Theorem xrltmininf 11656

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 11651	. . . 4 |- ( ( B e. RR* /\ C e. RR* ) -> inf ( { B , C } , RR* , < ) = -e sup ( { -e B , -e C } , RR* , < ) )
2	1	3adant1 1018	. . 3 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> inf ( { B , C } , RR* , < ) = -e sup ( { -e B , -e C } , RR* , < ) )
3	2	breq2d 4063	. 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 1001	. . . . 5 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> B e. RR* )
5	4	xnegcld 9997	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> -e B e. RR* )
6		simp3 1002	. . . . 5 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> C e. RR* )
7	6	xnegcld 9997	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> -e C e. RR* )
8		simp1 1000	. . . . 5 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> A e. RR* )
9	8	xnegcld 9997	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> -e A e. RR* )
10		xrmaxltsup 11644	. . . 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 1250	. . 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 11638	. . . . . . 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 9997	. . . . 5 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> -e sup ( { -e B , -e C } , RR* , < ) e. RR* )
15		xltneg 9978	. . . . 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 9975	. . . . . 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 4061	. . . 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 9978	. . . . 5 |- ( ( A e. RR* /\ B e. RR* ) -> ( A < B <-> -e B < -e A ) )
22	21	3adant3 1020	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( A < B <-> -e B < -e A ) )
23		xltneg 9978	. . . . 5 |- ( ( A e. RR* /\ C e. RR* ) -> ( A < C <-> -e C < -e A ) )
24	23	3adant2 1019	. . . 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 981   = wceq 1373   e. wcel 2177   {cpr 3639   class class class wbr 4051   supcsup 7099  infcinf 7100   RR*cxr 8126   < clt 8127   -ecxne 9911

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4167  ax-sep 4170  ax-nul 4178  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-setind 4593  ax-iinf 4644  ax-cnex 8036  ax-resscn 8037  ax-1cn 8038  ax-1re 8039  ax-icn 8040  ax-addcl 8041  ax-addrcl 8042  ax-mulcl 8043  ax-mulrcl 8044  ax-addcom 8045  ax-mulcom 8046  ax-addass 8047  ax-mulass 8048  ax-distr 8049  ax-i2m1 8050  ax-0lt1 8051  ax-1rid 8052  ax-0id 8053  ax-rnegex 8054  ax-precex 8055  ax-cnre 8056  ax-pre-ltirr 8057  ax-pre-ltwlin 8058  ax-pre-lttrn 8059  ax-pre-apti 8060  ax-pre-ltadd 8061  ax-pre-mulgt0 8062  ax-pre-mulext 8063  ax-arch 8064  ax-caucvg 8065

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-if 3576  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-int 3892  df-iun 3935  df-br 4052  df-opab 4114  df-mpt 4115  df-tr 4151  df-id 4348  df-po 4351  df-iso 4352  df-iord 4421  df-on 4423  df-ilim 4424  df-suc 4426  df-iom 4647  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696  df-iota 5241  df-fun 5282  df-fn 5283  df-f 5284  df-f1 5285  df-fo 5286  df-f1o 5287  df-fv 5288  df-isom 5289  df-riota 5912  df-ov 5960  df-oprab 5961  df-mpo 5962  df-1st 6239  df-2nd 6240  df-recs 6404  df-frec 6490  df-sup 7101  df-inf 7102  df-pnf 8129  df-mnf 8130  df-xr 8131  df-ltxr 8132  df-le 8133  df-sub 8265  df-neg 8266  df-reap 8668  df-ap 8675  df-div 8766  df-inn 9057  df-2 9115  df-3 9116  df-4 9117  df-n0 9316  df-z 9393  df-uz 9669  df-rp 9796  df-xneg 9914  df-seqfrec 10615  df-exp 10706  df-cj 11228  df-re 11229  df-im 11230  df-rsqrt 11384  df-abs 11385

This theorem is referenced by:  xrminrpcl  11660  iooinsup  11663  blininf  14971  bdxmet  15048  bdmopn  15051