Theorem limsuppnfdlem 39933

Description: If the restriction of a function to every upper interval is unbounded above, its limsup is +oo. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypotheses

Ref	Expression
limsuppnfdlem.a	|- ( ph -> A C_ RR )
limsuppnfdlem.f	|- ( ph -> F : A --> RR* )
limsuppnfdlem.u	|- ( ph -> A. x e. RR A. k e. RR E. j e. A ( k <_ j /\ x <_ ( F ` j ) ) )
limsuppnfdlem.g	|- G = ( k e. RR |-> sup ( ( ( F " ( k [,) +oo ) ) i^i RR* ) , RR* , < ) )

Assertion

Ref	Expression
limsuppnfdlem	|- ( ph -> ( limsup ` F ) = +oo )

Distinct variable groups:   j, F, k, x   ph, j, k, x

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

Proof of Theorem limsuppnfdlem

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		limsuppnfdlem.f	. . . 4 |- ( ph -> F : A --> RR* )
2		reex 10027	. . . . . 6 |- RR e. _V
3	2	a1i 11	. . . . 5 |- ( ph -> RR e. _V )
4		limsuppnfdlem.a	. . . . 5 |- ( ph -> A C_ RR )
5	3, 4	ssexd 4805	. . . 4 |- ( ph -> A e. _V )
6	1, 5	fexd 39296	. . 3 |- ( ph -> F e. _V )
7		limsuppnfdlem.g	. . . 4 |- G = ( k e. RR |-> sup ( ( ( F " ( k [,) +oo ) ) i^i RR* ) , RR* , < ) )
8	7	limsupval 14205	. . 3 |- ( F e. _V -> ( limsup ` F ) = inf ( ran G , RR* , < ) )
9	6, 8	syl 17	. 2 |- ( ph -> ( limsup ` F ) = inf ( ran G , RR* , < ) )
10	7	a1i 11	. . . . . 6 |- ( ph -> G = ( k e. RR |-> sup ( ( ( F " ( k [,) +oo ) ) i^i RR* ) , RR* , < ) ) )
11		limsuppnfdlem.u	. . . . . . . . . . . . 13 |- ( ph -> A. x e. RR A. k e. RR E. j e. A ( k <_ j /\ x <_ ( F ` j ) ) )
12	11	r19.21bi 2932	. . . . . . . . . . . 12 |- ( ( ph /\ x e. RR ) -> A. k e. RR E. j e. A ( k <_ j /\ x <_ ( F ` j ) ) )
13	12	r19.21bi 2932	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. RR ) /\ k e. RR ) -> E. j e. A ( k <_ j /\ x <_ ( F ` j ) ) )
14	13	an32s 846	. . . . . . . . . 10 |- ( ( ( ph /\ k e. RR ) /\ x e. RR ) -> E. j e. A ( k <_ j /\ x <_ ( F ` j ) ) )
15	1	ffund 6049	. . . . . . . . . . . . . . . . . . . 20 |- ( ph -> Fun F )
16	15	adantr 481	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ j e. A ) -> Fun F )
17		simpr 477	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ j e. A ) -> j e. A )
18	1	fdmd 39420	. . . . . . . . . . . . . . . . . . . . 21 |- ( ph -> dom F = A )
19	18	adantr 481	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ j e. A ) -> dom F = A )
20	17, 19	eleqtrrd 2704	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ j e. A ) -> j e. dom F )
21	16, 20	jca 554	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ j e. A ) -> ( Fun F /\ j e. dom F ) )
22	21	ad4ant13 1292	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ph /\ k e. RR ) /\ j e. A ) /\ k <_ j ) -> ( Fun F /\ j e. dom F ) )
23		simpllr 799	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( ph /\ k e. RR ) /\ j e. A ) /\ k <_ j ) -> k e. RR )
24	23	rexrd 10089	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ph /\ k e. RR ) /\ j e. A ) /\ k <_ j ) -> k e. RR* )
25		pnfxr 10092	. . . . . . . . . . . . . . . . . . 19 |- +oo e. RR*
