Theorem infrenegsupex 8682

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 7191	. . . . . 6 |- ( ( f e. RR /\ g e. RR ) -> ( f = g <-> ( -. f < g /\ -. g < f ) ) )
2	1	adantl 271	. . . . 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 6436	. . . 4 |- ( ph -> inf ( A , RR , < ) e. RR )
5	4	recnd 7147	. . 3 |- ( ph -> inf ( A , RR , < ) e. CC )
6	5	negnegd 7410	. 2 |- ( ph -> -u -uinf ( A , RR , < ) = inf ( A , RR , < ) )
7		negeq 7301	. . . . . . . . 9 |- ( w = z -> -u w = -u z )
8	7	cbvmptv 3873	. . . . . . . 8 |- ( w e. RR |-> -u w ) = ( z e. RR |-> -u z )
9	8	mptpreima 4834	. . . . . . 7 |- ( `' ( w e. RR |-> -u w ) " A ) = { z e. RR | -u z e. A }
10		eqid 2081	. . . . . . . . . 10 |- ( w e. RR |-> -u w ) = ( w e. RR |-> -u w )
11	10	negiso 8033	. . . . . . . . 9 |- ( ( w e. RR |-> -u w ) Isom < , `' < ( RR , RR ) /\ `' ( w e. RR |-> -u w ) = ( w e. RR |-> -u w ) )
12	11	simpri 111	. . . . . . . 8 |- `' ( w e. RR |-> -u w ) = ( w e. RR |-> -u w )
13	12	imaeq1i 4685	. . . . . . 7 |- ( `' ( w e. RR |-> -u w ) " A ) = ( ( w e. RR |-> -u w ) " A )
14	9, 13	eqtr3i 2103	. . . . . 6 |- { z e. RR | -u z e. A } = ( ( w e. RR |-> -u w ) " A )
15	14	supeq1i 6401	. . . . 5 |- sup ( { z e. RR | -u z e. A } , RR , < ) = sup ( ( ( w e. RR |-> -u w ) " A ) , RR , < )
16	11	simpli 109	. . . . . . . . 9 |- ( w e. RR |-> -u w ) Isom < , `' < ( RR , RR )
17		isocnv 5471	. . . . . . . . 9 |- ( ( w e. RR |-> -u w ) Isom < , `' < ( RR , RR ) -> `' ( w e. RR |-> -u w ) Isom `' < , < ( RR , RR ) )
18	16, 17	ax-mp 7	. . . . . . . 8 |- `' ( w e. RR |-> -u w ) Isom `' < , < ( RR , RR )
19		isoeq1 5461	. . . . . . . . 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 7	. . . . . . . 8 |- ( `' ( w e. RR |-> -u w ) Isom `' < , < ( RR , RR ) <-> ( w e. RR |-> -u w ) Isom `' < , < ( RR , RR ) )
21	18, 20	mpbi 143	. . . . . . 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 6431	. . . . . 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 6432	. . . . . 6 |- ( ( ph /\ ( f e. RR /\ g e. RR ) ) -> ( f = g <-> ( -. f `' < g /\ -. g `' < f ) ) )
26	22, 23, 24, 25	supisoti 6423	. . . . 5 |- ( ph -> sup ( ( ( w e. RR |-> -u w ) " A ) , RR , < ) = ( ( w e. RR |-> -u w ) ` sup ( A , RR , `' < ) ) )
27	15, 26	syl5eq 2125	. . . 4 |- ( ph -> sup ( { z e. RR | -u z e. A } , RR , < ) = ( ( w e. RR |-> -u w ) ` sup ( A , RR , `' < ) ) )
28		df-inf 6398	. . . . . . 7 |- inf ( A , RR , < ) = sup ( A , RR , `' < )
29	28	eqcomi 2085	. . . . . 6 |- sup ( A , RR , `' < ) = inf ( A , RR , < )
30	29	fveq2i 5201	. . . . 5 |- ( ( w e. RR |-> -u w ) ` sup ( A , RR , `' < ) ) = ( ( w e. RR |-> -u w ) ` inf ( A , RR , < ) )
31		eqidd 2082	. . . . . 6 |- ( ph -> ( w e. RR |-> -u w ) = ( w e. RR |-> -u w ) )
32		negeq 7301	. . . . . . 7 |- ( w = inf ( A , RR , < ) -> -u w = -uinf ( A , RR , < ) )
33	32	adantl 271	. . . . . 6 |- ( ( ph /\ w = inf ( A , RR , < ) ) -> -u w = -uinf ( A , RR , < ) )
34	5	negcld 7406	. . . . . 6 |- ( ph -> -uinf ( A , RR , < ) e. CC )
35	31, 33, 4, 34	fvmptd 5274	. . . . 5 |- ( ph -> ( ( w e. RR |-> -u w ) ` inf ( A , RR , < ) ) = -uinf ( A , RR , < ) )
36	30, 35	syl5eq 2125	. . . 4 |- ( ph -> ( ( w e. RR |-> -u w ) ` sup ( A , RR , `' < ) ) = -uinf ( A , RR , < ) )
37	27, 36	eqtr2d 2114	. . 3 |- ( ph -> -uinf ( A , RR , < ) = sup ( { z e. RR | -u z e. A } , RR , < ) )
38	37	negeqd 7303	. 2 |- ( ph -> -u -uinf ( A , RR , < ) = -u sup ( { z e. RR | -u z e. A } , RR , < ) )
39	6, 38	eqtr3d 2115	1 |- ( ph -> inf ( A , RR , < ) = -u sup ( { z e. RR | -u z e. A } , RR , < ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   e. wcel 1433   A.wral 2348   E.wrex 2349   {crab 2352   C_ wss 2973   class class class wbr 3785   |-> cmpt 3839   `'ccnv 4362   "cima 4366   ` cfv 4922   Isom wiso 4923   supcsup 6395  infcinf 6396   CCcc 6979   RRcr 6980   < clt 7153   -ucneg 7280

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-cnex 7067  ax-resscn 7068  ax-1cn 7069  ax-1re 7070  ax-icn 7071  ax-addcl 7072  ax-addrcl 7073  ax-mulcl 7074  ax-addcom 7076  ax-addass 7078  ax-distr 7080  ax-i2m1 7081  ax-0id 7084  ax-rnegex 7085  ax-cnre 7087  ax-pre-ltirr 7088  ax-pre-apti 7091  ax-pre-ltadd 7092

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-ral 2353  df-rex 2354  df-reu 2355  df-rmo 2356  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-mpt 3841  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-isom 4931  df-riota 5488  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-sup 6397  df-inf 6398  df-pnf 7155  df-mnf 7156  df-ltxr 7158  df-sub 7281  df-neg 7282

This theorem is referenced by:  supminfex  8685  minmax  10112  infssuzcldc  10347