  ax-rnegex 8076  ax-precex 8077  ax-cnre 8078  ax-pre-ltirr 8079  ax-pre-ltwlin 8080  ax-pre-lttrn 8081  ax-pre-apti 8082  ax-pre-ltadd 8083  ax-pre-mulgt0 8084  ax-pre-mulext 8085  ax-arch 8086  ax-caucvg 8087
This theorem depends on definitions:  df-bi 117  df-dc 839  df-3or 984  df-3an 985  df-tru 1378  df-fal 1381  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ne 2381  df-nel 2476  df-ral 2493  df-rex 2494  df-reu 2495  df-rmo 2496  df-rab 2497  df-v 2781  df-sbc 3009  df-csb 3105  df-dif 3179  df-un 3181  df-in 3183  df-ss 3190  df-nul 3472  df-if 3583  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-uni 3868  df-int 3903  df-iun 3946  df-br 4063  df-opab 4125  df-mpt 4126  df-tr 4162  df-id 4361  df-po 4364  df-iso 4365  df-iord 4434  df-on 4436  df-ilim 4437  df-suc 4439  df-iom 4660  df-xp 4702  df-rel 4703  df-cnv 4704  df-co 4705  df-dm 4706  df-rn 4707  df-res 4708  df-ima 4709  df-iota 5254  df-fun 5296  df-fn 5297  df-f 5298  df-f1 5299  df-fo 5300  df-f1o 5301  df-fv 5302  df-isom 5303  df-riota 5927  df-ov 5977  df-oprab 5978  df-mpo 5979  df-1st 6256  df-2nd 6257  df-recs 6421  df-frec 6507  df-sup 7119  df-inf 7120  df-pnf 8151  df-mnf 8152  df-xr 8153  df-ltxr 8154  df-le 8155  df-sub 8287  df-neg 8288  df-reap 8690  df-ap 8697  df-div 8788  df-inn 9079  df-2 9137  df-3 9138  df-4 9139  df-n0 9338  df-z 9415  df-uz 9691  df-rp 9818  df-seqfrec 10637  df-exp 10728  df-cj 11319  df-re 11320  df-im 11321  df-rsqrt 11475  df-abs 11476
This theorem is referenced by:  bdtri  11717  mincncf  15255
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