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Theorem xrminrecl 11821
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 10053 . . . . . . . 8 |- ( A e. RR -> -e A = -u A )
21adantr 276 . . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> -e A = -u A )
3 rexneg 10053 . . . . . . . 8 |- ( B e. RR -> -e B = -u B )
43adantl 277 . . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> -e B = -u B )
52, 4preq12d 3752 . . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> { -e A , -e B } = { -u A , -u B } )
65supeq1d 7175 . . . . 5 |- ( ( A e. RR /\ B e. RR ) -> sup ( { -e A , -e B } , RR* , < ) = sup ( { -u A , -u B } , RR* , < ) )
7 renegcl 8428 . . . . . 6 |- ( A e. RR -> -u A e. RR )
8 renegcl 8428 . . . . . 6 |- ( B e. RR -> -u B e. RR )
9 xrmaxrecl 11803 . . . . . 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 2262 . . . 4 |- ( ( A e. RR /\ B e. RR ) -> sup ( { -e A , -e B } , RR* , < ) = sup ( { -u A , -u B } , RR , < ) )
12 xnegeq 10050 . . . 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 11758 . . . . 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 10053 . . . 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 2262 . 2 |- ( ( A e. RR /\ B e. RR ) -> -e sup ( { -e A , -e B } , RR* , < ) = -u sup ( { -u A , -u B } , RR , < ) )
19 rexr 8213 . . 3 |- ( A e. RR -> A e. RR* )
20 rexr 8213 . . 3 |- ( B e. RR -> B e. RR* )
21 xrminmax 11813 . . 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 11778 . 2 |- ( ( A e. RR /\ B e. RR ) -> inf ( { A , B } , RR , < ) = -u sup ( { -u A , -u B } , RR , < ) )
2418, 22, 233eqtr4d 2272 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 1395   e. wcel 2200   {cpr 3668   supcsup 7170  infcinf 7171   RRcr 8019   RR*cxr 8201   < clt 8202   -ucneg 8339   -ecxne 9992
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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4200  ax-sep 4203  ax-nul 4211  ax-pow 4260  ax-pr 4295  ax-un 4526  ax-setind 4631  ax-iinf 4682  ax-cnex 8111  ax-resscn 8112  ax-1cn 8113  ax-1re 8114  ax-icn 8115  ax-addcl 8116  ax-addrcl 8117  ax-mulcl 8118  ax-mulrcl 8119  ax-addcom 8120  ax-mulcom 8121  ax-addass 8122  ax-mulass 8123  ax-distr 8124  ax-i2m1 8125  ax-0lt1 8126  ax-1rid 8127  ax-0id 8128  ax-rnegex 8129  ax-precex 8130  ax-cnre 8131  ax-pre-ltirr 8132  ax-pre-ltwlin 8133  ax-pre-lttrn 8134  ax-pre-apti 8135  ax-pre-ltadd 8136  ax-pre-mulgt0 8137  ax-pre-mulext 8138  ax-arch 8139  ax-caucvg 8140
This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3890  df-int 3925  df-iun 3968  df-br 4085  df-opab 4147  df-mpt 4148  df-tr 4184  df-id 4386  df-po 4389  df-iso 4390  df-iord 4459  df-on 4461  df-ilim 4462  df-suc 4464  df-iom 4685  df-xp 4727  df-rel 4728  df-cnv 4729  df-co 4730  df-dm 4731  df-rn 4732  df-res 4733  df-ima 4734  df-iota 5282  df-fun 5324  df-fn 5325  df-f 5326  df-f1 5327  df-fo 5328  df-f1o 5329  df-fv 5330  df-isom 5331  df-riota 5964  df-ov 6014  df-oprab 6015  df-mpo 6016  df-1st 6296  df-2nd 6297  df-recs 6464  df-frec 6550  df-sup 7172  df-inf 7173  df-pnf 8204  df-mnf 8205  df-xr 8206  df-ltxr 8207  df-le 8208  df-sub 8340  df-neg 8341  df-reap 8743  df-ap 8750  df-div 8841  df-inn 9132  df-2 9190  df-3 9191  df-4 9192  df-n0 9391  df-z 9468  df-uz 9744  df-rp 9877  df-xneg 9995  df-seqfrec 10698  df-exp 10789  df-cj 11390  df-re 11391  df-im 11392  df-rsqrt 11546  df-abs 11547
This theorem is referenced by:  xrbdtri  11824  qtopbas  15233
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