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Theorem xrminrecl 11416
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 9896 . . . . . . . 8 |- ( A e. RR -> -e A = -u A )
21adantr 276 . . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> -e A = -u A )
3 rexneg 9896 . . . . . . . 8 |- ( B e. RR -> -e B = -u B )
43adantl 277 . . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> -e B = -u B )
52, 4preq12d 3703 . . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> { -e A , -e B } = { -u A , -u B } )
65supeq1d 7046 . . . . 5 |- ( ( A e. RR /\ B e. RR ) -> sup ( { -e A , -e B } , RR* , < ) = sup ( { -u A , -u B } , RR* , < ) )
7 renegcl 8280 . . . . . 6 |- ( A e. RR -> -u A e. RR )
8 renegcl 8280 . . . . . 6 |- ( B e. RR -> -u B e. RR )
9 xrmaxrecl 11398 . . . . . 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 2226 . . . 4 |- ( ( A e. RR /\ B e. RR ) -> sup ( { -e A , -e B } , RR* , < ) = sup ( { -u A , -u B } , RR , < ) )
12 xnegeq 9893 . . . 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 11354 . . . . 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 9896 . . . 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 2226 . 2 |- ( ( A e. RR /\ B e. RR ) -> -e sup ( { -e A , -e B } , RR* , < ) = -u sup ( { -u A , -u B } , RR , < ) )
19 rexr 8065 . . 3 |- ( A e. RR -> A e. RR* )
20 rexr 8065 . . 3 |- ( B e. RR -> B e. RR* )
21 xrminmax 11408 . . 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 11373 . 2 |- ( ( A e. RR /\ B e. RR ) -> inf ( { A , B } , RR , < ) = -u sup ( { -u A , -u B } , RR , < ) )
2418, 22, 233eqtr4d 2236 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 1364   e. wcel 2164   {cpr 3619   supcsup 7041  infcinf 7042   RRcr 7871   RR*cxr 8053   < clt 8054   -ucneg 8191   -ecxne 9835
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-mulrcl 7971  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-precex 7982  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988  ax-pre-mulgt0 7989  ax-pre-mulext 7990  ax-arch 7991  ax-caucvg 7992
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-isom 5263  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-frec 6444  df-sup 7043  df-inf 7044  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-reap 8594  df-ap 8601  df-div 8692  df-inn 8983  df-2 9041  df-3 9042  df-4 9043  df-n0 9241  df-z 9318  df-uz 9593  df-rp 9720  df-xneg 9838  df-seqfrec 10519  df-exp 10610  df-cj 10986  df-re 10987  df-im 10988  df-rsqrt 11142  df-abs 11143
This theorem is referenced by:  xrbdtri  11419  qtopbas  14690
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