Step |
Hyp |
Ref |
Expression |
1 |
|
ntrivcvgfvn0.1 |
|- Z = ( ZZ>= ` M ) |
2 |
|
ntrivcvgfvn0.2 |
|- ( ph -> N e. Z ) |
3 |
|
ntrivcvgfvn0.3 |
|- ( ph -> seq M ( x. , F ) ~~> X ) |
4 |
|
ntrivcvgfvn0.4 |
|- ( ph -> X =/= 0 ) |
5 |
|
ntrivcvgfvn0.5 |
|- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC ) |
6 |
|
fclim |
|- ~~> : dom ~~> --> CC |
7 |
|
ffun |
|- ( ~~> : dom ~~> --> CC -> Fun ~~> ) |
8 |
6 7
|
ax-mp |
|- Fun ~~> |
9 |
|
funbrfv |
|- ( Fun ~~> -> ( seq M ( x. , F ) ~~> X -> ( ~~> ` seq M ( x. , F ) ) = X ) ) |
10 |
8 3 9
|
mpsyl |
|- ( ph -> ( ~~> ` seq M ( x. , F ) ) = X ) |
11 |
10
|
adantr |
|- ( ( ph /\ ( seq M ( x. , F ) ` N ) = 0 ) -> ( ~~> ` seq M ( x. , F ) ) = X ) |
12 |
|
eqid |
|- ( ZZ>= ` N ) = ( ZZ>= ` N ) |
13 |
|
uzssz |
|- ( ZZ>= ` M ) C_ ZZ |
14 |
1 13
|
eqsstri |
|- Z C_ ZZ |
15 |
14 2
|
sselid |
|- ( ph -> N e. ZZ ) |
16 |
15
|
adantr |
|- ( ( ph /\ ( seq M ( x. , F ) ` N ) = 0 ) -> N e. ZZ ) |
17 |
|
seqex |
|- seq M ( x. , F ) e. _V |
18 |
17
|
a1i |
|- ( ( ph /\ ( seq M ( x. , F ) ` N ) = 0 ) -> seq M ( x. , F ) e. _V ) |
19 |
|
0cnd |
|- ( ( ph /\ ( seq M ( x. , F ) ` N ) = 0 ) -> 0 e. CC ) |
20 |
|
fveqeq2 |
|- ( m = N -> ( ( seq M ( x. , F ) ` m ) = 0 <-> ( seq M ( x. , F ) ` N ) = 0 ) ) |
21 |
20
|
imbi2d |
|- ( m = N -> ( ( ( ph /\ ( seq M ( x. , F ) ` N ) = 0 ) -> ( seq M ( x. , F ) ` m ) = 0 ) <-> ( ( ph /\ ( seq M ( x. , F ) ` N ) = 0 ) -> ( seq M ( x. , F ) ` N ) = 0 ) ) ) |
22 |
|
fveqeq2 |
|- ( m = n -> ( ( seq M ( x. , F ) ` m ) = 0 <-> ( seq M ( x. , F ) ` n ) = 0 ) ) |
23 |
22
|
imbi2d |
|- ( m = n -> ( ( ( ph /\ ( seq M ( x. , F ) ` N ) = 0 ) -> ( seq M ( x. , F ) ` m ) = 0 ) <-> ( ( ph /\ ( seq M ( x. , F ) ` N ) = 0 ) -> ( seq M ( x. , F ) ` n ) = 0 ) ) ) |
24 |
|
fveqeq2 |
|- ( m = ( n + 1 ) -> ( ( seq M ( x. , F ) ` m ) = 0 <-> ( seq M ( x. , F ) ` ( n + 1 ) ) = 0 ) ) |
25 |
24
|
imbi2d |
|- ( m = ( n + 1 ) -> ( ( ( ph /\ ( seq M ( x. , F ) ` N ) = 0 ) -> ( seq M ( x. , F ) ` m ) = 0 ) <-> ( ( ph /\ ( seq M ( x. , F ) ` N ) = 0 ) -> ( seq M ( x. , F ) ` ( n + 1 ) ) = 0 ) ) ) |
