Step |
Hyp |
Ref |
Expression |
1 |
|
limsupvaluz2.m |
|- ( ph -> M e. ZZ ) |
2 |
|
limsupvaluz2.z |
|- Z = ( ZZ>= ` M ) |
3 |
|
limsupvaluz2.f |
|- ( ph -> F : Z --> RR ) |
4 |
|
limsupvaluz2.r |
|- ( ph -> ( limsup ` F ) e. RR ) |
5 |
3
|
frexr |
|- ( ph -> F : Z --> RR* ) |
6 |
1 2 5
|
limsupvaluz |
|- ( ph -> ( limsup ` F ) = inf ( ran ( n e. Z |-> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) ) , RR* , < ) ) |
7 |
3
|
adantr |
|- ( ( ph /\ n e. Z ) -> F : Z --> RR ) |
8 |
2
|
uzssd3 |
|- ( n e. Z -> ( ZZ>= ` n ) C_ Z ) |
9 |
8
|
adantl |
|- ( ( ph /\ n e. Z ) -> ( ZZ>= ` n ) C_ Z ) |
10 |
7 9
|
feqresmpt |
|- ( ( ph /\ n e. Z ) -> ( F |` ( ZZ>= ` n ) ) = ( m e. ( ZZ>= ` n ) |-> ( F ` m ) ) ) |
11 |
10
|
rneqd |
|- ( ( ph /\ n e. Z ) -> ran ( F |` ( ZZ>= ` n ) ) = ran ( m e. ( ZZ>= ` n ) |-> ( F ` m ) ) ) |
12 |
11
|
supeq1d |
|- ( ( ph /\ n e. Z ) -> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) = sup ( ran ( m e. ( ZZ>= ` n ) |-> ( F ` m ) ) , RR* , < ) ) |
13 |
|
nfcv |
|- F/_ m F |
14 |
4
|
renepnfd |
|- ( ph -> ( limsup ` F ) =/= +oo ) |
15 |
13 2 3 14
|
limsupubuz |
|- ( ph -> E. x e. RR A. m e. Z ( F ` m ) <_ x ) |
16 |
15
|
adantr |
|- ( ( ph /\ n e. Z ) -> E. x e. RR A. m e. Z ( F ` m ) <_ x ) |
17 |
|
ssralv |
|- ( ( ZZ>= ` n ) C_ Z -> ( A. m e. Z ( F ` m ) <_ x -> A. m e. ( ZZ>= ` n ) ( F ` m ) <_ x ) ) |
18 |
8 17
|
syl |
|- ( n e. Z -> ( A. m e. Z ( F ` m ) <_ x -> A. m e. ( ZZ>= ` n ) ( F ` m ) <_ x ) ) |
19 |
18
|
adantl |
|- ( ( ph /\ n e. Z ) -> ( A. m e. Z ( F ` m ) <_ x -> A. m e. ( ZZ>= ` n ) ( F ` m ) <_ x ) ) |
20 |
19
|
reximdv |
|- ( ( 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 ) ) |
21 |
16 20
|
mpd |
|- ( ( ph /\ n e. Z ) -> E. x e. RR A. m e. ( ZZ>= ` n ) ( F ` m ) <_ x ) |
22 |
|
nfv |
|- F/ m ( ph /\ n e. Z ) |
23 |
2
|
eluzelz2 |
|- ( n e. Z -> n e. ZZ ) |
24 |
|
uzid |
|- ( n e. ZZ -> n e. ( ZZ>= ` n ) ) |
25 |
|
ne0i |
|- ( n e. ( ZZ>= ` n ) -> ( ZZ>= ` n ) =/= (/) ) |
26 |
23 24 25
|
3syl |
|- ( n e. Z -> ( ZZ>= ` n ) =/= (/) ) |
27 |
26
|
adantl |
|- ( ( ph /\ n e. Z ) -> ( ZZ>= ` n ) =/= (/) ) |
28 |
7
|
adantr |
|- ( ( ( ph /\ n e. Z ) /\ m e. ( ZZ>= ` n ) ) -> F : Z --> RR ) |
29 |
9
|
sselda |
|- ( ( ( ph /\ n e. Z ) /\ m e. ( ZZ>= ` n ) ) -> m e. Z ) |
30 |
28 29
|
ffvelcdmd |
|- ( ( ( ph /\ n e. Z ) /\ m e. ( ZZ>= ` n ) ) -> ( F ` m ) e. RR ) |
31 |
22 27 30
|
supxrre3rnmpt |
|- ( ( 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 ) ) |
32 |
21 31
|
mpbird |
|- ( ( ph /\ n e. Z ) -> sup ( ran ( m e. ( ZZ>= ` n ) |-> ( F ` m ) ) , RR* , < ) e. RR ) |
33 |
12 32
|
eqeltrd |
|- ( ( ph /\ n e. Z ) -> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) e. RR ) |
34 |
33
|
