Theorem supcnvlimsup 39972

Description: If a function on a set of upper integers has a real superior limit, the supremum of the rightmost parts of the function, converges to that superior limit. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypotheses

Ref	Expression
supcnvlimsup.m	|- ( ph -> M e. ZZ )
supcnvlimsup.z	|- Z = ( ZZ>= ` M )
supcnvlimsup.f	|- ( ph -> F : Z --> RR )
supcnvlimsup.r	|- ( ph -> ( limsup ` F ) e. RR )

Assertion

Ref	Expression
supcnvlimsup	|- ( ph -> ( k e. Z |-> sup ( ran ( F |` ( ZZ>= ` k ) ) , RR* , < ) ) ~~> ( limsup ` F ) )

Distinct variable groups:   k, F   k, Z

Allowed substitution hints:   ph( k)   M( k)

Proof of Theorem supcnvlimsup

Dummy variables i j x n m y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		supcnvlimsup.z	. . 3 |- Z = ( ZZ>= ` M )
2		supcnvlimsup.m	. . 3 |- ( ph -> M e. ZZ )
3		supcnvlimsup.f	. . . . . . . . 9 |- ( ph -> F : Z --> RR )
4	3	adantr 481	. . . . . . . 8 |- ( ( ph /\ n e. Z ) -> F : Z --> RR )
5		id 22	. . . . . . . . . 10 |- ( n e. Z -> n e. Z )
6	1, 5	uzssd2 39644	. . . . . . . . 9 |- ( n e. Z -> ( ZZ>= ` n ) C_ Z )
7	6	adantl 482	. . . . . . . 8 |- ( ( ph /\ n e. Z ) -> ( ZZ>= ` n ) C_ Z )
8	4, 7	feqresmpt 6250	. . . . . . 7 |- ( ( ph /\ n e. Z ) -> ( F |` ( ZZ>= ` n ) ) = ( m e. ( ZZ>= ` n ) |-> ( F ` m ) ) )
9	8	rneqd 5353	. . . . . 6 |- ( ( ph /\ n e. Z ) -> ran ( F |` ( ZZ>= ` n ) ) = ran ( m e. ( ZZ>= ` n ) |-> ( F ` m ) ) )
10	9	supeq1d 8352	. . . . 5 |- ( ( ph /\ n e. Z ) -> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) = sup ( ran ( m e. ( ZZ>= ` n ) |-> ( F ` m ) ) , RR* , < ) )
11		nfcv 2764	. . . . . . . . 9 |- F/_ m F
12		supcnvlimsup.r	. . . . . . . . . 10 |- ( ph -> ( limsup ` F ) e. RR )
13	12	renepnfd 10090	. . . . . . . . 9 |- ( ph -> ( limsup ` F ) =/= +oo )
14	11, 1, 3, 13	limsupubuz 39945	. . . . . . . 8 |- ( ph -> E. x e. RR A. m e. Z ( F ` m ) <_ x )
15	14	adantr 481	. . . . . . 7 |- ( ( ph /\ n e. Z ) -> E. x e. RR A. m e. Z ( F ` m ) <_ x )
16		ssralv 3666	. . . . . . . . . 10 |- ( ( ZZ>= ` n ) C_ Z -> ( A. m e. Z ( F ` m ) <_ x -> A. m e. ( ZZ>= ` n ) ( F ` m ) <_ x ) )
17	6, 16	syl 17	. . . . . . . . 9 |- ( n e. Z -> ( A. m e. Z ( F ` m ) <_ x -> A. m e. ( ZZ>= ` n ) ( F ` m ) <_ x ) )
18	17	adantl 482	. . . . . . . 8 |- ( ( ph /\ n e. Z ) -> ( A. m e. Z ( F ` m ) <_ x -> A. m e. ( ZZ>= ` n ) ( F ` m ) <_ x ) )
19	18	reximdv 3016	. . . . . . 7 |- ( ( ph /\ n e. Z ) -> ( E. x e. RR A. m e. Z ( F ` m ) <_ x -> E. x e. RR A. m e. ( ZZ>= ` n ) ( F ` m ) <_ x ) )
20	15, 19	mpd 15	. . . . . 6 |- ( ( ph /\ n e. Z ) -> E. x e. RR A. m e. ( ZZ>= ` n ) ( F ` m ) <_ x )
