Theorem supminf 11775

Description: The supremum of a bounded-above set of reals is the negation of the infimum of that set's image under negation. (Contributed by Paul Chapman, 21-Mar-2011.) ( Revised by AV, 13-Sep-2020.)

Assertion

Ref	Expression
supminf	|- ( ( A C_ RR /\ A =/= (/) /\ E. x e. RR A. y e. A y <_ x ) -> sup ( A , RR , < ) = -uinf ( { z e. RR | -u z e. A } , RR , < ) )

Distinct variable group:   x, A, y, z

Proof of Theorem supminf

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		negn0 10459	. . . . 5 |- ( ( A C_ RR /\ A =/= (/) ) -> { z e. RR | -u z e. A } =/= (/) )
2		ublbneg 11773	. . . . 5 |- ( E. x e. RR A. y e. A y <_ x -> E. x e. RR A. y e. { z e. RR | -u z e. A } x <_ y )
3		ssrab2 3687	. . . . . 6 |- { z e. RR | -u z e. A } C_ RR
4		infrenegsup 11006	. . . . . 6 |- ( ( { z e. RR | -u z e. A } C_ RR /\ { z e. RR | -u z e. A } =/= (/) /\ E. x e. RR A. y e. { z e. RR | -u z e. A } x <_ y ) -> inf ( { z e. RR | -u z e. A } , RR , < ) = -u sup ( { w e. RR | -u w e. { z e. RR | -u z e. A } } , RR , < ) )
5	3, 4	mp3an1 1411	. . . . 5 |- ( ( { z e. RR | -u z e. A } =/= (/) /\ E. x e. RR A. y e. { z e. RR | -u z e. A } x <_ y ) -> inf ( { z e. RR | -u z e. A } , RR , < ) = -u sup ( { w e. RR | -u w e. { z e. RR | -u z e. A } } , RR , < ) )
6	1, 2, 5	syl2an 494	. . . 4 |- ( ( ( A C_ RR /\ A =/= (/) ) /\ E. x e. RR A. y e. A y <_ x ) -> inf ( { z e. RR | -u z e. A } , RR , < ) = -u sup ( { w e. RR | -u w e. { z e. RR | -u z e. A } } , RR , < ) )
7	6	3impa 1259	. . 3 |- ( ( A C_ RR /\ A =/= (/) /\ E. x e. RR A. y e. A y <_ x ) -> inf ( { z e. RR | -u z e. A } , RR , < ) = -u sup ( { w e. RR | -u w e. { z e. RR | -u z e. A } } , RR , < ) )
8		elrabi 3359	. . . . . . . 8 |- ( x e. { w e. RR | -u w e. { z e. RR | -u z e. A } } -> x e. RR )
9	8	adantl 482	. . . . . . 7 |- ( ( A C_ RR /\ x e. { w e. RR | -u w e. { z e. RR | -u z e. A } } ) -> x e. RR )
10		ssel2 3598	. . . . . . 7 |- ( ( A C_ RR /\ x e. A ) -> x e. RR )
11		negeq 10273	. . . . . . . . . . 11 |- ( w = x -> -u w = -u x )
12	11	eleq1d 2686	. . . . . . . . . 10 |- ( w = x -> ( -u w e. { z e. RR | -u z e. A } <-> -u x e. { z e. RR | -u z e. A } ) )
13	12	elrab3 3364	. . . . . . . . 9 |- ( x e. RR -> ( x e. { w e. RR | -u w e. { z e. RR | -u z e. A } } <-> -u x e. { z e. RR | -u z e. A } ) )
14		renegcl 10344	. . . . . . . . . 10 |- ( x e. RR -> -u x e. RR )
15		negeq 10273	. . . . . . . . . . . 12 |- ( z = -u x -> -u z = -u -u x )
16	15	eleq1d 2686	. . . . . . . . . . 11 |- ( z = -u x -> ( -u z e. A <-> -u -u x e. A ) )
17	16	elrab3 3364	. . . . . . . . . 10 |- ( -u x e. RR -> ( -u x e. { z e. RR | -u z e. A } <-> -u -u x e. A ) )
18	14, 17	syl 17	. . . . . . . . 9 |- ( x e. RR -> ( -u x e. { z e. RR | -u z e. A } <-> -u -u x e. A ) )
19		recn 10026	. . . . . . . . . . 11 |- ( x e. RR -> x e. CC )
20	19	negnegd 10383	. . . . . . . . . 10 |- ( x e. RR -> -u -u x = x )
21	20	eleq1d 2686	. . . . . . . . 9 |- ( x e. RR -> ( -u -u x e. A <-> x e. A ) )
22	13, 18, 21	3bitrd 294	. . . . . . . 8 |- ( x e. RR -> ( x e. { w e. RR | -u w e. { z e. RR | -u z e. A } } <-> x e. A ) )
