Theorem limsupval2 14211

Description: The superior limit, relativized to an unbounded set. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by AV, 12-Sep-2020.)

Hypotheses

Ref	Expression
limsupval.1	|- G = ( k e. RR |-> sup ( ( ( F " ( k [,) +oo ) ) i^i RR* ) , RR* , < ) )
limsupval2.1	|- ( ph -> F e. V )
limsupval2.2	|- ( ph -> A C_ RR )
limsupval2.3	|- ( ph -> sup ( A , RR* , < ) = +oo )

Assertion

Ref	Expression
limsupval2	|- ( ph -> ( limsup ` F ) = inf ( ( G " A ) , RR* , < ) )

Distinct variable groups:   k, F   A, k

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

Proof of Theorem limsupval2

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

Step	Hyp	Ref	Expression
1		limsupval2.1	. . 3 |- ( ph -> F e. V )
2		limsupval.1	. . . 4 |- G = ( k e. RR |-> sup ( ( ( F " ( k [,) +oo ) ) i^i RR* ) , RR* , < ) )
3	2	limsupval 14205	. . 3 |- ( F e. V -> ( limsup ` F ) = inf ( ran G , RR* , < ) )
4	1, 3	syl 17	. 2 |- ( ph -> ( limsup ` F ) = inf ( ran G , RR* , < ) )
5		imassrn 5477	. . . . 5 |- ( G " A ) C_ ran G
6	2	limsupgf 14206	. . . . . . 7 |- G : RR --> RR*
7		frn 6053	. . . . . . 7 |- ( G : RR --> RR* -> ran G C_ RR* )
8	6, 7	ax-mp 5	. . . . . 6 |- ran G C_ RR*
9		infxrlb 12164	. . . . . . 7 |- ( ( ran G C_ RR* /\ x e. ran G ) -> inf ( ran G , RR* , < ) <_ x )
10	9	ralrimiva 2966	. . . . . 6 |- ( ran G C_ RR* -> A. x e. ran Ginf ( ran G , RR* , < ) <_ x )
11	8, 10	mp1i 13	. . . . 5 |- ( ph -> A. x e. ran Ginf ( ran G , RR* , < ) <_ x )
12		ssralv 3666	. . . . 5 |- ( ( G " A ) C_ ran G -> ( A. x e. ran Ginf ( ran G , RR* , < ) <_ x -> A. x e. ( G " A )inf ( ran G , RR* , < ) <_ x ) )
13	5, 11, 12	mpsyl 68	. . . 4 |- ( ph -> A. x e. ( G " A )inf ( ran G , RR* , < ) <_ x )
14	5, 8	sstri 3612	. . . . 5 |- ( G " A ) C_ RR*
15		infxrcl 12163	. . . . . 6 |- ( ran G C_ RR* -> inf ( ran G , RR* , < ) e. RR* )
16	8, 15	ax-mp 5	. . . . 5 |- inf ( ran G , RR* , < ) e. RR*
17		infxrgelb 12165	. . . . 5 |- ( ( ( G " A ) C_ RR* /\ inf ( ran G , RR* , < ) e. RR* ) -> (inf ( ran G , RR* , < ) <_ inf ( ( G " A ) , RR* , < ) <-> A. x e. ( G " A )inf ( ran G , RR* , < ) <_ x ) )
18	14, 16, 17	mp2an 708	. . . 4 |- (inf ( ran G , RR* , < ) <_ inf ( ( G " A ) , RR* , < ) <-> A. x e. ( G " A )inf ( ran G , RR* , < ) <_ x )
19	13, 18	sylibr 224	. . 3 |- ( ph -> inf ( ran G , RR* , < ) <_ inf ( ( G " A ) , RR* , < ) )
20		limsupval2.3	. . . . . . 7 |- ( ph -> sup ( A , RR* , < ) = +oo )
21		limsupval2.2	. . . . . . . . 9 |- ( ph -> A C_ RR )
22		ressxr 10083	. . . . . . . . 9 |- RR C_ RR*
23	21, 22	syl6ss 3615	. . . . . . . 8 |- ( ph -> A C_ RR* )
24		supxrunb1 12149	. . . . . . . 8 |- ( A C_ RR* -> ( A. n e. RR E. x e. A n <_ x <-> sup ( A , RR* , < ) = +oo ) )
25	23, 24	syl 17	. . . . . . 7 |- ( ph -> ( A. n e. RR E. x e. A n <_ x <-> sup ( A , RR* , < ) = +oo ) )
