Step |
Hyp |
Ref |
Expression |
1 |
|
stoweidlem38.1 |
|- Q = { h e. A | ( ( h ` Z ) = 0 /\ A. t e. T ( 0 <_ ( h ` t ) /\ ( h ` t ) <_ 1 ) ) } |
2 |
|
stoweidlem38.2 |
|- P = ( t e. T |-> ( ( 1 / M ) x. sum_ i e. ( 1 ... M ) ( ( G ` i ) ` t ) ) ) |
3 |
|
stoweidlem38.3 |
|- ( ph -> M e. NN ) |
4 |
|
stoweidlem38.4 |
|- ( ph -> G : ( 1 ... M ) --> Q ) |
5 |
|
stoweidlem38.5 |
|- ( ( ph /\ f e. A ) -> f : T --> RR ) |
6 |
3
|
nnrecred |
|- ( ph -> ( 1 / M ) e. RR ) |
7 |
6
|
adantr |
|- ( ( ph /\ S e. T ) -> ( 1 / M ) e. RR ) |
8 |
|
fzfid |
|- ( ( ph /\ S e. T ) -> ( 1 ... M ) e. Fin ) |
9 |
1 4 5
|
stoweidlem15 |
|- ( ( ( ph /\ i e. ( 1 ... M ) ) /\ S e. T ) -> ( ( ( G ` i ) ` S ) e. RR /\ 0 <_ ( ( G ` i ) ` S ) /\ ( ( G ` i ) ` S ) <_ 1 ) ) |
10 |
9
|
simp1d |
|- ( ( ( ph /\ i e. ( 1 ... M ) ) /\ S e. T ) -> ( ( G ` i ) ` S ) e. RR ) |
11 |
10
|
an32s |
|- ( ( ( ph /\ S e. T ) /\ i e. ( 1 ... M ) ) -> ( ( G ` i ) ` S ) e. RR ) |
12 |
8 11
|
fsumrecl |
|- ( ( ph /\ S e. T ) -> sum_ i e. ( 1 ... M ) ( ( G ` i ) ` S ) e. RR ) |
13 |
|
1red |
|- ( ph -> 1 e. RR ) |
14 |
|
0le1 |
|- 0 <_ 1 |
15 |
14
|
a1i |
|- ( ph -> 0 <_ 1 ) |
16 |
3
|
nnred |
|- ( ph -> M e. RR ) |
17 |
3
|
nngt0d |
|- ( ph -> 0 < M ) |
18 |
|
divge0 |
|- ( ( ( 1 e. RR /\ 0 <_ 1 ) /\ ( M e. RR /\ 0 < M ) ) -> 0 <_ ( 1 / M ) ) |
19 |
13 15 16 17 18
|
syl22anc |
|- ( ph -> 0 <_ ( 1 / M ) ) |
20 |
19
|
adantr |
|- ( ( ph /\ S e. T ) -> 0 <_ ( 1 / M ) ) |
21 |
9
|
simp2d |
|- ( ( ( ph /\ i e. ( 1 ... M ) ) /\ S e. T ) -> 0 <_ ( ( G ` i ) ` S ) ) |
22 |
21
|
an32s |
|- ( ( ( ph /\ S e. T ) /\ i e. ( 1 ... M ) ) -> 0 <_ ( ( G ` i ) ` S ) ) |
23 |
8 11 22
|
fsumge0 |
|- ( ( ph /\ S e. T ) -> 0 <_ sum_ i e. ( 1 ... M ) ( ( G ` i ) ` S ) ) |
24 |
7 12 20 23
|
mulge0d |
|- ( ( ph /\ S e. T ) -> 0 <_ ( ( 1 / M ) x. sum_ i e. ( 1 ... M ) ( ( G ` i ) ` S ) ) ) |
25 |
1 2 3 4 5
|
stoweidlem30 |
|- ( ( ph /\ S e. T ) -> ( P ` S ) = ( ( 1 / M ) x. sum_ i e. ( 1 ... M ) ( ( G ` i ) ` S ) ) ) |
26 |
24 25
|
breqtrrd |
|- ( ( ph /\ S e. T ) -> 0 <_ ( P ` S ) ) |
27 |
|
1red |
|- ( ( ( ph /\ S e. T ) /\ i e. ( 1 ... M ) ) -> 1 e. RR ) |
28 |
9
|
simp3d |
|- ( ( ( ph /\ i e. ( 1 ... M ) ) /\ S e. T ) -> ( ( G ` i ) ` S ) <_ 1 ) |
29 |
28
|
an32s |
|- ( ( ( ph /\ S e. T ) /\ i e. ( 1 ... M ) ) -> ( ( G ` i ) ` S ) <_ 1 ) |
30 |
8 11 27 29
|
fsumle |
|- ( ( ph /\ S e. T ) -> sum_ i e. ( 1 ... M ) ( ( G ` i ) ` S ) <_ sum_ i e. ( 1 ... M ) 1 ) |
31 |
|
fzfid |
|- ( ph -> ( 1 ... M ) e. Fin ) |
32 |
|
