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Theorem xrminrpcl 11984
Description: The minimum of two positive reals is a positive real. (Contributed by Jim Kingdon, 4-May-2023.)
Assertion
Ref Expression
xrminrpcl |- ( ( A e. RR+ /\ B e. RR+ ) -> inf ( { A , B } , RR* , < ) e. RR+ )
Proof of Theorem xrminrpcl
StepHypRef Expression
1 rpxr 10012 . . . 4 |- ( A e. RR+ -> A e. RR* )
2 rpxr 10012 . . . 4 |- ( B e. RR+ -> B e. RR* )
3 xrminmax 11975 . . . 4 |- ( ( A e. RR* /\ B e. RR* ) -> inf ( { A , B } , RR* , < ) = -e sup ( { -e A , -e B } , RR* , < ) )
41, 2, 3syl2an 289 . . 3 |- ( ( A e. RR+ /\ B e. RR+ ) -> inf ( { A , B } , RR* , < ) = -e sup ( { -e A , -e B } , RR* , < ) )
5 rpre 10011 . . . . . . . 8 |- ( A e. RR+ -> A e. RR )
6 rexneg 10182 . . . . . . . . 9 |- ( A e. RR -> -e A = -u A )
7 renegcl 8550 . . . . . . . . 9 |- ( A e. RR -> -u A e. RR )
86, 7eqeltrd 2311 . . . . . . . 8 |- ( A e. RR -> -e A e. RR )
95, 8syl 14 . . . . . . 7 |- ( A e. RR+ -> -e A e. RR )
10 rpre 10011 . . . . . . . 8 |- ( B e. RR+ -> B e. RR )
11 rexneg 10182 . . . . . . . . 9 |- ( B e. RR -> -e B = -u B )
12 renegcl 8550 . . . . . . . . 9 |- ( B e. RR -> -u B e. RR )
1311, 12eqeltrd 2311 . . . . . . . 8 |- ( B e. RR -> -e B e. RR )
1410, 13syl 14 . . . . . . 7 |- ( B e. RR+ -> -e B e. RR )
15 xrmaxrecl 11965 . . . . . . 7 |- ( ( -e A e. RR /\ -e B e. RR ) -> sup ( { -e A , -e B } , RR* , < ) = sup ( { -e A , -e B } , RR , < ) )
169, 14, 15syl2an 289 . . . . . 6 |- ( ( A e. RR+ /\ B e. RR+ ) -> sup ( { -e A , -e B } , RR* , < ) = sup ( { -e A , -e B } , RR , < ) )
17 maxcl 11920 . . . . . . 7 |- ( ( -e A e. RR /\ -e B e. RR ) -> sup ( { -e A , -e B } , RR , < ) e. RR )
189, 14, 17syl2an 289 . . . . . 6 |- ( ( A e. RR+ /\ B e. RR+ ) -> sup ( { -e A , -e B } , RR , < ) e. RR )
1916, 18eqeltrd 2311 . . . . 5 |- ( ( A e. RR+ /\ B e. RR+ ) -> sup ( { -e A , -e B } , RR* , < ) e. RR )
20 rexneg 10182 . . . . 5 |- ( sup ( { -e A , -e B } , RR* , < ) e. RR -> -e sup ( { -e A , -e B } , RR* , < ) = -u sup ( { -e A , -e B } , RR* , < ) )
2119, 20syl 14 . . . 4 |- ( ( A e. RR+ /\ B e. RR+ ) -> -e sup ( { -e A , -e B } , RR* , < ) = -u sup ( { -e A , -e B } , RR* , < ) )
2219renegcld 8670 . . . 4 |- ( ( A e. RR+ /\ B e. RR+ ) -> -u sup ( { -e A , -e B } , RR* , < ) e. RR )
2321, 22eqeltrd 2311 . . 3 |- ( ( A e. RR+ /\ B e. RR+ ) -> -e sup ( { -e A , -e B } , RR* , < ) e. RR )
244, 23eqeltrd 2311 . 2 |- ( ( A e. RR+ /\ B e. RR+ ) -> inf ( { A , B } , RR* , < ) e. RR )
25 rpgt0 10016 . . . 4 |- ( A e. RR+ -> 0 < A )
26 rpgt0 10016 . . . 4 |- ( B e. RR+ -> 0 < B )
2725, 26anim12i 338 . . 3 |- ( ( A e. RR+ /\ B e. RR+ ) -> ( 0 < A /\ 0 < B ) )
28 0xr 8336 . . . 4 |- 0 e. RR*
29 xrltmininf 11980 . . . 4 |- ( ( 0 e. RR* /\ A e. RR* /\ B e. RR* ) -> ( 0 < inf ( { A , B } , RR* , < ) <-> ( 0 < A /\ 0 < B ) ) )
3028, 1, 2, 29mp3an3an 1380 . . 3 |- ( ( A e. RR+ /\ B e. RR+ ) -> ( 0 < inf ( { A , B } , RR* , < ) <-> ( 0 < A /\ 0 < B ) ) )
3127, 30mpbird 167 . 2 |- ( ( A e. RR+ /\ B e. RR+ ) -> 0 < inf ( { A , B } , RR* , < ) )
3224, 31elrpd 10044 1 |- ( ( A e. RR+ /\ B e. RR+ ) -> inf ( { A , B } , RR* , < ) e. RR+ )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2205   {cpr 3695   class class class wbr 4114   supcsup 7286  infcinf 7287   RRcr 8142   0cc0 8143   RR*cxr 8323   < clt 8324   -ucneg 8461   RR+crp 10004   -ecxne 10121
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 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261  ax-arch 8262  ax-caucvg 8263
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 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-isom 5366  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-sup 7288  df-inf 7289  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-n0 9514  df-z 9595  df-uz 9872  df-rp 10005  df-xneg 10124  df-seqfrec 10834  df-exp 10925  df-cj 11552  df-re 11553  df-im 11554  df-rsqrt 11708  df-abs 11709
This theorem is referenced by:  blin2  15423  xmettx  15501
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