Step |
Hyp |
Ref |
Expression |
1 |
|
ntrivcvgtail.1 |
|- Z = ( ZZ>= ` M ) |
2 |
|
ntrivcvgtail.2 |
|- ( ph -> N e. Z ) |
3 |
|
ntrivcvgtail.3 |
|- ( ph -> seq M ( x. , F ) ~~> X ) |
4 |
|
ntrivcvgtail.4 |
|- ( ph -> X =/= 0 ) |
5 |
|
ntrivcvgtail.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 4
|
eqnetrd |
|- ( ph -> ( ~~> ` seq M ( x. , F ) ) =/= 0 ) |
12 |
3 10
|
breqtrrd |
|- ( ph -> seq M ( x. , F ) ~~> ( ~~> ` seq M ( x. , F ) ) ) |
13 |
11 12
|
jca |
|- ( ph -> ( ( ~~> ` seq M ( x. , F ) ) =/= 0 /\ seq M ( x. , F ) ~~> ( ~~> ` seq M ( x. , F ) ) ) ) |
14 |
13
|
adantr |
|- ( ( ph /\ N = M ) -> ( ( ~~> ` seq M ( x. , F ) ) =/= 0 /\ seq M ( x. , F ) ~~> ( ~~> ` seq M ( x. , F ) ) ) ) |
15 |
|
seqeq1 |
|- ( N = M -> seq N ( x. , F ) = seq M ( x. , F ) ) |
16 |
15
|
fveq2d |
|- ( N = M -> ( ~~> ` seq N ( x. , F ) ) = ( ~~> ` seq M ( x. , F ) ) ) |
17 |
16
|
neeq1d |
|- ( N = M -> ( ( ~~> ` seq N ( x. , F ) ) =/= 0 <-> ( ~~> ` seq M ( x. , F ) ) =/= 0 ) ) |
18 |
15 16
|
breq12d |
|- ( N = M -> ( seq N ( x. , F ) ~~> ( ~~> ` seq N ( x. , F ) ) <-> seq M ( x. , F ) ~~> ( ~~> ` seq M ( x. , F ) ) ) ) |
19 |
17 18
|
anbi12d |
|- ( N = M -> ( ( ( ~~> ` seq N ( x. , F ) ) =/= 0 /\ seq N ( x. , F ) ~~> ( ~~> ` seq N ( x. , F ) ) ) <-> ( ( ~~> ` seq M ( x. , F ) ) =/= 0 /\ seq M ( x. , F ) ~~> ( ~~> ` seq M ( x. , F ) ) ) ) ) |
20 |
19
|
adantl |
|- ( ( ph /\ N = M ) -> ( ( ( ~~> ` seq N ( x. , F ) ) =/= 0 /\ seq N ( x. , F ) ~~> ( ~~> ` seq N ( x. , F ) ) ) <-> ( ( ~~> ` seq M ( x. , F ) ) =/= 0 /\ seq M ( x. , F ) ~~> ( ~~> ` seq M ( x. , F ) ) ) ) ) |
21 |
14 20
|
mpbird |
|- ( ( ph /\ N = M ) -> ( ( ~~> ` seq N ( x. , F ) ) =/= 0 /\ seq N ( x. , F ) ~~> ( ~~> ` seq N ( x. , F ) ) ) ) |
22 |
|
simpr |
|- ( ( ph /\ ( N - 1 ) e. ( ZZ>= ` M ) ) -> ( N - 1 ) e. ( ZZ>= ` M ) ) |
23 |
22 1
|
eleqtrrdi |
|- ( ( ph /\ ( N - 1 ) e. ( ZZ>= ` M ) ) -> ( N - 1 ) e. Z ) |
24 |
5
|
adantlr |
|- ( ( ( ph /\ ( N - 1 ) e. ( ZZ>= ` M ) ) /\ k e. Z ) -> ( F ` k ) e. CC ) |
25 |
3
|
adantr |
|- ( ( ph /\ ( N - 1 ) e. ( ZZ>= ` M ) ) -> seq M ( x. , F ) ~~> X ) |
26 |
4
|
adantr |
|- ( ( ph /\ ( N - 1 ) e. ( ZZ>= ` M ) ) -> X =/= 0 ) |
27 |
1 23 25 26 24
|
ntrivcvgfvn0 |
|- ( ( ph /\ ( N - 1 ) e. ( ZZ>= ` M ) ) -> ( seq M ( x. , F ) ` ( N - 1 ) ) =/= 0 ) |
28 |
1 23 24 25 27
|
clim2div |
|- ( ( ph /\ ( N - 1 ) e. ( ZZ>= ` M ) ) -> seq ( ( N - 1 ) + 1 ) ( x. , F ) ~~> ( X / ( seq M ( x. , F ) ` ( N - 1 ) ) ) ) |
29 |
|
funbrfv |
|- ( Fun ~~> -> ( seq ( ( N - 1 ) + 1 ) ( x. , F ) ~~> ( X / ( seq M ( x. , F ) ` ( N - 1 ) ) ) -> ( ~~> ` seq ( ( N - 1 ) + 1 ) ( x. , F ) ) = ( X / ( seq M ( x. , F ) ` ( N - 1 ) ) ) ) ) |
30 |
8 28 29
|
mpsyl |
|- ( ( ph /\ ( N - 1 ) e. ( ZZ>= ` M ) ) -> ( ~~> ` seq ( ( N - 1 ) + 1 ) ( x. , F ) ) = ( X / ( seq M ( x. , F ) ` ( N - 1 ) ) ) ) |
31 |
|
climcl |
|- ( seq M ( x. , F ) ~~> X -> X e. CC ) |
