Theorem liminfval2 40000

Description: The superior limit, relativized to an unbounded set. (Contributed by Glauco Siliprandi, 2-Jan-2022.)

Hypotheses

Ref	Expression
liminfval2.1	|- G = ( k e. RR |-> inf ( ( ( F " ( k [,) +oo ) ) i^i RR* ) , RR* , < ) )
liminfval2.2	|- ( ph -> F e. V )
liminfval2.3	|- ( ph -> A C_ RR )
liminfval2.4	|- ( ph -> sup ( A , RR* , < ) = +oo )

Assertion

Ref	Expression
liminfval2	|- ( ph -> (liminf ` F ) = sup ( ( G " A ) , RR* , < ) )

Distinct variable group:   k, F

Allowed substitution hints:   ph( k)   A( k)   G( k)   V( k)

Proof of Theorem liminfval2

Dummy variables x n j are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		liminfval2.2	. . 3 |- ( ph -> F e. V )
2		liminfval2.1	. . . . 5 |- G = ( k e. RR |-> inf ( ( ( F " ( k [,) +oo ) ) i^i RR* ) , RR* , < ) )
3		oveq1 6657	. . . . . . . . 9 |- ( k = j -> ( k [,) +oo ) = ( j [,) +oo ) )
4	3	imaeq2d 5466	. . . . . . . 8 |- ( k = j -> ( F " ( k [,) +oo ) ) = ( F " ( j [,) +oo ) ) )
5	4	ineq1d 3813	. . . . . . 7 |- ( k = j -> ( ( F " ( k [,) +oo ) ) i^i RR* ) = ( ( F " ( j [,) +oo ) ) i^i RR* ) )
6	5	infeq1d 8383	. . . . . 6 |- ( k = j -> inf ( ( ( F " ( k [,) +oo ) ) i^i RR* ) , RR* , < ) = inf ( ( ( F " ( j [,) +oo ) ) i^i RR* ) , RR* , < ) )
7	6	cbvmptv 4750	. . . . 5 |- ( k e. RR |-> inf ( ( ( F " ( k [,) +oo ) ) i^i RR* ) , RR* , < ) ) = ( j e. RR |-> inf ( ( ( F " ( j [,) +oo ) ) i^i RR* ) , RR* , < ) )
8	2, 7	eqtri 2644	. . . 4 |- G = ( j e. RR |-> inf ( ( ( F " ( j [,) +oo ) ) i^i RR* ) , RR* , < ) )
9	8	liminfval 39991	. . 3 |- ( F e. V -> (liminf ` F ) = sup ( ran G , RR* , < ) )
10	1, 9	syl 17	. 2 |- ( ph -> (liminf ` F ) = sup ( ran G , RR* , < ) )
11		liminfval2.4	. . . . . . 7 |- ( ph -> sup ( A , RR* , < ) = +oo )
12		liminfval2.3	. . . . . . . . 9 |- ( ph -> A C_ RR )
13	12	ssrexr 39659	. . . . . . . 8 |- ( ph -> A C_ RR* )
14		supxrunb1 12149	. . . . . . . 8 |- ( A C_ RR* -> ( A. n e. RR E. x e. A n <_ x <-> sup ( A , RR* , < ) = +oo ) )
15	13, 14	syl 17	. . . . . . 7 |- ( ph -> ( A. n e. RR E. x e. A n <_ x <-> sup ( A , RR* , < ) = +oo ) )
16	11, 15	mpbird 247	. . . . . 6 |- ( ph -> A. n e. RR E. x e. A n <_ x )
17	8	liminfgf 39990	. . . . . . . . . . 11 |- G : RR --> RR*
18	17	ffvelrni 6358	. . . . . . . . . 10 |- ( n e. RR -> ( G ` n ) e. RR* )
19	18	ad2antlr 763	. . . . . . . . 9 |- ( ( ( ph /\ n e. RR ) /\ ( x e. A /\ n <_ x ) ) -> ( G ` n ) e. RR* )
20		simpll 790	. . . . . . . . . 10 |- ( ( ( ph /\ n e. RR ) /\ ( x e. A /\ n <_ x ) ) -> ph )
21		simprl 794	. . . . . . . . . 10 |- ( ( ( ph /\ n e. RR ) /\ ( x e. A /\ n <_ x ) ) -> x e. A )
22	12	sselda 3603	. . . . . . . . . . 11 |- ( ( ph /\ x e. A ) -> x e. RR )
