Theorem minclpr 11581

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 9418 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 8335	. . . . 5 |- ( A e. RR -> -u A e. RR )
2		renegcl 8335	. . . . 5 |- ( B e. RR -> -u B e. RR )
3		maxcl 11554	. . . . 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 3653	. . . 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 11566	. . . 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 8103	. . . . . 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 8103	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> -u A e. CC )
12	9, 11	neg11ad 8381	. . . . 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 11574	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> inf ( { A , B } , RR , < ) = -u sup ( { -u A , -u B } , RR , < ) )
14	13	eqcomd 2211	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> -u sup ( { -u A , -u B } , RR , < ) = inf ( { A , B } , RR , < ) )
15		recn 8060	. . . . . . . 8 |- ( A e. RR -> A e. CC )
16	15	adantr 276	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> A e. CC )
17	16	negnegd 8376	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> -u -u A = A )
18	14, 17	eqeq12d 2220	. . . . 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 8103	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> -u B e. CC )
22	9, 21	neg11ad 8381	. . . . 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 8060	. . . . . . . 8 |- ( B e. RR -> B e. CC )
24	23	adantl 277	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> B e. CC )
25	24	negnegd 8376	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> -u -u B = B )
26	14, 25	eqeq12d 2220	. . . . 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 795	. . 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 11575	. . 3 |- ( ( A e. RR /\ B e. RR ) -> inf ( { A , B } , RR , < ) e. RR )
31		elprg 3653	. . 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 730	. . 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 8599	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( B <_ A <-> -u A <_ -u B ) )
37		leneg 8540	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( A <_ B <-> -u B <_ -u A ) )
38	36, 37	orbi12d 795	. . 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 710   = wceq 1373   e. wcel 2176   {cpr 3634   class class class wbr 4045   supcsup 7086  infcinf 7087   CCcc 7925   RRcr 7926   < clt 8109   <_ cle 8110   -ucneg 8246

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4160  ax-sep 4163  ax-nul 4171  ax-pow 4219  ax-pr 4254  ax-un 4481  ax-setind 4586  ax-iinf 4637  ax-cnex 8018  ax-resscn 8019  ax-1cn 8020  ax-1re 8021  ax-icn 8022  ax-addcl 8023  ax-addrcl 8024  ax-mulcl 8025  ax-mulrcl 8026  ax-addcom 8027  ax-mulcom 8028  ax-addass 8029  ax-mulass 8030  ax-distr 8031  ax-i2m1 8032  ax-0lt1 8033  ax-1rid 8034  ax-0id 8035  ax-rnegex 8036  ax-precex 8037  ax-cnre 8038  ax-pre-ltirr 8039  ax-pre-ltwlin 8040  ax-pre-lttrn 8041  ax-pre-apti 8042  ax-pre-ltadd 8043  ax-pre-mulgt0 8044  ax-pre-mulext 8045  ax-arch 8046  ax-caucvg 8047

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 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-if 3572  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4046  df-opab 4107  df-mpt 4108  df-tr 4144  df-id 4341  df-po 4344  df-iso 4345  df-iord 4414  df-on 4416  df-ilim 4417  df-suc 4419  df-iom 4640  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-res 4688  df-ima 4689  df-iota 5233  df-fun 5274  df-fn 5275  df-f 5276  df-f1 5277  df-fo 5278  df-f1o 5279  df-fv 5280  df-isom 5281  df-riota 5901  df-ov 5949  df-oprab 5950  df-mpo 5951  df-1st 6228  df-2nd 6229  df-recs 6393  df-frec 6479  df-sup 7088  df-inf 7089  df-pnf 8111  df-mnf 8112  df-xr 8113  df-ltxr 8114  df-le 8115  df-sub 8247  df-neg 8248  df-reap 8650  df-ap 8657  df-div 8748  df-inn 9039  df-2 9097  df-3 9098  df-4 9099  df-n0 9298  df-z 9375  df-uz 9651  df-rp 9778  df-seqfrec 10595  df-exp 10686  df-cj 11186  df-re 11187  df-im 11188  df-rsqrt 11342  df-abs 11343

This theorem is referenced by:  qtopbas  15027