Step |
Hyp |
Ref |
Expression |
1 |
|
dvntaylp.s |
|- ( ph -> S e. { RR , CC } ) |
2 |
|
dvntaylp.f |
|- ( ph -> F : A --> CC ) |
3 |
|
dvntaylp.a |
|- ( ph -> A C_ S ) |
4 |
|
dvntaylp.m |
|- ( ph -> M e. NN0 ) |
5 |
|
dvntaylp.n |
|- ( ph -> N e. NN0 ) |
6 |
|
dvntaylp.b |
|- ( ph -> B e. dom ( ( S Dn F ) ` ( N + M ) ) ) |
7 |
|
nn0uz |
|- NN0 = ( ZZ>= ` 0 ) |
8 |
4 7
|
eleqtrdi |
|- ( ph -> M e. ( ZZ>= ` 0 ) ) |
9 |
|
eluzfz2b |
|- ( M e. ( ZZ>= ` 0 ) <-> M e. ( 0 ... M ) ) |
10 |
8 9
|
sylib |
|- ( ph -> M e. ( 0 ... M ) ) |
11 |
|
fveq2 |
|- ( m = 0 -> ( ( CC Dn ( ( N + M ) ( S Tayl F ) B ) ) ` m ) = ( ( CC Dn ( ( N + M ) ( S Tayl F ) B ) ) ` 0 ) ) |
12 |
|
fveq2 |
|- ( m = 0 -> ( ( S Dn F ) ` m ) = ( ( S Dn F ) ` 0 ) ) |
13 |
12
|
oveq2d |
|- ( m = 0 -> ( S Tayl ( ( S Dn F ) ` m ) ) = ( S Tayl ( ( S Dn F ) ` 0 ) ) ) |
14 |
|
oveq2 |
|- ( m = 0 -> ( M - m ) = ( M - 0 ) ) |
15 |
14
|
oveq2d |
|- ( m = 0 -> ( N + ( M - m ) ) = ( N + ( M - 0 ) ) ) |
16 |
|
eqidd |
|- ( m = 0 -> B = B ) |
17 |
13 15 16
|
oveq123d |
|- ( m = 0 -> ( ( N + ( M - m ) ) ( S Tayl ( ( S Dn F ) ` m ) ) B ) = ( ( N + ( M - 0 ) ) ( S Tayl ( ( S Dn F ) ` 0 ) ) B ) ) |
18 |
11 17
|
eqeq12d |
|- ( m = 0 -> ( ( ( CC Dn ( ( N + M ) ( S Tayl F ) B ) ) ` m ) = ( ( N + ( M - m ) ) ( S Tayl ( ( S Dn F ) ` m ) ) B ) <-> ( ( CC Dn ( ( N + M ) ( S Tayl F ) B ) ) ` 0 ) = ( ( N + ( M - 0 ) ) ( S Tayl ( ( S Dn F ) ` 0 ) ) B ) ) ) |
19 |
18
|
imbi2d |
|- ( m = 0 -> ( ( ph -> ( ( CC Dn ( ( N + M ) ( S Tayl F ) B ) ) ` m ) = ( ( N + ( M - m ) ) ( S Tayl ( ( S Dn F ) ` m ) ) B ) ) <-> ( ph -> ( ( CC Dn ( ( N + M ) ( S Tayl F ) B ) ) ` 0 ) = ( ( N + ( M - 0 ) ) ( S Tayl ( ( S Dn F ) ` 0 ) ) B ) ) ) ) |
20 |
|
fveq2 |
|- ( m = n -> ( ( CC Dn ( ( N + M ) ( S Tayl F ) B ) ) ` m ) = ( ( CC Dn ( ( N + M ) ( S Tayl F ) B ) ) ` n ) ) |
21 |
|
fveq2 |
|- ( m = n -> ( ( S Dn F ) ` m ) = ( ( S Dn F ) ` n ) ) |
22 |
21
|
oveq2d |
|- ( m = n -> ( S Tayl ( ( S Dn F ) ` m ) ) = ( S Tayl ( ( S Dn F ) ` n ) ) ) |
23 |
|
oveq2 |
|- ( m = n -> ( M - m ) = ( M - n ) ) |
24 |
23
|
oveq2d |
|- ( m = n -> ( N + ( M - m ) ) = ( N + ( M - n ) ) ) |
25 |
|