23	22	adantl 482	. . . . . . 7 |- ( ( A C_ RR /\ x e. RR ) -> ( x e. { w e. RR | -u w e. { z e. RR | -u z e. A } } <-> x e. A ) )
24	9, 10, 23	eqrdav 2621	. . . . . 6 |- ( A C_ RR -> { w e. RR | -u w e. { z e. RR | -u z e. A } } = A )
25	24	supeq1d 8352	. . . . 5 |- ( A C_ RR -> sup ( { w e. RR | -u w e. { z e. RR | -u z e. A } } , RR , < ) = sup ( A , RR , < ) )
26	25	3ad2ant1 1082	. . . 4 |- ( ( A C_ RR /\ A =/= (/) /\ E. x e. RR A. y e. A y <_ x ) -> sup ( { w e. RR | -u w e. { z e. RR | -u z e. A } } , RR , < ) = sup ( A , RR , < ) )
27	26	negeqd 10275	. . 3 |- ( ( A C_ RR /\ A =/= (/) /\ E. x e. RR A. y e. A y <_ x ) -> -u sup ( { w e. RR | -u w e. { z e. RR | -u z e. A } } , RR , < ) = -u sup ( A , RR , < ) )
28	7, 27	eqtrd 2656	. 2 |- ( ( A C_ RR /\ A =/= (/) /\ E. x e. RR A. y e. A y <_ x ) -> inf ( { z e. RR | -u z e. A } , RR , < ) = -u sup ( A , RR , < ) )
29		infrecl 11005	. . . . . 6 |- ( ( { z e. RR | -u z e. A } C_ RR /\ { z e. RR | -u z e. A } =/= (/) /\ E. x e. RR A. y e. { z e. RR | -u z e. A } x <_ y ) -> inf ( { z e. RR | -u z e. A } , RR , < ) e. RR )
30	3, 29	mp3an1 1411	. . . . 5 |- ( ( { z e. RR | -u z e. A } =/= (/) /\ E. x e. RR A. y e. { z e. RR | -u z e. A } x <_ y ) -> inf ( { z e. RR | -u z e. A } , RR , < ) e. RR )
31	1, 2, 30	syl2an 494	. . . 4 |- ( ( ( A C_ RR /\ A =/= (/) ) /\ E. x e. RR A. y e. A y <_ x ) -> inf ( { z e. RR | -u z e. A } , RR , < ) e. RR )
32	31	3impa 1259	. . 3 |- ( ( A C_ RR /\ A =/= (/) /\ E. x e. RR A. y e. A y <_ x ) -> inf ( { z e. RR | -u z e. A } , RR , < ) e. RR )
33		suprcl 10983	. . 3 |- ( ( A C_ RR /\ A =/= (/) /\ E. x e. RR A. y e. A y <_ x ) -> sup ( A , RR , < ) e. RR )
34		recn 10026	. . . 4 |- (inf ( { z e. RR | -u z e. A } , RR , < ) e. RR -> inf ( { z e. RR | -u z e. A } , RR , < ) e. CC )
35		recn 10026	. . . 4 |- ( sup ( A , RR , < ) e. RR -> sup ( A , RR , < ) e. CC )
36		negcon2 10334	. . . 4 |- ( (inf ( { z e. RR | -u z e. A } , RR , < ) e. CC /\ sup ( A , RR , < ) e. CC ) -> (inf ( { z e. RR | -u z e. A } , RR , < ) = -u sup ( A , RR , < ) <-> sup ( A , RR , < ) = -uinf ( { z e. RR | -u z e. A } , RR , < ) ) )
37	34, 35, 36	syl2an 494	. . 3 |- ( (inf ( { z e. RR | -u z e. A } , RR , < ) e. RR /\ sup ( A , RR , < ) e. RR ) -> (inf ( { z e. RR | -u z e. A } , RR , < ) = -u sup ( A , RR , < ) <-> sup ( A , RR , < ) = -uinf ( { z e. RR | -u z e. A } , RR , < ) ) )
38	32, 33, 37	syl2anc 693	. 2 |- ( ( A C_ RR /\ A =/= (/) /\ E. x e. RR A. y e. A y <_ x ) -> (inf ( { z e. RR | -u z e. A } , RR , < ) = -u sup ( A , RR , < ) <-> sup ( A , RR , < ) = -uinf ( { z e. RR | -u z e. A } , RR , < ) ) )
39	28, 38	mpbid 222	1 |- ( ( A C_ RR /\ A =/= (/) /\ E. x e. RR A. y e. A y <_ x ) -> sup ( A , RR , < ) = -uinf ( { z e. RR | -u z e. A } , RR , < ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   {crab 2916   C_ wss 3574   (/)c0 3915   class class class wbr 4653   supcsup 8346  infcinf 8347   CCcc 9934   RRcr 9935   < clt 10074   <_ cle 10075   -ucneg 10267

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269

This theorem is referenced by:  supminfrnmpt  39672