21		nfv 1843	. . . . . . 7 |- F/ m ( ph /\ n e. Z )
22	1	eluzelz2 39627	. . . . . . . . 9 |- ( n e. Z -> n e. ZZ )
23		uzid 11702	. . . . . . . . 9 |- ( n e. ZZ -> n e. ( ZZ>= ` n ) )
24		ne0i 3921	. . . . . . . . 9 |- ( n e. ( ZZ>= ` n ) -> ( ZZ>= ` n ) =/= (/) )
25	22, 23, 24	3syl 18	. . . . . . . 8 |- ( n e. Z -> ( ZZ>= ` n ) =/= (/) )
26	25	adantl 482	. . . . . . 7 |- ( ( ph /\ n e. Z ) -> ( ZZ>= ` n ) =/= (/) )
27	4	adantr 481	. . . . . . . 8 |- ( ( ( ph /\ n e. Z ) /\ m e. ( ZZ>= ` n ) ) -> F : Z --> RR )
28	7	sselda 3603	. . . . . . . 8 |- ( ( ( ph /\ n e. Z ) /\ m e. ( ZZ>= ` n ) ) -> m e. Z )
29	27, 28	ffvelrnd 6360	. . . . . . 7 |- ( ( ( ph /\ n e. Z ) /\ m e. ( ZZ>= ` n ) ) -> ( F ` m ) e. RR )
30	21, 26, 29	supxrre3rnmpt 39656	. . . . . 6 |- ( ( ph /\ n e. Z ) -> ( sup ( ran ( m e. ( ZZ>= ` n ) |-> ( F ` m ) ) , RR* , < ) e. RR <-> E. x e. RR A. m e. ( ZZ>= ` n ) ( F ` m ) <_ x ) )
31	20, 30	mpbird 247	. . . . 5 |- ( ( ph /\ n e. Z ) -> sup ( ran ( m e. ( ZZ>= ` n ) |-> ( F ` m ) ) , RR* , < ) e. RR )
32	10, 31	eqeltrd 2701	. . . 4 |- ( ( ph /\ n e. Z ) -> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) e. RR )
33		eqid 2622	. . . 4 |- ( n e. Z |-> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) ) = ( n e. Z |-> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) )
34	32, 33	fmptd 6385	. . 3 |- ( ph -> ( n e. Z |-> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) ) : Z --> RR )
35		eqid 2622	. . . . . . . . . 10 |- ( ZZ>= ` i ) = ( ZZ>= ` i )
36	1	eluzelz2 39627	. . . . . . . . . 10 |- ( i e. Z -> i e. ZZ )
37	36	peano2zd 11485	. . . . . . . . . 10 |- ( i e. Z -> ( i + 1 ) e. ZZ )
38	36	zred 11482	. . . . . . . . . . 11 |- ( i e. Z -> i e. RR )
39		lep1 10862	. . . . . . . . . . 11 |- ( i e. RR -> i <_ ( i + 1 ) )
40	38, 39	syl 17	. . . . . . . . . 10 |- ( i e. Z -> i <_ ( i + 1 ) )
41	35, 36, 37, 40	eluzd 39635	. . . . . . . . 9 |- ( i e. Z -> ( i + 1 ) e. ( ZZ>= ` i ) )
42		uzss 11708	. . . . . . . . 9 |- ( ( i + 1 ) e. ( ZZ>= ` i ) -> ( ZZ>= ` ( i + 1 ) ) C_ ( ZZ>= ` i ) )
43	41, 42	syl 17	. . . . . . . 8 |- ( i e. Z -> ( ZZ>= ` ( i + 1 ) ) C_ ( ZZ>= ` i ) )
44		ssres2 5425	. . . . . . . 8 |- ( ( ZZ>= ` ( i + 1 ) ) C_ ( ZZ>= ` i ) -> ( F |` ( ZZ>= ` ( i + 1 ) ) ) C_ ( F |` ( ZZ>= ` i ) ) )
45	43, 44	syl 17	. . . . . . 7 |- ( i e. Z -> ( F |` ( ZZ>= ` ( i + 1 ) ) ) C_ ( F |` ( ZZ>= ` i ) ) )
46		rnss 5354	. . . . . . 7 |- ( ( F |` ( ZZ>= ` ( i + 1 ) ) ) C_ ( F |` ( ZZ>= ` i ) ) -> ran ( F |` ( ZZ>= ` ( i + 1 ) ) ) C_ ran ( F |` ( ZZ>= ` i ) ) )
47	45, 46	syl 17	. . . . . 6 |- ( i e. Z -> ran ( F |` ( ZZ>= ` ( i + 1 ) ) ) C_ ran ( F |` ( ZZ>= ` i ) ) )