26 |
|
fveqeq2 |
|- ( m = k -> ( ( seq M ( x. , F ) ` m ) = 0 <-> ( seq M ( x. , F ) ` k ) = 0 ) ) |
27 |
26
|
imbi2d |
|- ( m = k -> ( ( ( ph /\ ( seq M ( x. , F ) ` N ) = 0 ) -> ( seq M ( x. , F ) ` m ) = 0 ) <-> ( ( ph /\ ( seq M ( x. , F ) ` N ) = 0 ) -> ( seq M ( x. , F ) ` k ) = 0 ) ) ) |
28 |
|
simpr |
|- ( ( ph /\ ( seq M ( x. , F ) ` N ) = 0 ) -> ( seq M ( x. , F ) ` N ) = 0 ) |
29 |
2 1
|
eleqtrdi |
|- ( ph -> N e. ( ZZ>= ` M ) ) |
30 |
|
uztrn |
|- ( ( n e. ( ZZ>= ` N ) /\ N e. ( ZZ>= ` M ) ) -> n e. ( ZZ>= ` M ) ) |
31 |
29 30
|
sylan2 |
|- ( ( n e. ( ZZ>= ` N ) /\ ph ) -> n e. ( ZZ>= ` M ) ) |
32 |
31
|
3adant3 |
|- ( ( n e. ( ZZ>= ` N ) /\ ph /\ ( seq M ( x. , F ) ` n ) = 0 ) -> n e. ( ZZ>= ` M ) ) |
33 |
|
seqp1 |
|- ( n e. ( ZZ>= ` M ) -> ( seq M ( x. , F ) ` ( n + 1 ) ) = ( ( seq M ( x. , F ) ` n ) x. ( F ` ( n + 1 ) ) ) ) |
34 |
32 33
|
syl |
|- ( ( n e. ( ZZ>= ` N ) /\ ph /\ ( seq M ( x. , F ) ` n ) = 0 ) -> ( seq M ( x. , F ) ` ( n + 1 ) ) = ( ( seq M ( x. , F ) ` n ) x. ( F ` ( n + 1 ) ) ) ) |
35 |
|
oveq1 |
|- ( ( seq M ( x. , F ) ` n ) = 0 -> ( ( seq M ( x. , F ) ` n ) x. ( F ` ( n + 1 ) ) ) = ( 0 x. ( F ` ( n + 1 ) ) ) ) |
36 |
35
|
3ad2ant3 |
|- ( ( n e. ( ZZ>= ` N ) /\ ph /\ ( seq M ( x. , F ) ` n ) = 0 ) -> ( ( seq M ( x. , F ) ` n ) x. ( F ` ( n + 1 ) ) ) = ( 0 x. ( F ` ( n + 1 ) ) ) ) |
37 |
|
peano2uz |
|- ( n e. ( ZZ>= ` N ) -> ( n + 1 ) e. ( ZZ>= ` N ) ) |
38 |
1
|
uztrn2 |
|- ( ( N e. Z /\ ( n + 1 ) e. ( ZZ>= ` N ) ) -> ( n + 1 ) e. Z ) |
39 |
2 37 38
|
syl2an |
|- ( ( ph /\ n e. ( ZZ>= ` N ) ) -> ( n + 1 ) e. Z ) |
40 |
5
|
ralrimiva |
|- ( ph -> A. k e. Z ( F ` k ) e. CC ) |
41 |
|
fveq2 |
|- ( k = ( n + 1 ) -> ( F ` k ) = ( F ` ( n + 1 ) ) ) |
42 |
41
|
eleq1d |
|- ( k = ( n + 1 ) -> ( ( F ` k ) e. CC <-> ( F ` ( n + 1 ) ) e. CC ) ) |
43 |
42
|
rspcv |
|- ( ( n + 1 ) e. Z -> ( A. k e. Z ( F ` k ) e. CC -> ( F ` ( n + 1 ) ) e. CC ) ) |