eqidd |
|- ( m = n -> B = B ) |
26 |
22 24 25
|
oveq123d |
|- ( m = n -> ( ( N + ( M - m ) ) ( S Tayl ( ( S Dn F ) ` m ) ) B ) = ( ( N + ( M - n ) ) ( S Tayl ( ( S Dn F ) ` n ) ) B ) ) |
27 |
20 26
|
eqeq12d |
|- ( m = n -> ( ( ( CC Dn ( ( N + M ) ( S Tayl F ) B ) ) ` m ) = ( ( N + ( M - m ) ) ( S Tayl ( ( S Dn F ) ` m ) ) B ) <-> ( ( CC Dn ( ( N + M ) ( S Tayl F ) B ) ) ` n ) = ( ( N + ( M - n ) ) ( S Tayl ( ( S Dn F ) ` n ) ) B ) ) ) |
28 |
27
|
imbi2d |
|- ( m = n -> ( ( ph -> ( ( CC Dn ( ( N + M ) ( S Tayl F ) B ) ) ` m ) = ( ( N + ( M - m ) ) ( S Tayl ( ( S Dn F ) ` m ) ) B ) ) <-> ( ph -> ( ( CC Dn ( ( N + M ) ( S Tayl F ) B ) ) ` n ) = ( ( N + ( M - n ) ) ( S Tayl ( ( S Dn F ) ` n ) ) B ) ) ) ) |
29 |
|
fveq2 |
|- ( m = ( n + 1 ) -> ( ( CC Dn ( ( N + M ) ( S Tayl F ) B ) ) ` m ) = ( ( CC Dn ( ( N + M ) ( S Tayl F ) B ) ) ` ( n + 1 ) ) ) |
30 |
|
fveq2 |
|- ( m = ( n + 1 ) -> ( ( S Dn F ) ` m ) = ( ( S Dn F ) ` ( n + 1 ) ) ) |
31 |
30
|
oveq2d |
|- ( m = ( n + 1 ) -> ( S Tayl ( ( S Dn F ) ` m ) ) = ( S Tayl ( ( S Dn F ) ` ( n + 1 ) ) ) ) |
32 |
|
oveq2 |
|- ( m = ( n + 1 ) -> ( M - m ) = ( M - ( n + 1 ) ) ) |
33 |
32
|
oveq2d |
|- ( m = ( n + 1 ) -> ( N + ( M - m ) ) = ( N + ( M - ( n + 1 ) ) ) ) |
34 |
|
eqidd |
|- ( m = ( n + 1 ) -> B = B ) |
35 |
31 33 34
|
oveq123d |
|- ( m = ( n + 1 ) -> ( ( N + ( M - m ) ) ( S Tayl ( ( S Dn F ) ` m ) ) B ) = ( ( N + ( M - ( n + 1 ) ) ) ( S Tayl ( ( S Dn F ) ` ( n + 1 ) ) ) B ) ) |
36 |
29 35
|
eqeq12d |
|- ( m = ( n + 1 ) -> ( ( ( CC Dn ( ( N + M ) ( S Tayl F ) B ) ) ` m ) = ( ( N + ( M - m ) ) ( S Tayl ( ( S Dn F ) ` m ) ) B ) <-> ( ( CC Dn ( ( N + M ) ( S Tayl F ) B ) ) ` ( n + 1 ) ) = ( ( N + ( M - ( n + 1 ) ) ) ( S Tayl ( ( S Dn F ) ` ( n + 1 ) ) ) B ) ) ) |
37 |
36
|
imbi2d |
|- ( m = ( n + 1 ) -> ( ( ph -> ( ( CC Dn ( ( N + M ) ( S Tayl F ) B ) ) ` m ) = ( ( N + ( M - m ) ) ( S Tayl ( ( S Dn F ) ` m ) ) B ) ) <-> ( ph -> ( ( CC Dn ( ( N + M ) ( S Tayl F ) B ) ) ` ( n + 1 ) ) = ( ( N + ( M - ( n + 1 ) ) ) ( S Tayl ( ( S Dn F ) ` ( n + 1 ) ) ) B ) ) ) ) |
38 |
|
fveq2 |
|- ( m = M -> ( ( CC Dn ( ( N + M ) ( S Tayl F ) B ) ) ` m ) = ( ( CC Dn ( ( N + M ) ( S Tayl F ) B ) ) ` M ) ) |
