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Theorem xrminrpcl 11215
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 9597 . . . 4 |- ( A e. RR+ -> A e. RR* )
2 rpxr 9597 . . . 4 |- ( B e. RR+ -> B e. RR* )
3 xrminmax 11206 . . . 4 |- ( ( A e. RR* /\ B e. RR* ) -> inf ( { A , B } , RR* , < ) = -e sup ( { -e A , -e B } , RR* , < ) )
41, 2, 3syl2an 287 . . 3 |- ( ( A e. RR+ /\ B e. RR+ ) -> inf ( { A , B } , RR* , < ) = -e sup ( { -e A , -e B } , RR* , < ) )
5 rpre 9596 . . . . . . . 8 |- ( A e. RR+ -> A e. RR )
6 rexneg 9766 . . . . . . . . 9 |- ( A e. RR -> -e A = -u A )
7 renegcl 8159 . . . . . . . . 9 |- ( A e. RR -> -u A e. RR )
86, 7eqeltrd 2243 . . . . . . . 8 |- ( A e. RR -> -e A e. RR )
95, 8syl 14 . . . . . . 7 |- ( A e. RR+ -> -e A e. RR )
10 rpre 9596 . . . . . . . 8 |- ( B e. RR+ -> B e. RR )
11 rexneg 9766 . . . . . . . . 9 |- ( B e. RR -> -e B = -u B )
12 renegcl 8159 . . . . . . . . 9 |- ( B e. RR -> -u B e. RR )
1311, 12eqeltrd 2243 . . . . . . . 8 |- ( B e. RR -> -e B e. RR )
1410, 13syl 14 . . . . . . 7 |- ( B e. RR+ -> -e B e. RR )
15 xrmaxrecl 11196 . . . . . . 7 |- ( ( -e A e. RR /\ -e B e. RR ) -> sup ( { -e A , -e B } , RR* , < ) = sup ( { -e A , -e B } , RR , < ) )
169, 14, 15syl2an 287 . . . . . 6 |- ( ( A e. RR+ /\ B e. RR+ ) -> sup ( { -e A , -e B } , RR* , < ) = sup ( { -e A , -e B } , RR , < ) )
17 maxcl 11152 . . . . . . 7 |- ( ( -e A e. RR /\ -e B e. RR ) -> sup ( { -e A , -e B } , RR , < ) e. RR )
189, 14, 17syl2an 287 . . . . . 6 |- ( ( A e. RR+ /\ B e. RR+ ) -> sup ( { -e A , -e B } , RR , < ) e. RR )
1916, 18eqeltrd 2243 . . . . 5 |- ( ( A e. RR+ /\ B e. RR+ ) -> sup ( { -e A , -e B } , RR* , < ) e. RR )
20 rexneg 9766 . . . . 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 8278 . . . 4 |- ( ( A e. RR+ /\ B e. RR+ ) -> -u sup ( { -e A , -e B } , RR* , < ) e. RR )
2321, 22eqeltrd 2243 . . 3 |- ( ( A e. RR+ /\ B e. RR+ ) -> -e sup ( { -e A , -e B } , RR* , < ) e. RR )
244, 23eqeltrd 2243 . 2 |- ( ( A e. RR+ /\ B e. RR+ ) -> inf ( { A , B } , RR* , < ) e. RR )
25 rpgt0 9601 . . . 4 |- ( A e. RR+ -> 0 < A )
26 rpgt0 9601 . . . 4 |- ( B e. RR+ -> 0 < B )
2725, 26anim12i 336 . . 3 |- ( ( A e. RR+ /\ B e. RR+ ) -> ( 0 < A /\ 0 < B ) )
28 0xr 7945 . . . 4 |- 0 e. RR*
29 xrltmininf 11211 . . . 4 |- ( ( 0 e. RR* /\ A e. RR* /\ B e. RR* ) -> ( 0 < inf ( { A , B } , RR* , < ) <-> ( 0 < A /\ 0 < B ) ) )
3028, 1, 2, 29mp3an3an 1333 . . 3 |- ( ( A e. RR+ /\ B e. RR+ ) -> ( 0 < inf ( { A , B } , RR* , < ) <-> ( 0 < A /\ 0 < B ) ) )
3127, 30mpbird 166 . 2 |- ( ( A e. RR+ /\ B e. RR+ ) -> 0 < inf ( { A , B } , RR* , < ) )
3224, 31elrpd 9629 1 |- ( ( A e. RR+ /\ B e. RR+ ) -> inf ( { A , B } , RR* , < ) e. RR+ )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1343   e. wcel 2136   {cpr 3577   class class class wbr 3982   supcsup 6947  infcinf 6948   RRcr 7752   0cc0 7753   RR*cxr 7932   < clt 7933   -ucneg 8070   RR+crp 9589   -ecxne 9705
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871  ax-arch 7872  ax-caucvg 7873
This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-ilim 4347  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-isom 5197  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-frec 6359  df-sup 6949  df-inf 6950  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-inn 8858  df-2 8916  df-3 8917  df-4 8918  df-n0 9115  df-z 9192  df-uz 9467  df-rp 9590  df-xneg 9708  df-seqfrec 10381  df-exp 10455  df-cj 10784  df-re 10785  df-im 10786  df-rsqrt 10940  df-abs 10941
This theorem is referenced by:  blin2  13072  xmettx  13150
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