Theorem infrenegsupex 9889

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 8318	. . . . . 6 |- ( ( f e. RR /\ g e. RR ) -> ( f = g <-> ( -. f < g /\ -. g < f ) ) )
2	1	adantl 277	. . . . 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 7282	. . . 4 |- ( ph -> inf ( A , RR , < ) e. RR )
5	4	recnd 8267	. . 3 |- ( ph -> inf ( A , RR , < ) e. CC )
6	5	negnegd 8540	. 2 |- ( ph -> -u -uinf ( A , RR , < ) = inf ( A , RR , < ) )
7		negeq 8431	. . . . . . . . 9 |- ( w = z -> -u w = -u z )
8	7	cbvmptv 4190	. . . . . . . 8 |- ( w e. RR |-> -u w ) = ( z e. RR |-> -u z )
9	8	mptpreima 5237	. . . . . . 7 |- ( `' ( w e. RR |-> -u w ) " A ) = { z e. RR | -u z e. A }
10		eqid 2231	. . . . . . . . . 10 |- ( w e. RR |-> -u w ) = ( w e. RR |-> -u w )
11	10	negiso 9194	. . . . . . . . 9 |- ( ( w e. RR |-> -u w ) Isom < , `' < ( RR , RR ) /\ `' ( w e. RR |-> -u w ) = ( w e. RR |-> -u w ) )
12	11	simpri 113	. . . . . . . 8 |- `' ( w e. RR |-> -u w ) = ( w e. RR |-> -u w )
13	12	imaeq1i 5079	. . . . . . 7 |- ( `' ( w e. RR |-> -u w ) " A ) = ( ( w e. RR |-> -u w ) " A )
14	9, 13	eqtr3i 2254	. . . . . 6 |- { z e. RR | -u z e. A } = ( ( w e. RR |-> -u w ) " A )
15	14	supeq1i 7247	. . . . 5 |- sup ( { z e. RR | -u z e. A } , RR , < ) = sup ( ( ( w e. RR |-> -u w ) " A ) , RR , < )
16	11	simpli 111	. . . . . . . . 9 |- ( w e. RR |-> -u w ) Isom < , `' < ( RR , RR )
17		isocnv 5962	. . . . . . . . 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 5952	. . . . . . . . 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 145	. . . . . . 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 7277	. . . . . 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 7278	. . . . . 6 |- ( ( ph /\ ( f e. RR /\ g e. RR ) ) -> ( f = g <-> ( -. f `' < g /\ -. g `' < f ) ) )
26	22, 23, 24, 25	supisoti 7269	. . . . 5 |- ( ph -> sup ( ( ( w e. RR |-> -u w ) " A ) , RR , < ) = ( ( w e. RR |-> -u w ) ` sup ( A , RR , `' < ) ) )
27	15, 26	eqtrid 2276	. . . 4 |- ( ph -> sup ( { z e. RR | -u z e. A } , RR , < ) = ( ( w e. RR |-> -u w ) ` sup ( A , RR , `' < ) ) )
28		df-inf 7244	. . . . . . 7 |- inf ( A , RR , < ) = sup ( A , RR , `' < )
29	28	eqcomi 2235	. . . . . 6 |- sup ( A , RR , `' < ) = inf ( A , RR , < )
30	29	fveq2i 5651	. . . . 5 |- ( ( w e. RR |-> -u w ) ` sup ( A , RR , `' < ) ) = ( ( w e. RR |-> -u w ) ` inf ( A , RR , < ) )
31		eqidd 2232	. . . . . 6 |- ( ph -> ( w e. RR |-> -u w ) = ( w e. RR |-> -u w ) )
32		negeq 8431	. . . . . . 7 |- ( w = inf ( A , RR , < ) -> -u w = -uinf ( A , RR , < ) )
33	32	adantl 277	. . . . . 6 |- ( ( ph /\ w = inf ( A , RR , < ) ) -> -u w = -uinf ( A , RR , < ) )
34	5	negcld 8536	. . . . . 6 |- ( ph -> -uinf ( A , RR , < ) e. CC )
35	31, 33, 4, 34	fvmptd 5736	. . . . 5 |- ( ph -> ( ( w e. RR |-> -u w ) ` inf ( A , RR , < ) ) = -uinf ( A , RR , < ) )
36	30, 35	eqtrid 2276	. . . 4 |- ( ph -> ( ( w e. RR |-> -u w ) ` sup ( A , RR , `' < ) ) = -uinf ( A , RR , < ) )
37	27, 36	eqtr2d 2265	. . 3 |- ( ph -> -uinf ( A , RR , < ) = sup ( { z e. RR | -u z e. A } , RR , < ) )
38	37	negeqd 8433	. 2 |- ( ph -> -u -uinf ( A , RR , < ) = -u sup ( { z e. RR | -u z e. A } , RR , < ) )
39	6, 38	eqtr3d 2266	1 |- ( ph -> inf ( A , RR , < ) = -u sup ( { z e. RR | -u z e. A } , RR , < ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2202   A.wral 2511   E.wrex 2512   {crab 2515   C_ wss 3201   class class class wbr 4093   |-> cmpt 4155   `'ccnv 4730   "cima 4734   ` cfv 5333   Isom wiso 5334   supcsup 7241  infcinf 7242   CCcc 8090   RRcr 8091   < clt 8273   -ucneg 8410

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 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-addcom 8192  ax-addass 8194  ax-distr 8196  ax-i2m1 8197  ax-0id 8200  ax-rnegex 8201  ax-cnre 8203  ax-pre-ltirr 8204  ax-pre-apti 8207  ax-pre-ltadd 8208

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-isom 5342  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-sup 7243  df-inf 7244  df-pnf 8275  df-mnf 8276  df-ltxr 8278  df-sub 8411  df-neg 8412

This theorem is referenced by:  supminfex  9892  infssuzcldc  10558  minmax  11870