23	17	ffvelrni 6358	. . . . . . . . . . 11 |- ( x e. RR -> ( G ` x ) e. RR* )
24	22, 23	syl 17	. . . . . . . . . 10 |- ( ( ph /\ x e. A ) -> ( G ` x ) e. RR* )
25	20, 21, 24	syl2anc 693	. . . . . . . . 9 |- ( ( ( ph /\ n e. RR ) /\ ( x e. A /\ n <_ x ) ) -> ( G ` x ) e. RR* )
26		imassrn 5477	. . . . . . . . . . . 12 |- ( G " A ) C_ ran G
27		frn 6053	. . . . . . . . . . . . 13 |- ( G : RR --> RR* -> ran G C_ RR* )
28	17, 27	ax-mp 5	. . . . . . . . . . . 12 |- ran G C_ RR*
29	26, 28	sstri 3612	. . . . . . . . . . 11 |- ( G " A ) C_ RR*
30		supxrcl 12145	. . . . . . . . . . 11 |- ( ( G " A ) C_ RR* -> sup ( ( G " A ) , RR* , < ) e. RR* )
31	29, 30	ax-mp 5	. . . . . . . . . 10 |- sup ( ( G " A ) , RR* , < ) e. RR*
32	31	a1i 11	. . . . . . . . 9 |- ( ( ( ph /\ n e. RR ) /\ ( x e. A /\ n <_ x ) ) -> sup ( ( G " A ) , RR* , < ) e. RR* )
33		simplr 792	. . . . . . . . . . 11 |- ( ( ( ph /\ n e. RR ) /\ ( x e. A /\ n <_ x ) ) -> n e. RR )
34	20, 21, 22	syl2anc 693	. . . . . . . . . . 11 |- ( ( ( ph /\ n e. RR ) /\ ( x e. A /\ n <_ x ) ) -> x e. RR )
35		simprr 796	. . . . . . . . . . 11 |- ( ( ( ph /\ n e. RR ) /\ ( x e. A /\ n <_ x ) ) -> n <_ x )
36		liminfgord 39986	. . . . . . . . . . 11 |- ( ( n e. RR /\ x e. RR /\ n <_ x ) -> inf ( ( ( F " ( n [,) +oo ) ) i^i RR* ) , RR* , < ) <_ inf ( ( ( F " ( x [,) +oo ) ) i^i RR* ) , RR* , < ) )
37	33, 34, 35, 36	syl3anc 1326	. . . . . . . . . 10 |- ( ( ( ph /\ n e. RR ) /\ ( x e. A /\ n <_ x ) ) -> inf ( ( ( F " ( n [,) +oo ) ) i^i RR* ) , RR* , < ) <_ inf ( ( ( F " ( x [,) +oo ) ) i^i RR* ) , RR* , < ) )
38	8	liminfgval 39994	. . . . . . . . . . . . 13 |- ( n e. RR -> ( G ` n ) = inf ( ( ( F " ( n [,) +oo ) ) i^i RR* ) , RR* , < ) )
39	38	ad2antlr 763	. . . . . . . . . . . 12 |- ( ( ( ph /\ n e. RR ) /\ x e. A ) -> ( G ` n ) = inf ( ( ( F " ( n [,) +oo ) ) i^i RR* ) , RR* , < ) )
40	8	liminfgval 39994	. . . . . . . . . . . . . 14 |- ( x e. RR -> ( G ` x ) = inf ( ( ( F " ( x [,) +oo ) ) i^i RR* ) , RR* , < ) )
41	22, 40	syl 17	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. A ) -> ( G ` x ) = inf ( ( ( F " ( x [,) +oo ) ) i^i RR* ) , RR* , < ) )
42	41	adantlr 751	. . . . . . . . . . . 12 |- ( ( ( ph /\ n e. RR ) /\ x e. A ) -> ( G ` x ) = inf ( ( ( F " ( x [,) +oo ) ) i^i RR* ) , RR* , < ) )
43	39, 42	breq12d 4666	. . . . . . . . . . 11 |- ( ( ( ph /\ n e. RR ) /\ x e. A ) -> ( ( G ` n ) <_ ( G ` x ) <-> inf ( ( ( F " ( n [,) +oo ) ) i^i RR* ) , RR* , < ) <_ inf ( ( ( F " ( x [,) +oo ) ) i^i RR* ) , RR* , < ) ) )
