Step |
Hyp |
Ref |
Expression |
1 |
|
vonicclem1.x |
|- ( ph -> X e. Fin ) |
2 |
|
vonicclem1.a |
|- ( ph -> A : X --> RR ) |
3 |
|
vonicclem1.b |
|- ( ph -> B : X --> RR ) |
4 |
|
vonicclem1.u |
|- ( ph -> X =/= (/) ) |
5 |
|
vonicclem1.t |
|- ( ( ph /\ k e. X ) -> ( A ` k ) <_ ( B ` k ) ) |
6 |
|
vonicclem1.c |
|- C = ( n e. NN |-> ( k e. X |-> ( ( B ` k ) + ( 1 / n ) ) ) ) |
7 |
|
vonicclem1.d |
|- D = ( n e. NN |-> X_ k e. X ( ( A ` k ) [,) ( ( C ` n ) ` k ) ) ) |
8 |
|
vonicclem1.s |
|- S = ( n e. NN |-> ( ( voln ` X ) ` ( D ` n ) ) ) |
9 |
8
|
a1i |
|- ( ph -> S = ( n e. NN |-> ( ( voln ` X ) ` ( D ` n ) ) ) ) |
10 |
|
simpr |
|- ( ( ph /\ n e. NN ) -> n e. NN ) |
11 |
7
|
a1i |
|- ( ph -> D = ( n e. NN |-> X_ k e. X ( ( A ` k ) [,) ( ( C ` n ) ` k ) ) ) ) |
12 |
1
|
adantr |
|- ( ( ph /\ n e. NN ) -> X e. Fin ) |
13 |
|
eqid |
|- dom ( voln ` X ) = dom ( voln ` X ) |
14 |
2
|
adantr |
|- ( ( ph /\ n e. NN ) -> A : X --> RR ) |
15 |
3
|
ffvelrnda |
|- ( ( ph /\ k e. X ) -> ( B ` k ) e. RR ) |
16 |
15
|
adantlr |
|- ( ( ( ph /\ n e. NN ) /\ k e. X ) -> ( B ` k ) e. RR ) |
17 |
|
nnrecre |
|- ( n e. NN -> ( 1 / n ) e. RR ) |
18 |
17
|
ad2antlr |
|- ( ( ( ph /\ n e. NN ) /\ k e. X ) -> ( 1 / n ) e. RR ) |
19 |
16 18
|
readdcld |
|- ( ( ( ph /\ n e. NN ) /\ k e. X ) -> ( ( B ` k ) + ( 1 / n ) ) e. RR ) |
20 |
19
|
fmpttd |
|- ( ( ph /\ n e. NN ) -> ( k e. X |-> ( ( B ` k ) + ( 1 / n ) ) ) : X --> RR ) |
21 |
6
|
a1i |
|- ( ph -> C = ( n e. NN |-> ( k e. X |-> ( ( B ` k ) + ( 1 / n ) ) ) ) ) |
22 |
1
|
mptexd |
|- ( ph -> ( k e. X |-> ( ( B ` k ) + ( 1 / n ) ) ) e. _V ) |
23 |
22
|
adantr |
|- ( ( ph /\ n e. NN ) -> ( k e. X |-> ( ( B ` k ) + ( 1 / n ) ) ) e. _V ) |
24 |
21 23
|
fvmpt2d |
|- ( ( ph /\ n e. NN ) -> ( C ` n ) = ( k e. X |-> ( ( B ` k ) + ( 1 / n ) ) ) ) |
25 |
24
|
feq1d |
|- ( ( ph /\ n e. NN ) -> ( ( C ` n ) : X --> RR <-> ( k e. X |-> ( ( B ` k ) + ( 1 / n ) ) ) : X --> RR ) ) |
26 |
20 25
|
mpbird |
|- ( ( ph /\ n e. NN ) -> ( C ` n ) : X --> RR ) |
27 |
12 13 14 26
|
hoimbl |
|- ( ( ph /\ n e. NN ) -> X_ k e. X ( ( A ` k ) [,) ( ( C ` n ) ` k ) ) e. dom ( voln ` X ) ) |
28 |
27
|
elexd |
|- ( ( ph /\ n e. NN ) -> X_ k e. X ( ( A ` k ) [,) ( ( C ` n ) ` k ) ) e. _V ) |
29 |
11 28
|
fvmpt2d |
|- ( ( ph /\ n e. NN ) -> ( D ` n ) = X_ k e. X ( ( A ` k ) [,) ( ( C ` n ) ` k ) ) ) |
30 |
10 29
|
syldan |
