Theorem supminfex 9054

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 9052	. . . 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 3104	. . . . 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 9051	. . 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 2766	. . . . . . 7 |- ( x e. { z e. RR | -u z e. { w e. RR | -u w e. A } } -> x e. RR )
8	7	adantl 271	. . . . . 6 |- ( ( ph /\ x e. { z e. RR | -u z e. { w e. RR | -u w e. A } } ) -> x e. RR )
9	2	sselda 3023	. . . . . 6 |- ( ( ph /\ x e. A ) -> x e. RR )
10		negeq 7654	. . . . . . . . . 10 |- ( z = x -> -u z = -u x )
11	10	eleq1d 2156	. . . . . . . . 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 2770	. . . . . . . 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 7722	. . . . . . . . 9 |- ( x e. RR -> -u x e. RR )
14		negeq 7654	. . . . . . . . . . 11 |- ( w = -u x -> -u w = -u -u x )
15	14	eleq1d 2156	. . . . . . . . . 10 |- ( w = -u x -> ( -u w e. A <-> -u -u x e. A ) )
16	15	elrab3 2770	. . . . . . . . 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 7454	. . . . . . . . . 10 |- ( x e. RR -> x e. CC )
19	18	negnegd 7763	. . . . . . . . 9 |- ( x e. RR -> -u -u x = x )
20	19	eleq1d 2156	. . . . . . . 8 |- ( x e. RR -> ( -u -u x e. A <-> x e. A ) )
21	12, 17, 20	3bitrd 212	. . . . . . 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 271	. . . . . 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 2087	. . . . 5 |- ( ph -> { z e. RR | -u z e. { w e. RR | -u w e. A } } = A )
24	23	supeq1d 6661	. . . 4 |- ( ph -> sup ( { z e. RR | -u z e. { w e. RR | -u w e. A } } , RR , < ) = sup ( A , RR , < ) )
25	24	negeqd 7656	. . 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 2120	. 2 |- ( ph -> inf ( { w e. RR | -u w e. A } , RR , < ) = -u sup ( A , RR , < ) )
27		lttri3 7544	. . . . . 6 |- ( ( f e. RR /\ g e. RR ) -> ( f = g <-> ( -. f < g /\ -. g < f ) ) )
28	27	adantl 271	. . . . 5 |- ( ( ph /\ ( f e. RR /\ g e. RR ) ) -> ( f = g <-> ( -. f < g /\ -. g < f ) ) )
29	28, 3	infclti 6697	. . . 4 |- ( ph -> inf ( { w e. RR | -u w e. A } , RR , < ) e. RR )
30	29	recnd 7495	. . 3 |- ( ph -> inf ( { w e. RR | -u w e. A } , RR , < ) e. CC )
31	28, 1	supclti 6672	. . . 4 |- ( ph -> sup ( A , RR , < ) e. RR )
32	31	recnd 7495	. . 3 |- ( ph -> sup ( A , RR , < ) e. CC )
33		negcon2 7714	. . 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 403	. 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 145	1 |- ( ph -> sup ( A , RR , < ) = -uinf ( { w e. RR | -u w e. A } , RR , < ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1289   e. wcel 1438   A.wral 2359   E.wrex 2360   {crab 2363   C_ wss 2997   class class class wbr 3837   supcsup 6656  infcinf 6657   CCcc 7327   RRcr 7328   < clt 7501   -ucneg 7633

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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-cnex 7415  ax-resscn 7416  ax-1cn 7417  ax-1re 7418  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-addcom 7424  ax-addass 7426  ax-distr 7428  ax-i2m1 7429  ax-0id 7432  ax-rnegex 7433  ax-cnre 7435  ax-pre-ltirr 7436  ax-pre-apti 7439  ax-pre-ltadd 7440

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-mpt 3893  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-isom 5011  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-sup 6658  df-inf 6659  df-pnf 7503  df-mnf 7504  df-ltxr 7506  df-sub 7634  df-neg 7635

This theorem is referenced by: (None)