Theorem minclpr 11237

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 9292 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 8213	. . . . 5 |- ( A e. RR -> -u A e. RR )
2		renegcl 8213	. . . . 5 |- ( B e. RR -> -u B e. RR )
3		maxcl 11211	. . . . 5 |- ( ( -u A e. RR /\ -u B e. RR ) -> sup ( { -u A , -u B } , RR , < ) e. RR )
4	1, 2, 3	syl2an 289	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> sup ( { -u A , -u B } , RR , < ) e. RR )
5		elprg 3612	. . . 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 11223	. . . 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 289	. . 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 7981	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> sup ( { -u A , -u B } , RR , < ) e. CC )
10	1	adantr 276	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> -u A e. RR )
11	10	recnd 7981	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> -u A e. CC )
12	9, 11	neg11ad 8259	. . . . 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 11230	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> inf ( { A , B } , RR , < ) = -u sup ( { -u A , -u B } , RR , < ) )
14	13	eqcomd 2183	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> -u sup ( { -u A , -u B } , RR , < ) = inf ( { A , B } , RR , < ) )
15		recn 7940	. . . . . . . 8 |- ( A e. RR -> A e. CC )
16	15	adantr 276	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> A e. CC )
17	16	negnegd 8254	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> -u -u A = A )
18	14, 17	eqeq12d 2192	. . . . 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 190	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( sup ( { -u A , -u B } , RR , < ) = -u A <-> inf ( { A , B } , RR , < ) = A ) )
20	2	adantl 277	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> -u B e. RR )
21	20	recnd 7981	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> -u B e. CC )
22	9, 21	neg11ad 8259	. . . . 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 7940	. . . . . . . 8 |- ( B e. RR -> B e. CC )
24	23	adantl 277	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> B e. CC )
25	24	negnegd 8254	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> -u -u B = B )
26	14, 25	eqeq12d 2192	. . . . 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 190	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( sup ( { -u A , -u B } , RR , < ) = -u B <-> inf ( { A , B } , RR , < ) = B ) )
28	19, 27	orbi12d 793	. . 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 219	. 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 11231	. . 3 |- ( ( A e. RR /\ B e. RR ) -> inf ( { A , B } , RR , < ) e. RR )
31		elprg 3612	. . 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 728	. . 3 |- ( ( B <_ A \/ A <_ B ) <-> ( A <_ B \/ B <_ A ) )
34		simpr 110	. . . . 5 |- ( ( A e. RR /\ B e. RR ) -> B e. RR )
35		simpl 109	. . . . 5 |- ( ( A e. RR /\ B e. RR ) -> A e. RR )
36	34, 35	lenegd 8476	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( B <_ A <-> -u A <_ -u B ) )
37		leneg 8417	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( A <_ B <-> -u B <_ -u A ) )
38	36, 37	orbi12d 793	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( ( B <_ A \/ A <_ B ) <-> ( -u A <_ -u B \/ -u B <_ -u A ) ) )
39	33, 38	bitr3id 194	. 2 |- ( ( A e. RR /\ B e. RR ) -> ( ( A <_ B \/ B <_ A ) <-> ( -u A <_ -u B \/ -u B <_ -u A ) ) )
40	29, 32, 39	3bitr4d 220	1 |- ( ( A e. RR /\ B e. RR ) -> (inf ( { A , B } , RR , < ) e. { A , B } <-> ( A <_ B \/ B <_ A ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 708   = wceq 1353   e. wcel 2148   {cpr 3593   class class class wbr 4002   supcsup 6977  infcinf 6978   CCcc 7805   RRcr 7806   < clt 7987   <_ cle 7988   -ucneg 8124

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 4117  ax-sep 4120  ax-nul 4128  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-setind 4535  ax-iinf 4586  ax-cnex 7898  ax-resscn 7899  ax-1cn 7900  ax-1re 7901  ax-icn 7902  ax-addcl 7903  ax-addrcl 7904  ax-mulcl 7905  ax-mulrcl 7906  ax-addcom 7907  ax-mulcom 7908  ax-addass 7909  ax-mulass 7910  ax-distr 7911  ax-i2m1 7912  ax-0lt1 7913  ax-1rid 7914  ax-0id 7915  ax-rnegex 7916  ax-precex 7917  ax-cnre 7918  ax-pre-ltirr 7919  ax-pre-ltwlin 7920  ax-pre-lttrn 7921  ax-pre-apti 7922  ax-pre-ltadd 7923  ax-pre-mulgt0 7924  ax-pre-mulext 7925  ax-arch 7926  ax-caucvg 7927

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 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4003  df-opab 4064  df-mpt 4065  df-tr 4101  df-id 4292  df-po 4295  df-iso 4296  df-iord 4365  df-on 4367  df-ilim 4368  df-suc 4370  df-iom 4589  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638  df-iota 5176  df-fun 5216  df-fn 5217  df-f 5218  df-f1 5219  df-fo 5220  df-f1o 5221  df-fv 5222  df-isom 5223  df-riota 5827  df-ov 5874  df-oprab 5875  df-mpo 5876  df-1st 6137  df-2nd 6138  df-recs 6302  df-frec 6388  df-sup 6979  df-inf 6980  df-pnf 7989  df-mnf 7990  df-xr 7991  df-ltxr 7992  df-le 7993  df-sub 8125  df-neg 8126  df-reap 8527  df-ap 8534  df-div 8625  df-inn 8915  df-2 8973  df-3 8974  df-4 8975  df-n0 9172  df-z 9249  df-uz 9524  df-rp 9649  df-seqfrec 10440  df-exp 10514  df-cj 10843  df-re 10844  df-im 10845  df-rsqrt 10999  df-abs 11000

This theorem is referenced by:  qtopbas  13884