Theorem infrenegsupex 9523

Description: The infimum of a set of reals A is the negative of the supremum of the negatives of its elements. (Contributed by Jim Kingdon, 14-Jan-2022.)

Hypotheses

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

Assertion

Ref	Expression
infrenegsupex	|- ( ph -> inf ( A , RR , < ) = -u sup ( { z e. RR | -u z e. A } , RR , < ) )

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

Proof of Theorem infrenegsupex

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

Step	Hyp	Ref	Expression
1		lttri3 7969	. . . . . 6 |- ( ( f e. RR /\ g e. RR ) -> ( f = g <-> ( -. f < g /\ -. g < f ) ) )
2	1	adantl 275	. . . . 5 |- ( ( ph /\ ( f e. RR /\ g e. RR ) ) -> ( f = g <-> ( -. f < g /\ -. g < f ) ) )
3		infrenegsupex.ex	. . . . 5 |- ( ph -> E. x e. RR ( A. y e. A -. y < x /\ A. y e. RR ( x < y -> E. z e. A z < y ) ) )
4	2, 3	infclti 6979	. . . 4 |- ( ph -> inf ( A , RR , < ) e. RR )
5	4	recnd 7918	. . 3 |- ( ph -> inf ( A , RR , < ) e. CC )
6	5	negnegd 8191	. 2 |- ( ph -> -u -uinf ( A , RR , < ) = inf ( A , RR , < ) )
7		negeq 8082	. . . . . . . . 9 |- ( w = z -> -u w = -u z )
8	7	cbvmptv 4072	. . . . . . . 8 |- ( w e. RR |-> -u w ) = ( z e. RR |-> -u z )
9	8	mptpreima 5091	. . . . . . 7 |- ( `' ( w e. RR |-> -u w ) " A ) = { z e. RR | -u z e. A }
10		eqid 2164	. . . . . . . . . 10 |- ( w e. RR |-> -u w ) = ( w e. RR |-> -u w )
11	10	negiso 8841	. . . . . . . . 9 |- ( ( w e. RR |-> -u w ) Isom < , `' < ( RR , RR ) /\ `' ( w e. RR |-> -u w ) = ( w e. RR |-> -u w ) )
12	11	simpri 112	. . . . . . . 8 |- `' ( w e. RR |-> -u w ) = ( w e. RR |-> -u w )
13	12	imaeq1i 4937	. . . . . . 7 |- ( `' ( w e. RR |-> -u w ) " A ) = ( ( w e. RR |-> -u w ) " A )
14	9, 13	eqtr3i 2187	. . . . . 6 |- { z e. RR | -u z e. A } = ( ( w e. RR |-> -u w ) " A )
15	14	supeq1i 6944	. . . . 5 |- sup ( { z e. RR | -u z e. A } , RR , < ) = sup ( ( ( w e. RR |-> -u w ) " A ) , RR , < )
16	11	simpli 110	. . . . . . . . 9 |- ( w e. RR |-> -u w ) Isom < , `' < ( RR , RR )
17		isocnv 5773	. . . . . . . . 9 |- ( ( w e. RR |-> -u w ) Isom < , `' < ( RR , RR ) -> `' ( w e. RR |-> -u w ) Isom `' < , < ( RR , RR ) )
18	16, 17	ax-mp 5	. . . . . . . 8 |- `' ( w e. RR |-> -u w ) Isom `' < , < ( RR , RR )
19		isoeq1 5763	. . . . . . . . 9 |- ( `' ( w e. RR |-> -u w ) = ( w e. RR |-> -u w ) -> ( `' ( w e. RR |-> -u w ) Isom `' < , < ( RR , RR ) <-> ( w e. RR |-> -u w ) Isom `' < , < ( RR , RR ) ) )
20	12, 19	ax-mp 5	. . . . . . . 8 |- ( `' ( w e. RR |-> -u w ) Isom `' < , < ( RR , RR ) <-> ( w e. RR |-> -u w ) Isom `' < , < ( RR , RR ) )
21	18, 20	mpbi 144	. . . . . . 7 |- ( w e. RR |-> -u w ) Isom `' < , < ( RR , RR )
22	21	a1i 9	. . . . . 6 |- ( ph -> ( w e. RR |-> -u w ) Isom `' < , < ( RR , RR ) )
23		infrenegsupex.ss	. . . . . 6 |- ( ph -> A C_ RR )
24	3	cnvinfex 6974	. . . . . 6 |- ( ph -> E. x e. RR ( A. y e. A -. x `' < y /\ A. y e. RR ( y `' < x -> E. z e. A y `' < z ) ) )
25	2	cnvti 6975	. . . . . 6 |- ( ( ph /\ ( f e. RR /\ g e. RR ) ) -> ( f = g <-> ( -. f `' < g /\ -. g `' < f ) ) )
26	22, 23, 24, 25	supisoti 6966	. . . . 5 |- ( ph -> sup ( ( ( w e. RR |-> -u w ) " A ) , RR , < ) = ( ( w e. RR |-> -u w ) ` sup ( A , RR , `' < ) ) )
27	15, 26	syl5eq 2209	. . . 4 |- ( ph -> sup ( { z e. RR | -u z e. A } , RR , < ) = ( ( w e. RR |-> -u w ) ` sup ( A , RR , `' < ) ) )
28		df-inf 6941	. . . . . . 7 |- inf ( A , RR , < ) = sup ( A , RR , `' < )
29	28	eqcomi 2168	. . . . . 6 |- sup ( A , RR , `' < ) = inf ( A , RR , < )
30	29	fveq2i 5483	. . . . 5 |- ( ( w e. RR |-> -u w ) ` sup ( A , RR , `' < ) ) = ( ( w e. RR |-> -u w ) ` inf ( A , RR , < ) )
31		eqidd 2165	. . . . . 6 |- ( ph -> ( w e. RR |-> -u w ) = ( w e. RR |-> -u w ) )
32		negeq 8082	. . . . . . 7 |- ( w = inf ( A , RR , < ) -> -u w = -uinf ( A , RR , < ) )
33	32	adantl 275	. . . . . 6 |- ( ( ph /\ w = inf ( A , RR , < ) ) -> -u w = -uinf ( A , RR , < ) )
34	5	negcld 8187	. . . . . 6 |- ( ph -> -uinf ( A , RR , < ) e. CC )
35	31, 33, 4, 34	fvmptd 5561	. . . . 5 |- ( ph -> ( ( w e. RR |-> -u w ) ` inf ( A , RR , < ) ) = -uinf ( A , RR , < ) )
36	30, 35	syl5eq 2209	. . . 4 |- ( ph -> ( ( w e. RR |-> -u w ) ` sup ( A , RR , `' < ) ) = -uinf ( A , RR , < ) )
37	27, 36	eqtr2d 2198	. . 3 |- ( ph -> -uinf ( A , RR , < ) = sup ( { z e. RR | -u z e. A } , RR , < ) )
38	37	negeqd 8084	. 2 |- ( ph -> -u -uinf ( A , RR , < ) = -u sup ( { z e. RR | -u z e. A } , RR , < ) )
39	6, 38	eqtr3d 2199	1 |- ( ph -> inf ( A , RR , < ) = -u sup ( { z e. RR | -u z e. A } , RR , < ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1342   e. wcel 2135   A.wral 2442   E.wrex 2443   {crab 2446   C_ wss 3111   class class class wbr 3976   |-> cmpt 4037   `'ccnv 4597   "cima 4601   ` cfv 5182   Isom wiso 5183   supcsup 6938  infcinf 6939   CCcc 7742   RRcr 7743   < clt 7924   -ucneg 8061

