Step |
Hyp |
Ref |
Expression |
1 |
|
taylpfval.s |
|- ( ph -> S e. { RR , CC } ) |
2 |
|
taylpfval.f |
|- ( ph -> F : A --> CC ) |
3 |
|
taylpfval.a |
|- ( ph -> A C_ S ) |
4 |
|
taylpfval.n |
|- ( ph -> N e. NN0 ) |
5 |
|
taylpfval.b |
|- ( ph -> B e. dom ( ( S Dn F ) ` N ) ) |
6 |
|
taylpfval.t |
|- T = ( N ( S Tayl F ) B ) |
7 |
|
cnring |
|- CCfld e. Ring |
8 |
|
cnfldbas |
|- CC = ( Base ` CCfld ) |
9 |
8
|
subrgid |
|- ( CCfld e. Ring -> CC e. ( SubRing ` CCfld ) ) |
10 |
7 9
|
mp1i |
|- ( ph -> CC e. ( SubRing ` CCfld ) ) |
11 |
|
cnex |
|- CC e. _V |
12 |
11
|
a1i |
|- ( ph -> CC e. _V ) |
13 |
|
elpm2r |
|- ( ( ( CC e. _V /\ S e. { RR , CC } ) /\ ( F : A --> CC /\ A C_ S ) ) -> F e. ( CC ^pm S ) ) |
14 |
12 1 2 3 13
|
syl22anc |
|- ( ph -> F e. ( CC ^pm S ) ) |
15 |
|
dvnbss |
|- ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) /\ N e. NN0 ) -> dom ( ( S Dn F ) ` N ) C_ dom F ) |
16 |
1 14 4 15
|
syl3anc |
|- ( ph -> dom ( ( S Dn F ) ` N ) C_ dom F ) |
17 |
2 16
|
fssdmd |
|- ( ph -> dom ( ( S Dn F ) ` N ) C_ A ) |
18 |
|
recnprss |
|- ( S e. { RR , CC } -> S C_ CC ) |
19 |
1 18
|
syl |
|- ( ph -> S C_ CC ) |
20 |
3 19
|
sstrd |
|- ( ph -> A C_ CC ) |
21 |
17 20
|
sstrd |
|- ( ph -> dom ( ( S Dn F ) ` N ) C_ CC ) |
22 |
21 5
|
sseldd |
|- ( ph -> B e. CC ) |
23 |
1
|
adantr |
|- ( ( ph /\ k e. ( 0 ... N ) ) -> S e. { RR , CC } ) |
24 |
14
|
adantr |
|- ( ( ph /\ k e. ( 0 ... N ) ) -> F e. ( CC ^pm S ) ) |
25 |
|
elfznn0 |
|- ( k e. ( 0 ... N ) -> k e. NN0 ) |
26 |
25
|
adantl |
|- ( ( ph /\ k e. ( 0 ... N ) ) -> k e. NN0 ) |
27 |
|
dvnf |
|- ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) /\ k e. NN0 ) -> ( ( S Dn F ) ` k ) : dom ( ( S Dn F ) ` k ) --> CC ) |
28 |
23 24 26 27
|
syl3anc |
|- ( ( ph /\ k e. ( 0 ... N ) ) -> ( ( S Dn F ) ` k ) : dom ( ( S Dn F ) ` k ) --> CC ) |
29 |
|
simpr |
|- ( ( ph /\ k e. ( 0 ... N ) ) -> k e. ( 0 ... N ) ) |
30 |
|
dvn2bss |
|- ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) /\ k e. ( 0 ... N ) ) -> dom ( ( S Dn F ) ` N ) C_ dom ( ( S Dn F ) ` k ) ) |
31 |
23 24 29 30
|
syl3anc |
|- ( ( ph /\ k e. ( 0 ... N ) ) -> dom ( ( S Dn F ) ` N ) C_ dom ( ( S Dn F ) ` k ) ) |
32 |
5
|
adantr |
|- ( ( ph /\ k e. ( 0 ... N ) ) -> B e. dom ( ( S Dn F ) ` N ) ) |
33 |
31 32
|
sseldd |
|- ( ( ph /\ k e. ( 0 ... N ) ) -> B e. dom ( ( S Dn F ) ` k ) ) |
34 |
28 33
|
ffvelrnd |
|- ( ( ph /\ k e. ( 0 ... N ) ) -> ( ( ( S Dn F ) ` k ) ` B ) e. CC ) |
35 |
26
|
faccld |
|- ( ( ph /\ k e. ( 0 ... N ) ) -> ( ! ` k ) e. NN ) |
36 |
35
|
nncnd |
|- ( ( ph /\ k e. ( 0 ... N ) ) -> ( ! ` k ) e. CC ) |
37 |
35
|
nnne0d |
|- ( ( ph /\ k e. ( 0 ... N ) ) -> ( ! ` k ) =/= 0 ) |
38 |
34 36 37
|
divcld |
|- ( ( ph /\ k e. ( 0 ... N ) ) -> ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) e. CC ) |
39 |
1 2 3 4 5 6 10 22 38
|
taylply2 |
|- ( ph -> ( T e. ( Poly ` CC ) /\ ( deg ` T ) <_ N ) ) |