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Theorem xrminrecl 11236
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 9787 . . . . . . . 8 |- ( A e. RR -> -e A = -u A )
21adantr 274 . . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> -e A = -u A )
3 rexneg 9787 . . . . . . . 8 |- ( B e. RR -> -e B = -u B )
43adantl 275 . . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> -e B = -u B )
52, 4preq12d 3668 . . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> { -e A , -e B } = { -u A , -u B } )
65supeq1d 6964 . . . . 5 |- ( ( A e. RR /\ B e. RR ) -> sup ( { -e A , -e B } , RR* , < ) = sup ( { -u A , -u B } , RR* , < ) )
7 renegcl 8180 . . . . . 6 |- ( A e. RR -> -u A e. RR )
8 renegcl 8180 . . . . . 6 |- ( B e. RR -> -u B e. RR )
9 xrmaxrecl 11218 . . . . . 6 |- ( ( -u A e. RR /\ -u B e. RR ) -> sup ( { -u A , -u B } , RR* , < ) = sup ( { -u A , -u B } , RR , < ) )
107, 8, 9syl2an 287 . . . . 5 |- ( ( A e. RR /\ B e. RR ) -> sup ( { -u A , -u B } , RR* , < ) = sup ( { -u A , -u B } , RR , < ) )
116, 10eqtrd 2203 . . . 4 |- ( ( A e. RR /\ B e. RR ) -> sup ( { -e A , -e B } , RR* , < ) = sup ( { -u A , -u B } , RR , < ) )
12 xnegeq 9784 . . . 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 11174 . . . . 5 |- ( ( -u A e. RR /\ -u B e. RR ) -> sup ( { -u A , -u B } , RR , < ) e. RR )
157, 8, 14syl2an 287 . . . 4 |- ( ( A e. RR /\ B e. RR ) -> sup ( { -u A , -u B } , RR , < ) e. RR )
16 rexneg 9787 . . . 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 2203 . 2 |- ( ( A e. RR /\ B e. RR ) -> -e sup ( { -e A , -e B } , RR* , < ) = -u sup ( { -u A , -u B } , RR , < ) )
19 rexr 7965 . . 3 |- ( A e. RR -> A e. RR* )
20 rexr 7965 . . 3 |- ( B e. RR -> B e. RR* )
21 xrminmax 11228 . . 3 |- ( ( A e. RR* /\ B e. RR* ) -> inf ( { A , B } , RR* , < ) = -e sup ( { -e A , -e B } , RR* , < ) )
2219, 20, 21syl2an 287 . 2 |- ( ( A e. RR /\ B e. RR ) -> inf ( { A , B } , RR* , < ) = -e sup ( { -e A , -e B } , RR* , < ) )
23 minmax 11193 . 2 |- ( ( A e. RR /\ B e. RR ) -> inf ( { A , B } , RR , < ) = -u sup ( { -u A , -u B } , RR , < ) )
2418, 22, 233eqtr4d 2213 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 103   = wceq 1348   e. wcel 2141   {cpr 3584   supcsup 6959  infcinf 6960   RRcr 7773   RR*cxr 7953   < clt 7954   -ucneg 8091   -ecxne 9726
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893  ax-caucvg 7894
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-isom 5207  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-frec 6370  df-sup 6961  df-inf 6962  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-n0 9136  df-z 9213  df-uz 9488  df-rp 9611  df-xneg 9729  df-seqfrec 10402  df-exp 10476  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963
This theorem is referenced by:  xrbdtri  11239  qtopbas  13316
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