44	43	adantrr 753	. . . . . . . . . 10 |- ( ( ( ph /\ n e. RR ) /\ ( x e. A /\ n <_ x ) ) -> ( ( G ` n ) <_ ( G ` x ) <-> inf ( ( ( F " ( n [,) +oo ) ) i^i RR* ) , RR* , < ) <_ inf ( ( ( F " ( x [,) +oo ) ) i^i RR* ) , RR* , < ) ) )
45	37, 44	mpbird 247	. . . . . . . . 9 |- ( ( ( ph /\ n e. RR ) /\ ( x e. A /\ n <_ x ) ) -> ( G ` n ) <_ ( G ` x ) )
46	29	a1i 11	. . . . . . . . . . 11 |- ( ( ph /\ x e. A ) -> ( G " A ) C_ RR* )
47		nfv 1843	. . . . . . . . . . . . . 14 |- F/ j ph
48		inss2 3834	. . . . . . . . . . . . . . . 16 |- ( ( F " ( j [,) +oo ) ) i^i RR* ) C_ RR*
49		infxrcl 12163	. . . . . . . . . . . . . . . 16 |- ( ( ( F " ( j [,) +oo ) ) i^i RR* ) C_ RR* -> inf ( ( ( F " ( j [,) +oo ) ) i^i RR* ) , RR* , < ) e. RR* )
50	48, 49	ax-mp 5	. . . . . . . . . . . . . . 15 |- inf ( ( ( F " ( j [,) +oo ) ) i^i RR* ) , RR* , < ) e. RR*
51	50	a1i 11	. . . . . . . . . . . . . 14 |- ( ( ph /\ j e. RR ) -> inf ( ( ( F " ( j [,) +oo ) ) i^i RR* ) , RR* , < ) e. RR* )
52	47, 51, 8	fnmptd 39434	. . . . . . . . . . . . 13 |- ( ph -> G Fn RR )
53	52	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ x e. A ) -> G Fn RR )
54		simpr 477	. . . . . . . . . . . 12 |- ( ( ph /\ x e. A ) -> x e. A )
55	53, 22, 54	fnfvimad 39459	. . . . . . . . . . 11 |- ( ( ph /\ x e. A ) -> ( G ` x ) e. ( G " A ) )
56		supxrub 12154	. . . . . . . . . . 11 |- ( ( ( G " A ) C_ RR* /\ ( G ` x ) e. ( G " A ) ) -> ( G ` x ) <_ sup ( ( G " A ) , RR* , < ) )
57	46, 55, 56	syl2anc 693	. . . . . . . . . 10 |- ( ( ph /\ x e. A ) -> ( G ` x ) <_ sup ( ( G " A ) , RR* , < ) )
58	20, 21, 57	syl2anc 693	. . . . . . . . 9 |- ( ( ( ph /\ n e. RR ) /\ ( x e. A /\ n <_ x ) ) -> ( G ` x ) <_ sup ( ( G " A ) , RR* , < ) )
59	19, 25, 32, 45, 58	xrletrd 11993	. . . . . . . 8 |- ( ( ( ph /\ n e. RR ) /\ ( x e. A /\ n <_ x ) ) -> ( G ` n ) <_ sup ( ( G " A ) , RR* , < ) )
60	59	rexlimdvaa 3032	. . . . . . 7 |- ( ( ph /\ n e. RR ) -> ( E. x e. A n <_ x -> ( G ` n ) <_ sup ( ( G " A ) , RR* , < ) ) )
61	60	ralimdva 2962	. . . . . 6 |- ( ph -> ( A. n e. RR E. x e. A n <_ x -> A. n e. RR ( G ` n ) <_ sup ( ( G " A ) , RR* , < ) ) )
62	16, 61	mpd 15	. . . . 5 |- ( ph -> A. n e. RR ( G ` n ) <_ sup ( ( G " A ) , RR* , < ) )
63		xrltso 11974	. . . . . . . . 9 |- < Or RR*
64	63	infex 8399	. . . . . . . 8 |- inf ( ( ( F " ( j [,) +oo ) ) i^i RR* ) , RR* , < ) e. _V
65	64	rgenw 2924	. . . . . . 7 |- A. j e. RR inf ( ( ( F " ( j [,) +oo ) ) i^i RR* ) , RR* , < ) e. _V
66	8	fnmpt 6020	. . . . . . 7 |- ( A. j e. RR inf ( ( ( F " ( j [,) +oo ) ) i^i RR* ) , RR* , < ) e. _V -> G Fn RR )
67	65, 66	ax-mp 5	. . . . . 6 |- G Fn RR
68		breq1 4656	. . . . . . 7 |- ( x = ( G ` n ) -> ( x <_ sup ( ( G " A ) , RR* , < ) <-> ( G ` n ) <_ sup ( ( G " A ) , RR* , < ) ) )
