Theorem infxrnegsupex 11291

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 9817	. . . . 5 |- ( ( f e. RR* /\ g e. RR* ) -> ( f = g <-> ( -. f < g /\ -. g < f ) ) )
2	1	adantl 277	. . . 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 7042	. . 3 |- ( ph -> inf ( A , RR* , < ) e. RR* )
5		xnegneg 9853	. . 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 9847	. . . . . . . . 9 |- ( w = z -> -e w = -e z )
8	7	cbvmptv 4114	. . . . . . . 8 |- ( w e. RR* |-> -e w ) = ( z e. RR* |-> -e z )
9	8	mptpreima 5137	. . . . . . 7 |- ( `' ( w e. RR* |-> -e w ) " A ) = { z e. RR* | -e z e. A }
10		eqid 2189	. . . . . . . . . 10 |- ( w e. RR* |-> -e w ) = ( w e. RR* |-> -e w )
11	10	xrnegiso 11290	. . . . . . . . 9 |- ( ( w e. RR* |-> -e w ) Isom < , `' < ( RR* , RR* ) /\ `' ( w e. RR* |-> -e w ) = ( w e. RR* |-> -e w ) )
12	11	simpri 113	. . . . . . . 8 |- `' ( w e. RR* |-> -e w ) = ( w e. RR* |-> -e w )
13	12	imaeq1i 4982	. . . . . . 7 |- ( `' ( w e. RR* |-> -e w ) " A ) = ( ( w e. RR* |-> -e w ) " A )
14	9, 13	eqtr3i 2212	. . . . . 6 |- { z e. RR* | -e z e. A } = ( ( w e. RR* |-> -e w ) " A )
15	14	supeq1i 7007	. . . . 5 |- sup ( { z e. RR* | -e z e. A } , RR* , < ) = sup ( ( ( w e. RR* |-> -e w ) " A ) , RR* , < )
16	11	simpli 111	. . . . . . . . 9 |- ( w e. RR* |-> -e w ) Isom < , `' < ( RR* , RR* )
17		isocnv 5829	. . . . . . . . 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 5819	. . . . . . . . 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 145	. . . . . . 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 7037	. . . . . 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 7038	. . . . . 6 |- ( ( ph /\ ( f e. RR* /\ g e. RR* ) ) -> ( f = g <-> ( -. f `' < g /\ -. g `' < f ) ) )
26	22, 23, 24, 25	supisoti 7029	. . . . 5 |- ( ph -> sup ( ( ( w e. RR* |-> -e w ) " A ) , RR* , < ) = ( ( w e. RR* |-> -e w ) ` sup ( A , RR* , `' < ) ) )
27	15, 26	eqtrid 2234	. . . 4 |- ( ph -> sup ( { z e. RR* | -e z e. A } , RR* , < ) = ( ( w e. RR* |-> -e w ) ` sup ( A , RR* , `' < ) ) )
28		df-inf 7004	. . . . . . 7 |- inf ( A , RR* , < ) = sup ( A , RR* , `' < )
29	28	eqcomi 2193	. . . . . 6 |- sup ( A , RR* , `' < ) = inf ( A , RR* , < )
30	29	fveq2i 5534	. . . . 5 |- ( ( w e. RR* |-> -e w ) ` sup ( A , RR* , `' < ) ) = ( ( w e. RR* |-> -e w ) ` inf ( A , RR* , < ) )
31		eqidd 2190	. . . . . 6 |- ( ph -> ( w e. RR* |-> -e w ) = ( w e. RR* |-> -e w ) )
32		xnegeq 9847	. . . . . . 7 |- ( w = inf ( A , RR* , < ) -> -e w = -einf ( A , RR* , < ) )
33	32	adantl 277	. . . . . 6 |- ( ( ph /\ w = inf ( A , RR* , < ) ) -> -e w = -einf ( A , RR* , < ) )
34	4	xnegcld 9875	. . . . . 6 |- ( ph -> -einf ( A , RR* , < ) e. RR* )
35	31, 33, 4, 34	fvmptd 5614	. . . . 5 |- ( ph -> ( ( w e. RR* |-> -e w ) ` inf ( A , RR* , < ) ) = -einf ( A , RR* , < ) )
36	30, 35	eqtrid 2234	. . . 4 |- ( ph -> ( ( w e. RR* |-> -e w ) ` sup ( A , RR* , `' < ) ) = -einf ( A , RR* , < ) )
37	27, 36	eqtr2d 2223	. . 3 |- ( ph -> -einf ( A , RR* , < ) = sup ( { z e. RR* | -e z e. A } , RR* , < ) )
38		xnegeq 9847	. . 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 2224	1 |- ( ph -> inf ( A , RR* , < ) = -e sup ( { z e. RR* | -e z e. A } , RR* , < ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2160   A.wral 2468   E.wrex 2469   {crab 2472   C_ wss 3144   class class class wbr 4018   |-> cmpt 4079   `'ccnv 4640   "cima 4644   ` cfv 5232   Isom wiso 5233   supcsup 7001  infcinf 7002   RR*cxr 8011   < clt 8012   -ecxne 9789

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-cnex 7922  ax-resscn 7923  ax-1cn 7924  ax-1re 7925  ax-icn 7926  ax-addcl 7927  ax-addrcl 7928  ax-mulcl 7929  ax-addcom 7931  ax-addass 7933  ax-distr 7935  ax-i2m1 7936  ax-0id 7939  ax-rnegex 7940  ax-cnre 7942  ax-pre-ltirr 7943  ax-pre-apti 7946  ax-pre-ltadd 7947

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5234  df-fn 5235  df-f 5236  df-f1 5237  df-fo 5238  df-f1o 5239  df-fv 5240  df-isom 5241  df-riota 5848  df-ov 5895  df-oprab 5896  df-mpo 5897  df-sup 7003  df-inf 7004  df-pnf 8014  df-mnf 8015  df-xr 8016  df-ltxr 8017  df-sub 8150  df-neg 8151  df-xneg 9792

This theorem is referenced by:  xrminmax  11293