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Theorem xrminrpcl 11700
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 9818 . . . 4 |- ( A e. RR+ -> A e. RR* )
2 rpxr 9818 . . . 4 |- ( B e. RR+ -> B e. RR* )
3 xrminmax 11691 . . . 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 9817 . . . . . . . 8 |- ( A e. RR+ -> A e. RR )
6 rexneg 9987 . . . . . . . . 9 |- ( A e. RR -> -e A = -u A )
7 renegcl 8368 . . . . . . . . 9 |- ( A e. RR -> -u A e. RR )
86, 7eqeltrd 2284 . . . . . . . 8 |- ( A e. RR -> -e A e. RR )
95, 8syl 14 . . . . . . 7 |- ( A e. RR+ -> -e A e. RR )
10 rpre 9817 . . . . . . . 8 |- ( B e. RR+ -> B e. RR )
11 rexneg 9987 . . . . . . . . 9 |- ( B e. RR -> -e B = -u B )
12 renegcl 8368 . . . . . . . . 9 |- ( B e. RR -> -u B e. RR )
1311, 12eqeltrd 2284 . . . . . . . 8 |- ( B e. RR -> -e B e. RR )
1410, 13syl 14 . . . . . . 7 |- ( B e. RR+ -> -e B e. RR )
15 xrmaxrecl 11681 . . . . . . 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 11636 . . . . . . 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 2284 . . . . 5 |- ( ( A e. RR+ /\ B e. RR+ ) -> sup ( { -e A , -e B } , RR* , < ) e. RR )
20 rexneg 9987 . . . . 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 8487 . . . 4 |- ( ( A e. RR+ /\ B e. RR+ ) -> -u sup ( { -e A , -e B } , RR* , < ) e. RR )
2321, 22eqeltrd 2284 . . 3 |- ( ( A e. RR+ /\ B e. RR+ ) -> -e sup ( { -e A , -e B } , RR* , < ) e. RR )
244, 23eqeltrd 2284 . 2 |- ( ( A e. RR+ /\ B e. RR+ ) -> inf ( { A , B } , RR* , < ) e. RR )
25 rpgt0 9822 . . . 4 |- ( A e. RR+ -> 0 < A )
26 rpgt0 9822 . . . 4 |- ( B e. RR+ -> 0 < B )
2725, 26anim12i 338 . . 3 |- ( ( A e. RR+ /\ B e. RR+ ) -> ( 0 < A /\ 0 < B ) )
28 0xr 8154 . . . 4 |- 0 e. RR*
29 xrltmininf 11696 . . . 4 |- ( ( 0 e. RR* /\ A e. RR* /\ B e. RR* ) -> ( 0 < inf ( { A , B } , RR* , < ) <-> ( 0 < A /\ 0 < B ) ) )
3028, 1, 2, 29mp3an3an 1356 . . 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 9850 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 1373   e. wcel 2178   {cpr 3644   class class class wbr 4059   supcsup 7110  infcinf 7111   RRcr 7959   0cc0 7960   RR*cxr 8141   < clt 8142   -ucneg 8279   RR+crp 9810   -ecxne 9926
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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-mulrcl 8059  ax-addcom 8060  ax-mulcom 8061  ax-addass 8062  ax-mulass 8063  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-1rid 8067  ax-0id 8068  ax-rnegex 8069  ax-precex 8070  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-apti 8075  ax-pre-ltadd 8076  ax-pre-mulgt0 8077  ax-pre-mulext 8078  ax-arch 8079  ax-caucvg 8080
This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-if 3580  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-id 4358  df-po 4361  df-iso 4362  df-iord 4431  df-on 4433  df-ilim 4434  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-isom 5299  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-recs 6414  df-frec 6500  df-sup 7112  df-inf 7113  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-reap 8683  df-ap 8690  df-div 8781  df-inn 9072  df-2 9130  df-3 9131  df-4 9132  df-n0 9331  df-z 9408  df-uz 9684  df-rp 9811  df-xneg 9929  df-seqfrec 10630  df-exp 10721  df-cj 11268  df-re 11269  df-im 11270  df-rsqrt 11424  df-abs 11425
This theorem is referenced by:  blin2  15019  xmettx  15097
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