48	47	adantl 482	. . . . 5 |- ( ( ph /\ i e. Z ) -> ran ( F |` ( ZZ>= ` ( i + 1 ) ) ) C_ ran ( F |` ( ZZ>= ` i ) ) )
49		rnresss 39365	. . . . . . . 8 |- ran ( F |` ( ZZ>= ` i ) ) C_ ran F
50	49	a1i 11	. . . . . . 7 |- ( ( ph /\ i e. Z ) -> ran ( F |` ( ZZ>= ` i ) ) C_ ran F )
51	3	frnd 39426	. . . . . . . 8 |- ( ph -> ran F C_ RR )
52	51	adantr 481	. . . . . . 7 |- ( ( ph /\ i e. Z ) -> ran F C_ RR )
53	50, 52	sstrd 3613	. . . . . 6 |- ( ( ph /\ i e. Z ) -> ran ( F |` ( ZZ>= ` i ) ) C_ RR )
54		ressxr 10083	. . . . . . 7 |- RR C_ RR*
55	54	a1i 11	. . . . . 6 |- ( ( ph /\ i e. Z ) -> RR C_ RR* )
56	53, 55	sstrd 3613	. . . . 5 |- ( ( ph /\ i e. Z ) -> ran ( F |` ( ZZ>= ` i ) ) C_ RR* )
57		supxrss 12162	. . . . 5 |- ( ( ran ( F |` ( ZZ>= ` ( i + 1 ) ) ) C_ ran ( F |` ( ZZ>= ` i ) ) /\ ran ( F |` ( ZZ>= ` i ) ) C_ RR* ) -> sup ( ran ( F |` ( ZZ>= ` ( i + 1 ) ) ) , RR* , < ) <_ sup ( ran ( F |` ( ZZ>= ` i ) ) , RR* , < ) )
58	48, 56, 57	syl2anc 693	. . . 4 |- ( ( ph /\ i e. Z ) -> sup ( ran ( F |` ( ZZ>= ` ( i + 1 ) ) ) , RR* , < ) <_ sup ( ran ( F |` ( ZZ>= ` i ) ) , RR* , < ) )
59		eqidd 2623	. . . . . . 7 |- ( i e. Z -> ( n e. Z |-> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) ) = ( n e. Z |-> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) ) )
60		fveq2 6191	. . . . . . . . . . 11 |- ( n = ( i + 1 ) -> ( ZZ>= ` n ) = ( ZZ>= ` ( i + 1 ) ) )
61	60	reseq2d 5396	. . . . . . . . . 10 |- ( n = ( i + 1 ) -> ( F |` ( ZZ>= ` n ) ) = ( F |` ( ZZ>= ` ( i + 1 ) ) ) )
62	61	rneqd 5353	. . . . . . . . 9 |- ( n = ( i + 1 ) -> ran ( F |` ( ZZ>= ` n ) ) = ran ( F |` ( ZZ>= ` ( i + 1 ) ) ) )
63	62	supeq1d 8352	. . . . . . . 8 |- ( n = ( i + 1 ) -> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) = sup ( ran ( F |` ( ZZ>= ` ( i + 1 ) ) ) , RR* , < ) )
64	63	adantl 482	. . . . . . 7 |- ( ( i e. Z /\ n = ( i + 1 ) ) -> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) = sup ( ran ( F |` ( ZZ>= ` ( i + 1 ) ) ) , RR* , < ) )
65	1	peano2uzs 11742	. . . . . . 7 |- ( i e. Z -> ( i + 1 ) e. Z )
66		xrltso 11974	. . . . . . . . 9 |- < Or RR*
67	66	supex 8369	. . . . . . . 8 |- sup ( ran ( F |` ( ZZ>= ` ( i + 1 ) ) ) , RR* , < ) e. _V
68	67	a1i 11	. . . . . . 7 |- ( i e. Z -> sup ( ran ( F |` ( ZZ>= ` ( i + 1 ) ) ) , RR* , < ) e. _V )
69	59, 64, 65, 68	fvmptd 6288	. . . . . 6 |- ( i e. Z -> ( ( n e. Z |-> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) ) ` ( i + 1 ) ) = sup ( ran ( F |` ( ZZ>= ` ( i + 1 ) ) ) , RR* , < ) )
70	69	adantl 482	. . . . 5 |- ( ( ph /\ i e. Z ) -> ( ( n e. Z |-> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) ) ` ( i + 1 ) ) = sup ( ran ( F |` ( ZZ>= ` ( i + 1 ) ) ) , RR* , < ) )