32 |
3 31
|
syl |
|- ( ph -> X e. CC ) |
33 |
32
|
adantr |
|- ( ( ph /\ ( N - 1 ) e. ( ZZ>= ` M ) ) -> X e. CC ) |
34 |
|
eluzel2 |
|- ( N e. ( ZZ>= ` M ) -> M e. ZZ ) |
35 |
34 1
|
eleq2s |
|- ( N e. Z -> M e. ZZ ) |
36 |
2 35
|
syl |
|- ( ph -> M e. ZZ ) |
37 |
1 36 5
|
prodf |
|- ( ph -> seq M ( x. , F ) : Z --> CC ) |
38 |
1
|
feq2i |
|- ( seq M ( x. , F ) : Z --> CC <-> seq M ( x. , F ) : ( ZZ>= ` M ) --> CC ) |
39 |
37 38
|
sylib |
|- ( ph -> seq M ( x. , F ) : ( ZZ>= ` M ) --> CC ) |
40 |
39
|
ffvelrnda |
|- ( ( ph /\ ( N - 1 ) e. ( ZZ>= ` M ) ) -> ( seq M ( x. , F ) ` ( N - 1 ) ) e. CC ) |
41 |
33 40 26 27
|
divne0d |
|- ( ( ph /\ ( N - 1 ) e. ( ZZ>= ` M ) ) -> ( X / ( seq M ( x. , F ) ` ( N - 1 ) ) ) =/= 0 ) |
42 |
30 41
|
eqnetrd |
|- ( ( ph /\ ( N - 1 ) e. ( ZZ>= ` M ) ) -> ( ~~> ` seq ( ( N - 1 ) + 1 ) ( x. , F ) ) =/= 0 ) |
43 |
28 30
|
breqtrrd |
|- ( ( ph /\ ( N - 1 ) e. ( ZZ>= ` M ) ) -> seq ( ( N - 1 ) + 1 ) ( x. , F ) ~~> ( ~~> ` seq ( ( N - 1 ) + 1 ) ( x. , F ) ) ) |
44 |
|
uzssz |
|- ( ZZ>= ` M ) C_ ZZ |
45 |
1 44
|
eqsstri |
|- Z C_ ZZ |
46 |
45 2
|
sselid |
|- ( ph -> N e. ZZ ) |
47 |
46
|
zcnd |
|- ( ph -> N e. CC ) |
48 |
47
|
adantr |
|- ( ( ph /\ ( N - 1 ) e. ( ZZ>= ` M ) ) -> N e. CC ) |
49 |
|
1cnd |
|- ( ( ph /\ ( N - 1 ) e. ( ZZ>= ` M ) ) -> 1 e. CC ) |
50 |
48 49
|
npcand |
|- ( ( ph /\ ( N - 1 ) e. ( ZZ>= ` M ) ) -> ( ( N - 1 ) + 1 ) = N ) |
51 |
50
|
seqeq1d |
|- ( ( ph /\ ( N - 1 ) e. ( ZZ>= ` M ) ) -> seq ( ( N - 1 ) + 1 ) ( x. , F ) = seq N ( x. , F ) ) |
52 |
51
|
fveq2d |
|- ( ( ph /\ ( N - 1 ) e. ( ZZ>= ` M ) ) -> ( ~~> ` seq ( ( N - 1 ) + 1 ) ( x. , F ) ) = ( ~~> ` seq N ( x. , F ) ) ) |
53 |
52
|
neeq1d |
|- ( ( ph /\ ( N - 1 ) e. ( ZZ>= ` M ) ) -> ( ( ~~> ` seq ( ( N - 1 ) + 1 ) ( x. , F ) ) =/= 0 <-> ( ~~> ` seq N ( x. , F ) ) =/= 0 ) ) |
54 |
51 52
|
breq12d |
|- ( ( ph /\ ( N - 1 ) e. ( ZZ>= ` M ) ) -> ( seq ( ( N - 1 ) + 1 ) ( x. , F ) ~~> ( ~~> ` seq ( ( N - 1 ) + 1 ) ( x. , F ) ) <-> seq N ( x. , F ) ~~> ( ~~> ` seq N ( x. , F ) ) ) ) |
55 |
53 54
|
anbi12d |
|- ( ( ph /\ ( N - 1 ) e. ( ZZ>= ` M ) ) -> ( ( ( ~~> ` seq ( ( N - 1 ) + 1 ) ( x. , F ) ) =/= 0 /\ seq ( ( N - 1 ) + 1 ) ( x. , F ) ~~> ( ~~> ` seq ( ( N - 1 ) + 1 ) ( x. , F ) ) ) <-> ( ( ~~> ` seq N ( x. , F ) ) =/= 0 /\ seq N ( x. , F ) ~~> ( ~~> ` seq N ( x. , F ) ) ) ) ) |
56 |
42 43 55
|
mpbi2and |
|- ( ( ph /\ ( N - 1 ) e. ( ZZ>= ` M ) ) -> ( ( ~~> ` seq N ( x. , F ) ) =/= 0 /\ seq N ( x. , F ) ~~> ( ~~> ` seq N ( x. , F ) ) ) ) |
57 |
2 1
|
eleqtrdi |
|- ( ph -> N e. ( ZZ>= ` M ) ) |
58 |
|
uzm1 |
|- ( N e. ( ZZ>= ` M ) -> ( N = M \/ ( N - 1 ) e. ( ZZ>= ` M ) ) ) |
59 |
57 58
|
syl |
|- ( ph -> ( N = M \/ ( N - 1 ) e. ( ZZ>= ` M ) ) ) |
60 |
21 56 59
|
mpjaodan |
|- ( ph -> ( ( ~~> ` seq N ( x. , F ) ) =/= 0 /\ seq N ( x. , F ) ~~> ( ~~> ` seq N ( x. , F ) ) ) ) |