69	68	ralrn 6362	. . . . . 6 |- ( G Fn RR -> ( A. x e. ran G x <_ sup ( ( G " A ) , RR* , < ) <-> A. n e. RR ( G ` n ) <_ sup ( ( G " A ) , RR* , < ) ) )
70	67, 69	ax-mp 5	. . . . 5 |- ( A. x e. ran G x <_ sup ( ( G " A ) , RR* , < ) <-> A. n e. RR ( G ` n ) <_ sup ( ( G " A ) , RR* , < ) )
71	62, 70	sylibr 224	. . . 4 |- ( ph -> A. x e. ran G x <_ sup ( ( G " A ) , RR* , < ) )
72		supxrleub 12156	. . . . 5 |- ( ( ran G C_ RR* /\ sup ( ( G " A ) , RR* , < ) e. RR* ) -> ( sup ( ran G , RR* , < ) <_ sup ( ( G " A ) , RR* , < ) <-> A. x e. ran G x <_ sup ( ( G " A ) , RR* , < ) ) )
73	28, 31, 72	mp2an 708	. . . 4 |- ( sup ( ran G , RR* , < ) <_ sup ( ( G " A ) , RR* , < ) <-> A. x e. ran G x <_ sup ( ( G " A ) , RR* , < ) )
74	71, 73	sylibr 224	. . 3 |- ( ph -> sup ( ran G , RR* , < ) <_ sup ( ( G " A ) , RR* , < ) )
75	26	a1i 11	. . . 4 |- ( ph -> ( G " A ) C_ ran G )
76	28	a1i 11	. . . 4 |- ( ph -> ran G C_ RR* )
77		supxrss 12162	. . . 4 |- ( ( ( G " A ) C_ ran G /\ ran G C_ RR* ) -> sup ( ( G " A ) , RR* , < ) <_ sup ( ran G , RR* , < ) )
78	75, 76, 77	syl2anc 693	. . 3 |- ( ph -> sup ( ( G " A ) , RR* , < ) <_ sup ( ran G , RR* , < ) )
79		supxrcl 12145	. . . . 5 |- ( ran G C_ RR* -> sup ( ran G , RR* , < ) e. RR* )
80	28, 79	ax-mp 5	. . . 4 |- sup ( ran G , RR* , < ) e. RR*
81		xrletri3 11985	. . . 4 |- ( ( sup ( ran G , RR* , < ) e. RR* /\ sup ( ( G " A ) , RR* , < ) e. RR* ) -> ( sup ( ran G , RR* , < ) = sup ( ( G " A ) , RR* , < ) <-> ( sup ( ran G , RR* , < ) <_ sup ( ( G " A ) , RR* , < ) /\ sup ( ( G " A ) , RR* , < ) <_ sup ( ran G , RR* , < ) ) ) )
82	80, 31, 81	mp2an 708	. . 3 |- ( sup ( ran G , RR* , < ) = sup ( ( G " A ) , RR* , < ) <-> ( sup ( ran G , RR* , < ) <_ sup ( ( G " A ) , RR* , < ) /\ sup ( ( G " A ) , RR* , < ) <_ sup ( ran G , RR* , < ) ) )
83	74, 78, 82	sylanbrc 698	. 2 |- ( ph -> sup ( ran G , RR* , < ) = sup ( ( G " A ) , RR* , < ) )
84	10, 83	eqtrd 2656	1 |- ( ph -> (liminf ` F ) = sup ( ( G " A ) , RR* , < ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   _Vcvv 3200   i^i cin 3573   C_ wss 3574   class class class wbr 4653   |-> cmpt 4729   ran crn 5115   "cima 5117   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   supcsup 8346  infcinf 8347   RRcr 9935   +oocpnf 10071   RR*cxr 10073   < clt 10074   <_ cle 10075   [,)cico 12177  liminfclsi 39983

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-iun 4522  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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  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-ico 12181  df-liminf 39984

This theorem is referenced by:  liminfresico  40003