Step |
Hyp |
Ref |
Expression |
1 |
|
ntrivcvgmullem.1 |
|- Z = ( ZZ>= ` M ) |
2 |
|
ntrivcvgmullem.2 |
|- ( ph -> N e. Z ) |
3 |
|
ntrivcvgmullem.3 |
|- ( ph -> P e. Z ) |
4 |
|
ntrivcvgmullem.4 |
|- ( ph -> X =/= 0 ) |
5 |
|
ntrivcvgmullem.5 |
|- ( ph -> Y =/= 0 ) |
6 |
|
ntrivcvgmullem.6 |
|- ( ph -> seq N ( x. , F ) ~~> X ) |
7 |
|
ntrivcvgmullem.7 |
|- ( ph -> seq P ( x. , G ) ~~> Y ) |
8 |
|
ntrivcvgmullem.8 |
|- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC ) |
9 |
|
ntrivcvgmullem.9 |
|- ( ( ph /\ k e. Z ) -> ( G ` k ) e. CC ) |
10 |
|
ntrivcvgmullem.a |
|- ( ph -> N <_ P ) |
11 |
|
ntrivcvgmullem.b |
|- ( ( ph /\ k e. Z ) -> ( H ` k ) = ( ( F ` k ) x. ( G ` k ) ) ) |
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 |
14 3
|
sselid |
|- ( ph -> P e. ZZ ) |
17 |
|
eluz |
|- ( ( N e. ZZ /\ P e. ZZ ) -> ( P e. ( ZZ>= ` N ) <-> N <_ P ) ) |
18 |
15 16 17
|
syl2anc |
|- ( ph -> ( P e. ( ZZ>= ` N ) <-> N <_ P ) ) |
19 |
10 18
|
mpbird |
|- ( ph -> P e. ( ZZ>= ` N ) ) |
20 |
1
|
uztrn2 |
|- ( ( N e. Z /\ k e. ( ZZ>= ` N ) ) -> k e. Z ) |
21 |
2 20
|
sylan |
|- ( ( ph /\ k e. ( ZZ>= ` N ) ) -> k e. Z ) |
22 |
21 8
|
syldan |
|- ( ( ph /\ k e. ( ZZ>= ` N ) ) -> ( F ` k ) e. CC ) |
23 |
12 19 6 4 22
|
ntrivcvgtail |
|- ( ph -> ( ( ~~> ` seq P ( x. , F ) ) =/= 0 /\ seq P ( x. , F ) ~~> ( ~~> ` seq P ( x. , F ) ) ) ) |
24 |
23
|
simprd |
|- ( ph -> seq P ( x. , F ) ~~> ( ~~> ` seq P ( x. , F ) ) ) |
25 |
|
climcl |
|- ( seq P ( x. , F ) ~~> ( ~~> ` seq P ( x. , F ) ) -> ( ~~> ` seq P ( x. , F ) ) e. CC ) |
26 |
24 25
|
syl |
|- ( ph -> ( ~~> ` seq P ( x. , F ) ) e. CC ) |
27 |
|
climcl |
|- ( seq P ( x. , G ) ~~> Y -> Y e. CC ) |
28 |
7 27
|
syl |
|- ( ph -> Y e. CC ) |
29 |
23
|
simpld |
|- ( ph -> ( ~~> ` seq P ( x. , F ) ) =/= 0 ) |
30 |
26 28 29 5
|
mulne0d |
|- ( ph -> ( ( ~~> ` seq P ( x. , F ) ) x. Y ) =/= 0 ) |
31 |
|
eqid |
|- ( ZZ>= ` P ) = ( ZZ>= ` P ) |
32 |
|
seqex |
|- seq P ( x. , H ) e. _V |
33 |
32
|
a1i |
|- ( ph -> seq P ( x. , H ) e. _V ) |
34 |
1
|
uztrn2 |
|- ( ( P e. Z /\ k e. ( ZZ>= ` P ) ) -> k e. Z ) |
35 |
3 34
|
sylan |
|- ( ( ph /\ k e. ( ZZ>= ` P ) ) -> k e. Z ) |
36 |
35 8
|
syldan |
|- ( ( ph /\ k e. ( ZZ>= ` P ) ) -> ( F ` k ) e. CC ) |
37 |
31 16 36
|
prodf |
|- ( ph -> seq P ( x. , F ) : ( ZZ>= ` P ) --> CC ) |
38 |
37
|
ffvelrnda |
|- ( ( ph /\ j e. ( ZZ>= ` P ) ) -> ( seq P ( x. , F ) ` j ) e. CC ) |
39 |
35 9
|
syldan |
|- ( ( ph /\ k e. ( ZZ>= ` P ) ) -> ( G ` k ) e. CC ) |
40 |
31 16 39
|
prodf |
|- ( ph -> seq P ( x. , G ) : ( ZZ>= ` P ) --> CC ) |
41 |
40