26	25	a1i 11	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ph /\ k e. RR ) /\ j e. A ) /\ k <_ j ) -> +oo e. RR* )
27		ressxr 10083	. . . . . . . . . . . . . . . . . . . . . . 23 |- RR C_ RR*
28	27	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ph -> RR C_ RR* )
29	4, 28	sstrd 3613	. . . . . . . . . . . . . . . . . . . . 21 |- ( ph -> A C_ RR* )
30	29	adantr 481	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ j e. A ) -> A C_ RR* )
31	30, 17	sseldd 3604	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ j e. A ) -> j e. RR* )
32	31	ad4ant13 1292	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ph /\ k e. RR ) /\ j e. A ) /\ k <_ j ) -> j e. RR* )
33		simpr 477	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ph /\ k e. RR ) /\ j e. A ) /\ k <_ j ) -> k <_ j )
34	4	sselda 3603	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ j e. A ) -> j e. RR )
35		ltpnf 11954	. . . . . . . . . . . . . . . . . . . 20 |- ( j e. RR -> j < +oo )
36	34, 35	syl 17	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ j e. A ) -> j < +oo )
37	36	ad4ant13 1292	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ph /\ k e. RR ) /\ j e. A ) /\ k <_ j ) -> j < +oo )
38	24, 26, 32, 33, 37	elicod 12224	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ph /\ k e. RR ) /\ j e. A ) /\ k <_ j ) -> j e. ( k [,) +oo ) )
39		funfvima 6492	. . . . . . . . . . . . . . . . 17 |- ( ( Fun F /\ j e. dom F ) -> ( j e. ( k [,) +oo ) -> ( F ` j ) e. ( F " ( k [,) +oo ) ) ) )
40	22, 38, 39	sylc 65	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ph /\ k e. RR ) /\ j e. A ) /\ k <_ j ) -> ( F ` j ) e. ( F " ( k [,) +oo ) ) )
41	1	ffvelrnda 6359	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ j e. A ) -> ( F ` j ) e. RR* )
42	41	ad4ant13 1292	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ph /\ k e. RR ) /\ j e. A ) /\ k <_ j ) -> ( F ` j ) e. RR* )
43	40, 42	elind 3798	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ k e. RR ) /\ j e. A ) /\ k <_ j ) -> ( F ` j ) e. ( ( F " ( k [,) +oo ) ) i^i RR* ) )
44	43	adantllr 755	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ph /\ k e. RR ) /\ x e. RR ) /\ j e. A ) /\ k <_ j ) -> ( F ` j ) e. ( ( F " ( k [,) +oo ) ) i^i RR* ) )
45	44	adantrr 753	. . . . . . . . . . . . 13 |- ( ( ( ( ( ph /\ k e. RR ) /\ x e. RR ) /\ j e. A ) /\ ( k <_ j /\ x <_ ( F ` j ) ) ) -> ( F ` j ) e. ( ( F " ( k [,) +oo ) ) i^i RR* ) )
46		simprr 796	. . . . . . . . . . . . 13 |- ( ( ( ( ( ph /\ k e. RR ) /\ x e. RR ) /\ j e. A ) /\ ( k <_ j /\ x <_ ( F ` j ) ) ) -> x <_ ( F ` j ) )
47		nfv 1843	. . . . . . . . . . . . . 14 |- F/ y x <_ ( F ` j )
48		breq2 4657	. . . . . . . . . . . . . 14 |- ( y = ( F ` j ) -> ( x <_ y <-> x <_ ( F ` j ) ) )
49	47, 48	rspce 3304	. . . . . . . . . . . . 13 |- ( ( ( F ` j ) e. ( ( F " ( k [,) +oo ) ) i^i RR* ) /\ x <_ ( F ` j ) ) -> E. y e. ( ( F " ( k [,) +oo ) ) i^i RR* ) x <_ y )
