Theorem liminflelimsuplem 40007

Description: The superior limit is greater than or equal to the inferior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.)

Hypotheses

Ref	Expression
liminflelimsuplem.1	|- ( ph -> F e. V )
liminflelimsuplem.2	|- ( ph -> A. k e. RR E. j e. ( k [,) +oo ) ( ( F " ( j [,) +oo ) ) i^i RR* ) =/= (/) )

Assertion

Ref	Expression
liminflelimsuplem	|- ( ph -> (liminf ` F ) <_ ( limsup ` F ) )

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

Allowed substitution hints:   ph( k)   V( j, k)

Proof of Theorem liminflelimsuplem

Dummy variables i l y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 477	. . . . . . . . . . . 12 |- ( ( i e. RR /\ l e. RR ) -> l e. RR )
2		simpl 473	. . . . . . . . . . . 12 |- ( ( i e. RR /\ l e. RR ) -> i e. RR )
3	1, 2	ifcld 4131	. . . . . . . . . . 11 |- ( ( i e. RR /\ l e. RR ) -> if ( i <_ l , l , i ) e. RR )
4	3	adantll 750	. . . . . . . . . 10 |- ( ( ( ph /\ i e. RR ) /\ l e. RR ) -> if ( i <_ l , l , i ) e. RR )
5		liminflelimsuplem.2	. . . . . . . . . . 11 |- ( ph -> A. k e. RR E. j e. ( k [,) +oo ) ( ( F " ( j [,) +oo ) ) i^i RR* ) =/= (/) )
6	5	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ph /\ i e. RR ) /\ l e. RR ) -> A. k e. RR E. j e. ( k [,) +oo ) ( ( F " ( j [,) +oo ) ) i^i RR* ) =/= (/) )
7		oveq1 6657	. . . . . . . . . . . 12 |- ( k = if ( i <_ l , l , i ) -> ( k [,) +oo ) = ( if ( i <_ l , l , i ) [,) +oo ) )
8	7	rexeqdv 3145	. . . . . . . . . . 11 |- ( k = if ( i <_ l , l , i ) -> ( E. j e. ( k [,) +oo ) ( ( F " ( j [,) +oo ) ) i^i RR* ) =/= (/) <-> E. j e. ( if ( i <_ l , l , i ) [,) +oo ) ( ( F " ( j [,) +oo ) ) i^i RR* ) =/= (/) ) )
9	8	rspcva 3307	. . . . . . . . . 10 |- ( ( if ( i <_ l , l , i ) e. RR /\ A. k e. RR E. j e. ( k [,) +oo ) ( ( F " ( j [,) +oo ) ) i^i RR* ) =/= (/) ) -> E. j e. ( if ( i <_ l , l , i ) [,) +oo ) ( ( F " ( j [,) +oo ) ) i^i RR* ) =/= (/) )
10	4, 6, 9	syl2anc 693	. . . . . . . . 9 |- ( ( ( ph /\ i e. RR ) /\ l e. RR ) -> E. j e. ( if ( i <_ l , l , i ) [,) +oo ) ( ( F " ( j [,) +oo ) ) i^i RR* ) =/= (/) )
11		inss2 3834	. . . . . . . . . . . . . 14 |- ( ( F " ( i [,) +oo ) ) i^i RR* ) C_ RR*
12		infxrcl 12163	. . . . . . . . . . . . . 14 |- ( ( ( F " ( i [,) +oo ) ) i^i RR* ) C_ RR* -> inf ( ( ( F " ( i [,) +oo ) ) i^i RR* ) , RR* , < ) e. RR* )
13	11, 12	ax-mp 5	. . . . . . . . . . . . 13 |- inf ( ( ( F " ( i [,) +oo ) ) i^i RR* ) , RR* , < ) e. RR*
14	13	a1i 11	. . . . . . . . . . . 12 |- ( ( ( ( i e. RR /\ l e. RR ) /\ j e. ( if ( i <_ l , l , i ) [,) +oo ) ) /\ ( ( F " ( j [,) +oo ) ) i^i RR* ) =/= (/) ) -> inf ( ( ( F " ( i [,) +oo ) ) i^i RR* ) , RR* , < ) e. RR* )
