Theorem limsuplt 13061

Description: The defining property of the superior limit. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by Mario Carneiro, 7-May-2016.)

Hypothesis

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

Assertion

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

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

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

Proof of Theorem limsuplt

Step	Hyp	Ref	Expression
1		limsupval.1	. . . . 5 |- G = ( k e. RR |-> sup ( ( ( F " ( k [,) +oo ) ) i^i RR* ) , RR* , < ) )
2	1	limsuple 13060	. . . 4 |- ( ( B C_ RR /\ F : B --> RR* /\ A e. RR* ) -> ( A <_ ( limsup ` F ) <-> A. j e. RR A <_ ( G ` j ) ) )
3	2	notbid 294	. . 3 |- ( ( B C_ RR /\ F : B --> RR* /\ A e. RR* ) -> ( -. A <_ ( limsup ` F ) <-> -. A. j e. RR A <_ ( G ` j ) ) )
4		rexnal 2842	. . 3 |- ( E. j e. RR -. A <_ ( G ` j ) <-> -. A. j e. RR A <_ ( G ` j ) )
5	3, 4	syl6bbr 263	. 2 |- ( ( B C_ RR /\ F : B --> RR* /\ A e. RR* ) -> ( -. A <_ ( limsup ` F ) <-> E. j e. RR -. A <_ ( G ` j ) ) )
6		simp2 989	. . . . 5 |- ( ( B C_ RR /\ F : B --> RR* /\ A e. RR* ) -> F : B --> RR* )
7		reex 9476	. . . . . . 7 |- RR e. _V
8	7	ssex 4536	. . . . . 6 |- ( B C_ RR -> B e. _V )
9	8	3ad2ant1 1009	. . . . 5 |- ( ( B C_ RR /\ F : B --> RR* /\ A e. RR* ) -> B e. _V )
10		xrex 11091	. . . . . 6 |- RR* e. _V
11	10	a1i 11	. . . . 5 |- ( ( B C_ RR /\ F : B --> RR* /\ A e. RR* ) -> RR* e. _V )
12		fex2 6634	. . . . 5 |- ( ( F : B --> RR* /\ B e. _V /\ RR* e. _V ) -> F e. _V )
13	6, 9, 11, 12	syl3anc 1219	. . . 4 |- ( ( B C_ RR /\ F : B --> RR* /\ A e. RR* ) -> F e. _V )
14		limsupcl 13055	. . . 4 |- ( F e. _V -> ( limsup ` F ) e. RR* )
15	13, 14	syl 16	. . 3 |- ( ( B C_ RR /\ F : B --> RR* /\ A e. RR* ) -> ( limsup ` F ) e. RR* )
16		simp3 990	. . 3 |- ( ( B C_ RR /\ F : B --> RR* /\ A e. RR* ) -> A e. RR* )
17		xrltnle 9546	. . 3 |- ( ( ( limsup ` F ) e. RR* /\ A e. RR* ) -> ( ( limsup ` F ) < A <-> -. A <_ ( limsup ` F ) ) )
18	15, 16, 17	syl2anc 661	. 2 |- ( ( B C_ RR /\ F : B --> RR* /\ A e. RR* ) -> ( ( limsup ` F ) < A <-> -. A <_ ( limsup ` F ) ) )
19	1	limsupgf 13057	. . . . . 6 |- G : RR --> RR*
20	19	ffvelrni 5943	. . . . 5 |- ( j e. RR -> ( G ` j ) e. RR* )
21	20	adantl 466	. . . 4 |- ( ( ( B C_ RR /\ F : B --> RR* /\ A e. RR* ) /\ j e. RR ) -> ( G ` j ) e. RR* )
22		simpl3 993	. . . 4 |- ( ( ( B C_ RR /\ F : B --> RR* /\ A e. RR* ) /\ j e. RR ) -> A e. RR* )
23		xrltnle 9546	. . . 4 |- ( ( ( G ` j ) e. RR* /\ A e. RR* ) -> ( ( G ` j ) < A <-> -. A <_ ( G ` j ) ) )
24	21, 22, 23	syl2anc 661	. . 3 |- ( ( ( B C_ RR /\ F : B --> RR* /\ A e. RR* ) /\ j e. RR ) -> ( ( G ` j ) < A <-> -. A <_ ( G ` j ) ) )
25	24	rexbidva 2845	. 2 |- ( ( B C_ RR /\ F : B --> RR* /\ A e. RR* ) -> ( E. j e. RR ( G ` j ) < A <-> E. j e. RR -. A <_ ( G ` j ) ) )
26	5, 18, 25	3bitr4d 285	1 |- ( ( B C_ RR /\ F : B --> RR* /\ A e. RR* ) -> ( ( limsup ` F ) < A <-> E. j e. RR ( G ` j ) < A ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 965   = wceq 1370   e. wcel 1758   A.wral 2795   E.wrex 2796   _Vcvv 3070   i^i cin 3427   C_ wss 3428   class class class wbr 4392   |-> cmpt 4450   "cima 4943   -->wf 5514   ` cfv 5518  (class class class)co 6192   supcsup 7793   RRcr 9384   +oocpnf 9518   RR*cxr 9520   < clt 9521   <_ cle 9522   [,)cico 11405   limsupclsp 13052

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-cnex 9441  ax-resscn 9442  ax-1cn 9443  ax-icn 9444  ax-addcl 9445  ax-addrcl 9446  ax-mulcl 9447  ax-mulrcl 9448  ax-mulcom 9449  ax-addass 9450  ax-mulass 9451  ax-distr 9452  ax-i2m1 9453  ax-1ne0 9454  ax-1rid 9455  ax-rnegex 9456  ax-rrecex 9457  ax-cnre 9458  ax-pre-lttri 9459  ax-pre-lttrn 9460  ax-pre-ltadd 9461  ax-pre-mulgt0 9462  ax-pre-sup 9463

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-po 4741  df-so 4742  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-er 7203  df-en 7413  df-dom 7414  df-sdom 7415  df-sup 7794  df-pnf 9523  df-mnf 9524  df-xr 9525  df-ltxr 9526  df-le 9527  df-sub 9700  df-neg 9701  df-limsup 13053

This theorem is referenced by:  limsupgre  13063