39 |
|
fveq2 |
|- ( m = M -> ( ( S Dn F ) ` m ) = ( ( S Dn F ) ` M ) ) |
40 |
39
|
oveq2d |
|- ( m = M -> ( S Tayl ( ( S Dn F ) ` m ) ) = ( S Tayl ( ( S Dn F ) ` M ) ) ) |
41 |
|
oveq2 |
|- ( m = M -> ( M - m ) = ( M - M ) ) |
42 |
41
|
oveq2d |
|- ( m = M -> ( N + ( M - m ) ) = ( N + ( M - M ) ) ) |
43 |
|
eqidd |
|- ( m = M -> B = B ) |
44 |
40 42 43
|
oveq123d |
|- ( m = M -> ( ( N + ( M - m ) ) ( S Tayl ( ( S Dn F ) ` m ) ) B ) = ( ( N + ( M - M ) ) ( S Tayl ( ( S Dn F ) ` M ) ) B ) ) |
45 |
38 44
|
eqeq12d |
|- ( m = M -> ( ( ( CC Dn ( ( N + M ) ( S Tayl F ) B ) ) ` m ) = ( ( N + ( M - m ) ) ( S Tayl ( ( S Dn F ) ` m ) ) B ) <-> ( ( CC Dn ( ( N + M ) ( S Tayl F ) B ) ) ` M ) = ( ( N + ( M - M ) ) ( S Tayl ( ( S Dn F ) ` M ) ) B ) ) ) |
46 |
45
|
imbi2d |
|- ( m = M -> ( ( ph -> ( ( CC Dn ( ( N + M ) ( S Tayl F ) B ) ) ` m ) = ( ( N + ( M - m ) ) ( S Tayl ( ( S Dn F ) ` m ) ) B ) ) <-> ( ph -> ( ( CC Dn ( ( N + M ) ( S Tayl F ) B ) ) ` M ) = ( ( N + ( M - M ) ) ( S Tayl ( ( S Dn F ) ` M ) ) B ) ) ) ) |
47 |
|
ssidd |
|- ( ph -> CC C_ CC ) |
48 |
|
mapsspm |
|- ( CC ^m CC ) C_ ( CC ^pm CC ) |
49 |
5 4
|
nn0addcld |
|- ( ph -> ( N + M ) e. NN0 ) |
50 |
|
eqid |
|- ( ( N + M ) ( S Tayl F ) B ) = ( ( N + M ) ( S Tayl F ) B ) |
51 |
1 2 3 49 6 50
|
taylpf |
|- ( ph -> ( ( N + M ) ( S Tayl F ) B ) : CC --> CC ) |
52 |
|
cnex |
|- CC e. _V |
53 |
52 52
|
elmap |
|- ( ( ( N + M ) ( S Tayl F ) B ) e. ( CC ^m CC ) <-> ( ( N + M ) ( S Tayl F ) B ) : CC --> CC ) |
54 |
51 53
|
sylibr |
|- ( ph -> ( ( N + M ) ( S Tayl F ) B ) e. ( CC ^m CC ) ) |
55 |
48 54
|
sseldi |
|- ( ph -> ( ( N + M ) ( S Tayl F ) B ) e. ( CC ^pm CC ) ) |
56 |
|
dvn0 |
|- ( ( CC C_ CC /\ ( ( N + M ) ( S Tayl F ) B ) e. ( CC ^pm CC ) ) -> ( ( CC Dn ( ( N + M ) ( S Tayl F ) B ) ) ` 0 ) = ( ( N + M ) ( S Tayl F ) B ) ) |
57 |
47 55 56
|
syl2anc |
|- ( ph -> ( ( CC Dn ( ( N + M ) ( S Tayl F ) B ) ) ` 0 ) = ( ( N + M ) ( S Tayl F ) B ) ) |
58 |
|
recnprss |
|- ( S e. { RR , CC } -> S C_ CC ) |
59 |
1 58
|
syl |
|- ( ph -> S C_ CC ) |
60 |
52
|
a1i |
|- ( ph -> CC e. _V ) |
61 |
|
elpm2r |