fmpttd |
|- ( ph -> ( n e. Z |-> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) ) : Z --> RR ) |
35 |
34
|
frnd |
|- ( ph -> ran ( n e. Z |-> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) ) C_ RR ) |
36 |
|
nfv |
|- F/ n ph |
37 |
|
eqid |
|- ( n e. Z |-> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) ) = ( n e. Z |-> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) ) |
38 |
1 2
|
uzn0d |
|- ( ph -> Z =/= (/) ) |
39 |
36 33 37 38
|
rnmptn0 |
|- ( ph -> ran ( n e. Z |-> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) ) =/= (/) ) |
40 |
|
nfcv |
|- F/_ j F |
41 |
40 1 2 5
|
limsupre3uz |
|- ( 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 ) ) ) |
42 |
4 41
|
mpbid |
|- ( 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 ) ) |
43 |
42
|
simpld |
|- ( ph -> E. x e. RR A. i e. Z E. j e. ( ZZ>= ` i ) x <_ ( F ` j ) ) |
44 |
|
simp-4r |
|- ( ( ( ( ( ph /\ x e. RR ) /\ i e. Z ) /\ j e. ( ZZ>= ` i ) ) /\ x <_ ( F ` j ) ) -> x e. RR ) |
45 |
44
|
rexrd |
|- ( ( ( ( ( ph /\ x e. RR ) /\ i e. Z ) /\ j e. ( ZZ>= ` i ) ) /\ x <_ ( F ` j ) ) -> x e. RR* ) |
46 |
5
|
3ad2ant1 |
|- ( ( ph /\ i e. Z /\ j e. ( ZZ>= ` i ) ) -> F : Z --> RR* ) |
47 |
2
|
uztrn2 |
|- ( ( i e. Z /\ j e. ( ZZ>= ` i ) ) -> j e. Z ) |
48 |
47
|
3adant1 |
|- ( ( ph /\ i e. Z /\ j e. ( ZZ>= ` i ) ) -> j e. Z ) |
49 |
46 48
|
ffvelcdmd |
|- ( ( ph /\ i e. Z /\ j e. ( ZZ>= ` i ) ) -> ( F ` j ) e. RR* ) |
50 |
49
|
ad5ant134 |
|- ( ( ( ( ( ph /\ x e. RR ) /\ i e. Z ) /\ j e. ( ZZ>= ` i ) ) /\ x <_ ( F ` j ) ) -> ( F ` j ) e. RR* ) |
51 |
|
rnresss |
|- ran ( F |` ( ZZ>= ` i ) ) C_ ran F |
52 |
3
|
frnd |
|- ( ph -> ran F C_ RR ) |
53 |
52
|
adantr |
|- ( ( ph /\ i e. Z ) -> ran F C_ RR ) |
54 |
51 53
|
sstrid |
|- ( ( ph /\ i e. Z ) -> ran ( F |` ( ZZ>= ` i ) ) C_ RR ) |
55 |
54
|
ssrexr |
|- ( ( ph /\ i e. Z ) -> ran ( F |` ( ZZ>= ` i ) ) C_ RR* ) |
56 |
55
|
supxrcld |
|- ( ( ph /\ i e. Z ) -> sup ( ran ( F |` ( ZZ>= ` i ) ) , RR* , < ) e. RR* ) |
57 |
56
|
ad5ant13 |
|- ( ( ( ( ( ph /\ x e. RR ) /\ i e. Z ) /\ j e. ( ZZ>= ` i ) ) /\ x <_ ( F ` j ) ) -> sup ( ran ( F |` ( ZZ>= ` i ) ) , RR* , < ) e. RR* ) |
58 |
|
simpr |
|- ( ( ( ( ( ph /\ x e. RR ) /\ i e. Z ) /\ j e. ( ZZ>= ` i ) ) /\ x <_ ( F ` j ) ) -> x <_ ( F ` j ) ) |
59 |
55
|
3adant3 |
|- ( ( ph /\ i e. Z /\ j e. ( ZZ>= ` i ) ) -> ran ( F |` ( ZZ>= ` i ) ) C_ RR* ) |
60 |
|
fvres |
|- ( j e. ( ZZ>= ` i ) -> ( ( F |` ( ZZ>= ` i ) ) ` j ) = ( F ` j ) ) |
61 |
60
|
eqcomd |
|- ( j e. ( ZZ>= ` i ) -> ( F ` j ) = ( ( F |` ( ZZ>= ` i ) ) ` j ) ) |
62 |
61
|
3ad2ant3 |
|- ( ( ph /\ i e. Z /\ j e. ( ZZ>= ` i ) ) -> ( F ` j ) = ( ( F |` ( ZZ>= ` i ) ) ` j ) ) |
63 |
3
|
ffnd |
|- ( ph -> F Fn Z ) |