50	45, 46, 49	syl2anc 693	. . . . . . . . . . . 12 |- ( ( ( ( ( ph /\ k e. RR ) /\ x e. RR ) /\ j e. A ) /\ ( k <_ j /\ x <_ ( F ` j ) ) ) -> E. y e. ( ( F " ( k [,) +oo ) ) i^i RR* ) x <_ y )
51	50	ex 450	. . . . . . . . . . 11 |- ( ( ( ( ph /\ k e. RR ) /\ x e. RR ) /\ j e. A ) -> ( ( k <_ j /\ x <_ ( F ` j ) ) -> E. y e. ( ( F " ( k [,) +oo ) ) i^i RR* ) x <_ y ) )
52	51	rexlimdva 3031	. . . . . . . . . 10 |- ( ( ( ph /\ k e. RR ) /\ x e. RR ) -> ( E. j e. A ( k <_ j /\ x <_ ( F ` j ) ) -> E. y e. ( ( F " ( k [,) +oo ) ) i^i RR* ) x <_ y ) )
53	14, 52	mpd 15	. . . . . . . . 9 |- ( ( ( ph /\ k e. RR ) /\ x e. RR ) -> E. y e. ( ( F " ( k [,) +oo ) ) i^i RR* ) x <_ y )
54	53	ralrimiva 2966	. . . . . . . 8 |- ( ( ph /\ k e. RR ) -> A. x e. RR E. y e. ( ( F " ( k [,) +oo ) ) i^i RR* ) x <_ y )
55		inss2 3834	. . . . . . . . . 10 |- ( ( F " ( k [,) +oo ) ) i^i RR* ) C_ RR*
56	55	a1i 11	. . . . . . . . 9 |- ( ( ph /\ k e. RR ) -> ( ( F " ( k [,) +oo ) ) i^i RR* ) C_ RR* )
57		supxrunb3 39623	. . . . . . . . 9 |- ( ( ( F " ( k [,) +oo ) ) i^i RR* ) C_ RR* -> ( A. x e. RR E. y e. ( ( F " ( k [,) +oo ) ) i^i RR* ) x <_ y <-> sup ( ( ( F " ( k [,) +oo ) ) i^i RR* ) , RR* , < ) = +oo ) )
58	56, 57	syl 17	. . . . . . . 8 |- ( ( ph /\ k e. RR ) -> ( A. x e. RR E. y e. ( ( F " ( k [,) +oo ) ) i^i RR* ) x <_ y <-> sup ( ( ( F " ( k [,) +oo ) ) i^i RR* ) , RR* , < ) = +oo ) )
59	54, 58	mpbid 222	. . . . . . 7 |- ( ( ph /\ k e. RR ) -> sup ( ( ( F " ( k [,) +oo ) ) i^i RR* ) , RR* , < ) = +oo )
60	59	mpteq2dva 4744	. . . . . 6 |- ( ph -> ( k e. RR |-> sup ( ( ( F " ( k [,) +oo ) ) i^i RR* ) , RR* , < ) ) = ( k e. RR |-> +oo ) )
61	10, 60	eqtrd 2656	. . . . 5 |- ( ph -> G = ( k e. RR |-> +oo ) )
62	61	rneqd 5353	. . . 4 |- ( ph -> ran G = ran ( k e. RR |-> +oo ) )
63		eqid 2622	. . . . 5 |- ( k e. RR |-> +oo ) = ( k e. RR |-> +oo )
64	25	a1i 11	. . . . 5 |- ( ( ph /\ k e. RR ) -> +oo e. RR* )
65		ren0 39626	. . . . . 6 |- RR =/= (/)
66	65	a1i 11	. . . . 5 |- ( ph -> RR =/= (/) )
67	63, 64, 66	rnmptc 39353	. . . 4 |- ( ph -> ran ( k e. RR |-> +oo ) = { +oo } )
68	62, 67	eqtrd 2656	. . 3 |- ( ph -> ran G = { +oo } )
69	68	infeq1d 8383	. 2 |- ( ph -> inf ( ran G , RR* , < ) = inf ( { +oo } , RR* , < ) )
70		xrltso 11974	. . . . 5 |- < Or RR*
71	70, 25	pm3.2i 471	. . . 4 |- ( < Or RR* /\ +oo e. RR* )
72		infsn 8410	. . . 4 |- ( ( < Or RR* /\ +oo e. RR* ) -> inf ( { +oo } , RR* , < ) = +oo )
73	71, 72	ax-mp 5	. . 3 |- inf ( { +oo } , RR* , < ) = +oo
74	73	a1i 11	. 2 |- ( ph -> inf ( { +oo } , RR* , < ) = +oo )
75	9, 69, 74	3eqtrd 2660	1 |- ( ph -> ( limsup ` F ) = +oo )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   _Vcvv 3200   i^i cin 3573   C_ wss 3574   (/)c0 3915   {csn 4177   class class class wbr 4653   |-> cmpt 4729   Or wor 5034   dom cdm 5114   ran crn 5115   "cima 5117   Fun wfun 5882   -->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-rep 4771  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-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:  limsuppnfd  39934