|- ( ( ( CC e. _V /\ S e. { RR , CC } ) /\ ( F : A --> CC /\ A C_ S ) ) -> F e. ( CC ^pm S ) ) |
62 |
60 1 2 3 61
|
syl22anc |
|- ( ph -> F e. ( CC ^pm S ) ) |
63 |
|
dvn0 |
|- ( ( S C_ CC /\ F e. ( CC ^pm S ) ) -> ( ( S Dn F ) ` 0 ) = F ) |
64 |
59 62 63
|
syl2anc |
|- ( ph -> ( ( S Dn F ) ` 0 ) = F ) |
65 |
64
|
oveq2d |
|- ( ph -> ( S Tayl ( ( S Dn F ) ` 0 ) ) = ( S Tayl F ) ) |
66 |
4
|
nn0cnd |
|- ( ph -> M e. CC ) |
67 |
66
|
subid1d |
|- ( ph -> ( M - 0 ) = M ) |
68 |
67
|
oveq2d |
|- ( ph -> ( N + ( M - 0 ) ) = ( N + M ) ) |
69 |
|
eqidd |
|- ( ph -> B = B ) |
70 |
65 68 69
|
oveq123d |
|- ( ph -> ( ( N + ( M - 0 ) ) ( S Tayl ( ( S Dn F ) ` 0 ) ) B ) = ( ( N + M ) ( S Tayl F ) B ) ) |
71 |
57 70
|
eqtr4d |
|- ( ph -> ( ( CC Dn ( ( N + M ) ( S Tayl F ) B ) ) ` 0 ) = ( ( N + ( M - 0 ) ) ( S Tayl ( ( S Dn F ) ` 0 ) ) B ) ) |
72 |
71
|
a1i |
|- ( M e. ( ZZ>= ` 0 ) -> ( ph -> ( ( CC Dn ( ( N + M ) ( S Tayl F ) B ) ) ` 0 ) = ( ( N + ( M - 0 ) ) ( S Tayl ( ( S Dn F ) ` 0 ) ) B ) ) ) |
73 |
|
oveq2 |
|- ( ( ( CC Dn ( ( N + M ) ( S Tayl F ) B ) ) ` n ) = ( ( N + ( M - n ) ) ( S Tayl ( ( S Dn F ) ` n ) ) B ) -> ( CC _D ( ( CC Dn ( ( N + M ) ( S Tayl F ) B ) ) ` n ) ) = ( CC _D ( ( N + ( M - n ) ) ( S Tayl ( ( S Dn F ) ` n ) ) B ) ) ) |
74 |
|
ssidd |
|- ( ( ph /\ n e. ( 0 ..^ M ) ) -> CC C_ CC ) |
75 |
55
|
adantr |
|- ( ( ph /\ n e. ( 0 ..^ M ) ) -> ( ( N + M ) ( S Tayl F ) B ) e. ( CC ^pm CC ) ) |
76 |
|
elfzouz |
|- ( n e. ( 0 ..^ M ) -> n e. ( ZZ>= ` 0 ) ) |
77 |
76
|
adantl |
|- ( ( ph /\ n e. ( 0 ..^ M ) ) -> n e. ( ZZ>= ` 0 ) ) |
78 |
77 7
|
eleqtrrdi |
|- ( ( ph /\ n e. ( 0 ..^ M ) ) -> n e. NN0 ) |
79 |
|
dvnp1 |
|- ( ( CC C_ CC /\ ( ( N + M ) ( S Tayl F ) B ) e. ( CC ^pm CC ) /\ n e. NN0 ) -> ( ( CC Dn ( ( N + M ) ( S Tayl F ) B ) ) ` ( n + 1 ) ) = ( CC _D ( ( CC Dn ( ( N + M ) ( S Tayl F ) B ) ) ` n ) ) ) |
80 |
74 75 78 79
|
syl3anc |
|- ( ( ph /\ n e. ( 0 ..^ M ) ) -> ( ( CC Dn ( ( N + M ) ( S Tayl F ) B ) ) ` ( n + 1 ) ) = ( CC _D ( ( CC Dn ( ( N + M ) ( S Tayl F ) B ) ) ` n ) ) ) |
81 |
1
|
adantr |
|- ( ( ph /\ n e. ( 0 ..^ M ) ) -> S e. { RR , CC } ) |
82 |
62
|
adantr |