15		inss2 3834	. . . . . . . . . . . . . 14 |- ( ( F " ( j [,) +oo ) ) i^i RR* ) C_ RR*
16		infxrcl 12163	. . . . . . . . . . . . . 14 |- ( ( ( F " ( j [,) +oo ) ) i^i RR* ) C_ RR* -> inf ( ( ( F " ( j [,) +oo ) ) i^i RR* ) , RR* , < ) e. RR* )
17	15, 16	ax-mp 5	. . . . . . . . . . . . 13 |- inf ( ( ( F " ( j [,) +oo ) ) i^i RR* ) , RR* , < ) e. RR*
18	17	a1i 11	. . . . . . . . . . . 12 |- ( ( ( ( i e. RR /\ l e. RR ) /\ j e. ( if ( i <_ l , l , i ) [,) +oo ) ) /\ ( ( F " ( j [,) +oo ) ) i^i RR* ) =/= (/) ) -> inf ( ( ( F " ( j [,) +oo ) ) i^i RR* ) , RR* , < ) e. RR* )
19		inss2 3834	. . . . . . . . . . . . . 14 |- ( ( F " ( l [,) +oo ) ) i^i RR* ) C_ RR*
20		supxrcl 12145	. . . . . . . . . . . . . 14 |- ( ( ( F " ( l [,) +oo ) ) i^i RR* ) C_ RR* -> sup ( ( ( F " ( l [,) +oo ) ) i^i RR* ) , RR* , < ) e. RR* )
21	19, 20	ax-mp 5	. . . . . . . . . . . . 13 |- sup ( ( ( F " ( l [,) +oo ) ) i^i RR* ) , RR* , < ) e. RR*
22	21	a1i 11	. . . . . . . . . . . 12 |- ( ( ( ( i e. RR /\ l e. RR ) /\ j e. ( if ( i <_ l , l , i ) [,) +oo ) ) /\ ( ( F " ( j [,) +oo ) ) i^i RR* ) =/= (/) ) -> sup ( ( ( F " ( l [,) +oo ) ) i^i RR* ) , RR* , < ) e. RR* )
23		rexr 10085	. . . . . . . . . . . . . . . . . 18 |- ( i e. RR -> i e. RR* )
24	23	ad2antrr 762	. . . . . . . . . . . . . . . . 17 |- ( ( ( i e. RR /\ l e. RR ) /\ j e. ( if ( i <_ l , l , i ) [,) +oo ) ) -> i e. RR* )
25		pnfxr 10092	. . . . . . . . . . . . . . . . . 18 |- +oo e. RR*
26	25	a1i 11	. . . . . . . . . . . . . . . . 17 |- ( ( ( i e. RR /\ l e. RR ) /\ j e. ( if ( i <_ l , l , i ) [,) +oo ) ) -> +oo e. RR* )
27	3	rexrd 10089	. . . . . . . . . . . . . . . . . . 19 |- ( ( i e. RR /\ l e. RR ) -> if ( i <_ l , l , i ) e. RR* )
28	27	adantr 481	. . . . . . . . . . . . . . . . . 18 |- ( ( ( i e. RR /\ l e. RR ) /\ j e. ( if ( i <_ l , l , i ) [,) +oo ) ) -> if ( i <_ l , l , i ) e. RR* )
29		icossxr 12258	. . . . . . . . . . . . . . . . . . . 20 |- ( if ( i <_ l , l , i ) [,) +oo ) C_ RR*
30		id 22	. . . . . . . . . . . . . . . . . . . 20 |- ( j e. ( if ( i <_ l , l , i ) [,) +oo ) -> j e. ( if ( i <_ l , l , i ) [,) +oo ) )
31	29, 30	sseldi 3601	. . . . . . . . . . . . . . . . . . 19 |- ( j e. ( if ( i <_ l , l , i ) [,) +oo ) -> j e. RR* )
32	31	adantl 482	. . . . . . . . . . . . . . . . . 18 |- ( ( ( i e. RR /\ l e. RR ) /\ j e. ( if ( i <_ l , l , i ) [,) +oo ) ) -> j e. RR* )
33		max1 12016	. . . . . . . . . . . . . . . . . . 19 |- ( ( i e. RR /\ l e. RR ) -> i <_ if ( i <_ l , l , i ) )
