Theorem infrenegsupex 9382

Description: The infimum of a set of reals A is the negative of the supremum of the negatives of its elements. (Contributed by Jim Kingdon, 14-Jan-2022.)

Hypotheses

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

Assertion

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

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

Proof of Theorem infrenegsupex

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

Step	Hyp	Ref	Expression
1		lttri3 7837	. . . . . 6 |- ( ( f e. RR /\ g e. RR ) -> ( f = g <-> ( -. f < g /\ -. g < f ) ) )
2	1	adantl 275	. . . . 5 |- ( ( ph /\ ( f e. RR /\ g e. RR ) ) -> ( f = g <-> ( -. f < g /\ -. g < f ) ) )
3		infrenegsupex.ex	. . . . 5 |- ( 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 6903	. . . 4 |- ( ph -> inf ( A , RR , < ) e. RR )
5	4	recnd 7787	. . 3 |- ( ph -> inf ( A , RR , < ) e. CC )
6	5	negnegd 8057	. 2 |- ( ph -> -u -uinf ( A , RR , < ) = inf ( A , RR , < ) )
7		negeq 7948	. . . . . . . . 9 |- ( w = z -> -u w = -u z )
8	7	cbvmptv 4019	. . . . . . . 8 |- ( w e. RR |-> -u w ) = ( z e. RR |-> -u z )
9	8	mptpreima 5027	. . . . . . 7 |- ( `' ( w e. RR |-> -u w ) " A ) = { z e. RR | -u z e. A }
10		eqid 2137	. . . . . . . . . 10 |- ( w e. RR |-> -u w ) = ( w e. RR |-> -u w )
11	10	negiso 8706	. . . . . . . . 9 |- ( ( w e. RR |-> -u w ) Isom < , `' < ( RR , RR ) /\ `' ( w e. RR |-> -u w ) = ( w e. RR |-> -u w ) )
12	11	simpri 112	. . . . . . . 8 |- `' ( w e. RR |-> -u w ) = ( w e. RR |-> -u w )
13	12	imaeq1i 4873	. . . . . . 7 |- ( `' ( w e. RR |-> -u w ) " A ) = ( ( w e. RR |-> -u w ) " A )
14	9, 13	eqtr3i 2160	. . . . . 6 |- { z e. RR | -u z e. A } = ( ( w e. RR |-> -u w ) " A )
15	14	supeq1i 6868	. . . . 5 |- sup ( { z e. RR | -u z e. A } , RR , < ) = sup ( ( ( w e. RR |-> -u w ) " A ) , RR , < )
16	11	simpli 110	. . . . . . . . 9 |- ( w e. RR |-> -u w ) Isom < , `' < ( RR , RR )
17		isocnv 5705	. . . . . . . . 9 |- ( ( w e. RR |-> -u w ) Isom < , `' < ( RR , RR ) -> `' ( w e. RR |-> -u w ) Isom `' < , < ( RR , RR ) )
18	16, 17	ax-mp 5	. . . . . . . 8 |- `' ( w e. RR |-> -u w ) Isom `' < , < ( RR , RR )
19		isoeq1 5695	. . . . . . . . 9 |- ( `' ( w e. RR |-> -u w ) = ( w e. RR |-> -u w ) -> ( `' ( w e. RR |-> -u w ) Isom `' < , < ( RR , RR ) <-> ( w e. RR |-> -u w ) Isom `' < , < ( RR , RR ) ) )
20	12, 19	ax-mp 5	. . . . . . . 8 |- ( `' ( w e. RR |-> -u w ) Isom `' < , < ( RR , RR ) <-> ( w e. RR |-> -u w ) Isom `' < , < ( RR , RR ) )
21	18, 20	mpbi 144	. . . . . . 7 |- ( w e. RR |-> -u w ) Isom `' < , < ( RR , RR )
22	21	a1i 9	. . . . . 6 |- ( ph -> ( w e. RR |-> -u w ) Isom `' < , < ( RR , RR ) )
23		infrenegsupex.ss	. . . . . 6 |- ( ph -> A C_ RR )
