Theorem xrltmininf 11776

Description: Two ways of saying an extended real is less than the minimum of two others. (Contributed by NM, 7-Feb-2007.) (Revised by Jim Kingdon, 3-May-2023.)

Assertion

Ref	Expression
xrltmininf	|- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( A < inf ( { B , C } , RR* , < ) <-> ( A < B /\ A < C ) ) )

Proof of Theorem xrltmininf

Step	Hyp	Ref	Expression
1		xrminmax 11771	. . . 4 |- ( ( B e. RR* /\ C e. RR* ) -> inf ( { B , C } , RR* , < ) = -e sup ( { -e B , -e C } , RR* , < ) )
2	1	3adant1 1039	. . 3 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> inf ( { B , C } , RR* , < ) = -e sup ( { -e B , -e C } , RR* , < ) )
3	2	breq2d 4094	. 2 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( A < inf ( { B , C } , RR* , < ) <-> A < -e sup ( { -e B , -e C } , RR* , < ) ) )
4		simp2 1022	. . . . 5 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> B e. RR* )
5	4	xnegcld 10047	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> -e B e. RR* )
6		simp3 1023	. . . . 5 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> C e. RR* )
7	6	xnegcld 10047	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> -e C e. RR* )
8		simp1 1021	. . . . 5 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> A e. RR* )
9	8	xnegcld 10047	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> -e A e. RR* )
10		xrmaxltsup 11764	. . . 4 |- ( ( -e B e. RR* /\ -e C e. RR* /\ -e A e. RR* ) -> ( sup ( { -e B , -e C } , RR* , < ) < -e A <-> ( -e B < -e A /\ -e C < -e A ) ) )
11	5, 7, 9, 10	syl3anc 1271	. . 3 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( sup ( { -e B , -e C } , RR* , < ) < -e A <-> ( -e B < -e A /\ -e C < -e A ) ) )
12		xrmaxcl 11758	. . . . . . 7 |- ( ( -e B e. RR* /\ -e C e. RR* ) -> sup ( { -e B , -e C } , RR* , < ) e. RR* )
13	5, 7, 12	syl2anc 411	. . . . . 6 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> sup ( { -e B , -e C } , RR* , < ) e. RR* )
14	13	xnegcld 10047	. . . . 5 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> -e sup ( { -e B , -e C } , RR* , < ) e. RR* )
15		xltneg 10028	. . . . 5 |- ( ( A e. RR* /\ -e sup ( { -e B , -e C } , RR* , < ) e. RR* ) -> ( A < -e sup ( { -e B , -e C } , RR* , < ) <-> -e -e sup ( { -e B , -e C } , RR* , < ) < -e A ) )
16	8, 14, 15	syl2anc 411	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( A < -e sup ( { -e B , -e C } , RR* , < ) <-> -e -e sup ( { -e B , -e C } , RR* , < ) < -e A ) )
17		xnegneg 10025	. . . . . 6 |- ( sup ( { -e B , -e C } , RR* , < ) e. RR* -> -e -e sup ( { -e B , -e C } , RR* , < ) = sup ( { -e B , -e C } , RR* , < ) )
18	13, 17	syl 14	. . . . 5 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> -e -e sup ( { -e B , -e C } , RR* , < ) = sup ( { -e B , -e C } , RR* , < ) )
19	18	breq1d 4092	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( -e -e sup ( { -e B , -e C } , RR* , < ) < -e A <-> sup ( { -e B , -e C } , RR* , < ) < -e A ) )
20	16, 19	bitrd 188	. . 3 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( A < -e sup ( { -e B , -e C } , RR* , < ) <-> sup ( { -e B , -e C } , RR* , < ) < -e A ) )
21		xltneg 10028	. . . . 5 |- ( ( A e. RR* /\ B e. RR* ) -> ( A < B <-> -e B < -e A ) )
22	21	3adant3 1041	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( A < B <-> -e B < -e A ) )
23		xltneg 10028	. . . . 5 |- ( ( A e. RR* /\ C e. RR* ) -> ( A < C <-> -e C < -e A ) )
24	23	3adant2 1040	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( A < C <-> -e C < -e A ) )
25	22, 24	anbi12d 473	. . 3 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( ( A < B /\ A < C ) <-> ( -e B < -e A /\ -e C < -e A ) ) )
26	11, 20, 25	3bitr4d 220	. 2 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( A < -e sup ( { -e B , -e C } , RR* , < ) <-> ( A < B /\ A < C ) ) )
27	3, 26	bitrd 188	1 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( A < inf ( { B , C } , RR* , < ) <-> ( A < B /\ A < C ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1002   = wceq 1395   e. wcel 2200   {cpr 3667   class class class wbr 4082   supcsup 7145  infcinf 7146   RR*cxr 8176   < clt 8177   -ecxne 9961

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 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-mulrcl 8094  ax-addcom 8095  ax-mulcom 8096  ax-addass 8097  ax-mulass 8098  ax-distr 8099  ax-i2m1 8100  ax-0lt1 8101  ax-1rid 8102  ax-0id 8103  ax-rnegex 8104  ax-precex 8105  ax-cnre 8106  ax-pre-ltirr 8107  ax-pre-ltwlin 8108  ax-pre-lttrn 8109  ax-pre-apti 8110  ax-pre-ltadd 8111  ax-pre-mulgt0 8112  ax-pre-mulext 8113  ax-arch 8114  ax-caucvg 8115

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 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4383  df-po 4386  df-iso 4387  df-iord 4456  df-on 4458  df-ilim 4459  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-isom 5326  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-recs 6449  df-frec 6535  df-sup 7147  df-inf 7148  df-pnf 8179  df-mnf 8180  df-xr 8181  df-ltxr 8182  df-le 8183  df-sub 8315  df-neg 8316  df-reap 8718  df-ap 8725  df-div 8816  df-inn 9107  df-2 9165  df-3 9166  df-4 9167  df-n0 9366  df-z 9443  df-uz 9719  df-rp 9846  df-xneg 9964  df-seqfrec 10665  df-exp 10756  df-cj 11348  df-re 11349  df-im 11350  df-rsqrt 11504  df-abs 11505

This theorem is referenced by:  xrminrpcl  11780  iooinsup  11783  blininf  15092  bdxmet  15169  bdmopn  15172