Step |
Hyp |
Ref |
Expression |
1 |
|
itg2i1fseq.1 |
|- ( ph -> F e. MblFn ) |
2 |
|
itg2i1fseq.2 |
|- ( ph -> F : RR --> ( 0 [,) +oo ) ) |
3 |
|
itg2i1fseq.3 |
|- ( ph -> P : NN --> dom S.1 ) |
4 |
|
itg2i1fseq.4 |
|- ( ph -> A. n e. NN ( 0p oR <_ ( P ` n ) /\ ( P ` n ) oR <_ ( P ` ( n + 1 ) ) ) ) |
5 |
|
itg2i1fseq.5 |
|- ( ph -> A. x e. RR ( n e. NN |-> ( ( P ` n ) ` x ) ) ~~> ( F ` x ) ) |
6 |
|
itg2i1fseq.6 |
|- S = ( m e. NN |-> ( S.1 ` ( P ` m ) ) ) |
7 |
|
itg2i1fseq2.7 |
|- ( ph -> M e. RR ) |
8 |
|
itg2i1fseq2.8 |
|- ( ( ph /\ k e. NN ) -> ( S.1 ` ( P ` k ) ) <_ M ) |
9 |
|
nnuz |
|- NN = ( ZZ>= ` 1 ) |
10 |
|
1zzd |
|- ( ph -> 1 e. ZZ ) |
11 |
3
|
ffvelrnda |
|- ( ( ph /\ m e. NN ) -> ( P ` m ) e. dom S.1 ) |
12 |
|
itg1cl |
|- ( ( P ` m ) e. dom S.1 -> ( S.1 ` ( P ` m ) ) e. RR ) |
13 |
11 12
|
syl |
|- ( ( ph /\ m e. NN ) -> ( S.1 ` ( P ` m ) ) e. RR ) |
14 |
13 6
|
fmptd |
|- ( ph -> S : NN --> RR ) |
15 |
3
|
ffvelrnda |
|- ( ( ph /\ k e. NN ) -> ( P ` k ) e. dom S.1 ) |
16 |
|
peano2nn |
|- ( k e. NN -> ( k + 1 ) e. NN ) |
17 |
|
ffvelrn |
|- ( ( P : NN --> dom S.1 /\ ( k + 1 ) e. NN ) -> ( P ` ( k + 1 ) ) e. dom S.1 ) |
18 |
3 16 17
|
syl2an |
|- ( ( ph /\ k e. NN ) -> ( P ` ( k + 1 ) ) e. dom S.1 ) |
19 |
|
simpr |
|- ( ( 0p oR <_ ( P ` n ) /\ ( P ` n ) oR <_ ( P ` ( n + 1 ) ) ) -> ( P ` n ) oR <_ ( P ` ( n + 1 ) ) ) |
20 |
19
|
ralimi |
|- ( A. n e. NN ( 0p oR <_ ( P ` n ) /\ ( P ` n ) oR <_ ( P ` ( n + 1 ) ) ) -> A. n e. NN ( P ` n ) oR <_ ( P ` ( n + 1 ) ) ) |
21 |
4 20
|
syl |
|- ( ph -> A. n e. NN ( P ` n ) oR <_ ( P ` ( n + 1 ) ) ) |
22 |
|
fveq2 |
|- ( n = k -> ( P ` n ) = ( P ` k ) ) |
23 |
|
fvoveq1 |
|- ( n = k -> ( P ` ( n + 1 ) ) = ( P ` ( k + 1 ) ) ) |
24 |
22 23
|
breq12d |
|- ( n = k -> ( ( P ` n ) oR <_ ( P ` ( n + 1 ) ) <-> ( P ` k ) oR <_ ( P ` ( k + 1 ) ) ) ) |
25 |
24
|
rspccva |
|- ( ( A. n e. NN ( P ` n ) oR <_ ( P ` ( n + 1 ) ) /\ k e. NN ) -> ( P ` k ) oR <_ ( P ` ( k + 1 ) ) ) |
26 |
21 25
|
sylan |
|- ( ( ph /\ k e. NN ) -> ( P ` k ) oR <_ ( P ` ( k + 1 ) ) ) |
27 |
|
itg1le |
|- ( ( ( P ` k ) e. dom S.1 /\ ( P ` ( k + 1 ) ) e. dom S.1 /\ ( P ` k ) oR <_ ( P ` ( k + 1 ) ) ) -> ( S.1 ` ( P ` k ) ) <_ ( S.1 ` ( P ` ( k + 1 ) ) ) ) |
28 |
15 18 26 27
|
syl3anc |
|- ( ( ph /\ k e. NN ) -> ( S.1 ` ( P ` k ) ) <_ ( S.1 ` ( P ` ( k + 1 ) ) ) ) |
29 |
|
2fveq3 |
|- ( m = k -> ( S.1 ` ( P ` m ) ) = ( S.1 ` ( P ` k ) ) ) |
30 |
|
fvex |
|- ( S.1 ` ( P ` k ) ) e. _V |
