Theorem infxrnegsupex 11064

Description: The infimum of a set of extended reals A is the negative of the supremum of the negatives of its elements. (Contributed by Jim Kingdon, 2-May-2023.)

Hypotheses

Ref	Expression
infxrnegsupex.ex	|- ( ph -> E. x e. RR* ( A. y e. A -. y < x /\ A. y e. RR* ( x < y -> E. z e. A z < y ) ) )
infxrnegsupex.ss	|- ( ph -> A C_ RR* )

Assertion

Ref	Expression
infxrnegsupex	|- ( ph -> inf ( A , RR* , < ) = -e sup ( { z e. RR* | -e z e. A } , RR* , < ) )

Distinct variable groups:   x, A, y, z   ph, x, y, z

Proof of Theorem infxrnegsupex

Dummy variables f g w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xrlttri3 9613	. . . . 5 |- ( ( f e. RR* /\ g e. RR* ) -> ( f = g <-> ( -. f < g /\ -. g < f ) ) )
2	1	adantl 275	. . . 4 |- ( ( ph /\ ( f e. RR* /\ g e. RR* ) ) -> ( f = g <-> ( -. f < g /\ -. g < f ) ) )
3		infxrnegsupex.ex	. . . 4 |- ( ph -> E. x e. RR* ( A. y e. A -. y < x /\ A. y e. RR* ( x < y -> E. z e. A z < y ) ) )
4	2, 3	infclti 6918	. . 3 |- ( ph -> inf ( A , RR* , < ) e. RR* )
5		xnegneg 9646	. . 3 |- (inf ( A , RR* , < ) e. RR* -> -e -einf ( A , RR* , < ) = inf ( A , RR* , < ) )
6	4, 5	syl 14	. 2 |- ( ph -> -e -einf ( A , RR* , < ) = inf ( A , RR* , < ) )
7		xnegeq 9640	. . . . . . . . 9 |- ( w = z -> -e w = -e z )
8	7	cbvmptv 4032	. . . . . . . 8 |- ( w e. RR* |-> -e w ) = ( z e. RR* |-> -e z )
9	8	mptpreima 5040	. . . . . . 7 |- ( `' ( w e. RR* |-> -e w ) " A ) = { z e. RR* | -e z e. A }
10		eqid 2140	. . . . . . . . . 10 |- ( w e. RR* |-> -e w ) = ( w e. RR* |-> -e w )
11	10	xrnegiso 11063	. . . . . . . . 9 |- ( ( w e. RR* |-> -e w ) Isom < , `' < ( RR* , RR* ) /\ `' ( w e. RR* |-> -e w ) = ( w e. RR* |-> -e w ) )
12	11	simpri 112	. . . . . . . 8 |- `' ( w e. RR* |-> -e w ) = ( w e. RR* |-> -e w )
13	12	imaeq1i 4886	. . . . . . 7 |- ( `' ( w e. RR* |-> -e w ) " A ) = ( ( w e. RR* |-> -e w ) " A )
14	9, 13	eqtr3i 2163	. . . . . 6 |- { z e. RR* | -e z e. A } = ( ( w e. RR* |-> -e w ) " A )
15	14	supeq1i 6883	. . . . 5 |- sup ( { z e. RR* | -e z e. A } , RR* , < ) = sup ( ( ( w e. RR* |-> -e w ) " A ) , RR* , < )
16	11	simpli 110	. . . . . . . . 9 |- ( w e. RR* |-> -e w ) Isom < , `' < ( RR* , RR* )
17		isocnv 5720	. . . . . . . . 9 |- ( ( w e. RR* |-> -e w ) Isom < , `' < ( RR* , RR* ) -> `' ( w e. RR* |-> -e w ) Isom `' < , < ( RR* , RR* ) )
18	16, 17	ax-mp 5	. . . . . . . 8 |- `' ( w e. RR* |-> -e w ) Isom `' < , < ( RR* , RR* )
19		isoeq1 5710	. . . . . . . . 9 |- ( `' ( w e. RR* |-> -e w ) = ( w e. RR* |-> -e w ) -> ( `' ( w e. RR* |-> -e w ) Isom `' < , < ( RR* , RR* ) <-> ( w e. RR* |-> -e w ) Isom `' < , < ( RR* , RR* ) ) )
20	12, 19	ax-mp 5	. . . . . . . 8 |- ( `' ( w e. RR* |-> -e w ) Isom `' < , < ( RR* , RR* ) <-> ( w e. RR* |-> -e w ) Isom `' < , < ( RR* , RR* ) )
21	18, 20	mpbi 144	. . . . . . 7 |- ( w e. RR* |-> -e w ) Isom `' < , < ( RR* , RR* )
22	21	a1i 9	. . . . . 6 |- ( ph -> ( w e. RR* |-> -e w ) Isom `' < , < ( RR* , RR* ) )
23		infxrnegsupex.ss	. . . . . 6 |- ( ph -> A C_ RR* )