64 |
2
|
uzssd3 |
|- ( i e. Z -> ( ZZ>= ` i ) C_ Z ) |
65 |
|
fnssres |
|- ( ( F Fn Z /\ ( ZZ>= ` i ) C_ Z ) -> ( F |` ( ZZ>= ` i ) ) Fn ( ZZ>= ` i ) ) |
66 |
63 64 65
|
syl2an |
|- ( ( ph /\ i e. Z ) -> ( F |` ( ZZ>= ` i ) ) Fn ( ZZ>= ` i ) ) |
67 |
|
fnfvelrn |
|- ( ( ( F |` ( ZZ>= ` i ) ) Fn ( ZZ>= ` i ) /\ j e. ( ZZ>= ` i ) ) -> ( ( F |` ( ZZ>= ` i ) ) ` j ) e. ran ( F |` ( ZZ>= ` i ) ) ) |
68 |
66 67
|
stoic3 |
|- ( ( ph /\ i e. Z /\ j e. ( ZZ>= ` i ) ) -> ( ( F |` ( ZZ>= ` i ) ) ` j ) e. ran ( F |` ( ZZ>= ` i ) ) ) |
69 |
62 68
|
eqeltrd |
|- ( ( ph /\ i e. Z /\ j e. ( ZZ>= ` i ) ) -> ( F ` j ) e. ran ( F |` ( ZZ>= ` i ) ) ) |
70 |
|
eqid |
|- sup ( ran ( F |` ( ZZ>= ` i ) ) , RR* , < ) = sup ( ran ( F |` ( ZZ>= ` i ) ) , RR* , < ) |
71 |
59 69 70
|
supxrubd |
|- ( ( ph /\ i e. Z /\ j e. ( ZZ>= ` i ) ) -> ( F ` j ) <_ sup ( ran ( F |` ( ZZ>= ` i ) ) , RR* , < ) ) |
72 |
71
|
ad5ant134 |
|- ( ( ( ( ( ph /\ x e. RR ) /\ i e. Z ) /\ j e. ( ZZ>= ` i ) ) /\ x <_ ( F ` j ) ) -> ( F ` j ) <_ sup ( ran ( F |` ( ZZ>= ` i ) ) , RR* , < ) ) |
73 |
45 50 57 58 72
|
xrletrd |
|- ( ( ( ( ( ph /\ x e. RR ) /\ i e. Z ) /\ j e. ( ZZ>= ` i ) ) /\ x <_ ( F ` j ) ) -> x <_ sup ( ran ( F |` ( ZZ>= ` i ) ) , RR* , < ) ) |
74 |
73
|
rexlimdva2 |
|- ( ( ( ph /\ x e. RR ) /\ i e. Z ) -> ( E. j e. ( ZZ>= ` i ) x <_ ( F ` j ) -> x <_ sup ( ran ( F |` ( ZZ>= ` i ) ) , RR* , < ) ) ) |
75 |
74
|
ralimdva |
|- ( ( 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* , < ) ) ) |
76 |
75
|
reximdva |
|- ( 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* , < ) ) ) |
77 |
43 76
|
mpd |
|- ( ph -> E. x e. RR A. i e. Z x <_ sup ( ran ( F |` ( ZZ>= ` i ) ) , RR* , < ) ) |
78 |
|
fveq2 |
|- ( n = i -> ( ZZ>= ` n ) = ( ZZ>= ` i ) ) |
79 |
78
|
reseq2d |
|- ( n = i -> ( F |` ( ZZ>= ` n ) ) = ( F |` ( ZZ>= ` i ) ) ) |
80 |
79
|
rneqd |
|- ( n = i -> ran ( F |` ( ZZ>= ` n ) ) = ran ( F |` ( ZZ>= ` i ) ) ) |
81 |
80
|
supeq1d |
|- ( n = i -> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) = sup ( ran ( F |` ( ZZ>= ` i ) ) , RR* , < ) ) |
82 |
|
eqcom |
|- ( n = i <-> i = n ) |
83 |
|
eqcom |
|- ( sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) = sup ( ran ( F |` ( ZZ>= ` i ) ) , RR* , < ) <-> sup ( ran ( F |` ( ZZ>= ` i ) ) , RR* , < ) = sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) ) |
84 |
81 82 83
|
3imtr3i |
|- ( i = n -> sup ( ran ( F |` ( ZZ>= ` i ) ) , RR* , < ) = sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) ) |
85 |
84
|
breq2d |
|- ( i = n -> ( x <_ sup ( ran ( F |` ( ZZ>= ` i ) ) , RR* , < ) <-> x <_ sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) ) ) |
86 |
85
|
cbvralvw |
