Theorem supminfxrrnmpt 39701

Description: The indexed supremum of a set of reals is the negation of the indexed infimum of that set's image under negation. (Contributed by Glauco Siliprandi, 2-Jan-2022.)

Hypotheses

Ref	Expression
supminfxrrnmpt.x	|- F/ x ph
supminfxrrnmpt.b	|- ( ( ph /\ x e. A ) -> B e. RR* )

Assertion

Ref	Expression
supminfxrrnmpt	|- ( ph -> sup ( ran ( x e. A |-> B ) , RR* , < ) = -einf ( ran ( x e. A |-> -e B ) , RR* , < ) )

Distinct variable group:   x, A

Allowed substitution hints:   ph( x)   B( x)

Proof of Theorem supminfxrrnmpt

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		supminfxrrnmpt.x	. . . 4 |- F/ x ph
2		eqid 2622	. . . 4 |- ( x e. A |-> B ) = ( x e. A |-> B )
3		supminfxrrnmpt.b	. . . 4 |- ( ( ph /\ x e. A ) -> B e. RR* )
4	1, 2, 3	rnmptssd 39385	. . 3 |- ( ph -> ran ( x e. A |-> B ) C_ RR* )
5	4	supminfxr2 39699	. 2 |- ( ph -> sup ( ran ( x e. A |-> B ) , RR* , < ) = -einf ( { y e. RR* | -e y e. ran ( x e. A |-> B ) } , RR* , < ) )
6		xnegex 12039	. . . . . . . . . . . 12 |- -e y e. _V
7	2	elrnmpt 5372	. . . . . . . . . . . 12 |- ( -e y e. _V -> ( -e y e. ran ( x e. A |-> B ) <-> E. x e. A -e y = B ) )
8	6, 7	ax-mp 5	. . . . . . . . . . 11 |- ( -e y e. ran ( x e. A |-> B ) <-> E. x e. A -e y = B )
9	8	biimpi 206	. . . . . . . . . 10 |- ( -e y e. ran ( x e. A |-> B ) -> E. x e. A -e y = B )
10		eqid 2622	. . . . . . . . . . 11 |- ( x e. A |-> -e B ) = ( x e. A |-> -e B )
11		xnegneg 12045	. . . . . . . . . . . . . . . . 17 |- ( y e. RR* -> -e -e y = y )
12	11	eqcomd 2628	. . . . . . . . . . . . . . . 16 |- ( y e. RR* -> y = -e -e y )
13	12	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( y e. RR* /\ -e y = B ) -> y = -e -e y )
14		xnegeq 12038	. . . . . . . . . . . . . . . 16 |- ( -e y = B -> -e -e y = -e B )
15	14	adantl 482	. . . . . . . . . . . . . . 15 |- ( ( y e. RR* /\ -e y = B ) -> -e -e y = -e B )
16	13, 15	eqtrd 2656	. . . . . . . . . . . . . 14 |- ( ( y e. RR* /\ -e y = B ) -> y = -e B )
17	16	ex 450	. . . . . . . . . . . . 13 |- ( y e. RR* -> ( -e y = B -> y = -e B ) )
18	17	reximdv 3016	. . . . . . . . . . . 12 |- ( y e. RR* -> ( E. x e. A -e y = B -> E. x e. A y = -e B ) )
19	18	imp 445	. . . . . . . . . . 11 |- ( ( y e. RR* /\ E. x e. A -e y = B ) -> E. x e. A y = -e B )
20		simpl 473	. . . . . . . . . . 11 |- ( ( y e. RR* /\ E. x e. A -e y = B ) -> y e. RR* )
21	10, 19, 20	elrnmptd 39366	. . . . . . . . . 10 |- ( ( y e. RR* /\ E. x e. A -e y = B ) -> y e. ran ( x e. A |-> -e B ) )
22	9, 21	sylan2 491	. . . . . . . . 9 |- ( ( y e. RR* /\ -e y e. ran ( x e. A |-> B ) ) -> y e. ran ( x e. A |-> -e B ) )
23	22	ex 450	. . . . . . . 8 |- ( y e. RR* -> ( -e y e. ran ( x e. A |-> B ) -> y e. ran ( x e. A |-> -e B ) ) )
24	23	rgen 2922	. . . . . . 7 |- A. y e. RR* ( -e y e. ran ( x e. A |-> B ) -> y e. ran ( x e. A |-> -e B ) )