26	20, 25	mpbird 247	. . . . . 6 |- ( ph -> A. n e. RR E. x e. A n <_ x )
27		infxrcl 12163	. . . . . . . . . 10 |- ( ( G " A ) C_ RR* -> inf ( ( G " A ) , RR* , < ) e. RR* )
28	14, 27	mp1i 13	. . . . . . . . 9 |- ( ( ( ph /\ n e. RR ) /\ ( x e. A /\ n <_ x ) ) -> inf ( ( G " A ) , RR* , < ) e. RR* )
29	21	sselda 3603	. . . . . . . . . . 11 |- ( ( ph /\ x e. A ) -> x e. RR )
30	29	ad2ant2r 783	. . . . . . . . . 10 |- ( ( ( ph /\ n e. RR ) /\ ( x e. A /\ n <_ x ) ) -> x e. RR )
31	6	ffvelrni 6358	. . . . . . . . . 10 |- ( x e. RR -> ( G ` x ) e. RR* )
32	30, 31	syl 17	. . . . . . . . 9 |- ( ( ( ph /\ n e. RR ) /\ ( x e. A /\ n <_ x ) ) -> ( G ` x ) e. RR* )
33	6	ffvelrni 6358	. . . . . . . . . 10 |- ( n e. RR -> ( G ` n ) e. RR* )
34	33	ad2antlr 763	. . . . . . . . 9 |- ( ( ( ph /\ n e. RR ) /\ ( x e. A /\ n <_ x ) ) -> ( G ` n ) e. RR* )
35		ffn 6045	. . . . . . . . . . . 12 |- ( G : RR --> RR* -> G Fn RR )
36	6, 35	mp1i 13	. . . . . . . . . . 11 |- ( ( ( ph /\ n e. RR ) /\ ( x e. A /\ n <_ x ) ) -> G Fn RR )
37	21	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( ph /\ n e. RR ) /\ ( x e. A /\ n <_ x ) ) -> A C_ RR )
38		simprl 794	. . . . . . . . . . 11 |- ( ( ( ph /\ n e. RR ) /\ ( x e. A /\ n <_ x ) ) -> x e. A )
39		fnfvima 6496	. . . . . . . . . . 11 |- ( ( G Fn RR /\ A C_ RR /\ x e. A ) -> ( G ` x ) e. ( G " A ) )
40	36, 37, 38, 39	syl3anc 1326	. . . . . . . . . 10 |- ( ( ( ph /\ n e. RR ) /\ ( x e. A /\ n <_ x ) ) -> ( G ` x ) e. ( G " A ) )
41		infxrlb 12164	. . . . . . . . . 10 |- ( ( ( G " A ) C_ RR* /\ ( G ` x ) e. ( G " A ) ) -> inf ( ( G " A ) , RR* , < ) <_ ( G ` x ) )
42	14, 40, 41	sylancr 695	. . . . . . . . 9 |- ( ( ( ph /\ n e. RR ) /\ ( x e. A /\ n <_ x ) ) -> inf ( ( G " A ) , RR* , < ) <_ ( G ` x ) )
43		simplr 792	. . . . . . . . . . 11 |- ( ( ( ph /\ n e. RR ) /\ ( x e. A /\ n <_ x ) ) -> n e. RR )
44		simprr 796	. . . . . . . . . . 11 |- ( ( ( ph /\ n e. RR ) /\ ( x e. A /\ n <_ x ) ) -> n <_ x )
45		limsupgord 14203	. . . . . . . . . . 11 |- ( ( n e. RR /\ x e. RR /\ n <_ x ) -> sup ( ( ( F " ( x [,) +oo ) ) i^i RR* ) , RR* , < ) <_ sup ( ( ( F " ( n [,) +oo ) ) i^i RR* ) , RR* , < ) )
46	43, 30, 44, 45	syl3anc 1326	. . . . . . . . . 10 |- ( ( ( ph /\ n e. RR ) /\ ( x e. A /\ n <_ x ) ) -> sup ( ( ( F " ( x [,) +oo ) ) i^i RR* ) , RR* , < ) <_ sup ( ( ( F " ( n [,) +oo ) ) i^i RR* ) , RR* , < ) )
47	2	limsupgval 14207	. . . . . . . . . . 11 |- ( x e. RR -> ( G ` x ) = sup ( ( ( F " ( x [,) +oo ) ) i^i RR* ) , RR* , < ) )
48	30, 47	syl 17	. . . . . . . . . 10 |- ( ( ( ph /\ n e. RR ) /\ ( x e. A /\ n <_ x ) ) -> ( G ` x ) = sup ( ( ( F " ( x [,) +oo ) ) i^i RR* ) , RR* , < ) )
49	2	limsupgval 14207	. . . . . . . . . . 11 |- ( n e. RR -> ( G ` n ) = sup ( ( ( F " ( n [,) +oo ) ) i^i RR* ) , RR* , < ) )