34	33	adantr 481	. . . . . . . . . . . . . . . . . 18 |- ( ( ( i e. RR /\ l e. RR ) /\ j e. ( if ( i <_ l , l , i ) [,) +oo ) ) -> i <_ if ( i <_ l , l , i ) )
35		simpr 477	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( i e. RR /\ l e. RR ) /\ j e. ( if ( i <_ l , l , i ) [,) +oo ) ) -> j e. ( if ( i <_ l , l , i ) [,) +oo ) )
36	28, 26, 35	icogelbd 39785	. . . . . . . . . . . . . . . . . 18 |- ( ( ( i e. RR /\ l e. RR ) /\ j e. ( if ( i <_ l , l , i ) [,) +oo ) ) -> if ( i <_ l , l , i ) <_ j )
37	24, 28, 32, 34, 36	xrletrd 11993	. . . . . . . . . . . . . . . . 17 |- ( ( ( i e. RR /\ l e. RR ) /\ j e. ( if ( i <_ l , l , i ) [,) +oo ) ) -> i <_ j )
38	24, 26, 37	icossico2 39791	. . . . . . . . . . . . . . . 16 |- ( ( ( i e. RR /\ l e. RR ) /\ j e. ( if ( i <_ l , l , i ) [,) +oo ) ) -> ( j [,) +oo ) C_ ( i [,) +oo ) )
39	38	imass2d 39480	. . . . . . . . . . . . . . 15 |- ( ( ( i e. RR /\ l e. RR ) /\ j e. ( if ( i <_ l , l , i ) [,) +oo ) ) -> ( F " ( j [,) +oo ) ) C_ ( F " ( i [,) +oo ) ) )
40	39	ssrind 39333	. . . . . . . . . . . . . 14 |- ( ( ( i e. RR /\ l e. RR ) /\ j e. ( if ( i <_ l , l , i ) [,) +oo ) ) -> ( ( F " ( j [,) +oo ) ) i^i RR* ) C_ ( ( F " ( i [,) +oo ) ) i^i RR* ) )
41	11	a1i 11	. . . . . . . . . . . . . 14 |- ( ( ( i e. RR /\ l e. RR ) /\ j e. ( if ( i <_ l , l , i ) [,) +oo ) ) -> ( ( F " ( i [,) +oo ) ) i^i RR* ) C_ RR* )
42		infxrss 12169	. . . . . . . . . . . . . 14 |- ( ( ( ( F " ( j [,) +oo ) ) i^i RR* ) C_ ( ( F " ( i [,) +oo ) ) i^i RR* ) /\ ( ( F " ( i [,) +oo ) ) i^i RR* ) C_ RR* ) -> inf ( ( ( F " ( i [,) +oo ) ) i^i RR* ) , RR* , < ) <_ inf ( ( ( F " ( j [,) +oo ) ) i^i RR* ) , RR* , < ) )
43	40, 41, 42	syl2anc 693	. . . . . . . . . . . . 13 |- ( ( ( i e. RR /\ l e. RR ) /\ j e. ( if ( i <_ l , l , i ) [,) +oo ) ) -> inf ( ( ( F " ( i [,) +oo ) ) i^i RR* ) , RR* , < ) <_ inf ( ( ( F " ( j [,) +oo ) ) i^i RR* ) , RR* , < ) )
44	43	adantr 481	. . . . . . . . . . . 12 |- ( ( ( ( i e. RR /\ l e. RR ) /\ j e. ( if ( i <_ l , l , i ) [,) +oo ) ) /\ ( ( F " ( j [,) +oo ) ) i^i RR* ) =/= (/) ) -> inf ( ( ( F " ( i [,) +oo ) ) i^i RR* ) , RR* , < ) <_ inf ( ( ( F " ( j [,) +oo ) ) i^i RR* ) , RR* , < ) )
45		supxrcl 12145	. . . . . . . . . . . . . . 15 |- ( ( ( F " ( j [,) +oo ) ) i^i RR* ) C_ RR* -> sup ( ( ( F " ( j [,) +oo ) ) i^i RR* ) , RR* , < ) e. RR* )
46	15, 45	ax-mp 5	. . . . . . . . . . . . . 14 |- sup ( ( ( F " ( j [,) +oo ) ) i^i RR* ) , RR* , < ) e. RR*
47	46	a1i 11	. . . . . . . . . . . . 13 |- ( ( ( ( i e. RR /\ l e. RR ) /\ j e. ( if ( i <_ l , l , i ) [,) +oo ) ) /\ ( ( F " ( j [,) +oo ) ) i^i RR* ) =/= (/) ) -> sup ( ( ( F " ( j [,) +oo ) ) i^i RR* ) , RR* , < ) e. RR* )
