Theorem supminfrnmpt 39672

Description: The indexed supremum of a bounded-above 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
supminfrnmpt.x	|- F/ x ph
supminfrnmpt.a	|- ( ph -> A =/= (/) )
supminfrnmpt.b	|- ( ( ph /\ x e. A ) -> B e. RR )
supminfrnmpt.y	|- ( ph -> E. y e. RR A. x e. A B <_ y )

Assertion

Ref	Expression
supminfrnmpt	|- ( ph -> sup ( ran ( x e. A |-> B ) , RR , < ) = -uinf ( ran ( x e. A |-> -u B ) , RR , < ) )

Distinct variable groups:   x, A, y   y, B

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

Proof of Theorem supminfrnmpt

Dummy variables w z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		supminfrnmpt.x	. . . 4 |- F/ x ph
2		eqid 2622	. . . 4 |- ( x e. A |-> B ) = ( x e. A |-> B )
3		supminfrnmpt.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		supminfrnmpt.a	. . . 4 |- ( ph -> A =/= (/) )
6	1, 3, 2, 5	rnmptn0 39413	. . 3 |- ( ph -> ran ( x e. A |-> B ) =/= (/) )
7		supminfrnmpt.y	. . . 4 |- ( ph -> E. y e. RR A. x e. A B <_ y )
8	1, 3	rnmptbd 39471	. . . 4 |- ( ph -> ( E. y e. RR A. x e. A B <_ y <-> E. y e. RR A. z e. ran ( x e. A |-> B ) z <_ y ) )
9	7, 8	mpbid 222	. . 3 |- ( ph -> E. y e. RR A. z e. ran ( x e. A |-> B ) z <_ y )
10		supminf 11775	. . 3 |- ( ( ran ( x e. A |-> B ) C_ RR /\ ran ( x e. A |-> B ) =/= (/) /\ E. y e. RR A. z e. ran ( x e. A |-> B ) z <_ y ) -> sup ( ran ( x e. A |-> B ) , RR , < ) = -uinf ( { w e. RR | -u w e. ran ( x e. A |-> B ) } , RR , < ) )
11	4, 6, 9, 10	syl3anc 1326	. 2 |- ( ph -> sup ( ran ( x e. A |-> B ) , RR , < ) = -uinf ( { w e. RR | -u w e. ran ( x e. A |-> B ) } , RR , < ) )
12		eqid 2622	. . . . . . . . 9 |- ( x e. A |-> -u B ) = ( x e. A |-> -u B )
13		simpr 477	. . . . . . . . . . . 12 |- ( ( w e. RR /\ -u w e. ran ( x e. A |-> B ) ) -> -u w e. ran ( x e. A |-> B ) )
14		renegcl 10344	. . . . . . . . . . . . . 14 |- ( w e. RR -> -u w e. RR )
15	2	elrnmpt 5372	. . . . . . . . . . . . . 14 |- ( -u w e. RR -> ( -u w e. ran ( x e. A |-> B ) <-> E. x e. A -u w = B ) )
16	14, 15	syl 17	. . . . . . . . . . . . 13 |- ( w e. RR -> ( -u w e. ran ( x e. A |-> B ) <-> E. x e. A -u w = B ) )
17	16	adantr 481	. . . . . . . . . . . 12 |- ( ( w e. RR /\ -u w e. ran ( x e. A |-> B ) ) -> ( -u w e. ran ( x e. A |-> B ) <-> E. x e. A -u w = B ) )
18	13, 17	mpbid 222	. . . . . . . . . . 11 |- ( ( w e. RR /\ -u w e. ran ( x e. A |-> B ) ) -> E. x e. A -u w = B )
19	18	adantll 750	. . . . . . . . . 10 |- ( ( ( ph /\ w e. RR ) /\ -u w e. ran ( x e. A |-> B ) ) -> E. x e. A -u w = B )
20		nfv 1843	. . . . . . . . . . . . 13 |- F/ x w e. RR
