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Theorem minabs 11591
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 11585 . . 3 |- ( ( A e. RR /\ B e. RR ) -> inf ( { A , B } , RR , < ) = -u sup ( { -u A , -u B } , RR , < ) )
2 renegcl 8340 . . . . 5 |- ( A e. RR -> -u A e. RR )
3 renegcl 8340 . . . . 5 |- ( B e. RR -> -u B e. RR )
4 maxabs 11564 . . . . 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 8274 . . 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 2239 . 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 8108 . . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> A e. CC )
109negcld 8377 . . . . 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 8108 . . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> B e. CC )
1312negcld 8377 . . . . 5 |- ( ( A e. RR /\ B e. RR ) -> -u B e. CC )
1410, 13addcld 8099 . . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( -u A + -u B ) e. CC )
1510, 13subcld 8390 . . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( -u A - -u B ) e. CC )
1615abscld 11536 . . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( abs ` ( -u A - -u B ) ) e. RR )
1716recnd 8108 . . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( abs ` ( -u A - -u B ) ) e. CC )
1814, 17addcld 8099 . . 3 |- ( ( A e. RR /\ B e. RR ) -> ( ( -u A + -u B ) + ( abs ` ( -u A - -u B ) ) ) e. CC )
19 2cnd 9116 . . 3 |- ( ( A e. RR /\ B e. RR ) -> 2 e. CC )
20 2ap0 9136 . . . 4 |- 2 # 0
2120a1i 9 . . 3 |- ( ( A e. RR /\ B e. RR ) -> 2 # 0 )
2218, 19, 21divnegapd 8883 . 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 8404 . . . 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 8403 . . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> -u ( -u A + -u B ) = ( -u -u A + -u -u B ) )
259negnegd 8381 . . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> -u -u A = A )
2612negnegd 8381 . . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> -u -u B = B )
2725, 26oveq12d 5969 . . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( -u -u A + -u -u B ) = ( A + B ) )
2824, 27eqtrd 2239 . . . . 5 |- ( ( A e. RR /\ B e. RR ) -> -u ( -u A + -u B ) = ( A + B ) )
299, 12neg2subd 8407 . . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( -u A - -u B ) = ( B - A ) )
3029fveq2d 5587 . . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( abs ` ( -u A - -u B ) ) = ( abs ` ( B - A ) ) )
319, 12abssubd 11548 . . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( abs ` ( A - B ) ) = ( abs ` ( B - A ) ) )
3230, 31eqtr4d 2242 . . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( abs ` ( -u A - -u B ) ) = ( abs ` ( A - B ) ) )
3328, 32oveq12d 5969 . . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( -u ( -u A + -u B ) - ( abs ` ( -u A - -u B ) ) ) = ( ( A + B ) - ( abs ` ( A - B ) ) ) )
3423, 33eqtrd 2239 . . 3 |- ( ( A e. RR /\ B e. RR ) -> -u ( ( -u A + -u B ) + ( abs ` ( -u A - -u B ) ) ) = ( ( A + B ) - ( abs ` ( A - B ) ) ) )
3534oveq1d 5966 . 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 2243 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 1373   e. wcel 2177   {cpr 3635   class class class wbr 4047   ` cfv 5276  (class class class)co 5951   supcsup 7091  infcinf 7092   RRcr 7931   0cc0 7932   + caddc 7935   < clt 8114   - cmin 8250   -ucneg 8251   # cap 8661   / cdiv 8752   2c2 9094   abscabs 11352
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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4163  ax-sep 4166  ax-nul 4174  ax-pow 4222  ax-pr 4257  ax-un 4484  ax-setind 4589  ax-iinf 4640  ax-cnex 8023  ax-resscn 8024  ax-1cn 8025  ax-1re 8026  ax-icn 8027  ax-addcl 8028  ax-addrcl 8029  ax-mulcl 8030  ax-mulrcl 8031  ax-addcom 8032  ax-mulcom 8033  ax-addass 8034  ax-mulass 8035  ax-distr 8036  ax-i2m1 8037  ax-0lt1 8038  ax-1rid 8039  ax-0id 8040  ax-rnegex 8041  ax-precex 8042  ax-cnre 8043  ax-pre-ltirr 8044  ax-pre-ltwlin 8045  ax-pre-lttrn 8046  ax-pre-apti 8047  ax-pre-ltadd 8048  ax-pre-mulgt0 8049  ax-pre-mulext 8050  ax-arch 8051  ax-caucvg 8052
This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3000  df-csb 3095  df-dif 3169  df-un 3171  df-in 3173  df-ss 3180  df-nul 3462  df-if 3573  df-pw 3619  df-sn 3640  df-pr 3641  df-op 3643  df-uni 3853  df-int 3888  df-iun 3931  df-br 4048  df-opab 4110  df-mpt 4111  df-tr 4147  df-id 4344  df-po 4347  df-iso 4348  df-iord 4417  df-on 4419  df-ilim 4420  df-suc 4422  df-iom 4643  df-xp 4685  df-rel 4686  df-cnv 4687  df-co 4688  df-dm 4689  df-rn 4690  df-res 4691  df-ima 4692  df-iota 5237  df-fun 5278  df-fn 5279  df-f 5280  df-f1 5281  df-fo 5282  df-f1o 5283  df-fv 5284  df-isom 5285  df-riota 5906  df-ov 5954  df-oprab 5955  df-mpo 5956  df-1st 6233  df-2nd 6234  df-recs 6398  df-frec 6484  df-sup 7093  df-inf 7094  df-pnf 8116  df-mnf 8117  df-xr 8118  df-ltxr 8119  df-le 8120  df-sub 8252  df-neg 8253  df-reap 8655  df-ap 8662  df-div 8753  df-inn 9044  df-2 9102  df-3 9103  df-4 9104  df-n0 9303  df-z 9380  df-uz 9656  df-rp 9783  df-seqfrec 10600  df-exp 10691  df-cj 11197  df-re 11198  df-im 11199  df-rsqrt 11353  df-abs 11354
This theorem is referenced by:  bdtri  11595  mincncf  15132
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