48	15	a1i 11	. . . . . . . . . . . . . 14 |- ( ( ( ( i e. RR /\ l e. RR ) /\ j e. ( if ( i <_ l , l , i ) [,) +oo ) ) /\ ( ( F " ( j [,) +oo ) ) i^i RR* ) =/= (/) ) -> ( ( F " ( j [,) +oo ) ) i^i RR* ) C_ RR* )
49		simpr 477	. . . . . . . . . . . . . 14 |- ( ( ( ( i e. RR /\ l e. RR ) /\ j e. ( if ( i <_ l , l , i ) [,) +oo ) ) /\ ( ( F " ( j [,) +oo ) ) i^i RR* ) =/= (/) ) -> ( ( F " ( j [,) +oo ) ) i^i RR* ) =/= (/) )
50	48, 49	infxrlesupxr 39663	. . . . . . . . . . . . 13 |- ( ( ( ( i e. RR /\ l e. RR ) /\ j e. ( if ( i <_ l , l , i ) [,) +oo ) ) /\ ( ( F " ( j [,) +oo ) ) i^i RR* ) =/= (/) ) -> inf ( ( ( F " ( j [,) +oo ) ) i^i RR* ) , RR* , < ) <_ sup ( ( ( F " ( j [,) +oo ) ) i^i RR* ) , RR* , < ) )
51		rexr 10085	. . . . . . . . . . . . . . . . . . 19 |- ( l e. RR -> l e. RR* )
52	51	ad2antlr 763	. . . . . . . . . . . . . . . . . 18 |- ( ( ( i e. RR /\ l e. RR ) /\ j e. ( if ( i <_ l , l , i ) [,) +oo ) ) -> l e. RR* )
53		max2 12018	. . . . . . . . . . . . . . . . . . . 20 |- ( ( i e. RR /\ l e. RR ) -> l <_ if ( i <_ l , l , i ) )
54	53	adantr 481	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( i e. RR /\ l e. RR ) /\ j e. ( if ( i <_ l , l , i ) [,) +oo ) ) -> l <_ if ( i <_ l , l , i ) )
55	52, 28, 32, 54, 36	xrletrd 11993	. . . . . . . . . . . . . . . . . 18 |- ( ( ( i e. RR /\ l e. RR ) /\ j e. ( if ( i <_ l , l , i ) [,) +oo ) ) -> l <_ j )
56	52, 26, 55	icossico2 39791	. . . . . . . . . . . . . . . . 17 |- ( ( ( i e. RR /\ l e. RR ) /\ j e. ( if ( i <_ l , l , i ) [,) +oo ) ) -> ( j [,) +oo ) C_ ( l [,) +oo ) )
57	56	imass2d 39480	. . . . . . . . . . . . . . . 16 |- ( ( ( i e. RR /\ l e. RR ) /\ j e. ( if ( i <_ l , l , i ) [,) +oo ) ) -> ( F " ( j [,) +oo ) ) C_ ( F " ( l [,) +oo ) ) )
58	57	ssrind 39333	. . . . . . . . . . . . . . 15 |- ( ( ( i e. RR /\ l e. RR ) /\ j e. ( if ( i <_ l , l , i ) [,) +oo ) ) -> ( ( F " ( j [,) +oo ) ) i^i RR* ) C_ ( ( F " ( l [,) +oo ) ) i^i RR* ) )
59	19	a1i 11	. . . . . . . . . . . . . . 15 |- ( ( ( i e. RR /\ l e. RR ) /\ j e. ( if ( i <_ l , l , i ) [,) +oo ) ) -> ( ( F " ( l [,) +oo ) ) i^i RR* ) C_ RR* )
60		supxrss 12162	. . . . . . . . . . . . . . 15 |- ( ( ( ( F " ( j [,) +oo ) ) i^i RR* ) C_ ( ( F " ( l [,) +oo ) ) i^i RR* ) /\ ( ( F " ( l [,) +oo ) ) i^i RR* ) C_ RR* ) -> sup ( ( ( F " ( j [,) +oo ) ) i^i RR* ) , RR* , < ) <_ sup ( ( ( F " ( l [,) +oo ) ) i^i RR* ) , RR* , < ) )