71		fveq2 6191	. . . . . . . . . . 11 |- ( n = i -> ( ZZ>= ` n ) = ( ZZ>= ` i ) )
72	71	reseq2d 5396	. . . . . . . . . 10 |- ( n = i -> ( F |` ( ZZ>= ` n ) ) = ( F |` ( ZZ>= ` i ) ) )
73	72	rneqd 5353	. . . . . . . . 9 |- ( n = i -> ran ( F |` ( ZZ>= ` n ) ) = ran ( F |` ( ZZ>= ` i ) ) )
74	73	supeq1d 8352	. . . . . . . 8 |- ( n = i -> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) = sup ( ran ( F |` ( ZZ>= ` i ) ) , RR* , < ) )
75	74	adantl 482	. . . . . . 7 |- ( ( i e. Z /\ n = i ) -> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) = sup ( ran ( F |` ( ZZ>= ` i ) ) , RR* , < ) )
76		id 22	. . . . . . 7 |- ( i e. Z -> i e. Z )
77	66	supex 8369	. . . . . . . 8 |- sup ( ran ( F |` ( ZZ>= ` i ) ) , RR* , < ) e. _V
78	77	a1i 11	. . . . . . 7 |- ( i e. Z -> sup ( ran ( F |` ( ZZ>= ` i ) ) , RR* , < ) e. _V )
79	59, 75, 76, 78	fvmptd 6288	. . . . . 6 |- ( i e. Z -> ( ( n e. Z |-> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) ) ` i ) = sup ( ran ( F |` ( ZZ>= ` i ) ) , RR* , < ) )
80	79	adantl 482	. . . . 5 |- ( ( ph /\ i e. Z ) -> ( ( n e. Z |-> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) ) ` i ) = sup ( ran ( F |` ( ZZ>= ` i ) ) , RR* , < ) )
81	70, 80	breq12d 4666	. . . 4 |- ( ( ph /\ i e. Z ) -> ( ( ( n e. Z |-> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) ) ` ( i + 1 ) ) <_ ( ( n e. Z |-> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) ) ` i ) <-> sup ( ran ( F |` ( ZZ>= ` ( i + 1 ) ) ) , RR* , < ) <_ sup ( ran ( F |` ( ZZ>= ` i ) ) , RR* , < ) ) )
82	58, 81	mpbird 247	. . 3 |- ( ( ph /\ i e. Z ) -> ( ( n e. Z |-> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) ) ` ( i + 1 ) ) <_ ( ( n e. Z |-> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) ) ` i ) )
83		nfcv 2764	. . . . . . . 8 |- F/_ j F
84	3	frexr 39604	. . . . . . . 8 |- ( ph -> F : Z --> RR* )
85	83, 2, 1, 84	limsupre3uz 39968	. . . . . . 7 |- ( ph -> ( ( limsup ` F ) e. RR <-> ( E. x e. RR A. i e. Z E. j e. ( ZZ>= ` i ) x <_ ( F ` j ) /\ E. x e. RR E. i e. Z A. j e. ( ZZ>= ` i ) ( F ` j ) <_ x ) ) )
86	12, 85	mpbid 222	. . . . . 6 |- ( ph -> ( E. x e. RR A. i e. Z E. j e. ( ZZ>= ` i ) x <_ ( F ` j ) /\ E. x e. RR E. i e. Z A. j e. ( ZZ>= ` i ) ( F ` j ) <_ x ) )
87	86	simpld 475	. . . . 5 |- ( ph -> E. x e. RR A. i e. Z E. j e. ( ZZ>= ` i ) x <_ ( F ` j ) )
88		simp-4r 807	. . . . . . . . . . 11 |- ( ( ( ( ( ph /\ x e. RR ) /\ i e. Z ) /\ j e. ( ZZ>= ` i ) ) /\ x <_ ( F ` j ) ) -> x e. RR )
89	88	rexrd 10089	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ x e. RR ) /\ i e. Z ) /\ j e. ( ZZ>= ` i ) ) /\ x <_ ( F ` j ) ) -> x e. RR* )
