Theorem xrltmininf 11280

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 11275	. . . 4 |- ( ( B e. RR* /\ C e. RR* ) -> inf ( { B , C } , RR* , < ) = -e sup ( { -e B , -e C } , RR* , < ) )
2	1	3adant1 1015	. . 3 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> inf ( { B , C } , RR* , < ) = -e sup ( { -e B , -e C } , RR* , < ) )
3	2	breq2d 4017	. 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 998	. . . . 5 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> B e. RR* )
5	4	xnegcld 9857	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> -e B e. RR* )
6		simp3 999	. . . . 5 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> C e. RR* )
7	6	xnegcld 9857	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> -e C e. RR* )
8		simp1 997	. . . . 5 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> A e. RR* )
9	8	xnegcld 9857	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> -e A e. RR* )
10		xrmaxltsup 11268	. . . 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 1238	. . 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 11262	. . . . . . 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 9857	. . . . 5 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> -e sup ( { -e B , -e C } , RR* , < ) e. RR* )
15		xltneg 9838	. . . . 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 9835	. . . . . 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 4015	. . . 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 9838	. . . . 5 |- ( ( A e. RR* /\ B e. RR* ) -> ( A < B <-> -e B < -e A ) )
22	21	3adant3 1017	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( A < B <-> -e B < -e A ) )
23		xltneg 9838	. . . . 5 |- ( ( A e. RR* /\ C e. RR* ) -> ( A < C <-> -e C < -e A ) )
24	23	3adant2 1016	. . . 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 978   = wceq 1353   e. wcel 2148   {cpr 3595   class class class wbr 4005   supcsup 6983  infcinf 6984   RR*cxr 7993   < clt 7994   -ecxne 9771

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 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-mulrcl 7912  ax-addcom 7913  ax-mulcom 7914  ax-addass 7915  ax-mulass 7916  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-1rid 7920  ax-0id 7921  ax-rnegex 7922  ax-precex 7923  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-apti 7928  ax-pre-ltadd 7929  ax-pre-mulgt0 7930  ax-pre-mulext 7931  ax-arch 7932  ax-caucvg 7933

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 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-if 3537  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-ilim 4371  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-isom 5227  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-frec 6394  df-sup 6985  df-inf 6986  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-reap 8534  df-ap 8541  df-div 8632  df-inn 8922  df-2 8980  df-3 8981  df-4 8982  df-n0 9179  df-z 9256  df-uz 9531  df-rp 9656  df-xneg 9774  df-seqfrec 10448  df-exp 10522  df-cj 10853  df-re 10854  df-im 10855  df-rsqrt 11009  df-abs 11010

This theorem is referenced by:  xrminrpcl  11284  iooinsup  11287  blininf  14009  bdxmet  14086  bdmopn  14089