44 |
40 43
|
mpan9 |
|- ( ( ph /\ ( n + 1 ) e. Z ) -> ( F ` ( n + 1 ) ) e. CC ) |
45 |
39 44
|
syldan |
|- ( ( ph /\ n e. ( ZZ>= ` N ) ) -> ( F ` ( n + 1 ) ) e. CC ) |
46 |
45
|
ancoms |
|- ( ( n e. ( ZZ>= ` N ) /\ ph ) -> ( F ` ( n + 1 ) ) e. CC ) |
47 |
46
|
mul02d |
|- ( ( n e. ( ZZ>= ` N ) /\ ph ) -> ( 0 x. ( F ` ( n + 1 ) ) ) = 0 ) |
48 |
47
|
3adant3 |
|- ( ( n e. ( ZZ>= ` N ) /\ ph /\ ( seq M ( x. , F ) ` n ) = 0 ) -> ( 0 x. ( F ` ( n + 1 ) ) ) = 0 ) |
49 |
34 36 48
|
3eqtrd |
|- ( ( n e. ( ZZ>= ` N ) /\ ph /\ ( seq M ( x. , F ) ` n ) = 0 ) -> ( seq M ( x. , F ) ` ( n + 1 ) ) = 0 ) |
50 |
49
|
3exp |
|- ( n e. ( ZZ>= ` N ) -> ( ph -> ( ( seq M ( x. , F ) ` n ) = 0 -> ( seq M ( x. , F ) ` ( n + 1 ) ) = 0 ) ) ) |
51 |
50
|
adantrd |
|- ( n e. ( ZZ>= ` N ) -> ( ( ph /\ ( seq M ( x. , F ) ` N ) = 0 ) -> ( ( seq M ( x. , F ) ` n ) = 0 -> ( seq M ( x. , F ) ` ( n + 1 ) ) = 0 ) ) ) |
52 |
51
|
a2d |
|- ( n e. ( ZZ>= ` N ) -> ( ( ( ph /\ ( seq M ( x. , F ) ` N ) = 0 ) -> ( seq M ( x. , F ) ` n ) = 0 ) -> ( ( ph /\ ( seq M ( x. , F ) ` N ) = 0 ) -> ( seq M ( x. , F ) ` ( n + 1 ) ) = 0 ) ) ) |
53 |
21 23 25 27 28 52
|
uzind4i |
|- ( k e. ( ZZ>= ` N ) -> ( ( ph /\ ( seq M ( x. , F ) ` N ) = 0 ) -> ( seq M ( x. , F ) ` k ) = 0 ) ) |
54 |
53
|
impcom |
|- ( ( ( ph /\ ( seq M ( x. , F ) ` N ) = 0 ) /\ k e. ( ZZ>= ` N ) ) -> ( seq M ( x. , F ) ` k ) = 0 ) |
55 |
12 16 18 19 54
|
climconst |
|- ( ( ph /\ ( seq M ( x. , F ) ` N ) = 0 ) -> seq M ( x. , F ) ~~> 0 ) |
56 |
|
funbrfv |
|- ( Fun ~~> -> ( seq M ( x. , F ) ~~> 0 -> ( ~~> ` seq M ( x. , F ) ) = 0 ) ) |
57 |
8 55 56
|
mpsyl |
|- ( ( ph /\ ( seq M ( x. , F ) ` N ) = 0 ) -> ( ~~> ` seq M ( x. , F ) ) = 0 ) |
58 |
11 57
|
eqtr3d |
|- ( ( ph /\ ( seq M ( x. , F ) ` N ) = 0 ) -> X = 0 ) |
59 |
58
|
ex |
|- ( ph -> ( ( seq M ( x. , F ) ` N ) = 0 -> X = 0 ) ) |
60 |
59
|
necon3d |
|- ( ph -> ( X =/= 0 -> ( seq M ( x. , F ) ` N ) =/= 0 ) ) |
61 |
4 60
|
mpd |
|- ( ph -> ( seq M ( x. , F ) ` N ) =/= 0 ) |