21	1, 20	nfan 1828	. . . . . . . . . . . 12 |- F/ x ( ph /\ w e. RR )
22		negeq 10273	. . . . . . . . . . . . . . . . . . 19 |- ( -u w = B -> -u -u w = -u B )
23	22	eqcomd 2628	. . . . . . . . . . . . . . . . . 18 |- ( -u w = B -> -u B = -u -u w )
24	23	adantl 482	. . . . . . . . . . . . . . . . 17 |- ( ( w e. RR /\ -u w = B ) -> -u B = -u -u w )
25		recn 10026	. . . . . . . . . . . . . . . . . . 19 |- ( w e. RR -> w e. CC )
26	25	negnegd 10383	. . . . . . . . . . . . . . . . . 18 |- ( w e. RR -> -u -u w = w )
27	26	adantr 481	. . . . . . . . . . . . . . . . 17 |- ( ( w e. RR /\ -u w = B ) -> -u -u w = w )
28	24, 27	eqtr2d 2657	. . . . . . . . . . . . . . . 16 |- ( ( w e. RR /\ -u w = B ) -> w = -u B )
29	28	ex 450	. . . . . . . . . . . . . . 15 |- ( w e. RR -> ( -u w = B -> w = -u B ) )
30	29	adantl 482	. . . . . . . . . . . . . 14 |- ( ( ph /\ w e. RR ) -> ( -u w = B -> w = -u B ) )
31	30	adantr 481	. . . . . . . . . . . . 13 |- ( ( ( ph /\ w e. RR ) /\ x e. A ) -> ( -u w = B -> w = -u B ) )
32		negeq 10273	. . . . . . . . . . . . . . . . 17 |- ( w = -u B -> -u w = -u -u B )
33	32	adantl 482	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ x e. A ) /\ w = -u B ) -> -u w = -u -u B )
34	3	recnd 10068	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ x e. A ) -> B e. CC )
35	34	negnegd 10383	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ x e. A ) -> -u -u B = B )
36	35	adantr 481	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ x e. A ) /\ w = -u B ) -> -u -u B = B )
37	33, 36	eqtrd 2656	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ x e. A ) /\ w = -u B ) -> -u w = B )
38	37	ex 450	. . . . . . . . . . . . . 14 |- ( ( ph /\ x e. A ) -> ( w = -u B -> -u w = B ) )
39	38	adantlr 751	. . . . . . . . . . . . 13 |- ( ( ( ph /\ w e. RR ) /\ x e. A ) -> ( w = -u B -> -u w = B ) )
40	31, 39	impbid 202	. . . . . . . . . . . 12 |- ( ( ( ph /\ w e. RR ) /\ x e. A ) -> ( -u w = B <-> w = -u B ) )
41	21, 40	rexbida 3047	. . . . . . . . . . 11 |- ( ( ph /\ w e. RR ) -> ( E. x e. A -u w = B <-> E. x e. A w = -u B ) )
42	41	adantr 481	. . . . . . . . . 10 |- ( ( ( ph /\ w e. RR ) /\ -u w e. ran ( x e. A |-> B ) ) -> ( E. x e. A -u w = B <-> E. x e. A w = -u B ) )
43	19, 42	mpbid 222	. . . . . . . . 9 |- ( ( ( ph /\ w e. RR ) /\ -u w e. ran ( x e. A |-> B ) ) -> E. x e. A w = -u B )
44		simplr 792	. . . . . . . . 9 |- ( ( ( ph /\ w e. RR ) /\ -u w e. ran ( x e. A |-> B ) ) -> w e. RR )
45	12, 43, 44	elrnmptd 39366	. . . . . . . 8 |- ( ( ( ph /\ w e. RR ) /\ -u w e. ran ( x e. A |-> B ) ) -> w e. ran ( x e. A |-> -u B ) )
