Theorem minclpr 11178

Description: The minimum of two real numbers is one of those numbers if and only if dichotomy ( A <_ B \/ B <_ A) holds. For example, this can be combined with zletric 9235 if one is dealing with integers, but real number dichotomy in general does not follow from our axioms. (Contributed by Jim Kingdon, 23-May-2023.)

Assertion

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

Proof of Theorem minclpr

Step	Hyp	Ref	Expression
1		renegcl 8159	. . . . 5 |- ( A e. RR -> -u A e. RR )
2		renegcl 8159	. . . . 5 |- ( B e. RR -> -u B e. RR )
3		maxcl 11152	. . . . 5 |- ( ( -u A e. RR /\ -u B e. RR ) -> sup ( { -u A , -u B } , RR , < ) e. RR )
4	1, 2, 3	syl2an 287	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> sup ( { -u A , -u B } , RR , < ) e. RR )
5		elprg 3596	. . . 4 |- ( sup ( { -u A , -u B } , RR , < ) e. RR -> ( sup ( { -u A , -u B } , RR , < ) e. { -u A , -u B } <-> ( sup ( { -u A , -u B } , RR , < ) = -u A \/ sup ( { -u A , -u B } , RR , < ) = -u B ) ) )
6	4, 5	syl 14	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( sup ( { -u A , -u B } , RR , < ) e. { -u A , -u B } <-> ( sup ( { -u A , -u B } , RR , < ) = -u A \/ sup ( { -u A , -u B } , RR , < ) = -u B ) ) )
7		maxclpr 11164	. . . 4 |- ( ( -u A e. RR /\ -u B e. RR ) -> ( sup ( { -u A , -u B } , RR , < ) e. { -u A , -u B } <-> ( -u A <_ -u B \/ -u B <_ -u A ) ) )
8	1, 2, 7	syl2an 287	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( sup ( { -u A , -u B } , RR , < ) e. { -u A , -u B } <-> ( -u A <_ -u B \/ -u B <_ -u A ) ) )
9	4	recnd 7927	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> sup ( { -u A , -u B } , RR , < ) e. CC )
10	1	adantr 274	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> -u A e. RR )
11	10	recnd 7927	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> -u A e. CC )
12	9, 11	neg11ad 8205	. . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( -u sup ( { -u A , -u B } , RR , < ) = -u -u A <-> sup ( { -u A , -u B } , RR , < ) = -u A ) )
13		minmax 11171	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> inf ( { A , B } , RR , < ) = -u sup ( { -u A , -u B } , RR , < ) )
14	13	eqcomd 2171	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> -u sup ( { -u A , -u B } , RR , < ) = inf ( { A , B } , RR , < ) )
15		recn 7886	. . . . . . . 8 |- ( A e. RR -> A e. CC )
16	15	adantr 274	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> A e. CC )
17	16	negnegd 8200	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> -u -u A = A )
18	14, 17	eqeq12d 2180	. . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( -u sup ( { -u A , -u B } , RR , < ) = -u -u A <-> inf ( { A , B } , RR , < ) = A ) )
19	12, 18	bitr3d 189	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( sup ( { -u A , -u B } , RR , < ) = -u A <-> inf ( { A , B } , RR , < ) = A ) )
20	2	adantl 275	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> -u B e. RR )
21	20	recnd 7927	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> -u B e. CC )
22	9, 21	neg11ad 8205	. . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( -u sup ( { -u A , -u B } , RR , < ) = -u -u B <-> sup ( { -u A , -u B } , RR , < ) = -u B ) )
23		recn 7886	. . . . . . . 8 |- ( B e. RR -> B e. CC )
24	23	adantl 275	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> B e. CC )
25	24	negnegd 8200	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> -u -u B = B )
26	14, 25	eqeq12d 2180	. . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( -u sup ( { -u A , -u B } , RR , < ) = -u -u B <-> inf ( { A , B } , RR , < ) = B ) )
27	22, 26	bitr3d 189	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( sup ( { -u A , -u B } , RR , < ) = -u B <-> inf ( { A , B } , RR , < ) = B ) )
28	19, 27	orbi12d 783	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( ( sup ( { -u A , -u B } , RR , < ) = -u A \/ sup ( { -u A , -u B } , RR , < ) = -u B ) <-> (inf ( { A , B } , RR , < ) = A \/ inf ( { A , B } , RR , < ) = B ) ) )
29	6, 8, 28	3bitr3rd 218	. 2 |- ( ( A e. RR /\ B e. RR ) -> ( (inf ( { A , B } , RR , < ) = A \/ inf ( { A , B } , RR , < ) = B ) <-> ( -u A <_ -u B \/ -u B <_ -u A ) ) )
30		mincl 11172	. . 3 |- ( ( A e. RR /\ B e. RR ) -> inf ( { A , B } , RR , < ) e. RR )
31		elprg 3596	. . 3 |- (inf ( { A , B } , RR , < ) e. RR -> (inf ( { A , B } , RR , < ) e. { A , B } <-> (inf ( { A , B } , RR , < ) = A \/ inf ( { A , B } , RR , < ) = B ) ) )
32	30, 31	syl 14	. 2 |- ( ( A e. RR /\ B e. RR ) -> (inf ( { A , B } , RR , < ) e. { A , B } <-> (inf ( { A , B } , RR , < ) = A \/ inf ( { A , B } , RR , < ) = B ) ) )
33		orcom 718	. . 3 |- ( ( B <_ A \/ A <_ B ) <-> ( A <_ B \/ B <_ A ) )
34		simpr 109	. . . . 5 |- ( ( A e. RR /\ B e. RR ) -> B e. RR )
35		simpl 108	. . . . 5 |- ( ( A e. RR /\ B e. RR ) -> A e. RR )
36	34, 35	lenegd 8422	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( B <_ A <-> -u A <_ -u B ) )
37		leneg 8363	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( A <_ B <-> -u B <_ -u A ) )
38	36, 37	orbi12d 783	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( ( B <_ A \/ A <_ B ) <-> ( -u A <_ -u B \/ -u B <_ -u A ) ) )
39	33, 38	bitr3id 193	. 2 |- ( ( A e. RR /\ B e. RR ) -> ( ( A <_ B \/ B <_ A ) <-> ( -u A <_ -u B \/ -u B <_ -u A ) ) )
40	29, 32, 39	3bitr4d 219	1 |- ( ( A e. RR /\ B e. RR ) -> (inf ( { A , B } , RR , < ) e. { A , B } <-> ( A <_ B \/ B <_ A ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 698   = wceq 1343   e. wcel 2136   {cpr 3577   class class class wbr 3982   supcsup 6947  infcinf 6948   CCcc 7751   RRcr 7752   < clt 7933   <_ cle 7934   -ucneg 8070

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871  ax-arch 7872  ax-caucvg 7873

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-ilim 4347  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-isom 5197  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-frec 6359  df-sup 6949  df-inf 6950  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-inn 8858  df-2 8916  df-3 8917  df-4 8918  df-n0 9115  df-z 9192  df-uz 9467  df-rp 9590  df-seqfrec 10381  df-exp 10455  df-cj 10784  df-re 10785  df-im 10786  df-rsqrt 10940  df-abs 10941

This theorem is referenced by:  qtopbas  13162