25		rabss 3679	. . . . . . . 8 |- ( { y e. RR* | -e y e. ran ( x e. A |-> B ) } C_ ran ( x e. A |-> -e B ) <-> A. y e. RR* ( -e y e. ran ( x e. A |-> B ) -> y e. ran ( x e. A |-> -e B ) ) )
26	25	biimpri 218	. . . . . . 7 |- ( A. y e. RR* ( -e y e. ran ( x e. A |-> B ) -> y e. ran ( x e. A |-> -e B ) ) -> { y e. RR* | -e y e. ran ( x e. A |-> B ) } C_ ran ( x e. A |-> -e B ) )
27	24, 26	ax-mp 5	. . . . . 6 |- { y e. RR* | -e y e. ran ( x e. A |-> B ) } C_ ran ( x e. A |-> -e B )
28	27	a1i 11	. . . . 5 |- ( ph -> { y e. RR* | -e y e. ran ( x e. A |-> B ) } C_ ran ( x e. A |-> -e B ) )
29		nfcv 2764	. . . . . . . 8 |- F/_ x -e y
30		nfmpt1 4747	. . . . . . . . 9 |- F/_ x ( x e. A |-> B )
31	30	nfrn 5368	. . . . . . . 8 |- F/_ x ran ( x e. A |-> B )
32	29, 31	nfel 2777	. . . . . . 7 |- F/ x -e y e. ran ( x e. A |-> B )
33		nfcv 2764	. . . . . . 7 |- F/_ x RR*
34	32, 33	nfrab 3123	. . . . . 6 |- F/_ x { y e. RR* | -e y e. ran ( x e. A |-> B ) }
35		xnegeq 12038	. . . . . . . 8 |- ( y = -e B -> -e y = -e -e B )
36	35	eleq1d 2686	. . . . . . 7 |- ( y = -e B -> ( -e y e. ran ( x e. A |-> B ) <-> -e -e B e. ran ( x e. A |-> B ) ) )
37	3	xnegcld 12130	. . . . . . 7 |- ( ( ph /\ x e. A ) -> -e B e. RR* )
38		xnegneg 12045	. . . . . . . . 9 |- ( B e. RR* -> -e -e B = B )
39	3, 38	syl 17	. . . . . . . 8 |- ( ( ph /\ x e. A ) -> -e -e B = B )
40		simpr 477	. . . . . . . . 9 |- ( ( ph /\ x e. A ) -> x e. A )
41	2, 40, 3	elrnmpt1d 39435	. . . . . . . 8 |- ( ( ph /\ x e. A ) -> B e. ran ( x e. A |-> B ) )
42	39, 41	eqeltrd 2701	. . . . . . 7 |- ( ( ph /\ x e. A ) -> -e -e B e. ran ( x e. A |-> B ) )
43	36, 37, 42	elrabd 3365	. . . . . 6 |- ( ( ph /\ x e. A ) -> -e B e. { y e. RR* | -e y e. ran ( x e. A |-> B ) } )
44	1, 34, 10, 43	rnmptssdf 39469	. . . . 5 |- ( ph -> ran ( x e. A |-> -e B ) C_ { y e. RR* | -e y e. ran ( x e. A |-> B ) } )
45	28, 44	eqssd 3620	. . . 4 |- ( ph -> { y e. RR* | -e y e. ran ( x e. A |-> B ) } = ran ( x e. A |-> -e B ) )
46	45	infeq1d 8383	. . 3 |- ( ph -> inf ( { y e. RR* | -e y e. ran ( x e. A |-> B ) } , RR* , < ) = inf ( ran ( x e. A |-> -e B ) , RR* , < ) )
47	46	xnegeqd 39664	. 2 |- ( ph -> -einf ( { y e. RR* | -e y e. ran ( x e. A |-> B ) } , RR* , < ) = -einf ( ran ( x e. A |-> -e B ) , RR* , < ) )
48	5, 47	eqtrd 2656	1 |- ( ph -> sup ( ran ( x e. A |-> B ) , RR* , < ) = -einf ( ran ( x e. A |-> -e B ) , RR* , < ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   F/wnf 1708   e. wcel 1990   A.wral 2912   E.wrex 2913   {crab 2916   _Vcvv 3200   C_ wss 3574   |-> cmpt 4729   ran crn 5115   supcsup 8346  infcinf 8347   RR*cxr 10073   < clt 10074   -ecxne 11943

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-xneg 11946

This theorem is referenced by:  liminfvalxr  40015