90	84	3ad2ant1 1082	. . . . . . . . . . . 12 |- ( ( ph /\ i e. Z /\ j e. ( ZZ>= ` i ) ) -> F : Z --> RR* )
91	1	uztrn2 11705	. . . . . . . . . . . . 13 |- ( ( i e. Z /\ j e. ( ZZ>= ` i ) ) -> j e. Z )
92	91	3adant1 1079	. . . . . . . . . . . 12 |- ( ( ph /\ i e. Z /\ j e. ( ZZ>= ` i ) ) -> j e. Z )
93	90, 92	ffvelrnd 6360	. . . . . . . . . . 11 |- ( ( ph /\ i e. Z /\ j e. ( ZZ>= ` i ) ) -> ( F ` j ) e. RR* )
94	93	ad5ant134 1313	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ x e. RR ) /\ i e. Z ) /\ j e. ( ZZ>= ` i ) ) /\ x <_ ( F ` j ) ) -> ( F ` j ) e. RR* )
95	56	supxrcld 39290	. . . . . . . . . . 11 |- ( ( ph /\ i e. Z ) -> sup ( ran ( F |` ( ZZ>= ` i ) ) , RR* , < ) e. RR* )
96	95	ad5ant13 1301	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ x e. RR ) /\ i e. Z ) /\ j e. ( ZZ>= ` i ) ) /\ x <_ ( F ` j ) ) -> sup ( ran ( F |` ( ZZ>= ` i ) ) , RR* , < ) e. RR* )
97		simpr 477	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ x e. RR ) /\ i e. Z ) /\ j e. ( ZZ>= ` i ) ) /\ x <_ ( F ` j ) ) -> x <_ ( F ` j ) )
98	56	3adant3 1081	. . . . . . . . . . . 12 |- ( ( ph /\ i e. Z /\ j e. ( ZZ>= ` i ) ) -> ran ( F |` ( ZZ>= ` i ) ) C_ RR* )
99		fvres 6207	. . . . . . . . . . . . . . 15 |- ( j e. ( ZZ>= ` i ) -> ( ( F |` ( ZZ>= ` i ) ) ` j ) = ( F ` j ) )
100	99	eqcomd 2628	. . . . . . . . . . . . . 14 |- ( j e. ( ZZ>= ` i ) -> ( F ` j ) = ( ( F |` ( ZZ>= ` i ) ) ` j ) )
101	100	3ad2ant3 1084	. . . . . . . . . . . . 13 |- ( ( ph /\ i e. Z /\ j e. ( ZZ>= ` i ) ) -> ( F ` j ) = ( ( F |` ( ZZ>= ` i ) ) ` j ) )
102	3	ffnd 6046	. . . . . . . . . . . . . . . . 17 |- ( ph -> F Fn Z )
103	102	adantr 481	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ i e. Z ) -> F Fn Z )
104	1, 76	uzssd2 39644	. . . . . . . . . . . . . . . . 17 |- ( i e. Z -> ( ZZ>= ` i ) C_ Z )
105	104	adantl 482	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ i e. Z ) -> ( ZZ>= ` i ) C_ Z )
106		fnssres 6004	. . . . . . . . . . . . . . . 16 |- ( ( F Fn Z /\ ( ZZ>= ` i ) C_ Z ) -> ( F |` ( ZZ>= ` i ) ) Fn ( ZZ>= ` i ) )
107	103, 105, 106	syl2anc 693	. . . . . . . . . . . . . . 15 |- ( ( ph /\ i e. Z ) -> ( F |` ( ZZ>= ` i ) ) Fn ( ZZ>= ` i ) )
108	107	3adant3 1081	. . . . . . . . . . . . . 14 |- ( ( ph /\ i e. Z /\ j e. ( ZZ>= ` i ) ) -> ( F |` ( ZZ>= ` i ) ) Fn ( ZZ>= ` i ) )
109		simp3 1063	. . . . . . . . . . . . . 14 |- ( ( ph /\ i e. Z /\ j e. ( ZZ>= ` i ) ) -> j e. ( ZZ>= ` i ) )
110		fnfvelrn 6356	. . . . . . . . . . . . . 14 |- ( ( ( F |` ( ZZ>= ` i ) ) Fn ( ZZ>= ` i ) /\ j e. ( ZZ>= ` i ) ) -> ( ( F |` ( ZZ>= ` i ) ) ` j ) e. ran ( F |` ( ZZ>= ` i ) ) )