61	58, 59, 60	syl2anc 693	. . . . . . . . . . . . . 14 |- ( ( ( i e. RR /\ l e. RR ) /\ j e. ( if ( i <_ l , l , i ) [,) +oo ) ) -> sup ( ( ( F " ( j [,) +oo ) ) i^i RR* ) , RR* , < ) <_ sup ( ( ( F " ( l [,) +oo ) ) i^i RR* ) , RR* , < ) )
62	61	adantr 481	. . . . . . . . . . . . 13 |- ( ( ( ( i e. RR /\ l e. RR ) /\ j e. ( if ( i <_ l , l , i ) [,) +oo ) ) /\ ( ( F " ( j [,) +oo ) ) i^i RR* ) =/= (/) ) -> sup ( ( ( F " ( j [,) +oo ) ) i^i RR* ) , RR* , < ) <_ sup ( ( ( F " ( l [,) +oo ) ) i^i RR* ) , RR* , < ) )
63	18, 47, 22, 50, 62	xrletrd 11993	. . . . . . . . . . . 12 |- ( ( ( ( i e. RR /\ l e. RR ) /\ j e. ( if ( i <_ l , l , i ) [,) +oo ) ) /\ ( ( F " ( j [,) +oo ) ) i^i RR* ) =/= (/) ) -> inf ( ( ( F " ( j [,) +oo ) ) i^i RR* ) , RR* , < ) <_ sup ( ( ( F " ( l [,) +oo ) ) i^i RR* ) , RR* , < ) )
64	14, 18, 22, 44, 63	xrletrd 11993	. . . . . . . . . . 11 |- ( ( ( ( i e. RR /\ l e. RR ) /\ j e. ( if ( i <_ l , l , i ) [,) +oo ) ) /\ ( ( F " ( j [,) +oo ) ) i^i RR* ) =/= (/) ) -> inf ( ( ( F " ( i [,) +oo ) ) i^i RR* ) , RR* , < ) <_ sup ( ( ( F " ( l [,) +oo ) ) i^i RR* ) , RR* , < ) )
65	64	ad5ant2345 1317	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ i e. RR ) /\ l e. RR ) /\ j e. ( if ( i <_ l , l , i ) [,) +oo ) ) /\ ( ( F " ( j [,) +oo ) ) i^i RR* ) =/= (/) ) -> inf ( ( ( F " ( i [,) +oo ) ) i^i RR* ) , RR* , < ) <_ sup ( ( ( F " ( l [,) +oo ) ) i^i RR* ) , RR* , < ) )
66	65	rexlimdva2 39339	. . . . . . . . 9 |- ( ( ( ph /\ i e. RR ) /\ l e. RR ) -> ( E. j e. ( if ( i <_ l , l , i ) [,) +oo ) ( ( F " ( j [,) +oo ) ) i^i RR* ) =/= (/) -> inf ( ( ( F " ( i [,) +oo ) ) i^i RR* ) , RR* , < ) <_ sup ( ( ( F " ( l [,) +oo ) ) i^i RR* ) , RR* , < ) ) )
67	10, 66	mpd 15	. . . . . . . 8 |- ( ( ( ph /\ i e. RR ) /\ l e. RR ) -> inf ( ( ( F " ( i [,) +oo ) ) i^i RR* ) , RR* , < ) <_ sup ( ( ( F " ( l [,) +oo ) ) i^i RR* ) , RR* , < ) )
68	67	ralrimiva 2966	. . . . . . 7 |- ( ( ph /\ i e. RR ) -> A. l e. RR inf ( ( ( F " ( i [,) +oo ) ) i^i RR* ) , RR* , < ) <_ sup ( ( ( F " ( l [,) +oo ) ) i^i RR* ) , RR* , < ) )
69		nfv 1843	. . . . . . . . 9 |- F/ l ph
70		xrltso 11974	. . . . . . . . . . 11 |- < Or RR*
71	70	supex 8369	. . . . . . . . . 10 |- sup ( ( ( F " ( l [,) +oo ) ) i^i RR* ) , RR* , < ) e. _V
72	71	a1i 11	. . . . . . . . 9 |- ( ( ph /\ l e. RR ) -> sup ( ( ( F " ( l [,) +oo ) ) i^i RR* ) , RR* , < ) e. _V )
73		breq2 4657	. . . . . . . . 9 |- ( y = sup ( ( ( F " ( l [,) +oo ) ) i^i RR* ) , RR* , < ) -> (inf ( ( ( F " ( i [,) +oo ) ) i^i RR* ) , RR* , < ) <_ y <-> inf ( ( ( F " ( i [,) +oo ) ) i^i RR* ) , RR* , < ) <_ sup ( ( ( F " ( l [,) +oo ) ) i^i RR* ) , RR* , < ) ) )
