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Theorem minclpr 11756
Description: The minimum of two real numbers is one of those numbers if and only if dichotomy ( A <_ B \/ B <_ A) holds. For example, this can be combined with zletric 9498 if one is dealing with integers, but real number dichotomy in general does not follow from our axioms. (Contributed by Jim Kingdon, 23-May-2023.)
Assertion
Ref Expression
minclpr |- ( ( A e. RR /\ B e. RR ) -> (inf ( { A , B } , RR , < ) e. { A , B } <-> ( A <_ B \/ B <_ A ) ) )
Proof of Theorem minclpr
StepHypRef Expression
1 renegcl 8415 . . . . 5 |- ( A e. RR -> -u A e. RR )
2 renegcl 8415 . . . . 5 |- ( B e. RR -> -u B e. RR )
3 maxcl 11729 . . . . 5 |- ( ( -u A e. RR /\ -u B e. RR ) -> sup ( { -u A , -u B } , RR , < ) e. RR )
41, 2, 3syl2an 289 . . . 4 |- ( ( A e. RR /\ B e. RR ) -> sup ( { -u A , -u B } , RR , < ) e. RR )
5 elprg 3686 . . . 4 |- ( sup ( { -u A , -u B } , RR , < ) e. RR -> ( sup ( { -u A , -u B } , RR , < ) e. { -u A , -u B } <-> ( sup ( { -u A , -u B } , RR , < ) = -u A \/ sup ( { -u A , -u B } , RR , < ) = -u B ) ) )
64, 5syl 14 . . 3 |- ( ( A e. RR /\ B e. RR ) -> ( sup ( { -u A , -u B } , RR , < ) e. { -u A , -u B } <-> ( sup ( { -u A , -u B } , RR , < ) = -u A \/ sup ( { -u A , -u B } , RR , < ) = -u B ) ) )
7 maxclpr 11741 . . . 4 |- ( ( -u A e. RR /\ -u B e. RR ) -> ( sup ( { -u A , -u B } , RR , < ) e. { -u A , -u B } <-> ( -u A <_ -u B \/ -u B <_ -u A ) ) )
81, 2, 7syl2an 289 . . 3 |- ( ( A e. RR /\ B e. RR ) -> ( sup ( { -u A , -u B } , RR , < ) e. { -u A , -u B } <-> ( -u A <_ -u B \/ -u B <_ -u A ) ) )
94recnd 8183 . . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> sup ( { -u A , -u B } , RR , < ) e. CC )
101adantr 276 . . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> -u A e. RR )
1110recnd 8183 . . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> -u A e. CC )
129, 11neg11ad 8461 . . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( -u sup ( { -u A , -u B } , RR , < ) = -u -u A <-> sup ( { -u A , -u B } , RR , < ) = -u A ) )
13 minmax 11749 . . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> inf ( { A , B } , RR , < ) = -u sup ( { -u A , -u B } , RR , < ) )
1413eqcomd 2235 . . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> -u sup ( { -u A , -u B } , RR , < ) = inf ( { A , B } , RR , < ) )
15 recn 8140 . . . . . . . 8 |- ( A e. RR -> A e. CC )
1615adantr 276 . . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> A e. CC )
1716negnegd 8456 . . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> -u -u A = A )
1814, 17eqeq12d 2244 . . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( -u sup ( { -u A , -u B } , RR , < ) = -u -u A <-> inf ( { A , B } , RR , < ) = A ) )
1912, 18bitr3d 190 . . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( sup ( { -u A , -u B } , RR , < ) = -u A <-> inf ( { A , B } , RR , < ) = A ) )
202adantl 277 . . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> -u B e. RR )
2120recnd 8183 . . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> -u B e. CC )
229, 21neg11ad 8461 . . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( -u sup ( { -u A , -u B } , RR , < ) = -u -u B <-> sup ( { -u A , -u B } , RR , < ) = -u B ) )
23 recn 8140 . . . . . . . 8 |- ( B e. RR -> B e. CC )
2423adantl 277 . . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> B e. CC )
