| 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 , < ) ) |