74	69, 72, 73	ralrnmpt3 39474	. . . . . . . 8 |- ( ph -> ( A. y e. ran ( l e. RR |-> sup ( ( ( F " ( l [,) +oo ) ) i^i RR* ) , RR* , < ) )inf ( ( ( F " ( i [,) +oo ) ) i^i RR* ) , RR* , < ) <_ y <-> A. l e. RR inf ( ( ( F " ( i [,) +oo ) ) i^i RR* ) , RR* , < ) <_ sup ( ( ( F " ( l [,) +oo ) ) i^i RR* ) , RR* , < ) ) )
75	74	adantr 481	. . . . . . 7 |- ( ( ph /\ i e. RR ) -> ( A. y e. ran ( l e. RR |-> sup ( ( ( F " ( l [,) +oo ) ) i^i RR* ) , RR* , < ) )inf ( ( ( F " ( i [,) +oo ) ) i^i RR* ) , RR* , < ) <_ y <-> A. l e. RR inf ( ( ( F " ( i [,) +oo ) ) i^i RR* ) , RR* , < ) <_ sup ( ( ( F " ( l [,) +oo ) ) i^i RR* ) , RR* , < ) ) )
76	68, 75	mpbird 247	. . . . . 6 |- ( ( ph /\ i e. RR ) -> A. y e. ran ( l e. RR |-> sup ( ( ( F " ( l [,) +oo ) ) i^i RR* ) , RR* , < ) )inf ( ( ( F " ( i [,) +oo ) ) i^i RR* ) , RR* , < ) <_ y )
77		oveq1 6657	. . . . . . . . . . . . 13 |- ( l = i -> ( l [,) +oo ) = ( i [,) +oo ) )
78	77	imaeq2d 5466	. . . . . . . . . . . 12 |- ( l = i -> ( F " ( l [,) +oo ) ) = ( F " ( i [,) +oo ) ) )
79	78	ineq1d 3813	. . . . . . . . . . 11 |- ( l = i -> ( ( F " ( l [,) +oo ) ) i^i RR* ) = ( ( F " ( i [,) +oo ) ) i^i RR* ) )
80	79	supeq1d 8352	. . . . . . . . . 10 |- ( l = i -> sup ( ( ( F " ( l [,) +oo ) ) i^i RR* ) , RR* , < ) = sup ( ( ( F " ( i [,) +oo ) ) i^i RR* ) , RR* , < ) )
81	80	cbvmptv 4750	. . . . . . . . 9 |- ( l e. RR |-> sup ( ( ( F " ( l [,) +oo ) ) i^i RR* ) , RR* , < ) ) = ( i e. RR |-> sup ( ( ( F " ( i [,) +oo ) ) i^i RR* ) , RR* , < ) )
82	81	rneqi 5352	. . . . . . . 8 |- ran ( l e. RR |-> sup ( ( ( F " ( l [,) +oo ) ) i^i RR* ) , RR* , < ) ) = ran ( i e. RR |-> sup ( ( ( F " ( i [,) +oo ) ) i^i RR* ) , RR* , < ) )
83	82	raleqi 3142	. . . . . . 7 |- ( A. y e. ran ( l e. RR |-> sup ( ( ( F " ( l [,) +oo ) ) i^i RR* ) , RR* , < ) )inf ( ( ( F " ( i [,) +oo ) ) i^i RR* ) , RR* , < ) <_ y <-> A. y e. ran ( i e. RR |-> sup ( ( ( F " ( i [,) +oo ) ) i^i RR* ) , RR* , < ) )inf ( ( ( F " ( i [,) +oo ) ) i^i RR* ) , RR* , < ) <_ y )
84	83	a1i 11	. . . . . 6 |- ( ( ph /\ i e. RR ) -> ( A. y e. ran ( l e. RR |-> sup ( ( ( F " ( l [,) +oo ) ) i^i RR* ) , RR* , < ) )inf ( ( ( F " ( i [,) +oo ) ) i^i RR* ) , RR* , < ) <_ y <-> A. y e. ran ( i e. RR |-> sup ( ( ( F " ( i [,) +oo ) ) i^i RR* ) , RR* , < ) )inf ( ( ( F " ( i [,) +oo ) ) i^i RR* ) , RR* , < ) <_ y ) )
85	76, 84	mpbid 222	. . . . 5 |- ( ( ph /\ i e. RR ) -> A. y e. ran ( i e. RR |-> sup ( ( ( F " ( i [,) +oo ) ) i^i RR* ) , RR* , < ) )inf ( ( ( F " ( i [,) +oo ) ) i^i RR* ) , RR* , < ) <_ y )
86		supxrcl 12145	. . . . . . . . . 10 |- ( ( ( F " ( i [,) +oo ) ) i^i RR* ) C_ RR* -> sup ( ( ( F " ( i [,) +oo ) ) i^i RR* ) , RR* , < ) e. RR* )
