Theorem supminfex 9556

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 9554	. . . 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 3232	. . . . 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 9553	. . 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 2883	. . . . . . 7 |- ( x e. { z e. RR | -u z e. { w e. RR | -u w e. A } } -> x e. RR )
8	7	adantl 275	. . . . . 6 |- ( ( ph /\ x e. { z e. RR | -u z e. { w e. RR | -u w e. A } } ) -> x e. RR )
9	2	sselda 3147	. . . . . 6 |- ( ( ph /\ x e. A ) -> x e. RR )
10		negeq 8112	. . . . . . . . . 10 |- ( z = x -> -u z = -u x )
11	10	eleq1d 2239	. . . . . . . . 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 2887	. . . . . . . 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 8180	. . . . . . . . 9 |- ( x e. RR -> -u x e. RR )
14		negeq 8112	. . . . . . . . . . 11 |- ( w = -u x -> -u w = -u -u x )
15	14	eleq1d 2239	. . . . . . . . . 10 |- ( w = -u x -> ( -u w e. A <-> -u -u x e. A ) )
16	15	elrab3 2887	. . . . . . . . 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 7907	. . . . . . . . . 10 |- ( x e. RR -> x e. CC )
19	18	negnegd 8221	. . . . . . . . 9 |- ( x e. RR -> -u -u x = x )
20	19	eleq1d 2239	. . . . . . . 8 |- ( x e. RR -> ( -u -u x e. A <-> x e. A ) )
21	12, 17, 20	3bitrd 213	. . . . . . 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 275	. . . . . 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 2169	. . . . 5 |- ( ph -> { z e. RR | -u z e. { w e. RR | -u w e. A } } = A )
24	23	supeq1d 6964	. . . 4 |- ( ph -> sup ( { z e. RR | -u z e. { w e. RR | -u w e. A } } , RR , < ) = sup ( A , RR , < ) )
25	24	negeqd 8114	. . 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 2203	. 2 |- ( ph -> inf ( { w e. RR | -u w e. A } , RR , < ) = -u sup ( A , RR , < ) )
27		lttri3 7999	. . . . . 6 |- ( ( f e. RR /\ g e. RR ) -> ( f = g <-> ( -. f < g /\ -. g < f ) ) )
28	27	adantl 275	. . . . 5 |- ( ( ph /\ ( f e. RR /\ g e. RR ) ) -> ( f = g <-> ( -. f < g /\ -. g < f ) ) )
29	28, 3	infclti 7000	. . . 4 |- ( ph -> inf ( { w e. RR | -u w e. A } , RR , < ) e. RR )
30	29	recnd 7948	. . 3 |- ( ph -> inf ( { w e. RR | -u w e. A } , RR , < ) e. CC )
31	28, 1	supclti 6975	. . . 4 |- ( ph -> sup ( A , RR , < ) e. RR )
32	31	recnd 7948	. . 3 |- ( ph -> sup ( A , RR , < ) e. CC )
33		negcon2 8172	. . 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 409	. 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 146	1 |- ( ph -> sup ( A , RR , < ) = -uinf ( { w e. RR | -u w e. A } , RR , < ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1348   e. wcel 2141   A.wral 2448   E.wrex 2449   {crab 2452   C_ wss 3121   class class class wbr 3989   supcsup 6959  infcinf 6960   CCcc 7772   RRcr 7773   < clt 7954   -ucneg 8091

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-addcom 7874  ax-addass 7876  ax-distr 7878  ax-i2m1 7879  ax-0id 7882  ax-rnegex 7883  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-apti 7889  ax-pre-ltadd 7890

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-isom 5207  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-sup 6961  df-inf 6962  df-pnf 7956  df-mnf 7957  df-ltxr 7959  df-sub 8092  df-neg 8093

This theorem is referenced by: (None)