Theorem supminfex 9627

Description: A supremum is the negation of the infimum of that set's image under negation. (Contributed by Jim Kingdon, 14-Jan-2022.)

Hypotheses

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

Assertion

Ref	Expression
supminfex	|- ( ph -> sup ( A , RR , < ) = -uinf ( { w e. RR | -u w e. A } , RR , < ) )

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

Allowed substitution hint:   ph( w)

Proof of Theorem supminfex

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

Step	Hyp	Ref	Expression
1		supminfex.ex	. . . . 5 |- ( ph -> E. x e. RR ( A. y e. A -. x < y /\ A. y e. RR ( y < x -> E. z e. A y < z ) ) )
2		supminfex.ss	. . . . 5 |- ( ph -> A C_ RR )
3	1, 2	supinfneg 9625	. . . 4 |- ( ph -> E. x e. RR ( A. y e. { w e. RR | -u w e. A } -. y < x /\ A. y e. RR ( x < y -> E. z e. { w e. RR | -u w e. A } z < y ) ) )
4		ssrab2 3255	. . . . 5 |- { w e. RR | -u w e. A } C_ RR
5	4	a1i 9	. . . 4 |- ( ph -> { w e. RR | -u w e. A } C_ RR )
6	3, 5	infrenegsupex 9624	. . 3 |- ( ph -> inf ( { w e. RR | -u w e. A } , RR , < ) = -u sup ( { z e. RR | -u z e. { w e. RR | -u w e. A } } , RR , < ) )
7		elrabi 2905	. . . . . . 7 |- ( x e. { z e. RR | -u z e. { w e. RR | -u w e. A } } -> x e. RR )
8	7	adantl 277	. . . . . 6 |- ( ( ph /\ x e. { z e. RR | -u z e. { w e. RR | -u w e. A } } ) -> x e. RR )
9	2	sselda 3170	. . . . . 6 |- ( ( ph /\ x e. A ) -> x e. RR )
10		negeq 8180	. . . . . . . . . 10 |- ( z = x -> -u z = -u x )
11	10	eleq1d 2258	. . . . . . . . 9 |- ( z = x -> ( -u z e. { w e. RR | -u w e. A } <-> -u x e. { w e. RR | -u w e. A } ) )
12	11	elrab3 2909	. . . . . . . 8 |- ( x e. RR -> ( x e. { z e. RR | -u z e. { w e. RR | -u w e. A } } <-> -u x e. { w e. RR | -u w e. A } ) )
13		renegcl 8248	. . . . . . . . 9 |- ( x e. RR -> -u x e. RR )
14		negeq 8180	. . . . . . . . . . 11 |- ( w = -u x -> -u w = -u -u x )
15	14	eleq1d 2258	. . . . . . . . . 10 |- ( w = -u x -> ( -u w e. A <-> -u -u x e. A ) )
16	15	elrab3 2909	. . . . . . . . 9 |- ( -u x e. RR -> ( -u x e. { w e. RR | -u w e. A } <-> -u -u x e. A ) )
17	13, 16	syl 14	. . . . . . . 8 |- ( x e. RR -> ( -u x e. { w e. RR | -u w e. A } <-> -u -u x e. A ) )
18		recn 7974	. . . . . . . . . 10 |- ( x e. RR -> x e. CC )
19	18	negnegd 8289	. . . . . . . . 9 |- ( x e. RR -> -u -u x = x )
20	19	eleq1d 2258	. . . . . . . 8 |- ( x e. RR -> ( -u -u x e. A <-> x e. A ) )
21	12, 17, 20	3bitrd 214	. . . . . . 7 |- ( x e. RR -> ( x e. { z e. RR | -u z e. { w e. RR | -u w e. A } } <-> x e. A ) )
22	21	adantl 277	. . . . . 6 |- ( ( ph /\ x e. RR ) -> ( x e. { z e. RR | -u z e. { w e. RR | -u w e. A } } <-> x e. A ) )
23	8, 9, 22	eqrdav 2188	. . . . 5 |- ( ph -> { z e. RR | -u z e. { w e. RR | -u w e. A } } = A )
24	23	supeq1d 7016	. . . 4 |- ( ph -> sup ( { z e. RR | -u z e. { w e. RR | -u w e. A } } , RR , < ) = sup ( A , RR , < ) )
25	24	negeqd 8182	. . 3 |- ( ph -> -u sup ( { z e. RR | -u z e. { w e. RR | -u w e. A } } , RR , < ) = -u sup ( A , RR , < ) )
26	6, 25	eqtrd 2222	. 2 |- ( ph -> inf ( { w e. RR | -u w e. A } , RR , < ) = -u sup ( A , RR , < ) )
27		lttri3 8067	. . . . . 6 |- ( ( f e. RR /\ g e. RR ) -> ( f = g <-> ( -. f < g /\ -. g < f ) ) )
28	27	adantl 277	. . . . 5 |- ( ( ph /\ ( f e. RR /\ g e. RR ) ) -> ( f = g <-> ( -. f < g /\ -. g < f ) ) )
29	28, 3	infclti 7052	. . . 4 |- ( ph -> inf ( { w e. RR | -u w e. A } , RR , < ) e. RR )
30	29	recnd 8016	. . 3 |- ( ph -> inf ( { w e. RR | -u w e. A } , RR , < ) e. CC )
31	28, 1	supclti 7027	. . . 4 |- ( ph -> sup ( A , RR , < ) e. RR )
32	31	recnd 8016	. . 3 |- ( ph -> sup ( A , RR , < ) e. CC )
33		negcon2 8240	. . 3 |- ( (inf ( { w e. RR | -u w e. A } , RR , < ) e. CC /\ sup ( A , RR , < ) e. CC ) -> (inf ( { w e. RR | -u w e. A } , RR , < ) = -u sup ( A , RR , < ) <-> sup ( A , RR , < ) = -uinf ( { w e. RR | -u w e. A } , RR , < ) ) )
34	30, 32, 33	syl2anc 411	. 2 |- ( ph -> (inf ( { w e. RR | -u w e. A } , RR , < ) = -u sup ( A , RR , < ) <-> sup ( A , RR , < ) = -uinf ( { w e. RR | -u w e. A } , RR , < ) ) )
35	26, 34	mpbid 147	1 |- ( ph -> sup ( A , RR , < ) = -uinf ( { w e. RR | -u w e. A } , RR , < ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2160   A.wral 2468   E.wrex 2469   {crab 2472   C_ wss 3144   class class class wbr 4018   supcsup 7011  infcinf 7012   CCcc 7839   RRcr 7840   < clt 8022   -ucneg 8159

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7932  ax-resscn 7933  ax-1cn 7934  ax-1re 7935  ax-icn 7936  ax-addcl 7937  ax-addrcl 7938  ax-mulcl 7939  ax-addcom 7941  ax-addass 7943  ax-distr 7945  ax-i2m1 7946  ax-0id 7949  ax-rnegex 7950  ax-cnre 7952  ax-pre-ltirr 7953  ax-pre-apti 7956  ax-pre-ltadd 7957

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-isom 5244  df-riota 5852  df-ov 5899  df-oprab 5900  df-mpo 5901  df-sup 7013  df-inf 7014  df-pnf 8024  df-mnf 8025  df-ltxr 8027  df-sub 8160  df-neg 8161

This theorem is referenced by: (None)