87	11, 86	ax-mp 5	. . . . . . . . 9 |- sup ( ( ( F " ( i [,) +oo ) ) i^i RR* ) , RR* , < ) e. RR*
88	87	rgenw 2924	. . . . . . . 8 |- A. i e. RR sup ( ( ( F " ( i [,) +oo ) ) i^i RR* ) , RR* , < ) e. RR*
89		eqid 2622	. . . . . . . . 9 |- ( i e. RR |-> sup ( ( ( F " ( i [,) +oo ) ) i^i RR* ) , RR* , < ) ) = ( i e. RR |-> sup ( ( ( F " ( i [,) +oo ) ) i^i RR* ) , RR* , < ) )
90	89	rnmptss 6392	. . . . . . . 8 |- ( A. i e. RR sup ( ( ( F " ( i [,) +oo ) ) i^i RR* ) , RR* , < ) e. RR* -> ran ( i e. RR |-> sup ( ( ( F " ( i [,) +oo ) ) i^i RR* ) , RR* , < ) ) C_ RR* )
91	88, 90	ax-mp 5	. . . . . . 7 |- ran ( i e. RR |-> sup ( ( ( F " ( i [,) +oo ) ) i^i RR* ) , RR* , < ) ) C_ RR*
92	91	a1i 11	. . . . . 6 |- ( ( ph /\ i e. RR ) -> ran ( i e. RR |-> sup ( ( ( F " ( i [,) +oo ) ) i^i RR* ) , RR* , < ) ) C_ RR* )
93	13	a1i 11	. . . . . 6 |- ( ( ph /\ i e. RR ) -> inf ( ( ( F " ( i [,) +oo ) ) i^i RR* ) , RR* , < ) e. RR* )
94		infxrgelb 12165	. . . . . 6 |- ( ( ran ( i e. RR |-> sup ( ( ( F " ( i [,) +oo ) ) i^i RR* ) , RR* , < ) ) C_ RR* /\ inf ( ( ( F " ( i [,) +oo ) ) i^i RR* ) , RR* , < ) e. RR* ) -> (inf ( ( ( F " ( i [,) +oo ) ) i^i RR* ) , RR* , < ) <_ inf ( ran ( i e. RR |-> sup ( ( ( F " ( i [,) +oo ) ) i^i RR* ) , RR* , < ) ) , RR* , < ) <-> A. y e. ran ( i e. RR |-> sup ( ( ( F " ( i [,) +oo ) ) i^i RR* ) , RR* , < ) )inf ( ( ( F " ( i [,) +oo ) ) i^i RR* ) , RR* , < ) <_ y ) )
95	92, 93, 94	syl2anc 693	. . . . 5 |- ( ( ph /\ i e. RR ) -> (inf ( ( ( F " ( i [,) +oo ) ) i^i RR* ) , RR* , < ) <_ inf ( ran ( i e. RR |-> sup ( ( ( F " ( i [,) +oo ) ) i^i RR* ) , RR* , < ) ) , RR* , < ) <-> A. y e. ran ( i e. RR |-> sup ( ( ( F " ( i [,) +oo ) ) i^i RR* ) , RR* , < ) )inf ( ( ( F " ( i [,) +oo ) ) i^i RR* ) , RR* , < ) <_ y ) )
96	85, 95	mpbird 247	. . . 4 |- ( ( ph /\ i e. RR ) -> inf ( ( ( F " ( i [,) +oo ) ) i^i RR* ) , RR* , < ) <_ inf ( ran ( i e. RR |-> sup ( ( ( F " ( i [,) +oo ) ) i^i RR* ) , RR* , < ) ) , RR* , < ) )
97	96	ralrimiva 2966	. . 3 |- ( ph -> A. i e. RR inf ( ( ( F " ( i [,) +oo ) ) i^i RR* ) , RR* , < ) <_ inf ( ran ( i e. RR |-> sup ( ( ( F " ( i [,) +oo ) ) i^i RR* ) , RR* , < ) ) , RR* , < ) )
98		nfv 1843	. . . 4 |- F/ i ph
99		nfcv 2764	. . . 4 |- F/_ i RR
100		nfmpt1 4747	. . . . . 6 |- F/_ i ( i e. RR |-> sup ( ( ( F " ( i [,) +oo ) ) i^i RR* ) , RR* , < ) )
101	100	nfrn 5368	. . . . 5 |- F/_ i ran ( i e. RR |-> sup ( ( ( F " ( i [,) +oo ) ) i^i RR* ) , RR* , < ) )
102		nfcv 2764	. . . . 5 |- F/_ i RR*
103		nfcv 2764	. . . . 5 |- F/_ i <
