Theorem infrenegsupex 9668

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 8106	. . . . . 6 |- ( ( f e. RR /\ g e. RR ) -> ( f = g <-> ( -. f < g /\ -. g < f ) ) )
2	1	adantl 277	. . . . 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 7089	. . . 4 |- ( ph -> inf ( A , RR , < ) e. RR )
5	4	recnd 8055	. . 3 |- ( ph -> inf ( A , RR , < ) e. CC )
6	5	negnegd 8328	. 2 |- ( ph -> -u -uinf ( A , RR , < ) = inf ( A , RR , < ) )
7		negeq 8219	. . . . . . . . 9 |- ( w = z -> -u w = -u z )
8	7	cbvmptv 4129	. . . . . . . 8 |- ( w e. RR |-> -u w ) = ( z e. RR |-> -u z )
9	8	mptpreima 5163	. . . . . . 7 |- ( `' ( w e. RR |-> -u w ) " A ) = { z e. RR | -u z e. A }
10		eqid 2196	. . . . . . . . . 10 |- ( w e. RR |-> -u w ) = ( w e. RR |-> -u w )
11	10	negiso 8982	. . . . . . . . 9 |- ( ( w e. RR |-> -u w ) Isom < , `' < ( RR , RR ) /\ `' ( w e. RR |-> -u w ) = ( w e. RR |-> -u w ) )
12	11	simpri 113	. . . . . . . 8 |- `' ( w e. RR |-> -u w ) = ( w e. RR |-> -u w )
13	12	imaeq1i 5006	. . . . . . 7 |- ( `' ( w e. RR |-> -u w ) " A ) = ( ( w e. RR |-> -u w ) " A )
14	9, 13	eqtr3i 2219	. . . . . 6 |- { z e. RR | -u z e. A } = ( ( w e. RR |-> -u w ) " A )
15	14	supeq1i 7054	. . . . 5 |- sup ( { z e. RR | -u z e. A } , RR , < ) = sup ( ( ( w e. RR |-> -u w ) " A ) , RR , < )
16	11	simpli 111	. . . . . . . . 9 |- ( w e. RR |-> -u w ) Isom < , `' < ( RR , RR )
17		isocnv 5858	. . . . . . . . 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 5848	. . . . . . . . 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 145	. . . . . . 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 7084	. . . . . 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 7085	. . . . . 6 |- ( ( ph /\ ( f e. RR /\ g e. RR ) ) -> ( f = g <-> ( -. f `' < g /\ -. g `' < f ) ) )
26	22, 23, 24, 25	supisoti 7076	. . . . 5 |- ( ph -> sup ( ( ( w e. RR |-> -u w ) " A ) , RR , < ) = ( ( w e. RR |-> -u w ) ` sup ( A , RR , `' < ) ) )
27	15, 26	eqtrid 2241	. . . 4 |- ( ph -> sup ( { z e. RR | -u z e. A } , RR , < ) = ( ( w e. RR |-> -u w ) ` sup ( A , RR , `' < ) ) )
28		df-inf 7051	. . . . . . 7 |- inf ( A , RR , < ) = sup ( A , RR , `' < )
29	28	eqcomi 2200	. . . . . 6 |- sup ( A , RR , `' < ) = inf ( A , RR , < )
30	29	fveq2i 5561	. . . . 5 |- ( ( w e. RR |-> -u w ) ` sup ( A , RR , `' < ) ) = ( ( w e. RR |-> -u w ) ` inf ( A , RR , < ) )
31		eqidd 2197	. . . . . 6 |- ( ph -> ( w e. RR |-> -u w ) = ( w e. RR |-> -u w ) )
32		negeq 8219	. . . . . . 7 |- ( w = inf ( A , RR , < ) -> -u w = -uinf ( A , RR , < ) )
33	32	adantl 277	. . . . . 6 |- ( ( ph /\ w = inf ( A , RR , < ) ) -> -u w = -uinf ( A , RR , < ) )
34	5	negcld 8324	. . . . . 6 |- ( ph -> -uinf ( A , RR , < ) e. CC )
35	31, 33, 4, 34	fvmptd 5642	. . . . 5 |- ( ph -> ( ( w e. RR |-> -u w ) ` inf ( A , RR , < ) ) = -uinf ( A , RR , < ) )
36	30, 35	eqtrid 2241	. . . 4 |- ( ph -> ( ( w e. RR |-> -u w ) ` sup ( A , RR , `' < ) ) = -uinf ( A , RR , < ) )
37	27, 36	eqtr2d 2230	. . 3 |- ( ph -> -uinf ( A , RR , < ) = sup ( { z e. RR | -u z e. A } , RR , < ) )
38	37	negeqd 8221	. 2 |- ( ph -> -u -uinf ( A , RR , < ) = -u sup ( { z e. RR | -u z e. A } , RR , < ) )
39	6, 38	eqtr3d 2231	1 |- ( ph -> inf ( A , RR , < ) = -u sup ( { z e. RR | -u z e. A } , RR , < ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2167   A.wral 2475   E.wrex 2476   {crab 2479   C_ wss 3157   class class class wbr 4033   |-> cmpt 4094   `'ccnv 4662   "cima 4666   ` cfv 5258   Isom wiso 5259   supcsup 7048  infcinf 7049   CCcc 7877   RRcr 7878   < clt 8061   -ucneg 8198

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-distr 7983  ax-i2m1 7984  ax-0id 7987  ax-rnegex 7988  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-apti 7994  ax-pre-ltadd 7995

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-isom 5267  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-sup 7050  df-inf 7051  df-pnf 8063  df-mnf 8064  df-ltxr 8066  df-sub 8199  df-neg 8200

This theorem is referenced by:  supminfex  9671  infssuzcldc  10325  minmax  11395