Theorem limsupgle 14208

Description: The defining property of the superior limit function. (Contributed by Mario Carneiro, 5-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
limsupgle	|- ( ( ( B C_ RR /\ F : B --> RR* ) /\ C e. RR /\ A e. RR* ) -> ( ( G ` C ) <_ A <-> A. j e. B ( C <_ j -> ( F ` j ) <_ A ) ) )

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

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

Proof of Theorem limsupgle

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		limsupval.1	. . . . 5 |- G = ( k e. RR |-> sup ( ( ( F " ( k [,) +oo ) ) i^i RR* ) , RR* , < ) )
2	1	limsupgval 14207	. . . 4 |- ( C e. RR -> ( G ` C ) = sup ( ( ( F " ( C [,) +oo ) ) i^i RR* ) , RR* , < ) )
3	2	3ad2ant2 1083	. . 3 |- ( ( ( B C_ RR /\ F : B --> RR* ) /\ C e. RR /\ A e. RR* ) -> ( G ` C ) = sup ( ( ( F " ( C [,) +oo ) ) i^i RR* ) , RR* , < ) )
4	3	breq1d 4663	. 2 |- ( ( ( B C_ RR /\ F : B --> RR* ) /\ C e. RR /\ A e. RR* ) -> ( ( G ` C ) <_ A <-> sup ( ( ( F " ( C [,) +oo ) ) i^i RR* ) , RR* , < ) <_ A ) )
5		inss2 3834	. . 3 |- ( ( F " ( C [,) +oo ) ) i^i RR* ) C_ RR*
6		simp3 1063	. . 3 |- ( ( ( B C_ RR /\ F : B --> RR* ) /\ C e. RR /\ A e. RR* ) -> A e. RR* )
7		supxrleub 12156	. . 3 |- ( ( ( ( F " ( C [,) +oo ) ) i^i RR* ) C_ RR* /\ A e. RR* ) -> ( sup ( ( ( F " ( C [,) +oo ) ) i^i RR* ) , RR* , < ) <_ A <-> A. x e. ( ( F " ( C [,) +oo ) ) i^i RR* ) x <_ A ) )
8	5, 6, 7	sylancr 695	. 2 |- ( ( ( B C_ RR /\ F : B --> RR* ) /\ C e. RR /\ A e. RR* ) -> ( sup ( ( ( F " ( C [,) +oo ) ) i^i RR* ) , RR* , < ) <_ A <-> A. x e. ( ( F " ( C [,) +oo ) ) i^i RR* ) x <_ A ) )
9		imassrn 5477	. . . . . . 7 |- ( F " ( C [,) +oo ) ) C_ ran F
10		simp1r 1086	. . . . . . . 8 |- ( ( ( B C_ RR /\ F : B --> RR* ) /\ C e. RR /\ A e. RR* ) -> F : B --> RR* )
11		frn 6053	. . . . . . . 8 |- ( F : B --> RR* -> ran F C_ RR* )
12	10, 11	syl 17	. . . . . . 7 |- ( ( ( B C_ RR /\ F : B --> RR* ) /\ C e. RR /\ A e. RR* ) -> ran F C_ RR* )
13	9, 12	syl5ss 3614	. . . . . 6 |- ( ( ( B C_ RR /\ F : B --> RR* ) /\ C e. RR /\ A e. RR* ) -> ( F " ( C [,) +oo ) ) C_ RR* )
14		df-ss 3588	. . . . . 6 |- ( ( F " ( C [,) +oo ) ) C_ RR* <-> ( ( F " ( C [,) +oo ) ) i^i RR* ) = ( F " ( C [,) +oo ) ) )
15	13, 14	sylib 208	. . . . 5 |- ( ( ( B C_ RR /\ F : B --> RR* ) /\ C e. RR /\ A e. RR* ) -> ( ( F " ( C [,) +oo ) ) i^i RR* ) = ( F " ( C [,) +oo ) ) )
16		imadmres 5627	. . . . 5 |- ( F " dom ( F |` ( C [,) +oo ) ) ) = ( F " ( C [,) +oo ) )
