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Theorem xrminrpcl 12040
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 10062 . . . 4 |- ( A e. RR+ -> A e. RR* )
2 rpxr 10062 . . . 4 |- ( B e. RR+ -> B e. RR* )
3 xrminmax 12031 . . . 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 10061 . . . . . . . 8 |- ( A e. RR+ -> A e. RR )
6 rexneg 10232 . . . . . . . . 9 |- ( A e. RR -> -e A = -u A )
7 renegcl 8587 . . . . . . . . 9 |- ( A e. RR -> -u A e. RR )
86, 7eqeltrd 2315 . . . . . . . 8 |- ( A e. RR -> -e A e. RR )
95, 8syl 14 . . . . . . 7 |- ( A e. RR+ -> -e A e. RR )
10 rpre 10061 . . . . . . . 8 |- ( B e. RR+ -> B e. RR )
11 rexneg 10232 . . . . . . . . 9 |- ( B e. RR -> -e B = -u B )
12 renegcl 8587 . . . . . . . . 9 |- ( B e. RR -> -u B e. RR )
1311, 12eqeltrd 2315 . . . . . . . 8 |- ( B e. RR -> -e B e. RR )
1410, 13syl 14 . . . . . . 7 |- ( B e. RR+ -> -e B e. RR )
15 xrmaxrecl 12021 . . . . . . 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 11976 . . . . . . 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 2315 . . . . 5 |- ( ( A e. RR+ /\ B e. RR+ ) -> sup ( { -e A , -e B } , RR* , < ) e. RR )
20 rexneg 10232 . . . . 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 8707 . . . 4 |- ( ( A e. RR+ /\ B e. RR+ ) -> -u sup ( { -e A , -e B } , RR* , < ) e. RR )
2321, 22eqeltrd 2315 . . 3 |- ( ( A e. RR+ /\ B e. RR+ ) -> -e sup ( { -e A , -e B } , RR* , < ) e. RR )
244, 23eqeltrd 2315 . 2 |- ( ( A e. RR+ /\ B e. RR+ ) -> inf ( { A , B } , RR* , < ) e. RR )
25 rpgt0 10066 . . . 4 |- ( A e. RR+ -> 0 < A )
26 rpgt0 10066 . . . 4 |- ( B e. RR+ -> 0 < B )
2725, 26anim12i 338 . . 3 |- ( ( A e. RR+ /\ B e. RR+ ) -> ( 0 < A /\ 0 < B ) )
28 0xr 8372 . . . 4 |- 0 e. RR*
29 xrltmininf 12036 . . . 4 |- ( ( 0 e. RR* /\ A e. RR* /\ B e. RR* ) -> ( 0 < inf ( { A , B } , RR* , < ) <-> ( 0 < A /\ 0 < B ) ) )
3028, 1, 2, 29mp3an3an 1384 . . 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 10094 1 |- ( ( A e. RR+ /\ B e. RR+ ) -> inf ( { A , B } , RR* , < ) e. RR+ )
Colors of variables:    wff set class
This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   {cpr 3710   class class class wbr 4130   supcsup 7322  infcinf 7323   RRcr 8178   0cc0 8179   RR*cxr 8359   < clt 8360   -ucneg 8498   RR+crp 10054   -ecxne 10171
This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298  ax-caucvg 8299
This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-isom 5386  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-sup 7324  df-inf 7325  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-div 9003  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-n0 9564  df-z 9645  df-uz 9922  df-rp 10055  df-xneg 10174  df-seqfrec 10885  df-exp 10976  df-cj 11607  df-re 11608  df-im 11609  df-rsqrt 11764  df-abs 11765
This theorem is used by:  blin2  15533  xmettx  15611
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