Theorem supminfex 9929

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 9927	. . . 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 3323	. . . . 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 9926	. . 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 2970	. . . . . . 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 3238	. . . . . 6 |- ( ( ph /\ x e. A ) -> x e. RR )
10		negeq 8466	. . . . . . . . . 10 |- ( z = x -> -u z = -u x )
11	10	eleq1d 2301	. . . . . . . . 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 2974	. . . . . . . 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 8534	. . . . . . . . 9 |- ( x e. RR -> -u x e. RR )
14		negeq 8466	. . . . . . . . . . 11 |- ( w = -u x -> -u w = -u -u x )
15	14	eleq1d 2301	. . . . . . . . . 10 |- ( w = -u x -> ( -u w e. A <-> -u -u x e. A ) )
16	15	elrab3 2974	. . . . . . . . 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 8260	. . . . . . . . . 10 |- ( x e. RR -> x e. CC )
19	18	negnegd 8575	. . . . . . . . 9 |- ( x e. RR -> -u -u x = x )
20	19	eleq1d 2301	. . . . . . . 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 2231	. . . . 5 |- ( ph -> { z e. RR | -u z e. { w e. RR | -u w e. A } } = A )
24	23	supeq1d 7278	. . . 4 |- ( ph -> sup ( { z e. RR | -u z e. { w e. RR | -u w e. A } } , RR , < ) = sup ( A , RR , < ) )
25	24	negeqd 8468	. . 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 2265	. 2 |- ( ph -> inf ( { w e. RR | -u w e. A } , RR , < ) = -u sup ( A , RR , < ) )
27		lttri3 8353	. . . . . 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 7314	. . . 4 |- ( ph -> inf ( { w e. RR | -u w e. A } , RR , < ) e. RR )
30	29	recnd 8302	. . 3 |- ( ph -> inf ( { w e. RR | -u w e. A } , RR , < ) e. CC )
31	28, 1	supclti 7289	. . . 4 |- ( ph -> sup ( A , RR , < ) e. RR )
32	31	recnd 8302	. . 3 |- ( ph -> sup ( A , RR , < ) e. CC )
33		negcon2 8526	. . 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 1398   e. wcel 2203   A.wral 2520   E.wrex 2521   {crab 2524   C_ wss 3211   class class class wbr 4109   supcsup 7273  infcinf 7274   CCcc 8125   RRcr 8126   < clt 8308   -ucneg 8445

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-addass 8229  ax-distr 8231  ax-i2m1 8232  ax-0id 8235  ax-rnegex 8236  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-apti 8242  ax-pre-ltadd 8243

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-isom 5361  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-sup 7275  df-inf 7276  df-pnf 8310  df-mnf 8311  df-ltxr 8313  df-sub 8446  df-neg 8447

This theorem is referenced by: (None)