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Theorem xrminrecl 11252
Description: The minimum of two real numbers is the same when taken as extended reals or as reals. (Contributed by Jim Kingdon, 18-May-2023.)
Assertion
Ref Expression
xrminrecl  |-  ( ( A  e.  RR  /\  B  e.  RR )  -> inf ( { A ,  B } ,  RR* ,  <  )  = inf ( { A ,  B } ,  RR ,  <  ) )

Proof of Theorem xrminrecl
StepHypRef Expression
1 rexneg 9804 . . . . . . . 8  |-  ( A  e.  RR  ->  -e
A  =  -u A
)
21adantr 276 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  -> 
-e A  = 
-u A )
3 rexneg 9804 . . . . . . . 8  |-  ( B  e.  RR  ->  -e
B  =  -u B
)
43adantl 277 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  -> 
-e B  = 
-u B )
52, 4preq12d 3676 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  {  -e A ,  -e B }  =  { -u A ,  -u B }
)
65supeq1d 6979 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  sup ( {  -e
A ,  -e
B } ,  RR* ,  <  )  =  sup ( { -u A ,  -u B } ,  RR* ,  <  ) )
7 renegcl 8195 . . . . . 6  |-  ( A  e.  RR  ->  -u A  e.  RR )
8 renegcl 8195 . . . . . 6  |-  ( B  e.  RR  ->  -u B  e.  RR )
9 xrmaxrecl 11234 . . . . . 6  |-  ( (
-u A  e.  RR  /\  -u B  e.  RR )  ->  sup ( { -u A ,  -u B } ,  RR* ,  <  )  =  sup ( { -u A ,  -u B } ,  RR ,  <  )
)
107, 8, 9syl2an 289 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  sup ( { -u A ,  -u B } ,  RR* ,  <  )  =  sup ( { -u A ,  -u B } ,  RR ,  <  )
)
116, 10eqtrd 2210 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  sup ( {  -e
A ,  -e
B } ,  RR* ,  <  )  =  sup ( { -u A ,  -u B } ,  RR ,  <  ) )
12 xnegeq 9801 . . . 4  |-  ( sup ( {  -e
A ,  -e
B } ,  RR* ,  <  )  =  sup ( { -u A ,  -u B } ,  RR ,  <  )  ->  -e sup ( {  -e
A ,  -e
B } ,  RR* ,  <  )  =  -e sup ( { -u A ,  -u B } ,  RR ,  <  )
)
1311, 12syl 14 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  -> 
-e sup ( {  -e A ,  -e B } ,  RR* ,  <  )  = 
-e sup ( { -u A ,  -u B } ,  RR ,  <  ) )
14 maxcl 11190 . . . . 5  |-  ( (
-u A  e.  RR  /\  -u B  e.  RR )  ->  sup ( { -u A ,  -u B } ,  RR ,  <  )  e.  RR )
157, 8, 14syl2an 289 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  sup ( { -u A ,  -u B } ,  RR ,  <  )  e.  RR )
16 rexneg 9804 . . . 4  |-  ( sup ( { -u A ,  -u B } ,  RR ,  <  )  e.  RR  ->  -e sup ( { -u A ,  -u B } ,  RR ,  <  )  = 
-u sup ( { -u A ,  -u B } ,  RR ,  <  )
)
1715, 16syl 14 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  -> 
-e sup ( { -u A ,  -u B } ,  RR ,  <  )  =  -u sup ( { -u A ,  -u B } ,  RR ,  <  ) )
1813, 17eqtrd 2210 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  -> 
-e sup ( {  -e A ,  -e B } ,  RR* ,  <  )  = 
-u sup ( { -u A ,  -u B } ,  RR ,  <  )
)
19 rexr 7980 . . 3  |-  ( A  e.  RR  ->  A  e.  RR* )
20 rexr 7980 . . 3  |-  ( B  e.  RR  ->  B  e.  RR* )
21 xrminmax 11244 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  -> inf ( { A ,  B } ,  RR* ,  <  )  =  -e sup ( {  -e A ,  -e B } ,  RR* ,  <  ) )
2219, 20, 21syl2an 289 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  -> inf ( { A ,  B } ,  RR* ,  <  )  =  -e sup ( {  -e
A ,  -e
B } ,  RR* ,  <  ) )
23 minmax 11209 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  -> inf ( { A ,  B } ,  RR ,  <  )  =  -u sup ( { -u A ,  -u B } ,  RR ,  <  ) )
2418, 22, 233eqtr4d 2220 1  |-  ( ( A  e.  RR  /\  B  e.  RR )  -> inf ( { A ,  B } ,  RR* ,  <  )  = inf ( { A ,  B } ,  RR ,  <  ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1353    e. wcel 2148   {cpr 3592   supcsup 6974  infcinf 6975   RRcr 7788   RR*cxr 7968    < clt 7969   -ucneg 8106    -ecxne 9743
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-setind 4532  ax-iinf 4583  ax-cnex 7880  ax-resscn 7881  ax-1cn 7882  ax-1re 7883  ax-icn 7884  ax-addcl 7885  ax-addrcl 7886  ax-mulcl 7887  ax-mulrcl 7888  ax-addcom 7889  ax-mulcom 7890  ax-addass 7891  ax-mulass 7892  ax-distr 7893  ax-i2m1 7894  ax-0lt1 7895  ax-1rid 7896  ax-0id 7897  ax-rnegex 7898  ax-precex 7899  ax-cnre 7900  ax-pre-ltirr 7901  ax-pre-ltwlin 7902  ax-pre-lttrn 7903  ax-pre-apti 7904  ax-pre-ltadd 7905  ax-pre-mulgt0 7906  ax-pre-mulext 7907  ax-arch 7908  ax-caucvg 7909
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4289  df-po 4292  df-iso 4293  df-iord 4362  df-on 4364  df-ilim 4365  df-suc 4367  df-iom 4586  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-f1 5216  df-fo 5217  df-f1o 5218  df-fv 5219  df-isom 5220  df-riota 5824  df-ov 5871  df-oprab 5872  df-mpo 5873  df-1st 6134  df-2nd 6135  df-recs 6299  df-frec 6385  df-sup 6976  df-inf 6977  df-pnf 7971  df-mnf 7972  df-xr 7973  df-ltxr 7974  df-le 7975  df-sub 8107  df-neg 8108  df-reap 8509  df-ap 8516  df-div 8606  df-inn 8896  df-2 8954  df-3 8955  df-4 8956  df-n0 9153  df-z 9230  df-uz 9505  df-rp 9628  df-xneg 9746  df-seqfrec 10419  df-exp 10493  df-cj 10822  df-re 10823  df-im 10824  df-rsqrt 10978  df-abs 10979
This theorem is referenced by:  xrbdtri  11255  qtopbas  13655
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