Theorem supminfex 9662

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 9660	. . . 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 3264	. . . . 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 9659	. . 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 2913	. . . . . . 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 3179	. . . . . 6 |- ( ( ph /\ x e. A ) -> x e. RR )
10		negeq 8212	. . . . . . . . . 10 |- ( z = x -> -u z = -u x )
11	10	eleq1d 2262	. . . . . . . . 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 2917	. . . . . . . 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 8280	. . . . . . . . 9 |- ( x e. RR -> -u x e. RR )
14		negeq 8212	. . . . . . . . . . 11 |- ( w = -u x -> -u w = -u -u x )
15	14	eleq1d 2262	. . . . . . . . . 10 |- ( w = -u x -> ( -u w e. A <-> -u -u x e. A ) )
16	15	elrab3 2917	. . . . . . . . 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 8005	. . . . . . . . . 10 |- ( x e. RR -> x e. CC )
19	18	negnegd 8321	. . . . . . . . 9 |- ( x e. RR -> -u -u x = x )
20	19	eleq1d 2262	. . . . . . . 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 2192	. . . . 5 |- ( ph -> { z e. RR | -u z e. { w e. RR | -u w e. A } } = A )
24	23	supeq1d 7046	. . . 4 |- ( ph -> sup ( { z e. RR | -u z e. { w e. RR | -u w e. A } } , RR , < ) = sup ( A , RR , < ) )
25	24	negeqd 8214	. . 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 2226	. 2 |- ( ph -> inf ( { w e. RR | -u w e. A } , RR , < ) = -u sup ( A , RR , < ) )
27		lttri3 8099	. . . . . 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 7082	. . . 4 |- ( ph -> inf ( { w e. RR | -u w e. A } , RR , < ) e. RR )
30	29	recnd 8048	. . 3 |- ( ph -> inf ( { w e. RR | -u w e. A } , RR , < ) e. CC )
31	28, 1	supclti 7057	. . . 4 |- ( ph -> sup ( A , RR , < ) e. RR )
32	31	recnd 8048	. . 3 |- ( ph -> sup ( A , RR , < ) e. CC )
33		negcon2 8272	. . 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 2164   A.wral 2472   E.wrex 2473   {crab 2476   C_ wss 3153   class class class wbr 4029   supcsup 7041  infcinf 7042   CCcc 7870   RRcr 7871   < clt 8054   -ucneg 8191

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 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-addass 7974  ax-distr 7976  ax-i2m1 7977  ax-0id 7980  ax-rnegex 7981  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-apti 7987  ax-pre-ltadd 7988

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 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-isom 5263  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-sup 7043  df-inf 7044  df-pnf 8056  df-mnf 8057  df-ltxr 8059  df-sub 8192  df-neg 8193

This theorem is referenced by: (None)