|- ( ( ph /\ n e. NN ) -> ( D ` n ) = X_ k e. X ( ( A ` k ) [,) ( ( C ` n ) ` k ) ) ) |
31 |
30
|
fveq2d |
|- ( ( ph /\ n e. NN ) -> ( ( voln ` X ) ` ( D ` n ) ) = ( ( voln ` X ) ` X_ k e. X ( ( A ` k ) [,) ( ( C ` n ) ` k ) ) ) ) |
32 |
4
|
adantr |
|- ( ( ph /\ n e. NN ) -> X =/= (/) ) |
33 |
10 26
|
syldan |
|- ( ( ph /\ n e. NN ) -> ( C ` n ) : X --> RR ) |
34 |
|
eqid |
|- X_ k e. X ( ( A ` k ) [,) ( ( C ` n ) ` k ) ) = X_ k e. X ( ( A ` k ) [,) ( ( C ` n ) ` k ) ) |
35 |
12 32 14 33 34
|
vonn0hoi |
|- ( ( ph /\ n e. NN ) -> ( ( voln ` X ) ` X_ k e. X ( ( A ` k ) [,) ( ( C ` n ) ` k ) ) ) = prod_ k e. X ( vol ` ( ( A ` k ) [,) ( ( C ` n ) ` k ) ) ) ) |
36 |
14
|
ffvelrnda |
|- ( ( ( ph /\ n e. NN ) /\ k e. X ) -> ( A ` k ) e. RR ) |
37 |
10 36
|
syldanl |
|- ( ( ( ph /\ n e. NN ) /\ k e. X ) -> ( A ` k ) e. RR ) |
38 |
33
|
ffvelrnda |
|- ( ( ( ph /\ n e. NN ) /\ k e. X ) -> ( ( C ` n ) ` k ) e. RR ) |
39 |
|
volico |
|- ( ( ( A ` k ) e. RR /\ ( ( C ` n ) ` k ) e. RR ) -> ( vol ` ( ( A ` k ) [,) ( ( C ` n ) ` k ) ) ) = if ( ( A ` k ) < ( ( C ` n ) ` k ) , ( ( ( C ` n ) ` k ) - ( A ` k ) ) , 0 ) ) |
40 |
37 38 39
|
syl2anc |
|- ( ( ( ph /\ n e. NN ) /\ k e. X ) -> ( vol ` ( ( A ` k ) [,) ( ( C ` n ) ` k ) ) ) = if ( ( A ` k ) < ( ( C ` n ) ` k ) , ( ( ( C ` n ) ` k ) - ( A ` k ) ) , 0 ) ) |
41 |
10 16
|
syldanl |
|- ( ( ( ph /\ n e. NN ) /\ k e. X ) -> ( B ` k ) e. RR ) |
42 |
5
|
adantlr |
|- ( ( ( ph /\ n e. NN ) /\ k e. X ) -> ( A ` k ) <_ ( B ` k ) ) |
43 |
|
nnrp |
|- ( n e. NN -> n e. RR+ ) |
44 |
43
|
rpreccld |
|- ( n e. NN -> ( 1 / n ) e. RR+ ) |
45 |
44
|
ad2antlr |
|- ( ( ( ph /\ n e. NN ) /\ k e. X ) -> ( 1 / n ) e. RR+ ) |
46 |
41 45
|
ltaddrpd |
|- ( ( ( ph /\ n e. NN ) /\ k e. X ) -> ( B ` k ) < ( ( B ` k ) + ( 1 / n ) ) ) |
47 |
19
|
elexd |
|- ( ( ( ph /\ n e. NN ) /\ k e. X ) -> ( ( B ` k ) + ( 1 / n ) ) e. _V ) |
48 |
24 47
|
fvmpt2d |
|- ( ( ( ph /\ n e. NN ) /\ k e. X ) -> ( ( C ` n ) ` k ) = ( ( B ` k ) + ( 1 / n ) ) ) |
49 |
10 48
|
syldanl |
|- ( ( ( ph /\ n e. NN ) /\ k e. X ) -> ( ( C ` n ) ` k ) = ( ( B ` k ) + ( 1 / n ) ) ) |
50 |
46 49
|
breqtrrd |
|- ( ( ( ph /\ n e. NN ) /\ k e. X ) -> ( B ` k ) < ( ( C ` n ) ` k ) ) |
51 |
37 41 38 42 50
|
lelttrd |
|- ( ( ( ph /\ n e. NN ) /\ k e. X ) -> ( A ` k ) < ( ( C ` n ) ` k ) ) |
52 |
51
|
iftrued |
|- ( ( ( ph /\ n e. NN ) /\ k e. X ) -> if ( ( A ` k ) < ( ( C ` n ) ` k ) , ( ( ( C ` n ) ` k ) - ( A ` k ) ) , 0 ) = ( ( ( C ` n ) ` k ) - ( A ` k ) ) ) |