|- ( A. i e. Z x <_ sup ( ran ( F |` ( ZZ>= ` i ) ) , RR* , < ) <-> A. n e. Z x <_ sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) ) |
87 |
86
|
rexbii |
|- ( E. x e. RR A. i e. Z x <_ sup ( ran ( F |` ( ZZ>= ` i ) ) , RR* , < ) <-> E. x e. RR A. n e. Z x <_ sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) ) |
88 |
77 87
|
sylib |
|- ( ph -> E. x e. RR A. n e. Z x <_ sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) ) |
89 |
36 33
|
rnmptbd2 |
|- ( ph -> ( E. x e. RR A. n e. Z x <_ sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) <-> E. x e. RR A. y e. ran ( n e. Z |-> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) ) x <_ y ) ) |
90 |
88 89
|
mpbid |
|- ( ph -> E. x e. RR A. y e. ran ( n e. Z |-> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) ) x <_ y ) |
91 |
|
infxrre |
|- ( ( ran ( n e. Z |-> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) ) C_ RR /\ ran ( n e. Z |-> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) ) =/= (/) /\ E. x e. RR A. y e. ran ( n e. Z |-> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) ) x <_ y ) -> inf ( ran ( n e. Z |-> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) ) , RR* , < ) = inf ( ran ( n e. Z |-> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) ) , RR , < ) ) |
92 |
35 39 90 91
|
syl3anc |
|- ( ph -> inf ( ran ( n e. Z |-> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) ) , RR* , < ) = inf ( ran ( n e. Z |-> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) ) , RR , < ) ) |
93 |
|
fveq2 |
|- ( n = k -> ( ZZ>= ` n ) = ( ZZ>= ` k ) ) |
94 |
93
|
reseq2d |
|- ( n = k -> ( F |` ( ZZ>= ` n ) ) = ( F |` ( ZZ>= ` k ) ) ) |
95 |
94
|
rneqd |
|- ( n = k -> ran ( F |` ( ZZ>= ` n ) ) = ran ( F |` ( ZZ>= ` k ) ) ) |
96 |
95
|
supeq1d |
|- ( n = k -> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) = sup ( ran ( F |` ( ZZ>= ` k ) ) , RR* , < ) ) |
97 |
96
|
cbvmptv |
|- ( n e. Z |-> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) ) = ( k e. Z |-> sup ( ran ( F |` ( ZZ>= ` k ) ) , RR* , < ) ) |
98 |
97
|
rneqi |
|- ran ( n e. Z |-> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) ) = ran ( k e. Z |-> sup ( ran ( F |` ( ZZ>= ` k ) ) , RR* , < ) ) |
99 |
98
|
infeq1i |
|- inf ( ran ( n e. Z |-> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) ) , RR , < ) = inf ( ran ( k e. Z |-> sup ( ran ( F |` ( ZZ>= ` k ) ) , RR* , < ) ) , RR , < ) |
100 |
99
|
a1i |
|- ( ph -> inf ( ran ( n e. Z |-> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) ) , RR , < ) = inf ( ran ( k e. Z |-> sup ( ran ( F |` ( ZZ>= ` k ) ) , RR* , < ) ) , RR , < ) ) |
101 |
6 92 100
|
3eqtrd |
|- ( ph -> ( limsup ` F ) = inf ( ran ( k e. Z |-> sup ( ran ( F |` ( ZZ>= ` k ) ) , RR* , < ) ) , RR , < ) ) |