Theorem infxrnegsupex 11973

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 10149	. . . . 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 7327	. . 3 |- ( ph -> inf ( A , RR* , < ) e. RR* )
5		xnegneg 10185	. . 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 10179	. . . . . . . . 9 |- ( w = z -> -e w = -e z )
8	7	cbvmptv 4211	. . . . . . . 8 |- ( w e. RR* |-> -e w ) = ( z e. RR* |-> -e z )
9	8	mptpreima 5261	. . . . . . 7 |- ( `' ( w e. RR* |-> -e w ) " A ) = { z e. RR* | -e z e. A }
10		eqid 2234	. . . . . . . . . 10 |- ( w e. RR* |-> -e w ) = ( w e. RR* |-> -e w )
11	10	xrnegiso 11972	. . . . . . . . 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 5103	. . . . . . 7 |- ( `' ( w e. RR* |-> -e w ) " A ) = ( ( w e. RR* |-> -e w ) " A )
14	9, 13	eqtr3i 2257	. . . . . 6 |- { z e. RR* | -e z e. A } = ( ( w e. RR* |-> -e w ) " A )
15	14	supeq1i 7292	. . . . 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 5990	. . . . . . . . 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 5980	. . . . . . . . 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 7322	. . . . . 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 7323	. . . . . 6 |- ( ( ph /\ ( f e. RR* /\ g e. RR* ) ) -> ( f = g <-> ( -. f `' < g /\ -. g `' < f ) ) )
26	22, 23, 24, 25	supisoti 7314	. . . . 5 |- ( ph -> sup ( ( ( w e. RR* |-> -e w ) " A ) , RR* , < ) = ( ( w e. RR* |-> -e w ) ` sup ( A , RR* , `' < ) ) )
27	15, 26	eqtrid 2279	. . . 4 |- ( ph -> sup ( { z e. RR* | -e z e. A } , RR* , < ) = ( ( w e. RR* |-> -e w ) ` sup ( A , RR* , `' < ) ) )
28		df-inf 7289	. . . . . . 7 |- inf ( A , RR* , < ) = sup ( A , RR* , `' < )
29	28	eqcomi 2238	. . . . . 6 |- sup ( A , RR* , `' < ) = inf ( A , RR* , < )
30	29	fveq2i 5678	. . . . 5 |- ( ( w e. RR* |-> -e w ) ` sup ( A , RR* , `' < ) ) = ( ( w e. RR* |-> -e w ) ` inf ( A , RR* , < ) )
31		eqidd 2235	. . . . . 6 |- ( ph -> ( w e. RR* |-> -e w ) = ( w e. RR* |-> -e w ) )
32		xnegeq 10179	. . . . . . 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 10207	. . . . . 6 |- ( ph -> -einf ( A , RR* , < ) e. RR* )
35	31, 33, 4, 34	fvmptd 5763	. . . . 5 |- ( ph -> ( ( w e. RR* |-> -e w ) ` inf ( A , RR* , < ) ) = -einf ( A , RR* , < ) )
36	30, 35	eqtrid 2279	. . . 4 |- ( ph -> ( ( w e. RR* |-> -e w ) ` sup ( A , RR* , `' < ) ) = -einf ( A , RR* , < ) )
37	27, 36	eqtr2d 2268	. . 3 |- ( ph -> -einf ( A , RR* , < ) = sup ( { z e. RR* | -e z e. A } , RR* , < ) )
38		xnegeq 10179	. . 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 2269	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 1398   e. wcel 2205   A.wral 2522   E.wrex 2523   {crab 2526   C_ wss 3214   class class class wbr 4114   |-> cmpt 4176   `'ccnv 4753   "cima 4757   ` cfv 5357   Isom wiso 5358   supcsup 7286  infcinf 7287   RR*cxr 8323   < clt 8324   -ecxne 10121

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-distr 8247  ax-i2m1 8248  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-apti 8258  ax-pre-ltadd 8259

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-isom 5366  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-sup 7288  df-inf 7289  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-sub 8462  df-neg 8463  df-xneg 10124

This theorem is referenced by:  xrminmax  11975