53 |
40 52
|
eqtrd |
|- ( ( ( ph /\ n e. NN ) /\ k e. X ) -> ( vol ` ( ( A ` k ) [,) ( ( C ` n ) ` k ) ) ) = ( ( ( C ` n ) ` k ) - ( A ` k ) ) ) |
54 |
53
|
prodeq2dv |
|- ( ( ph /\ n e. NN ) -> prod_ k e. X ( vol ` ( ( A ` k ) [,) ( ( C ` n ) ` k ) ) ) = prod_ k e. X ( ( ( C ` n ) ` k ) - ( A ` k ) ) ) |
55 |
31 35 54
|
3eqtrd |
|- ( ( ph /\ n e. NN ) -> ( ( voln ` X ) ` ( D ` n ) ) = prod_ k e. X ( ( ( C ` n ) ` k ) - ( A ` k ) ) ) |
56 |
48
|
oveq1d |
|- ( ( ( ph /\ n e. NN ) /\ k e. X ) -> ( ( ( C ` n ) ` k ) - ( A ` k ) ) = ( ( ( B ` k ) + ( 1 / n ) ) - ( A ` k ) ) ) |
57 |
16
|
recnd |
|- ( ( ( ph /\ n e. NN ) /\ k e. X ) -> ( B ` k ) e. CC ) |
58 |
18
|
recnd |
|- ( ( ( ph /\ n e. NN ) /\ k e. X ) -> ( 1 / n ) e. CC ) |
59 |
36
|
recnd |
|- ( ( ( ph /\ n e. NN ) /\ k e. X ) -> ( A ` k ) e. CC ) |
60 |
57 58 59
|
addsubd |
|- ( ( ( ph /\ n e. NN ) /\ k e. X ) -> ( ( ( B ` k ) + ( 1 / n ) ) - ( A ` k ) ) = ( ( ( B ` k ) - ( A ` k ) ) + ( 1 / n ) ) ) |
61 |
56 60
|
eqtrd |
|- ( ( ( ph /\ n e. NN ) /\ k e. X ) -> ( ( ( C ` n ) ` k ) - ( A ` k ) ) = ( ( ( B ` k ) - ( A ` k ) ) + ( 1 / n ) ) ) |
62 |
61
|
prodeq2dv |
|- ( ( ph /\ n e. NN ) -> prod_ k e. X ( ( ( C ` n ) ` k ) - ( A ` k ) ) = prod_ k e. X ( ( ( B ` k ) - ( A ` k ) ) + ( 1 / n ) ) ) |
63 |
55 62
|
eqtrd |
|- ( ( ph /\ n e. NN ) -> ( ( voln ` X ) ` ( D ` n ) ) = prod_ k e. X ( ( ( B ` k ) - ( A ` k ) ) + ( 1 / n ) ) ) |
64 |
63
|
mpteq2dva |
|- ( ph -> ( n e. NN |-> ( ( voln ` X ) ` ( D ` n ) ) ) = ( n e. NN |-> prod_ k e. X ( ( ( B ` k ) - ( A ` k ) ) + ( 1 / n ) ) ) ) |
65 |
9 64
|
eqtrd |
|- ( ph -> S = ( n e. NN |-> prod_ k e. X ( ( ( B ` k ) - ( A ` k ) ) + ( 1 / n ) ) ) ) |
66 |
|
nfv |
|- F/ k ph |
67 |
2
|
ffvelrnda |
|- ( ( ph /\ k e. X ) -> ( A ` k ) e. RR ) |
68 |
15 67
|
resubcld |
|- ( ( ph /\ k e. X ) -> ( ( B ` k ) - ( A ` k ) ) e. RR ) |
69 |
68
|
recnd |
|- ( ( ph /\ k e. X ) -> ( ( B ` k ) - ( A ` k ) ) e. CC ) |
70 |
|
eqid |
|- ( n e. NN |-> prod_ k e. X ( ( ( B ` k ) - ( A ` k ) ) + ( 1 / n ) ) ) = ( n e. NN |-> prod_ k e. X ( ( ( B ` k ) - ( A ` k ) ) + ( 1 / n ) ) ) |
71 |
66 1 69 70
|
fprodaddrecnncnv |
|- ( ph -> ( n e. NN |-> prod_ k e. X ( ( ( B ` k ) - ( A ` k ) ) + ( 1 / n ) ) ) ~~> prod_ k e. X ( ( B ` k ) - ( A ` k ) ) ) |
72 |
65 71
|
eqbrtrd |
|- ( ph -> S ~~> prod_ k e. X ( ( B ` k ) - ( A ` k ) ) ) |