2524negnegd 8456 . . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> -u -u B = B )
2614, 25eqeq12d 2244 . . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( -u sup ( { -u A , -u B } , RR , < ) = -u -u B <-> inf ( { A , B } , RR , < ) = B ) )
2722, 26bitr3d 190 . . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( sup ( { -u A , -u B } , RR , < ) = -u B <-> inf ( { A , B } , RR , < ) = B ) )
2819, 27orbi12d 798 . . 3 |- ( ( A e. RR /\ B e. RR ) -> ( ( sup ( { -u A , -u B } , RR , < ) = -u A \/ sup ( { -u A , -u B } , RR , < ) = -u B ) <-> (inf ( { A , B } , RR , < ) = A \/ inf ( { A , B } , RR , < ) = B ) ) )
296, 8, 283bitr3rd 219 . 2 |- ( ( A e. RR /\ B e. RR ) -> ( (inf ( { A , B } , RR , < ) = A \/ inf ( { A , B } , RR , < ) = B ) <-> ( -u A <_ -u B \/ -u B <_ -u A ) ) )
30 mincl 11750 . . 3 |- ( ( A e. RR /\ B e. RR ) -> inf ( { A , B } , RR , < ) e. RR )
31 elprg 3686 . . 3 |- (inf ( { A , B } , RR , < ) e. RR -> (inf ( { A , B } , RR , < ) e. { A , B } <-> (inf ( { A , B } , RR , < ) = A \/ inf ( { A , B } , RR , < ) = B ) ) )
3230, 31syl 14 . 2 |- ( ( A e. RR /\ B e. RR ) -> (inf ( { A , B } , RR , < ) e. { A , B } <-> (inf ( { A , B } , RR , < ) = A \/ inf ( { A , B } , RR , < ) = B ) ) )
33 orcom 733 . . 3 |- ( ( B <_ A \/ A <_ B ) <-> ( A <_ B \/ B <_ A ) )
34 simpr 110 . . . . 5 |- ( ( A e. RR /\ B e. RR ) -> B e. RR )
35 simpl 109 . . . . 5 |- ( ( A e. RR /\ B e. RR ) -> A e. RR )
3634, 35lenegd 8679 . . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( B <_ A <-> -u A <_ -u B ) )
37 leneg 8620 . . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( A <_ B <-> -u B <_ -u A ) )
3836, 37orbi12d 798 . . 3 |- ( ( A e. RR /\ B e. RR ) -> ( ( B <_ A \/ A <_ B ) <-> ( -u A <_ -u B \/ -u B <_ -u A ) ) )
3933, 38bitr3id 194 . 2 |- ( ( A e. RR /\ B e. RR ) -> ( ( A <_ B \/ B <_ A ) <-> ( -u A <_ -u B \/ -u B <_ -u A ) ) )
4029, 32, 393bitr4d 220 1 |- ( ( A e. RR /\ B e. RR ) -> (inf ( { A , B } , RR , < ) e. { A , B } <-> ( A <_ B \/ B <_ A ) ) )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 713   = wceq 1395   e. wcel 2200   {cpr 3667   class class class wbr 4083   supcsup 7157  infcinf 7158   CCcc 8005   RRcr 8006   < clt 8189   <_ cle 8190   -ucneg 8326
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 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8098  ax-resscn 8099  ax-1cn 8100  ax-1re 8101  ax-icn 8102  ax-addcl 8103  ax-addrcl 8104  ax-mulcl 8105  ax-mulrcl 8106  ax-addcom 8107  ax-mulcom 8108  ax-addass 8109  ax-mulass 8110  ax-distr 8111  ax-i2m1 8112  ax-0lt1 8113  ax-1rid 8114  ax-0id 8115  ax-rnegex 8116  ax-precex 8117  ax-cnre 8118  ax-pre-ltirr 8119  ax-pre-ltwlin 8120  ax-pre-lttrn 8121  ax-pre-apti 8122  ax-pre-ltadd 8123  ax-pre-mulgt0 8124  ax-pre-mulext 8125  ax-arch 8126  ax-caucvg 8127
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 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-isom 5327  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-recs 6457  df-frec 6543  df-sup 7159  df-inf 7160  df-pnf 8191  df-mnf 8192  df-xr 8193  df-ltxr 8194  df-le 8195  df-sub 8327  df-neg 8328  df-reap 8730  df-ap 8737  df-div 8828  df-inn 9119  df-2 9177  df-3 9178  df-4 9179  df-n0 9378  df-z 9455  df-uz 9731  df-rp 9858  df-seqfrec 10678  df-exp 10769  df-cj 11361  df-re 11362  df-im 11363  df-rsqrt 11517  df-abs 11518
This theorem is referenced by:  qtopbas  15204
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