|
ffvelrnda |
|- ( ( ph /\ j e. ( ZZ>= ` P ) ) -> ( seq P ( x. , G ) ` j ) e. CC ) |
42 |
|
simpr |
|- ( ( ph /\ j e. ( ZZ>= ` P ) ) -> j e. ( ZZ>= ` P ) ) |
43 |
|
simpll |
|- ( ( ( ph /\ j e. ( ZZ>= ` P ) ) /\ k e. ( P ... j ) ) -> ph ) |
44 |
3
|
adantr |
|- ( ( ph /\ j e. ( ZZ>= ` P ) ) -> P e. Z ) |
45 |
|
elfzuz |
|- ( k e. ( P ... j ) -> k e. ( ZZ>= ` P ) ) |
46 |
44 45 34
|
syl2an |
|- ( ( ( ph /\ j e. ( ZZ>= ` P ) ) /\ k e. ( P ... j ) ) -> k e. Z ) |
47 |
43 46 8
|
syl2anc |
|- ( ( ( ph /\ j e. ( ZZ>= ` P ) ) /\ k e. ( P ... j ) ) -> ( F ` k ) e. CC ) |
48 |
43 46 9
|
syl2anc |
|- ( ( ( ph /\ j e. ( ZZ>= ` P ) ) /\ k e. ( P ... j ) ) -> ( G ` k ) e. CC ) |
49 |
43 46 11
|
syl2anc |
|- ( ( ( ph /\ j e. ( ZZ>= ` P ) ) /\ k e. ( P ... j ) ) -> ( H ` k ) = ( ( F ` k ) x. ( G ` k ) ) ) |
50 |
42 47 48 49
|
prodfmul |
|- ( ( ph /\ j e. ( ZZ>= ` P ) ) -> ( seq P ( x. , H ) ` j ) = ( ( seq P ( x. , F ) ` j ) x. ( seq P ( x. , G ) ` j ) ) ) |
51 |
31 16 24 33 7 38 41 50
|
climmul |
|- ( ph -> seq P ( x. , H ) ~~> ( ( ~~> ` seq P ( x. , F ) ) x. Y ) ) |
52 |
|
ovex |
|- ( ( ~~> ` seq P ( x. , F ) ) x. Y ) e. _V |
53 |
|
neeq1 |
|- ( w = ( ( ~~> ` seq P ( x. , F ) ) x. Y ) -> ( w =/= 0 <-> ( ( ~~> ` seq P ( x. , F ) ) x. Y ) =/= 0 ) ) |
54 |
|
breq2 |
|- ( w = ( ( ~~> ` seq P ( x. , F ) ) x. Y ) -> ( seq P ( x. , H ) ~~> w <-> seq P ( x. , H ) ~~> ( ( ~~> ` seq P ( x. , F ) ) x. Y ) ) ) |
55 |
53 54
|
anbi12d |
|- ( w = ( ( ~~> ` seq P ( x. , F ) ) x. Y ) -> ( ( w =/= 0 /\ seq P ( x. , H ) ~~> w ) <-> ( ( ( ~~> ` seq P ( x. , F ) ) x. Y ) =/= 0 /\ seq P ( x. , H ) ~~> ( ( ~~> ` seq P ( x. , F ) ) x. Y ) ) ) ) |
56 |
52 55
|
spcev |
|- ( ( ( ( ~~> ` seq P ( x. , F ) ) x. Y ) =/= 0 /\ seq P ( x. , H ) ~~> ( ( ~~> ` seq P ( x. , F ) ) x. Y ) ) -> E. w ( w =/= 0 /\ seq P ( x. , H ) ~~> w ) ) |
57 |
30 51 56
|
syl2anc |
|- ( ph -> E. w ( w =/= 0 /\ seq P ( x. , H ) ~~> w ) ) |
58 |
|
seqeq1 |
|- ( q = P -> seq q ( x. , H ) = seq P ( x. , H ) ) |
59 |
58
|
breq1d |
|- ( q = P -> ( seq q ( x. , H ) ~~> w <-> seq P ( x. , H ) ~~> w ) ) |
60 |
59
|
anbi2d |
|- ( q = P -> ( ( w =/= 0 /\ seq q ( x. , H ) ~~> w ) <-> ( w =/= 0 /\ seq P ( x. , H ) ~~> w ) ) ) |
61 |
60
|
exbidv |
|- ( q = P -> ( E. w ( w =/= 0 /\ seq q ( x. , H ) ~~> w ) <-> E. w ( w =/= 0 /\ seq P ( x. , H ) ~~> w ) ) ) |
62 |
61
|
rspcev |
|- ( ( P e. Z /\ E. w ( w =/= 0 /\ seq P ( x. , H ) ~~> w ) ) -> E. q e. Z E. w ( w =/= 0 /\ seq q ( x. , H ) ~~> w ) ) |
63 |
3 57 62
|
syl2anc |
|- ( ph -> E. q e. Z E. w ( w =/= 0 /\ seq q ( x. , H ) ~~> w ) ) |