|- ( ( ph /\ n e. ( 0 ..^ M ) ) -> F e. ( CC ^pm S ) ) |
83 |
|
dvnf |
|- ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) /\ n e. NN0 ) -> ( ( S Dn F ) ` n ) : dom ( ( S Dn F ) ` n ) --> CC ) |
84 |
81 82 78 83
|
syl3anc |
|- ( ( ph /\ n e. ( 0 ..^ M ) ) -> ( ( S Dn F ) ` n ) : dom ( ( S Dn F ) ` n ) --> CC ) |
85 |
|
dvnbss |
|- ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) /\ n e. NN0 ) -> dom ( ( S Dn F ) ` n ) C_ dom F ) |
86 |
81 82 78 85
|
syl3anc |
|- ( ( ph /\ n e. ( 0 ..^ M ) ) -> dom ( ( S Dn F ) ` n ) C_ dom F ) |
87 |
2
|
fdmd |
|- ( ph -> dom F = A ) |
88 |
87
|
adantr |
|- ( ( ph /\ n e. ( 0 ..^ M ) ) -> dom F = A ) |
89 |
86 88
|
sseqtrd |
|- ( ( ph /\ n e. ( 0 ..^ M ) ) -> dom ( ( S Dn F ) ` n ) C_ A ) |
90 |
3
|
adantr |
|- ( ( ph /\ n e. ( 0 ..^ M ) ) -> A C_ S ) |
91 |
89 90
|
sstrd |
|- ( ( ph /\ n e. ( 0 ..^ M ) ) -> dom ( ( S Dn F ) ` n ) C_ S ) |
92 |
5
|
adantr |
|- ( ( ph /\ n e. ( 0 ..^ M ) ) -> N e. NN0 ) |
93 |
|
fzofzp1 |
|- ( n e. ( 0 ..^ M ) -> ( n + 1 ) e. ( 0 ... M ) ) |
94 |
93
|
adantl |
|- ( ( ph /\ n e. ( 0 ..^ M ) ) -> ( n + 1 ) e. ( 0 ... M ) ) |
95 |
|
fznn0sub |
|- ( ( n + 1 ) e. ( 0 ... M ) -> ( M - ( n + 1 ) ) e. NN0 ) |
96 |
94 95
|
syl |
|- ( ( ph /\ n e. ( 0 ..^ M ) ) -> ( M - ( n + 1 ) ) e. NN0 ) |
97 |
92 96
|
nn0addcld |
|- ( ( ph /\ n e. ( 0 ..^ M ) ) -> ( N + ( M - ( n + 1 ) ) ) e. NN0 ) |
98 |
6
|
adantr |
|- ( ( ph /\ n e. ( 0 ..^ M ) ) -> B e. dom ( ( S Dn F ) ` ( N + M ) ) ) |
99 |
|
elfzofz |
|- ( n e. ( 0 ..^ M ) -> n e. ( 0 ... M ) ) |
100 |
99
|
adantl |
|- ( ( ph /\ n e. ( 0 ..^ M ) ) -> n e. ( 0 ... M ) ) |
101 |
|
fznn0sub |
|- ( n e. ( 0 ... M ) -> ( M - n ) e. NN0 ) |
102 |
100 101
|
syl |
|- ( ( ph /\ n e. ( 0 ..^ M ) ) -> ( M - n ) e. NN0 ) |
103 |
92 102
|
nn0addcld |
|- ( ( ph /\ n e. ( 0 ..^ M ) ) -> ( N + ( M - n ) ) e. NN0 ) |
104 |
|
dvnadd |
|- ( ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) ) /\ ( n e. NN0 /\ ( N + ( M - n ) ) e. NN0 ) ) -> ( ( S Dn ( ( S Dn F ) ` n ) ) ` ( N + ( M - n ) ) ) = ( ( S Dn F ) ` ( n + ( N + ( M - n ) ) ) ) ) |
105 |
81 82 78 103 104
|
syl22anc |
