ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  minabs Unicode version
Theorem minabs 11929
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 11923 . . 3 |- ( ( A e. RR /\ B e. RR ) -> inf ( { A , B } , RR , < ) = -u sup ( { -u A , -u B } , RR , < ) )
2 renegcl 8539 . . . . 5 |- ( A e. RR -> -u A e. RR )
3 renegcl 8539 . . . . 5 |- ( B e. RR -> -u B e. RR )
4 maxabs 11902 . . . . 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 8473 . . 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 2267 . 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 8307 . . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> A e. CC )
109negcld 8576 . . . . 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 8307 . . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> B e. CC )
1312negcld 8576 . . . . 5 |- ( ( A e. RR /\ B e. RR ) -> -u B e. CC )
1410, 13addcld 8298 . . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( -u A + -u B ) e. CC )
1510, 13subcld 8589 . . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( -u A - -u B ) e. CC )
1615abscld 11874 . . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( abs ` ( -u A - -u B ) ) e. RR )
1716recnd 8307 . . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( abs ` ( -u A - -u B ) ) e. CC )
1814, 17addcld 8298 . . 3 |- ( ( A e. RR /\ B e. RR ) -> ( ( -u A + -u B ) + ( abs ` ( -u A - -u B ) ) ) e. CC )
19 2cnd 9315 . . 3 |- ( ( A e. RR /\ B e. RR ) -> 2 e. CC )
20 2ap0 9335 . . . 4 |- 2 # 0
2120a1i 9 . . 3 |- ( ( A e. RR /\ B e. RR ) -> 2 # 0 )
2218, 19, 21divnegapd 9082 . 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 8603 . . . 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 8602 . . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> -u ( -u A + -u B ) = ( -u -u A + -u -u B ) )
259negnegd 8580 . . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> -u -u A = A )
2612negnegd 8580 . . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> -u -u B = B )
2725, 26oveq12d 6070 . . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( -u -u A + -u -u B ) = ( A + B ) )
2824, 27eqtrd 2267 . . . . 5 |- ( ( A e. RR /\ B e. RR ) -> -u ( -u A + -u B ) = ( A + B ) )
299, 12neg2subd 8606 . . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( -u A - -u B ) = ( B - A ) )
3029fveq2d 5676 . . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( abs ` ( -u A - -u B ) ) = ( abs ` ( B - A ) ) )
319, 12abssubd 11886 . . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( abs ` ( A - B ) ) = ( abs ` ( B - A ) ) )
3230, 31eqtr4d 2270 . . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( abs ` ( -u A - -u B ) ) = ( abs ` ( A - B ) ) )
3328, 32oveq12d 6070 . . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( -u ( -u A + -u B ) - ( abs ` ( -u A - -u B ) ) ) = ( ( A + B ) - ( abs ` ( A - B ) ) ) )
3423, 33eqtrd 2267 . . 3 |- ( ( A e. RR /\ B e. RR ) -> -u ( ( -u A + -u B ) + ( abs ` ( -u A - -u B ) ) ) = ( ( A + B ) - ( abs ` ( A - B ) ) ) )
3534oveq1d 6067 . 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 2271 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 1398   e. wcel 2205   {cpr 3692   class class class wbr 4111   ` cfv 5354  (class class class)co 6052   supcsup 7275  infcinf 7276   RRcr 8131   0cc0 8132   + caddc 8135   < clt 8313   - cmin 8449   -ucneg 8450   # cap 8860   / cdiv 8951   2c2 9293   abscabs 11690
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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-iinf 4712  ax-cnex 8223  ax-resscn 8224  ax-1cn 8225  ax-1re 8226  ax-icn 8227  ax-addcl 8228  ax-addrcl 8229  ax-mulcl 8230  ax-mulrcl 8231  ax-addcom 8232  ax-mulcom 8233  ax-addass 8234  ax-mulass 8235  ax-distr 8236  ax-i2m1 8237  ax-0lt1 8238  ax-1rid 8239  ax-0id 8240  ax-rnegex 8241  ax-precex 8242  ax-cnre 8243  ax-pre-ltirr 8244  ax-pre-ltwlin 8245  ax-pre-lttrn 8246  ax-pre-apti 8247  ax-pre-ltadd 8248  ax-pre-mulgt0 8249  ax-pre-mulext 8250  ax-arch 8251  ax-caucvg 8252
This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-if 3623  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-id 4416  df-po 4419  df-iso 4420  df-iord 4489  df-on 4491  df-ilim 4492  df-suc 4494  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-isom 5363  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-recs 6538  df-frec 6624  df-sup 7277  df-inf 7278  df-pnf 8315  df-mnf 8316  df-xr 8317  df-ltxr 8318  df-le 8319  df-sub 8451  df-neg 8452  df-reap 8854  df-ap 8861  df-div 8952  df-inn 9243  df-2 9301  df-3 9302  df-4 9303  df-n0 9502  df-z 9583  df-uz 9860  df-rp 9993  df-seqfrec 10817  df-exp 10908  df-cj 11535  df-re 11536  df-im 11537  df-rsqrt 11691  df-abs 11692
This theorem is referenced by:  bdtri  11933  mincncf  15530
  Copyright terms: Public domain W3C validator