46	45	ex 450	. . . . . . 7 |- ( ( ph /\ w e. RR ) -> ( -u w e. ran ( x e. A |-> B ) -> w e. ran ( x e. A |-> -u B ) ) )
47	46	ralrimiva 2966	. . . . . 6 |- ( ph -> A. w e. RR ( -u w e. ran ( x e. A |-> B ) -> w e. ran ( x e. A |-> -u B ) ) )
48		rabss 3679	. . . . . 6 |- ( { w e. RR | -u w e. ran ( x e. A |-> B ) } C_ ran ( x e. A |-> -u B ) <-> A. w e. RR ( -u w e. ran ( x e. A |-> B ) -> w e. ran ( x e. A |-> -u B ) ) )
49	47, 48	sylibr 224	. . . . 5 |- ( ph -> { w e. RR | -u w e. ran ( x e. A |-> B ) } C_ ran ( x e. A |-> -u B ) )
50		nfcv 2764	. . . . . . . 8 |- F/_ x -u w
51		nfmpt1 4747	. . . . . . . . 9 |- F/_ x ( x e. A |-> B )
52	51	nfrn 5368	. . . . . . . 8 |- F/_ x ran ( x e. A |-> B )
53	50, 52	nfel 2777	. . . . . . 7 |- F/ x -u w e. ran ( x e. A |-> B )
54		nfcv 2764	. . . . . . 7 |- F/_ x RR
55	53, 54	nfrab 3123	. . . . . 6 |- F/_ x { w e. RR | -u w e. ran ( x e. A |-> B ) }
56	32	eleq1d 2686	. . . . . . 7 |- ( w = -u B -> ( -u w e. ran ( x e. A |-> B ) <-> -u -u B e. ran ( x e. A |-> B ) ) )
57	3	renegcld 10457	. . . . . . 7 |- ( ( ph /\ x e. A ) -> -u B e. RR )
58		simpr 477	. . . . . . . . 9 |- ( ( ph /\ x e. A ) -> x e. A )
59	2	elrnmpt1 5374	. . . . . . . . 9 |- ( ( x e. A /\ B e. RR ) -> B e. ran ( x e. A |-> B ) )
60	58, 3, 59	syl2anc 693	. . . . . . . 8 |- ( ( ph /\ x e. A ) -> B e. ran ( x e. A |-> B ) )
61	35, 60	eqeltrd 2701	. . . . . . 7 |- ( ( ph /\ x e. A ) -> -u -u B e. ran ( x e. A |-> B ) )
62	56, 57, 61	elrabd 3365	. . . . . 6 |- ( ( ph /\ x e. A ) -> -u B e. { w e. RR | -u w e. ran ( x e. A |-> B ) } )
63	1, 55, 12, 62	rnmptssdf 39469	. . . . 5 |- ( ph -> ran ( x e. A |-> -u B ) C_ { w e. RR | -u w e. ran ( x e. A |-> B ) } )
64	49, 63	eqssd 3620	. . . 4 |- ( ph -> { w e. RR | -u w e. ran ( x e. A |-> B ) } = ran ( x e. A |-> -u B ) )
65	64	infeq1d 8383	. . 3 |- ( ph -> inf ( { w e. RR | -u w e. ran ( x e. A |-> B ) } , RR , < ) = inf ( ran ( x e. A |-> -u B ) , RR , < ) )
66	65	negeqd 10275	. 2 |- ( ph -> -uinf ( { w e. RR | -u w e. ran ( x e. A |-> B ) } , RR , < ) = -uinf ( ran ( x e. A |-> -u B ) , RR , < ) )
67	11, 66	eqtrd 2656	1 |- ( ph -> sup ( ran ( x e. A |-> B ) , RR , < ) = -uinf ( ran ( x e. A |-> -u B ) , RR , < ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   F/wnf 1708   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   {crab 2916   C_ wss 3574   (/)c0 3915   class class class wbr 4653   |-> cmpt 4729   ran crn 5115   supcsup 8346  infcinf 8347   RRcr 9935   < clt 10074   <_ cle 10075   -ucneg 10267

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-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

This theorem is referenced by: (None)