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Theorem minabs 11985
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 11979 . . 3 |- ( ( A e. RR /\ B e. RR ) -> inf ( { A , B } , RR , < ) = -u sup ( { -u A , -u B } , RR , < ) )
2 renegcl 8581 . . . . 5 |- ( A e. RR -> -u A e. RR )
3 renegcl 8581 . . . . 5 |- ( B e. RR -> -u B e. RR )
4 maxabs 11958 . . . . 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 8515 . . 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 2271 . 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 8348 . . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> A e. CC )
109negcld 8618 . . . . 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 8348 . . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> B e. CC )
1312negcld 8618 . . . . 5 |- ( ( A e. RR /\ B e. RR ) -> -u B e. CC )
1410, 13addcld 8339 . . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( -u A + -u B ) e. CC )
1510, 13subcld 8631 . . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( -u A - -u B ) e. CC )
1615abscld 11930 . . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( abs ` ( -u A - -u B ) ) e. RR )
1716recnd 8348 . . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( abs ` ( -u A - -u B ) ) e. CC )
1814, 17addcld 8339 . . 3 |- ( ( A e. RR /\ B e. RR ) -> ( ( -u A + -u B ) + ( abs ` ( -u A - -u B ) ) ) e. CC )
19 2cnd 9360 . . 3 |- ( ( A e. RR /\ B e. RR ) -> 2 e. CC )
20 2ap0 9380 . . . 4 |- 2 # 0
2120a1i 9 . . 3 |- ( ( A e. RR /\ B e. RR ) -> 2 # 0 )
2218, 19, 21divnegapd 9127 . 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 8645 . . . 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 8644 . . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> -u ( -u A + -u B ) = ( -u -u A + -u -u B ) )
259negnegd 8622 . . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> -u -u A = A )
2612negnegd 8622 . . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> -u -u B = B )
2725, 26oveq12d 6097 . . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( -u -u A + -u -u B ) = ( A + B ) )
2824, 27eqtrd 2271 . . . . 5 |- ( ( A e. RR /\ B e. RR ) -> -u ( -u A + -u B ) = ( A + B ) )
299, 12neg2subd 8648 . . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( -u A - -u B ) = ( B - A ) )
3029fveq2d 5697 . . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( abs ` ( -u A - -u B ) ) = ( abs ` ( B - A ) ) )
319, 12abssubd 11942 . . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( abs ` ( A - B ) ) = ( abs ` ( B - A ) ) )
3230, 31eqtr4d 2274 . . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( abs ` ( -u A - -u B ) ) = ( abs ` ( A - B ) ) )
3328, 32oveq12d 6097 . . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( -u ( -u A + -u B ) - ( abs ` ( -u A - -u B ) ) ) = ( ( A + B ) - ( abs ` ( A - B ) ) ) )
3423, 33eqtrd 2271 . . 3 |- ( ( A e. RR /\ B e. RR ) -> -u ( ( -u A + -u B ) + ( abs ` ( -u A - -u B ) ) ) = ( ( A + B ) - ( abs ` ( A - B ) ) ) )
3534oveq1d 6094 . 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 2275 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 1402   e. wcel 2209   {cpr 3709   class class class wbr 4128   ` cfv 5375  (class class class)co 6079   supcsup 7316  infcinf 7317   RRcr 8172   0cc0 8173   + caddc 8176   < clt 8354   - cmin 8491   -ucneg 8492   # cap 8903   / cdiv 8996   2c2 9338   abscabs 11746
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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-mulrcl 8272  ax-addcom 8273  ax-mulcom 8274  ax-addass 8275  ax-mulass 8276  ax-distr 8277  ax-i2m1 8278  ax-0lt1 8279  ax-1rid 8280  ax-0id 8281  ax-rnegex 8282  ax-precex 8283  ax-cnre 8284  ax-pre-ltirr 8285  ax-pre-ltwlin 8286  ax-pre-lttrn 8287  ax-pre-apti 8288  ax-pre-ltadd 8289  ax-pre-mulgt0 8290  ax-pre-mulext 8291  ax-arch 8292  ax-caucvg 8293
This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-id 4436  df-po 4439  df-iso 4440  df-iord 4509  df-on 4511  df-ilim 4512  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-isom 5384  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-recs 6570  df-frec 6656  df-sup 7318  df-inf 7319  df-pnf 8356  df-mnf 8357  df-xr 8358  df-ltxr 8359  df-le 8360  df-sub 8493  df-neg 8494  df-reap 8897  df-ap 8904  df-div 8997  df-inn 9288  df-2 9346  df-3 9347  df-4 9348  df-n0 9547  df-z 9628  df-uz 9905  df-rp 10038  df-seqfrec 10868  df-exp 10959  df-cj 11590  df-re 11591  df-im 11592  df-rsqrt 11747  df-abs 11748
This theorem is referenced by:  bdtri  11989  mincncf  15700
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