111	108, 109, 110	syl2anc 693	. . . . . . . . . . . . 13 |- ( ( ph /\ i e. Z /\ j e. ( ZZ>= ` i ) ) -> ( ( F |` ( ZZ>= ` i ) ) ` j ) e. ran ( F |` ( ZZ>= ` i ) ) )
112	101, 111	eqeltrd 2701	. . . . . . . . . . . 12 |- ( ( ph /\ i e. Z /\ j e. ( ZZ>= ` i ) ) -> ( F ` j ) e. ran ( F |` ( ZZ>= ` i ) ) )
113		eqid 2622	. . . . . . . . . . . 12 |- sup ( ran ( F |` ( ZZ>= ` i ) ) , RR* , < ) = sup ( ran ( F |` ( ZZ>= ` i ) ) , RR* , < )
114	98, 112, 113	supxrubd 39297	. . . . . . . . . . 11 |- ( ( ph /\ i e. Z /\ j e. ( ZZ>= ` i ) ) -> ( F ` j ) <_ sup ( ran ( F |` ( ZZ>= ` i ) ) , RR* , < ) )
115	114	ad5ant134 1313	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ x e. RR ) /\ i e. Z ) /\ j e. ( ZZ>= ` i ) ) /\ x <_ ( F ` j ) ) -> ( F ` j ) <_ sup ( ran ( F |` ( ZZ>= ` i ) ) , RR* , < ) )
116	89, 94, 96, 97, 115	xrletrd 11993	. . . . . . . . 9 |- ( ( ( ( ( ph /\ x e. RR ) /\ i e. Z ) /\ j e. ( ZZ>= ` i ) ) /\ x <_ ( F ` j ) ) -> x <_ sup ( ran ( F |` ( ZZ>= ` i ) ) , RR* , < ) )
117	116	ex 450	. . . . . . . 8 |- ( ( ( ( ph /\ x e. RR ) /\ i e. Z ) /\ j e. ( ZZ>= ` i ) ) -> ( x <_ ( F ` j ) -> x <_ sup ( ran ( F |` ( ZZ>= ` i ) ) , RR* , < ) ) )
118	117	rexlimdva 3031	. . . . . . 7 |- ( ( ( ph /\ x e. RR ) /\ i e. Z ) -> ( E. j e. ( ZZ>= ` i ) x <_ ( F ` j ) -> x <_ sup ( ran ( F |` ( ZZ>= ` i ) ) , RR* , < ) ) )
119	118	ralimdva 2962	. . . . . 6 |- ( ( ph /\ x e. RR ) -> ( A. i e. Z E. j e. ( ZZ>= ` i ) x <_ ( F ` j ) -> A. i e. Z x <_ sup ( ran ( F |` ( ZZ>= ` i ) ) , RR* , < ) ) )
120	119	reximdva 3017	. . . . 5 |- ( ph -> ( E. x e. RR A. i e. Z E. j e. ( ZZ>= ` i ) x <_ ( F ` j ) -> E. x e. RR A. i e. Z x <_ sup ( ran ( F |` ( ZZ>= ` i ) ) , RR* , < ) ) )
121	87, 120	mpd 15	. . . 4 |- ( ph -> E. x e. RR A. i e. Z x <_ sup ( ran ( F |` ( ZZ>= ` i ) ) , RR* , < ) )
122		simpl 473	. . . . . . 7 |- ( ( y = x /\ i e. Z ) -> y = x )
123	79	adantl 482	. . . . . . 7 |- ( ( y = x /\ i e. Z ) -> ( ( n e. Z |-> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) ) ` i ) = sup ( ran ( F |` ( ZZ>= ` i ) ) , RR* , < ) )
124	122, 123	breq12d 4666	. . . . . 6 |- ( ( y = x /\ i e. Z ) -> ( y <_ ( ( n e. Z |-> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) ) ` i ) <-> x <_ sup ( ran ( F |` ( ZZ>= ` i ) ) , RR* , < ) ) )
125	124	ralbidva 2985	. . . . 5 |- ( y = x -> ( A. i e. Z y <_ ( ( n e. Z |-> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) ) ` i ) <-> A. i e. Z x <_ sup ( ran ( F |` ( ZZ>= ` i ) ) , RR* , < ) ) )
126	125	cbvrexv 3172	. . . 4 |- ( E. y e. RR A. i e. Z y <_ ( ( n e. Z |-> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) ) ` i ) <-> E. x e. RR A. i e. Z x <_ sup ( ran ( F |` ( ZZ>= ` i ) ) , RR* , < ) )
