Theorem infxrnegsupex 12054

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 10216	. . . . 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 7369	. . 3 |- ( ph -> inf ( A , RR* , < ) e. RR* )
5		xnegneg 10252	. . 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 10246	. . . . . . . . 9 |- ( w = z -> -e w = -e z )
8	7	cbvmptv 4229	. . . . . . . 8 |- ( w e. RR* |-> -e w ) = ( z e. RR* |-> -e z )
9	8	mptpreima 5284	. . . . . . 7 |- ( `' ( w e. RR* |-> -e w ) " A ) = { z e. RR* | -e z e. A }
10		eqid 2238	. . . . . . . . . 10 |- ( w e. RR* |-> -e w ) = ( w e. RR* |-> -e w )
11	10	xrnegiso 12053	. . . . . . . . 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 5126	. . . . . . 7 |- ( `' ( w e. RR* |-> -e w ) " A ) = ( ( w e. RR* |-> -e w ) " A )
14	9, 13	eqtr3i 2261	. . . . . 6 |- { z e. RR* | -e z e. A } = ( ( w e. RR* |-> -e w ) " A )
15	14	supeq1i 7334	. . . . 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 6020	. . . . . . . . 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 6010	. . . . . . . . 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 7364	. . . . . 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 7365	. . . . . 6 |- ( ( ph /\ ( f e. RR* /\ g e. RR* ) ) -> ( f = g <-> ( -. f `' < g /\ -. g `' < f ) ) )
26	22, 23, 24, 25	supisoti 7356	. . . . 5 |- ( ph -> sup ( ( ( w e. RR* |-> -e w ) " A ) , RR* , < ) = ( ( w e. RR* |-> -e w ) ` sup ( A , RR* , `' < ) ) )
27	15, 26	eqtrid 2283	. . . 4 |- ( ph -> sup ( { z e. RR* | -e z e. A } , RR* , < ) = ( ( w e. RR* |-> -e w ) ` sup ( A , RR* , `' < ) ) )
28		df-inf 7331	. . . . . . 7 |- inf ( A , RR* , < ) = sup ( A , RR* , `' < )
29	28	eqcomi 2242	. . . . . 6 |- sup ( A , RR* , `' < ) = inf ( A , RR* , < )
30	29	fveq2i 5701	. . . . 5 |- ( ( w e. RR* |-> -e w ) ` sup ( A , RR* , `' < ) ) = ( ( w e. RR* |-> -e w ) ` inf ( A , RR* , < ) )
31		eqidd 2239	. . . . . 6 |- ( ph -> ( w e. RR* |-> -e w ) = ( w e. RR* |-> -e w ) )
32		xnegeq 10246	. . . . . . 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 10274	. . . . . 6 |- ( ph -> -einf ( A , RR* , < ) e. RR* )
35	31, 33, 4, 34	fvmptd 5789	. . . . 5 |- ( ph -> ( ( w e. RR* |-> -e w ) ` inf ( A , RR* , < ) ) = -einf ( A , RR* , < ) )
36	30, 35	eqtrid 2283	. . . 4 |- ( ph -> ( ( w e. RR* |-> -e w ) ` sup ( A , RR* , `' < ) ) = -einf ( A , RR* , < ) )
37	27, 36	eqtr2d 2272	. . 3 |- ( ph -> -einf ( A , RR* , < ) = sup ( { z e. RR* | -e z e. A } , RR* , < ) )
38		xnegeq 10246	. . 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 2273	1 |- ( ph -> inf ( A , RR* , < ) = -e sup ( { z e. RR* | -e z e. A } , RR* , < ) )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   A.wral 2528   E.wrex 2529   {crab 2532   C_ wss 3220   class class class wbr 4131   |-> cmpt 4193   `'ccnv 4775   "cima 4779   ` cfv 5380   Isom wiso 5381   supcsup 7328  infcinf 7329   RR*cxr 8365   < clt 8366   -ecxne 10188

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4251  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-addcom 8285  ax-addass 8287  ax-distr 8289  ax-i2m1 8290  ax-0id 8293  ax-rnegex 8294  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-apti 8300  ax-pre-ltadd 8301

This proof depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-isom 5389  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-sup 7330  df-inf 7331  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-sub 8506  df-neg 8507  df-xneg 10191

This theorem is used by:  xrminmax  12056