Theorem limsuple 14209

Description: The defining property of the superior limit. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by AV, 12-Sep-2020.)

Hypothesis

Ref	Expression
limsupval.1	|- G = ( k e. RR |-> sup ( ( ( F " ( k [,) +oo ) ) i^i RR* ) , RR* , < ) )

Assertion

Ref	Expression
limsuple	|- ( ( B C_ RR /\ F : B --> RR* /\ A e. RR* ) -> ( A <_ ( limsup ` F ) <-> A. j e. RR A <_ ( G ` j ) ) )

Distinct variable groups:   k, F   A, j   B, j   j, k, F   j, G

Allowed substitution hints:   A( k)   B( k)   G( k)

Proof of Theorem limsuple

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp2 1062	. . . . 5 |- ( ( B C_ RR /\ F : B --> RR* /\ A e. RR* ) -> F : B --> RR* )
2		reex 10027	. . . . . . 7 |- RR e. _V
3	2	ssex 4802	. . . . . 6 |- ( B C_ RR -> B e. _V )
4	3	3ad2ant1 1082	. . . . 5 |- ( ( B C_ RR /\ F : B --> RR* /\ A e. RR* ) -> B e. _V )
5		xrex 11829	. . . . . 6 |- RR* e. _V
6	5	a1i 11	. . . . 5 |- ( ( B C_ RR /\ F : B --> RR* /\ A e. RR* ) -> RR* e. _V )
7		fex2 7121	. . . . 5 |- ( ( F : B --> RR* /\ B e. _V /\ RR* e. _V ) -> F e. _V )
8	1, 4, 6, 7	syl3anc 1326	. . . 4 |- ( ( B C_ RR /\ F : B --> RR* /\ A e. RR* ) -> F e. _V )
9		limsupval.1	. . . . 5 |- G = ( k e. RR |-> sup ( ( ( F " ( k [,) +oo ) ) i^i RR* ) , RR* , < ) )
10	9	limsupval 14205	. . . 4 |- ( F e. _V -> ( limsup ` F ) = inf ( ran G , RR* , < ) )
11	8, 10	syl 17	. . 3 |- ( ( B C_ RR /\ F : B --> RR* /\ A e. RR* ) -> ( limsup ` F ) = inf ( ran G , RR* , < ) )
12	11	breq2d 4665	. 2 |- ( ( B C_ RR /\ F : B --> RR* /\ A e. RR* ) -> ( A <_ ( limsup ` F ) <-> A <_ inf ( ran G , RR* , < ) ) )
13	9	limsupgf 14206	. . . . 5 |- G : RR --> RR*
14		frn 6053	. . . . 5 |- ( G : RR --> RR* -> ran G C_ RR* )
15	13, 14	ax-mp 5	. . . 4 |- ran G C_ RR*
16		simp3 1063	. . . 4 |- ( ( B C_ RR /\ F : B --> RR* /\ A e. RR* ) -> A e. RR* )
17		infxrgelb 12165	. . . 4 |- ( ( ran G C_ RR* /\ A e. RR* ) -> ( A <_ inf ( ran G , RR* , < ) <-> A. x e. ran G A <_ x ) )
18	15, 16, 17	sylancr 695	. . 3 |- ( ( B C_ RR /\ F : B --> RR* /\ A e. RR* ) -> ( A <_ inf ( ran G , RR* , < ) <-> A. x e. ran G A <_ x ) )
19		ffn 6045	. . . . 5 |- ( G : RR --> RR* -> G Fn RR )
20	13, 19	ax-mp 5	. . . 4 |- G Fn RR
21		breq2 4657	. . . . 5 |- ( x = ( G ` j ) -> ( A <_ x <-> A <_ ( G ` j ) ) )
22	21	ralrn 6362	. . . 4 |- ( G Fn RR -> ( A. x e. ran G A <_ x <-> A. j e. RR A <_ ( G ` j ) ) )
23	20, 22	ax-mp 5	. . 3 |- ( A. x e. ran G A <_ x <-> A. j e. RR A <_ ( G ` j ) )
24	18, 23	syl6bb 276	. 2 |- ( ( B C_ RR /\ F : B --> RR* /\ A e. RR* ) -> ( A <_ inf ( ran G , RR* , < ) <-> A. j e. RR A <_ ( G ` j ) ) )
25	12, 24	bitrd 268	1 |- ( ( B C_ RR /\ F : B --> RR* /\ A e. RR* ) -> ( A <_ ( limsup ` F ) <-> A. j e. RR A <_ ( G ` j ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   _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   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-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-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-limsup 14202

This theorem is referenced by:  limsuplt  14210  limsupbnd1  14213  limsupbnd2  14214  mbflimsup  23433  limsupge  39993