ax-1cn |
|- 1 e. CC |
33 |
|
fsumconst |
|- ( ( ( 1 ... M ) e. Fin /\ 1 e. CC ) -> sum_ i e. ( 1 ... M ) 1 = ( ( # ` ( 1 ... M ) ) x. 1 ) ) |
34 |
31 32 33
|
sylancl |
|- ( ph -> sum_ i e. ( 1 ... M ) 1 = ( ( # ` ( 1 ... M ) ) x. 1 ) ) |
35 |
3
|
nnnn0d |
|- ( ph -> M e. NN0 ) |
36 |
|
hashfz1 |
|- ( M e. NN0 -> ( # ` ( 1 ... M ) ) = M ) |
37 |
35 36
|
syl |
|- ( ph -> ( # ` ( 1 ... M ) ) = M ) |
38 |
37
|
oveq1d |
|- ( ph -> ( ( # ` ( 1 ... M ) ) x. 1 ) = ( M x. 1 ) ) |
39 |
3
|
nncnd |
|- ( ph -> M e. CC ) |
40 |
39
|
mulid1d |
|- ( ph -> ( M x. 1 ) = M ) |
41 |
34 38 40
|
3eqtrd |
|- ( ph -> sum_ i e. ( 1 ... M ) 1 = M ) |
42 |
41
|
adantr |
|- ( ( ph /\ S e. T ) -> sum_ i e. ( 1 ... M ) 1 = M ) |
43 |
30 42
|
breqtrd |
|- ( ( ph /\ S e. T ) -> sum_ i e. ( 1 ... M ) ( ( G ` i ) ` S ) <_ M ) |
44 |
16
|
adantr |
|- ( ( ph /\ S e. T ) -> M e. RR ) |
45 |
|
1red |
|- ( ( ph /\ S e. T ) -> 1 e. RR ) |
46 |
|
0lt1 |
|- 0 < 1 |
47 |
46
|
a1i |
|- ( ( ph /\ S e. T ) -> 0 < 1 ) |
48 |
16 17
|
jca |
|- ( ph -> ( M e. RR /\ 0 < M ) ) |
49 |
48
|
adantr |
|- ( ( ph /\ S e. T ) -> ( M e. RR /\ 0 < M ) ) |
50 |
|
divgt0 |
|- ( ( ( 1 e. RR /\ 0 < 1 ) /\ ( M e. RR /\ 0 < M ) ) -> 0 < ( 1 / M ) ) |
51 |
45 47 49 50
|
syl21anc |
|- ( ( ph /\ S e. T ) -> 0 < ( 1 / M ) ) |
52 |
|
lemul2 |
|- ( ( sum_ i e. ( 1 ... M ) ( ( G ` i ) ` S ) e. RR /\ M e. RR /\ ( ( 1 / M ) e. RR /\ 0 < ( 1 / M ) ) ) -> ( sum_ i e. ( 1 ... M ) ( ( G ` i ) ` S ) <_ M <-> ( ( 1 / M ) x. sum_ i e. ( 1 ... M ) ( ( G ` i ) ` S ) ) <_ ( ( 1 / M ) x. M ) ) ) |
53 |
12 44 7 51 52
|
syl112anc |
|- ( ( ph /\ S e. T ) -> ( sum_ i e. ( 1 ... M ) ( ( G ` i ) ` S ) <_ M <-> ( ( 1 / M ) x. sum_ i e. ( 1 ... M ) ( ( G ` i ) ` S ) ) <_ ( ( 1 / M ) x. M ) ) ) |
54 |
43 53
|
mpbid |
|- ( ( ph /\ S e. T ) -> ( ( 1 / M ) x. sum_ i e. ( 1 ... M ) ( ( G ` i ) ` S ) ) <_ ( ( 1 / M ) x. M ) ) |
55 |
25 54
|
eqbrtrd |
|- ( ( ph /\ S e. T ) -> ( P ` S ) <_ ( ( 1 / M ) x. M ) ) |
56 |
32
|
a1i |
|- ( ph -> 1 e. CC ) |
57 |
3
|
nnne0d |
|- ( ph -> M =/= 0 ) |
58 |
56 39 57
|
3jca |
|- ( ph -> ( 1 e. CC /\ M e. CC /\ M =/= 0 ) ) |
59 |
58
|
adantr |
|- ( ( ph /\ S e. T ) -> ( 1 e. CC /\ M e. CC /\ M =/= 0 ) ) |
60 |
|
divcan1 |
|- ( ( 1 e. CC /\ M e. CC /\ M =/= 0 ) -> ( ( 1 / M ) x. M ) = 1 ) |
61 |
59 60
|
syl |
|- ( ( ph /\ S e. T ) -> ( ( 1 / M ) x. M ) = 1 ) |
62 |
55 61
|
breqtrd |
|- ( ( ph /\ S e. T ) -> ( P ` S ) <_ 1 ) |
63 |
26 62
|
jca |
|- ( ( ph /\ S e. T ) -> ( 0 <_ ( P ` S ) /\ ( P ` S ) <_ 1 ) ) |