50	49	ad2antlr 763	. . . . . . . . . 10 |- ( ( ( ph /\ n e. RR ) /\ ( x e. A /\ n <_ x ) ) -> ( G ` n ) = sup ( ( ( F " ( n [,) +oo ) ) i^i RR* ) , RR* , < ) )
51	46, 48, 50	3brtr4d 4685	. . . . . . . . 9 |- ( ( ( ph /\ n e. RR ) /\ ( x e. A /\ n <_ x ) ) -> ( G ` x ) <_ ( G ` n ) )
52	28, 32, 34, 42, 51	xrletrd 11993	. . . . . . . 8 |- ( ( ( ph /\ n e. RR ) /\ ( x e. A /\ n <_ x ) ) -> inf ( ( G " A ) , RR* , < ) <_ ( G ` n ) )
53	52	rexlimdvaa 3032	. . . . . . 7 |- ( ( ph /\ n e. RR ) -> ( E. x e. A n <_ x -> inf ( ( G " A ) , RR* , < ) <_ ( G ` n ) ) )
54	53	ralimdva 2962	. . . . . 6 |- ( ph -> ( A. n e. RR E. x e. A n <_ x -> A. n e. RR inf ( ( G " A ) , RR* , < ) <_ ( G ` n ) ) )
55	26, 54	mpd 15	. . . . 5 |- ( ph -> A. n e. RR inf ( ( G " A ) , RR* , < ) <_ ( G ` n ) )
56	6, 35	ax-mp 5	. . . . . 6 |- G Fn RR
57		breq2 4657	. . . . . . 7 |- ( x = ( G ` n ) -> (inf ( ( G " A ) , RR* , < ) <_ x <-> inf ( ( G " A ) , RR* , < ) <_ ( G ` n ) ) )
58	57	ralrn 6362	. . . . . 6 |- ( G Fn RR -> ( A. x e. ran Ginf ( ( G " A ) , RR* , < ) <_ x <-> A. n e. RR inf ( ( G " A ) , RR* , < ) <_ ( G ` n ) ) )
59	56, 58	ax-mp 5	. . . . 5 |- ( A. x e. ran Ginf ( ( G " A ) , RR* , < ) <_ x <-> A. n e. RR inf ( ( G " A ) , RR* , < ) <_ ( G ` n ) )
60	55, 59	sylibr 224	. . . 4 |- ( ph -> A. x e. ran Ginf ( ( G " A ) , RR* , < ) <_ x )
61	14, 27	ax-mp 5	. . . . 5 |- inf ( ( G " A ) , RR* , < ) e. RR*
62		infxrgelb 12165	. . . . 5 |- ( ( ran G C_ RR* /\ inf ( ( G " A ) , RR* , < ) e. RR* ) -> (inf ( ( G " A ) , RR* , < ) <_ inf ( ran G , RR* , < ) <-> A. x e. ran Ginf ( ( G " A ) , RR* , < ) <_ x ) )
63	8, 61, 62	mp2an 708	. . . 4 |- (inf ( ( G " A ) , RR* , < ) <_ inf ( ran G , RR* , < ) <-> A. x e. ran Ginf ( ( G " A ) , RR* , < ) <_ x )
64	60, 63	sylibr 224	. . 3 |- ( ph -> inf ( ( G " A ) , RR* , < ) <_ inf ( ran G , RR* , < ) )
65		xrletri3 11985	. . . 4 |- ( (inf ( ran G , RR* , < ) e. RR* /\ inf ( ( G " A ) , RR* , < ) e. RR* ) -> (inf ( ran G , RR* , < ) = inf ( ( G " A ) , RR* , < ) <-> (inf ( ran G , RR* , < ) <_ inf ( ( G " A ) , RR* , < ) /\ inf ( ( G " A ) , RR* , < ) <_ inf ( ran G , RR* , < ) ) ) )
66	16, 61, 65	mp2an 708	. . 3 |- (inf ( ran G , RR* , < ) = inf ( ( G " A ) , RR* , < ) <-> (inf ( ran G , RR* , < ) <_ inf ( ( G " A ) , RR* , < ) /\ inf ( ( G " A ) , RR* , < ) <_ inf ( ran G , RR* , < ) ) )
67	19, 64, 66	sylanbrc 698	. 2 |- ( ph -> inf ( ran G , RR* , < ) = inf ( ( G " A ) , RR* , < ) )
68	4, 67	eqtrd 2656	1 |- ( ph -> ( limsup ` F ) = inf ( ( G " A ) , RR* , < ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   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   limsupclsp 14201

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-limsup 14202

This theorem is referenced by:  mbflimsup  23433  limsupresico  39932  limsupvaluz  39940