24	3	cnvinfex 6898	. . . . . 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 6899	. . . . . 6 |- ( ( ph /\ ( f e. RR /\ g e. RR ) ) -> ( f = g <-> ( -. f `' < g /\ -. g `' < f ) ) )
26	22, 23, 24, 25	supisoti 6890	. . . . 5 |- ( ph -> sup ( ( ( w e. RR |-> -u w ) " A ) , RR , < ) = ( ( w e. RR |-> -u w ) ` sup ( A , RR , `' < ) ) )
27	15, 26	syl5eq 2182	. . . 4 |- ( ph -> sup ( { z e. RR | -u z e. A } , RR , < ) = ( ( w e. RR |-> -u w ) ` sup ( A , RR , `' < ) ) )
28		df-inf 6865	. . . . . . 7 |- inf ( A , RR , < ) = sup ( A , RR , `' < )
29	28	eqcomi 2141	. . . . . 6 |- sup ( A , RR , `' < ) = inf ( A , RR , < )
30	29	fveq2i 5417	. . . . 5 |- ( ( w e. RR |-> -u w ) ` sup ( A , RR , `' < ) ) = ( ( w e. RR |-> -u w ) ` inf ( A , RR , < ) )
31		eqidd 2138	. . . . . 6 |- ( ph -> ( w e. RR |-> -u w ) = ( w e. RR |-> -u w ) )
32		negeq 7948	. . . . . . 7 |- ( w = inf ( A , RR , < ) -> -u w = -uinf ( A , RR , < ) )
33	32	adantl 275	. . . . . 6 |- ( ( ph /\ w = inf ( A , RR , < ) ) -> -u w = -uinf ( A , RR , < ) )
34	5	negcld 8053	. . . . . 6 |- ( ph -> -uinf ( A , RR , < ) e. CC )
35	31, 33, 4, 34	fvmptd 5495	. . . . 5 |- ( ph -> ( ( w e. RR |-> -u w ) ` inf ( A , RR , < ) ) = -uinf ( A , RR , < ) )
36	30, 35	syl5eq 2182	. . . 4 |- ( ph -> ( ( w e. RR |-> -u w ) ` sup ( A , RR , `' < ) ) = -uinf ( A , RR , < ) )
37	27, 36	eqtr2d 2171	. . 3 |- ( ph -> -uinf ( A , RR , < ) = sup ( { z e. RR | -u z e. A } , RR , < ) )
38	37	negeqd 7950	. 2 |- ( ph -> -u -uinf ( A , RR , < ) = -u sup ( { z e. RR | -u z e. A } , RR , < ) )
39	6, 38	eqtr3d 2172	1 |- ( ph -> inf ( A , RR , < ) = -u sup ( { z e. RR | -u z e. A } , RR , < ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1331   e. wcel 1480   A.wral 2414   E.wrex 2415   {crab 2418   C_ wss 3066   class class class wbr 3924   |-> cmpt 3984   `'ccnv 4533   "cima 4537   ` cfv 5118   Isom wiso 5119   supcsup 6862  infcinf 6863   CCcc 7611   RRcr 7612   < clt 7793   -ucneg 7927

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-cnex 7704  ax-resscn 7705  ax-1cn 7706  ax-1re 7707  ax-icn 7708  ax-addcl 7709  ax-addrcl 7710  ax-mulcl 7711  ax-addcom 7713  ax-addass 7715  ax-distr 7717  ax-i2m1 7718  ax-0id 7721  ax-rnegex 7722  ax-cnre 7724  ax-pre-ltirr 7725  ax-pre-apti 7728  ax-pre-ltadd 7729

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-reu 2421  df-rmo 2422  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-isom 5127  df-riota 5723  df-ov 5770  df-oprab 5771  df-mpo 5772  df-sup 6864  df-inf 6865  df-pnf 7795  df-mnf 7796  df-ltxr 7798  df-sub 7928  df-neg 7929

This theorem is referenced by:  supminfex  9385  minmax  10994  infssuzcldc  11633