Theorem limsuplt 13384

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 13383	. . . 4 |- ( ( B C_ RR /\ F : B --> RR* /\ A e. RR* ) -> ( A <_ ( limsup ` F ) <-> A. j e. RR A <_ ( G ` j ) ) )
3	2	notbid 292	. . 3 |- ( ( B C_ RR /\ F : B --> RR* /\ A e. RR* ) -> ( -. A <_ ( limsup ` F ) <-> -. A. j e. RR A <_ ( G ` j ) ) )
4		rexnal 2902	. . 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 995	. . . . 5 |- ( ( B C_ RR /\ F : B --> RR* /\ A e. RR* ) -> F : B --> RR* )
7		reex 9572	. . . . . . 7 |- RR e. _V
8	7	ssex 4581	. . . . . 6 |- ( B C_ RR -> B e. _V )
9	8	3ad2ant1 1015	. . . . 5 |- ( ( B C_ RR /\ F : B --> RR* /\ A e. RR* ) -> B e. _V )
10		xrex 11218	. . . . . 6 |- RR* e. _V
11	10	a1i 11	. . . . 5 |- ( ( B C_ RR /\ F : B --> RR* /\ A e. RR* ) -> RR* e. _V )
12		fex2 6728	. . . . 5 |- ( ( F : B --> RR* /\ B e. _V /\ RR* e. _V ) -> F e. _V )
13	6, 9, 11, 12	syl3anc 1226	. . . 4 |- ( ( B C_ RR /\ F : B --> RR* /\ A e. RR* ) -> F e. _V )
14		limsupcl 13378	. . . 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 996	. . 3 |- ( ( B C_ RR /\ F : B --> RR* /\ A e. RR* ) -> A e. RR* )
17		xrltnle 9642	. . 3 |- ( ( ( limsup ` F ) e. RR* /\ A e. RR* ) -> ( ( limsup ` F ) < A <-> -. A <_ ( limsup ` F ) ) )
18	15, 16, 17	syl2anc 659	. 2 |- ( ( B C_ RR /\ F : B --> RR* /\ A e. RR* ) -> ( ( limsup ` F ) < A <-> -. A <_ ( limsup ` F ) ) )
19	1	limsupgf 13380	. . . . . 6 |- G : RR --> RR*
20	19	ffvelrni 6006	. . . . 5 |- ( j e. RR -> ( G ` j ) e. RR* )
21	20	adantl 464	. . . 4 |- ( ( ( B C_ RR /\ F : B --> RR* /\ A e. RR* ) /\ j e. RR ) -> ( G ` j ) e. RR* )
22		simpl3 999	. . . 4 |- ( ( ( B C_ RR /\ F : B --> RR* /\ A e. RR* ) /\ j e. RR ) -> A e. RR* )
23		xrltnle 9642	. . . 4 |- ( ( ( G ` j ) e. RR* /\ A e. RR* ) -> ( ( G ` j ) < A <-> -. A <_ ( G ` j ) ) )
24	21, 22, 23	syl2anc 659	. . 3 |- ( ( ( B C_ RR /\ F : B --> RR* /\ A e. RR* ) /\ j e. RR ) -> ( ( G ` j ) < A <-> -. A <_ ( G ` j ) ) )
25	24	rexbidva 2962	. 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 367   /\ w3a 971   = wceq 1398   e. wcel 1823   A.wral 2804   E.wrex 2805   _Vcvv 3106   i^i cin 3460   C_ wss 3461   class class class wbr 4439   |-> cmpt 4497   "cima 4991   -->wf 5566   ` cfv 5570  (class class class)co 6270   supcsup 7892   RRcr 9480   +oocpnf 9614   RR*cxr 9616   < clt 9617   <_ cle 9618   [,)cico 11534   limsupclsp 13375

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-po 4789  df-so 4790  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-sup 7893  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-limsup 13376

This theorem is referenced by:  limsupgre  13386