Theorem ltmininf 11256

Description: Two ways of saying a number is less than the minimum of two others. (Contributed by Jim Kingdon, 10-Feb-2022.)

Assertion

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

Proof of Theorem ltmininf

Step	Hyp	Ref	Expression
1		simp2 999	. . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> B e. RR )
2	1	renegcld 8350	. . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> -u B e. RR )
3		simp3 1000	. . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> C e. RR )
4	3	renegcld 8350	. . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> -u C e. RR )
5		simp1 998	. . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> A e. RR )
6	5	renegcld 8350	. . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> -u A e. RR )
7		maxltsup 11240	. . 3 |- ( ( -u B e. RR /\ -u C e. RR /\ -u A e. RR ) -> ( sup ( { -u B , -u C } , RR , < ) < -u A <-> ( -u B < -u A /\ -u C < -u A ) ) )
8	2, 4, 6, 7	syl3anc 1248	. 2 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( sup ( { -u B , -u C } , RR , < ) < -u A <-> ( -u B < -u A /\ -u C < -u A ) ) )
9		minmax 11251	. . . . 5 |- ( ( B e. RR /\ C e. RR ) -> inf ( { B , C } , RR , < ) = -u sup ( { -u B , -u C } , RR , < ) )
10	9	breq2d 4027	. . . 4 |- ( ( B e. RR /\ C e. RR ) -> ( A < inf ( { B , C } , RR , < ) <-> A < -u sup ( { -u B , -u C } , RR , < ) ) )
11	10	3adant1 1016	. . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < inf ( { B , C } , RR , < ) <-> A < -u sup ( { -u B , -u C } , RR , < ) ) )
12		maxcl 11232	. . . . 5 |- ( ( -u B e. RR /\ -u C e. RR ) -> sup ( { -u B , -u C } , RR , < ) e. RR )
13	2, 4, 12	syl2anc 411	. . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> sup ( { -u B , -u C } , RR , < ) e. RR )
14		ltnegcon2 8434	. . . 4 |- ( ( A e. RR /\ sup ( { -u B , -u C } , RR , < ) e. RR ) -> ( A < -u sup ( { -u B , -u C } , RR , < ) <-> sup ( { -u B , -u C } , RR , < ) < -u A ) )
15	5, 13, 14	syl2anc 411	. . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < -u sup ( { -u B , -u C } , RR , < ) <-> sup ( { -u B , -u C } , RR , < ) < -u A ) )
16	11, 15	bitrd 188	. 2 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < inf ( { B , C } , RR , < ) <-> sup ( { -u B , -u C } , RR , < ) < -u A ) )
17	5, 1	ltnegd 8493	. . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < B <-> -u B < -u A ) )
18	5, 3	ltnegd 8493	. . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < C <-> -u C < -u A ) )
19	17, 18	anbi12d 473	. 2 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A < B /\ A < C ) <-> ( -u B < -u A /\ -u C < -u A ) ) )
20	8, 16, 19	3bitr4d 220	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 979   e. wcel 2158   {cpr 3605   class class class wbr 4015   supcsup 6994  infcinf 6995   RRcr 7823   < clt 8005   -ucneg 8142

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599  ax-cnex 7915  ax-resscn 7916  ax-1cn 7917  ax-1re 7918  ax-icn 7919  ax-addcl 7920  ax-addrcl 7921  ax-mulcl 7922  ax-mulrcl 7923  ax-addcom 7924  ax-mulcom 7925  ax-addass 7926  ax-mulass 7927  ax-distr 7928  ax-i2m1 7929  ax-0lt1 7930  ax-1rid 7931  ax-0id 7932  ax-rnegex 7933  ax-precex 7934  ax-cnre 7935  ax-pre-ltirr 7936  ax-pre-ltwlin 7937  ax-pre-lttrn 7938  ax-pre-apti 7939  ax-pre-ltadd 7940  ax-pre-mulgt0 7941  ax-pre-mulext 7942  ax-arch 7943  ax-caucvg 7944

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-if 3547  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-id 4305  df-po 4308  df-iso 4309  df-iord 4378  df-on 4380  df-ilim 4381  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-isom 5237  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6154  df-2nd 6155  df-recs 6319  df-frec 6405  df-sup 6996  df-inf 6997  df-pnf 8007  df-mnf 8008  df-xr 8009  df-ltxr 8010  df-le 8011  df-sub 8143  df-neg 8144  df-reap 8545  df-ap 8552  df-div 8643  df-inn 8933  df-2 8991  df-3 8992  df-4 8993  df-n0 9190  df-z 9267  df-uz 9542  df-rp 9667  df-seqfrec 10459  df-exp 10533  df-cj 10864  df-re 10865  df-im 10866  df-rsqrt 11020  df-abs 11021

This theorem is referenced by:  rpmincl  11259  mul0inf  11262  reccn2ap  11334  addcncntoplem  14291  mulcncflem  14330  suplociccreex  14342  dveflem  14427