127	121, 126	sylibr 224	. . 3 |- ( ph -> E. y e. RR A. i e. Z y <_ ( ( n e. Z |-> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) ) ` i ) )
128	1, 2, 34, 82, 127	climinf 39838	. 2 |- ( ph -> ( n e. Z |-> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) ) ~~> inf ( ran ( n e. Z |-> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) ) , RR , < ) )
129		fveq2 6191	. . . . . . . 8 |- ( n = k -> ( ZZ>= ` n ) = ( ZZ>= ` k ) )
130	129	reseq2d 5396	. . . . . . 7 |- ( n = k -> ( F |` ( ZZ>= ` n ) ) = ( F |` ( ZZ>= ` k ) ) )
131	130	rneqd 5353	. . . . . 6 |- ( n = k -> ran ( F |` ( ZZ>= ` n ) ) = ran ( F |` ( ZZ>= ` k ) ) )
132	131	supeq1d 8352	. . . . 5 |- ( n = k -> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) = sup ( ran ( F |` ( ZZ>= ` k ) ) , RR* , < ) )
133	132	cbvmptv 4750	. . . 4 |- ( n e. Z |-> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) ) = ( k e. Z |-> sup ( ran ( F |` ( ZZ>= ` k ) ) , RR* , < ) )
134	133	a1i 11	. . 3 |- ( ph -> ( n e. Z |-> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) ) = ( k e. Z |-> sup ( ran ( F |` ( ZZ>= ` k ) ) , RR* , < ) ) )
135	2, 1, 3, 12	limsupvaluz2 39970	. . . 4 |- ( ph -> ( limsup ` F ) = inf ( ran ( n e. Z |-> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) ) , RR , < ) )
136	135	eqcomd 2628	. . 3 |- ( ph -> inf ( ran ( n e. Z |-> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) ) , RR , < ) = ( limsup ` F ) )
137	134, 136	breq12d 4666	. 2 |- ( ph -> ( ( n e. Z |-> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) ) ~~> inf ( ran ( n e. Z |-> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) ) , RR , < ) <-> ( k e. Z |-> sup ( ran ( F |` ( ZZ>= ` k ) ) , RR* , < ) ) ~~> ( limsup ` F ) ) )
138	128, 137	mpbid 222	1 |- ( ph -> ( k e. Z |-> sup ( ran ( F |` ( ZZ>= ` k ) ) , RR* , < ) ) ~~> ( limsup ` F ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   _Vcvv 3200   C_ wss 3574   (/)c0 3915   class class class wbr 4653   |-> cmpt 4729   ran crn 5115   |` cres 5116   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   supcsup 8346  infcinf 8347   RRcr 9935   1c1 9937   + caddc 9939   RR*cxr 10073   < clt 10074   <_ cle 10075   ZZcz 11377   ZZ>=cuz 11687   limsupclsp 14201   ~~> cli 14215

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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  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-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-ico 12181  df-fz 12327  df-fl 12593  df-ceil 12594  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219

This theorem is referenced by:  supcnvlimsupmpt  39973