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 604  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-sep 4094  ax-pow 4147  ax-pr 4181  ax-un 4405  ax-setind 4508  ax-cnex 7835  ax-resscn 7836  ax-1cn 7837  ax-1re 7838  ax-icn 7839  ax-addcl 7840  ax-addrcl 7841  ax-mulcl 7842  ax-addcom 7844  ax-addass 7846  ax-distr 7848  ax-i2m1 7849  ax-0id 7852  ax-rnegex 7853  ax-cnre 7855  ax-pre-ltirr 7856  ax-pre-apti 7859  ax-pre-ltadd 7860

This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-nel 2430  df-ral 2447  df-rex 2448  df-reu 2449  df-rmo 2450  df-rab 2451  df-v 2723  df-sbc 2947  df-csb 3041  df-dif 3113  df-un 3115  df-in 3117  df-ss 3124  df-pw 3555  df-sn 3576  df-pr 3577  df-op 3579  df-uni 3784  df-br 3977  df-opab 4038  df-mpt 4039  df-id 4265  df-xp 4604  df-rel 4605  df-cnv 4606  df-co 4607  df-dm 4608  df-rn 4609  df-res 4610  df-ima 4611  df-iota 5147  df-fun 5184  df-fn 5185  df-f 5186  df-f1 5187  df-fo 5188  df-f1o 5189  df-fv 5190  df-isom 5191  df-riota 5792  df-ov 5839  df-oprab 5840  df-mpo 5841  df-sup 6940  df-inf 6941  df-pnf 7926  df-mnf 7927  df-ltxr 7929  df-sub 8062  df-neg 8063

This theorem is referenced by:  supminfex  9526  minmax  11157  infssuzcldc  11869