24	3	cnvinfex 6913	. . . . . 6 |- ( ph -> E. x e. RR* ( A. y e. A -. x `' < y /\ A. y e. RR* ( y `' < x -> E. z e. A y `' < z ) ) )
25	2	cnvti 6914	. . . . . 6 |- ( ( ph /\ ( f e. RR* /\ g e. RR* ) ) -> ( f = g <-> ( -. f `' < g /\ -. g `' < f ) ) )
26	22, 23, 24, 25	supisoti 6905	. . . . 5 |- ( ph -> sup ( ( ( w e. RR* |-> -e w ) " A ) , RR* , < ) = ( ( w e. RR* |-> -e w ) ` sup ( A , RR* , `' < ) ) )
27	15, 26	syl5eq 2185	. . . 4 |- ( ph -> sup ( { z e. RR* | -e z e. A } , RR* , < ) = ( ( w e. RR* |-> -e w ) ` sup ( A , RR* , `' < ) ) )
28		df-inf 6880	. . . . . . 7 |- inf ( A , RR* , < ) = sup ( A , RR* , `' < )
29	28	eqcomi 2144	. . . . . 6 |- sup ( A , RR* , `' < ) = inf ( A , RR* , < )
30	29	fveq2i 5432	. . . . 5 |- ( ( w e. RR* |-> -e w ) ` sup ( A , RR* , `' < ) ) = ( ( w e. RR* |-> -e w ) ` inf ( A , RR* , < ) )
31		eqidd 2141	. . . . . 6 |- ( ph -> ( w e. RR* |-> -e w ) = ( w e. RR* |-> -e w ) )
32		xnegeq 9640	. . . . . . 7 |- ( w = inf ( A , RR* , < ) -> -e w = -einf ( A , RR* , < ) )
33	32	adantl 275	. . . . . 6 |- ( ( ph /\ w = inf ( A , RR* , < ) ) -> -e w = -einf ( A , RR* , < ) )
34	4	xnegcld 9668	. . . . . 6 |- ( ph -> -einf ( A , RR* , < ) e. RR* )
35	31, 33, 4, 34	fvmptd 5510	. . . . 5 |- ( ph -> ( ( w e. RR* |-> -e w ) ` inf ( A , RR* , < ) ) = -einf ( A , RR* , < ) )
36	30, 35	syl5eq 2185	. . . 4 |- ( ph -> ( ( w e. RR* |-> -e w ) ` sup ( A , RR* , `' < ) ) = -einf ( A , RR* , < ) )
37	27, 36	eqtr2d 2174	. . 3 |- ( ph -> -einf ( A , RR* , < ) = sup ( { z e. RR* | -e z e. A } , RR* , < ) )
38		xnegeq 9640	. . 3 |- ( -einf ( A , RR* , < ) = sup ( { z e. RR* | -e z e. A } , RR* , < ) -> -e -einf ( A , RR* , < ) = -e sup ( { z e. RR* | -e z e. A } , RR* , < ) )
39	37, 38	syl 14	. 2 |- ( ph -> -e -einf ( A , RR* , < ) = -e sup ( { z e. RR* | -e z e. A } , RR* , < ) )
40	6, 39	eqtr3d 2175	1 |- ( ph -> inf ( A , RR* , < ) = -e sup ( { z e. RR* | -e z e. A } , RR* , < ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1332   e. wcel 1481   A.wral 2417   E.wrex 2418   {crab 2421   C_ wss 3076   class class class wbr 3937   |-> cmpt 3997   `'ccnv 4546   "cima 4550   ` cfv 5131   Isom wiso 5132   supcsup 6877  infcinf 6878   RR*cxr 7823   < clt 7824   -ecxne 9586

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-cnex 7735  ax-resscn 7736  ax-1cn 7737  ax-1re 7738  ax-icn 7739  ax-addcl 7740  ax-addrcl 7741  ax-mulcl 7742  ax-addcom 7744  ax-addass 7746  ax-distr 7748  ax-i2m1 7749  ax-0id 7752  ax-rnegex 7753  ax-cnre 7755  ax-pre-ltirr 7756  ax-pre-apti 7759  ax-pre-ltadd 7760

This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rmo 2425  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-if 3480  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-mpt 3999  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-isom 5140  df-riota 5738  df-ov 5785  df-oprab 5786  df-mpo 5787  df-sup 6879  df-inf 6880  df-pnf 7826  df-mnf 7827  df-xr 7828  df-ltxr 7829  df-sub 7959  df-neg 7960  df-xneg 9589

This theorem is referenced by:  xrminmax  11066