31 |
29 6 30
|
fvmpt |
|- ( k e. NN -> ( S ` k ) = ( S.1 ` ( P ` k ) ) ) |
32 |
31
|
adantl |
|- ( ( ph /\ k e. NN ) -> ( S ` k ) = ( S.1 ` ( P ` k ) ) ) |
33 |
|
2fveq3 |
|- ( m = ( k + 1 ) -> ( S.1 ` ( P ` m ) ) = ( S.1 ` ( P ` ( k + 1 ) ) ) ) |
34 |
|
fvex |
|- ( S.1 ` ( P ` ( k + 1 ) ) ) e. _V |
35 |
33 6 34
|
fvmpt |
|- ( ( k + 1 ) e. NN -> ( S ` ( k + 1 ) ) = ( S.1 ` ( P ` ( k + 1 ) ) ) ) |
36 |
16 35
|
syl |
|- ( k e. NN -> ( S ` ( k + 1 ) ) = ( S.1 ` ( P ` ( k + 1 ) ) ) ) |
37 |
36
|
adantl |
|- ( ( ph /\ k e. NN ) -> ( S ` ( k + 1 ) ) = ( S.1 ` ( P ` ( k + 1 ) ) ) ) |
38 |
28 32 37
|
3brtr4d |
|- ( ( ph /\ k e. NN ) -> ( S ` k ) <_ ( S ` ( k + 1 ) ) ) |
39 |
32 8
|
eqbrtrd |
|- ( ( ph /\ k e. NN ) -> ( S ` k ) <_ M ) |
40 |
39
|
ralrimiva |
|- ( ph -> A. k e. NN ( S ` k ) <_ M ) |
41 |
|
brralrspcev |
|- ( ( M e. RR /\ A. k e. NN ( S ` k ) <_ M ) -> E. z e. RR A. k e. NN ( S ` k ) <_ z ) |
42 |
7 40 41
|
syl2anc |
|- ( ph -> E. z e. RR A. k e. NN ( S ` k ) <_ z ) |
43 |
9 10 14 38 42
|
climsup |
|- ( ph -> S ~~> sup ( ran S , RR , < ) ) |
44 |
1 2 3 4 5 6
|
itg2i1fseq |
|- ( ph -> ( S.2 ` F ) = sup ( ran S , RR* , < ) ) |
45 |
14
|
frnd |
|- ( ph -> ran S C_ RR ) |
46 |
6 13
|
dmmptd |
|- ( ph -> dom S = NN ) |
47 |
|
1nn |
|- 1 e. NN |
48 |
|
ne0i |
|- ( 1 e. NN -> NN =/= (/) ) |
49 |
47 48
|
mp1i |
|- ( ph -> NN =/= (/) ) |
50 |
46 49
|
eqnetrd |
|- ( ph -> dom S =/= (/) ) |
51 |
|
dm0rn0 |
|- ( dom S = (/) <-> ran S = (/) ) |
52 |
51
|
necon3bii |
|- ( dom S =/= (/) <-> ran S =/= (/) ) |
53 |
50 52
|
sylib |
|- ( ph -> ran S =/= (/) ) |
54 |
|
ffn |
|- ( S : NN --> RR -> S Fn NN ) |
55 |
|
breq1 |
|- ( y = ( S ` k ) -> ( y <_ z <-> ( S ` k ) <_ z ) ) |
56 |
55
|
ralrn |
|- ( S Fn NN -> ( A. y e. ran S y <_ z <-> A. k e. NN ( S ` k ) <_ z ) ) |
57 |
14 54 56
|
3syl |
|- ( ph -> ( A. y e. ran S y <_ z <-> A. k e. NN ( S ` k ) <_ z ) ) |
58 |
57
|
rexbidv |
|- ( ph -> ( E. z e. RR A. y e. ran S y <_ z <-> E. z e. RR A. k e. NN ( S ` k ) <_ z ) ) |
59 |
42 58
|
mpbird |
|- ( ph -> E. z e. RR A. y e. ran S y <_ z ) |
60 |
|
supxrre |
|- ( ( ran S C_ RR /\ ran S =/= (/) /\ E. z e. RR A. y e. ran S y <_ z ) -> sup ( ran S , RR* , < ) = sup ( ran S , RR , < ) ) |
61 |
45 53 59 60
|
syl3anc |
|- ( ph -> sup ( ran S , RR* , < ) = sup ( ran S , RR , < ) ) |
62 |
44 61
|
eqtrd |
|- ( ph -> ( S.2 ` F ) = sup ( ran S , RR , < ) ) |
63 |
43 62
|
breqtrrd |
|- ( ph -> S ~~> ( S.2 ` F ) ) |