17	15, 16	syl6eqr 2674	. . . 4 |- ( ( ( B C_ RR /\ F : B --> RR* ) /\ C e. RR /\ A e. RR* ) -> ( ( F " ( C [,) +oo ) ) i^i RR* ) = ( F " dom ( F |` ( C [,) +oo ) ) ) )
18	17	raleqdv 3144	. . 3 |- ( ( ( B C_ RR /\ F : B --> RR* ) /\ C e. RR /\ A e. RR* ) -> ( A. x e. ( ( F " ( C [,) +oo ) ) i^i RR* ) x <_ A <-> A. x e. ( F " dom ( F |` ( C [,) +oo ) ) ) x <_ A ) )
19		ffn 6045	. . . . 5 |- ( F : B --> RR* -> F Fn B )
20	10, 19	syl 17	. . . 4 |- ( ( ( B C_ RR /\ F : B --> RR* ) /\ C e. RR /\ A e. RR* ) -> F Fn B )
21		fdm 6051	. . . . . . . 8 |- ( F : B --> RR* -> dom F = B )
22	10, 21	syl 17	. . . . . . 7 |- ( ( ( B C_ RR /\ F : B --> RR* ) /\ C e. RR /\ A e. RR* ) -> dom F = B )
23	22	ineq2d 3814	. . . . . 6 |- ( ( ( B C_ RR /\ F : B --> RR* ) /\ C e. RR /\ A e. RR* ) -> ( ( C [,) +oo ) i^i dom F ) = ( ( C [,) +oo ) i^i B ) )
24		dmres 5419	. . . . . 6 |- dom ( F |` ( C [,) +oo ) ) = ( ( C [,) +oo ) i^i dom F )
25		incom 3805	. . . . . 6 |- ( B i^i ( C [,) +oo ) ) = ( ( C [,) +oo ) i^i B )
26	23, 24, 25	3eqtr4g 2681	. . . . 5 |- ( ( ( B C_ RR /\ F : B --> RR* ) /\ C e. RR /\ A e. RR* ) -> dom ( F |` ( C [,) +oo ) ) = ( B i^i ( C [,) +oo ) ) )
27		inss1 3833	. . . . 5 |- ( B i^i ( C [,) +oo ) ) C_ B
28	26, 27	syl6eqss 3655	. . . 4 |- ( ( ( B C_ RR /\ F : B --> RR* ) /\ C e. RR /\ A e. RR* ) -> dom ( F |` ( C [,) +oo ) ) C_ B )
29		breq1 4656	. . . . 5 |- ( x = ( F ` j ) -> ( x <_ A <-> ( F ` j ) <_ A ) )
30	29	ralima 6498	. . . 4 |- ( ( F Fn B /\ dom ( F |` ( C [,) +oo ) ) C_ B ) -> ( A. x e. ( F " dom ( F |` ( C [,) +oo ) ) ) x <_ A <-> A. j e. dom ( F |` ( C [,) +oo ) ) ( F ` j ) <_ A ) )
31	20, 28, 30	syl2anc 693	. . 3 |- ( ( ( B C_ RR /\ F : B --> RR* ) /\ C e. RR /\ A e. RR* ) -> ( A. x e. ( F " dom ( F |` ( C [,) +oo ) ) ) x <_ A <-> A. j e. dom ( F |` ( C [,) +oo ) ) ( F ` j ) <_ A ) )
32	26	eleq2d 2687	. . . . . . . 8 |- ( ( ( B C_ RR /\ F : B --> RR* ) /\ C e. RR /\ A e. RR* ) -> ( j e. dom ( F |` ( C [,) +oo ) ) <-> j e. ( B i^i ( C [,) +oo ) ) ) )
33		elin 3796	. . . . . . . 8 |- ( j e. ( B i^i ( C [,) +oo ) ) <-> ( j e. B /\ j e. ( C [,) +oo ) ) )
34	32, 33	syl6bb 276	. . . . . . 7 |- ( ( ( B C_ RR /\ F : B --> RR* ) /\ C e. RR /\ A e. RR* ) -> ( j e. dom ( F |` ( C [,) +oo ) ) <-> ( j e. B /\ j e. ( C [,) +oo ) ) ) )
35		simpl2 1065	. . . . . . . . 9 |- ( ( ( ( B C_ RR /\ F : B --> RR* ) /\ C e. RR /\ A e. RR* ) /\ j e. B ) -> C e. RR )
36		simp1l 1085	. . . . . . . . . 10 |- ( ( ( B C_ RR /\ F : B --> RR* ) /\ C e. RR /\ A e. RR* ) -> B C_ RR )
