Theorem infxrnegsupex 11237

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 9766	. . . . 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 7012	. . 3 |- ( ph -> inf ( A , RR* , < ) e. RR* )
5		xnegneg 9802	. . 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 9796	. . . . . . . . 9 |- ( w = z -> -e w = -e z )
8	7	cbvmptv 4094	. . . . . . . 8 |- ( w e. RR* |-> -e w ) = ( z e. RR* |-> -e z )
9	8	mptpreima 5114	. . . . . . 7 |- ( `' ( w e. RR* |-> -e w ) " A ) = { z e. RR* | -e z e. A }
10		eqid 2175	. . . . . . . . . 10 |- ( w e. RR* |-> -e w ) = ( w e. RR* |-> -e w )
11	10	xrnegiso 11236	. . . . . . . . 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 4960	. . . . . . 7 |- ( `' ( w e. RR* |-> -e w ) " A ) = ( ( w e. RR* |-> -e w ) " A )
14	9, 13	eqtr3i 2198	. . . . . 6 |- { z e. RR* | -e z e. A } = ( ( w e. RR* |-> -e w ) " A )
15	14	supeq1i 6977	. . . . 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 5802	. . . . . . . . 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 5792	. . . . . . . . 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 7007	. . . . . 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 7008	. . . . . 6 |- ( ( ph /\ ( f e. RR* /\ g e. RR* ) ) -> ( f = g <-> ( -. f `' < g /\ -. g `' < f ) ) )
26	22, 23, 24, 25	supisoti 6999	. . . . 5 |- ( ph -> sup ( ( ( w e. RR* |-> -e w ) " A ) , RR* , < ) = ( ( w e. RR* |-> -e w ) ` sup ( A , RR* , `' < ) ) )
27	15, 26	eqtrid 2220	. . . 4 |- ( ph -> sup ( { z e. RR* | -e z e. A } , RR* , < ) = ( ( w e. RR* |-> -e w ) ` sup ( A , RR* , `' < ) ) )
28		df-inf 6974	. . . . . . 7 |- inf ( A , RR* , < ) = sup ( A , RR* , `' < )
29	28	eqcomi 2179	. . . . . 6 |- sup ( A , RR* , `' < ) = inf ( A , RR* , < )
30	29	fveq2i 5510	. . . . 5 |- ( ( w e. RR* |-> -e w ) ` sup ( A , RR* , `' < ) ) = ( ( w e. RR* |-> -e w ) ` inf ( A , RR* , < ) )
31		eqidd 2176	. . . . . 6 |- ( ph -> ( w e. RR* |-> -e w ) = ( w e. RR* |-> -e w ) )
32		xnegeq 9796	. . . . . . 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 9824	. . . . . 6 |- ( ph -> -einf ( A , RR* , < ) e. RR* )
35	31, 33, 4, 34	fvmptd 5589	. . . . 5 |- ( ph -> ( ( w e. RR* |-> -e w ) ` inf ( A , RR* , < ) ) = -einf ( A , RR* , < ) )
36	30, 35	eqtrid 2220	. . . 4 |- ( ph -> ( ( w e. RR* |-> -e w ) ` sup ( A , RR* , `' < ) ) = -einf ( A , RR* , < ) )
37	27, 36	eqtr2d 2209	. . 3 |- ( ph -> -einf ( A , RR* , < ) = sup ( { z e. RR* | -e z e. A } , RR* , < ) )
38		xnegeq 9796	. . 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 2210	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 1353   e. wcel 2146   A.wral 2453   E.wrex 2454   {crab 2457   C_ wss 3127   class class class wbr 3998   |-> cmpt 4059   `'ccnv 4619   "cima 4623   ` cfv 5208   Isom wiso 5209   supcsup 6971  infcinf 6972   RR*cxr 7965   < clt 7966   -ecxne 9738

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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-cnex 7877  ax-resscn 7878  ax-1cn 7879  ax-1re 7880  ax-icn 7881  ax-addcl 7882  ax-addrcl 7883  ax-mulcl 7884  ax-addcom 7886  ax-addass 7888  ax-distr 7890  ax-i2m1 7891  ax-0id 7894  ax-rnegex 7895  ax-cnre 7897  ax-pre-ltirr 7898  ax-pre-apti 7901  ax-pre-ltadd 7902

This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-nel 2441  df-ral 2458  df-rex 2459  df-reu 2460  df-rmo 2461  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-if 3533  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-br 3999  df-opab 4060  df-mpt 4061  df-id 4287  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-isom 5217  df-riota 5821  df-ov 5868  df-oprab 5869  df-mpo 5870  df-sup 6973  df-inf 6974  df-pnf 7968  df-mnf 7969  df-xr 7970  df-ltxr 7971  df-sub 8104  df-neg 8105  df-xneg 9741

This theorem is referenced by:  xrminmax  11239