104	101, 102, 103	nfinf 8388	. . . 4 |- F/_ iinf ( ran ( i e. RR |-> sup ( ( ( F " ( i [,) +oo ) ) i^i RR* ) , RR* , < ) ) , RR* , < )
105		infxrcl 12163	. . . . . 6 |- ( ran ( i e. RR |-> sup ( ( ( F " ( i [,) +oo ) ) i^i RR* ) , RR* , < ) ) C_ RR* -> inf ( ran ( i e. RR |-> sup ( ( ( F " ( i [,) +oo ) ) i^i RR* ) , RR* , < ) ) , RR* , < ) e. RR* )
106	91, 105	ax-mp 5	. . . . 5 |- inf ( ran ( i e. RR |-> sup ( ( ( F " ( i [,) +oo ) ) i^i RR* ) , RR* , < ) ) , RR* , < ) e. RR*
107	106	a1i 11	. . . 4 |- ( ph -> inf ( ran ( i e. RR |-> sup ( ( ( F " ( i [,) +oo ) ) i^i RR* ) , RR* , < ) ) , RR* , < ) e. RR* )
108	98, 99, 104, 93, 107	supxrleubrnmptf 39680	. . 3 |- ( ph -> ( sup ( ran ( i e. RR |-> inf ( ( ( F " ( i [,) +oo ) ) i^i RR* ) , RR* , < ) ) , RR* , < ) <_ inf ( ran ( i e. RR |-> sup ( ( ( F " ( i [,) +oo ) ) i^i RR* ) , RR* , < ) ) , RR* , < ) <-> A. i e. RR inf ( ( ( F " ( i [,) +oo ) ) i^i RR* ) , RR* , < ) <_ inf ( ran ( i e. RR |-> sup ( ( ( F " ( i [,) +oo ) ) i^i RR* ) , RR* , < ) ) , RR* , < ) ) )
109	97, 108	mpbird 247	. 2 |- ( ph -> sup ( ran ( i e. RR |-> inf ( ( ( F " ( i [,) +oo ) ) i^i RR* ) , RR* , < ) ) , RR* , < ) <_ inf ( ran ( i e. RR |-> sup ( ( ( F " ( i [,) +oo ) ) i^i RR* ) , RR* , < ) ) , RR* , < ) )
110		liminflelimsuplem.1	. . . 4 |- ( ph -> F e. V )
111		eqid 2622	. . . 4 |- ( i e. RR |-> inf ( ( ( F " ( i [,) +oo ) ) i^i RR* ) , RR* , < ) ) = ( i e. RR |-> inf ( ( ( F " ( i [,) +oo ) ) i^i RR* ) , RR* , < ) )
112	110, 111	liminfvald 39996	. . 3 |- ( ph -> (liminf ` F ) = sup ( ran ( i e. RR |-> inf ( ( ( F " ( i [,) +oo ) ) i^i RR* ) , RR* , < ) ) , RR* , < ) )
113	110, 89	limsupvald 39987	. . 3 |- ( ph -> ( limsup ` F ) = inf ( ran ( i e. RR |-> sup ( ( ( F " ( i [,) +oo ) ) i^i RR* ) , RR* , < ) ) , RR* , < ) )
114	112, 113	breq12d 4666	. 2 |- ( ph -> ( (liminf ` F ) <_ ( limsup ` F ) <-> sup ( ran ( i e. RR |-> inf ( ( ( F " ( i [,) +oo ) ) i^i RR* ) , RR* , < ) ) , RR* , < ) <_ inf ( ran ( i e. RR |-> sup ( ( ( F " ( i [,) +oo ) ) i^i RR* ) , RR* , < ) ) , RR* , < ) ) )
115	109, 114	mpbird 247	1 |- ( ph -> (liminf ` F ) <_ ( limsup ` F ) )

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   ifcif 4086   class class class wbr 4653   |-> cmpt 4729   ran crn 5115   "cima 5117   ` 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  liminfclsi 39983

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-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-1st 7168  df-2nd 7169  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  df-liminf 39984

This theorem is referenced by:  liminflelimsup  40008