37	36	sselda 3603	. . . . . . . . 9 |- ( ( ( ( B C_ RR /\ F : B --> RR* ) /\ C e. RR /\ A e. RR* ) /\ j e. B ) -> j e. RR )
38		elicopnf 12269	. . . . . . . . . 10 |- ( C e. RR -> ( j e. ( C [,) +oo ) <-> ( j e. RR /\ C <_ j ) ) )
39	38	baibd 948	. . . . . . . . 9 |- ( ( C e. RR /\ j e. RR ) -> ( j e. ( C [,) +oo ) <-> C <_ j ) )
40	35, 37, 39	syl2anc 693	. . . . . . . 8 |- ( ( ( ( B C_ RR /\ F : B --> RR* ) /\ C e. RR /\ A e. RR* ) /\ j e. B ) -> ( j e. ( C [,) +oo ) <-> C <_ j ) )
41	40	pm5.32da 673	. . . . . . 7 |- ( ( ( B C_ RR /\ F : B --> RR* ) /\ C e. RR /\ A e. RR* ) -> ( ( j e. B /\ j e. ( C [,) +oo ) ) <-> ( j e. B /\ C <_ j ) ) )
42	34, 41	bitrd 268	. . . . . 6 |- ( ( ( B C_ RR /\ F : B --> RR* ) /\ C e. RR /\ A e. RR* ) -> ( j e. dom ( F |` ( C [,) +oo ) ) <-> ( j e. B /\ C <_ j ) ) )
43	42	imbi1d 331	. . . . 5 |- ( ( ( B C_ RR /\ F : B --> RR* ) /\ C e. RR /\ A e. RR* ) -> ( ( j e. dom ( F |` ( C [,) +oo ) ) -> ( F ` j ) <_ A ) <-> ( ( j e. B /\ C <_ j ) -> ( F ` j ) <_ A ) ) )
44		impexp 462	. . . . 5 |- ( ( ( j e. B /\ C <_ j ) -> ( F ` j ) <_ A ) <-> ( j e. B -> ( C <_ j -> ( F ` j ) <_ A ) ) )
45	43, 44	syl6bb 276	. . . 4 |- ( ( ( B C_ RR /\ F : B --> RR* ) /\ C e. RR /\ A e. RR* ) -> ( ( j e. dom ( F |` ( C [,) +oo ) ) -> ( F ` j ) <_ A ) <-> ( j e. B -> ( C <_ j -> ( F ` j ) <_ A ) ) ) )
46	45	ralbidv2 2984	. . 3 |- ( ( ( B C_ RR /\ F : B --> RR* ) /\ C e. RR /\ A e. RR* ) -> ( A. j e. dom ( F |` ( C [,) +oo ) ) ( F ` j ) <_ A <-> A. j e. B ( C <_ j -> ( F ` j ) <_ A ) ) )
47	18, 31, 46	3bitrd 294	. 2 |- ( ( ( B C_ RR /\ F : B --> RR* ) /\ C e. RR /\ A e. RR* ) -> ( A. x e. ( ( F " ( C [,) +oo ) ) i^i RR* ) x <_ A <-> A. j e. B ( C <_ j -> ( F ` j ) <_ A ) ) )
48	4, 8, 47	3bitrd 294	1 |- ( ( ( B C_ RR /\ F : B --> RR* ) /\ C e. RR /\ A e. RR* ) -> ( ( G ` C ) <_ A <-> A. j e. B ( C <_ j -> ( F ` j ) <_ A ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   i^i cin 3573   C_ wss 3574   class class class wbr 4653   |-> cmpt 4729   dom cdm 5114   ran crn 5115   |` cres 5116   "cima 5117   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   supcsup 8346   RRcr 9935   +oocpnf 10071   RR*cxr 10073   < clt 10074   <_ cle 10075   [,)cico 12177

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-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-ico 12181

This theorem is referenced by:  limsupgre  14212  limsupbnd1  14213  limsupbnd2  14214  mbflimsup  23433