|- ( ( ph /\ n e. ( 0 ..^ M ) ) -> ( ( S Dn ( ( S Dn F ) ` n ) ) ` ( N + ( M - n ) ) ) = ( ( S Dn F ) ` ( n + ( N + ( M - n ) ) ) ) ) |
106 |
5
|
nn0cnd |
|- ( ph -> N e. CC ) |
107 |
106
|
adantr |
|- ( ( ph /\ n e. ( 0 ..^ M ) ) -> N e. CC ) |
108 |
96
|
nn0cnd |
|- ( ( ph /\ n e. ( 0 ..^ M ) ) -> ( M - ( n + 1 ) ) e. CC ) |
109 |
|
1cnd |
|- ( ( ph /\ n e. ( 0 ..^ M ) ) -> 1 e. CC ) |
110 |
107 108 109
|
addassd |
|- ( ( ph /\ n e. ( 0 ..^ M ) ) -> ( ( N + ( M - ( n + 1 ) ) ) + 1 ) = ( N + ( ( M - ( n + 1 ) ) + 1 ) ) ) |
111 |
66
|
adantr |
|- ( ( ph /\ n e. ( 0 ..^ M ) ) -> M e. CC ) |
112 |
78
|
nn0cnd |
|- ( ( ph /\ n e. ( 0 ..^ M ) ) -> n e. CC ) |
113 |
111 112 109
|
nppcan2d |
|- ( ( ph /\ n e. ( 0 ..^ M ) ) -> ( ( M - ( n + 1 ) ) + 1 ) = ( M - n ) ) |
114 |
113
|
oveq2d |
|- ( ( ph /\ n e. ( 0 ..^ M ) ) -> ( N + ( ( M - ( n + 1 ) ) + 1 ) ) = ( N + ( M - n ) ) ) |
115 |
110 114
|
eqtrd |
|- ( ( ph /\ n e. ( 0 ..^ M ) ) -> ( ( N + ( M - ( n + 1 ) ) ) + 1 ) = ( N + ( M - n ) ) ) |
116 |
115
|
fveq2d |
|- ( ( ph /\ n e. ( 0 ..^ M ) ) -> ( ( S Dn ( ( S Dn F ) ` n ) ) ` ( ( N + ( M - ( n + 1 ) ) ) + 1 ) ) = ( ( S Dn ( ( S Dn F ) ` n ) ) ` ( N + ( M - n ) ) ) ) |
117 |
112 111
|
pncan3d |
|- ( ( ph /\ n e. ( 0 ..^ M ) ) -> ( n + ( M - n ) ) = M ) |
118 |
117
|
oveq2d |
|- ( ( ph /\ n e. ( 0 ..^ M ) ) -> ( N + ( n + ( M - n ) ) ) = ( N + M ) ) |
119 |
111 112
|
subcld |
|- ( ( ph /\ n e. ( 0 ..^ M ) ) -> ( M - n ) e. CC ) |
120 |
107 112 119
|
add12d |
|- ( ( ph /\ n e. ( 0 ..^ M ) ) -> ( N + ( n + ( M - n ) ) ) = ( n + ( N + ( M - n ) ) ) ) |
121 |
118 120
|
eqtr3d |
|- ( ( ph /\ n e. ( 0 ..^ M ) ) -> ( N + M ) = ( n + ( N + ( M - n ) ) ) ) |
122 |
121
|
fveq2d |
|- ( ( ph /\ n e. ( 0 ..^ M ) ) -> ( ( S Dn F ) ` ( N + M ) ) = ( ( S Dn F ) ` ( n + ( N + ( M - n ) ) ) ) ) |
123 |
105 116 122
|
3eqtr4d |
|- ( ( ph /\ n e. ( 0 ..^ M ) ) -> ( ( S Dn ( ( S Dn F ) ` n ) ) ` ( ( N + ( M - ( n + 1 ) ) ) + 1 ) ) = ( ( S Dn F ) ` ( N + M ) ) ) |
124 |
123
|
dmeqd |
|- ( ( ph /\ n e. ( 0 ..^ M ) ) -> dom ( ( S Dn ( ( S Dn F ) ` n ) ) ` ( ( N + ( M - ( n + 1 ) ) ) + 1 ) ) = dom ( ( S Dn F ) ` ( N + M ) ) ) |
125 |
98 124
|
eleqtrrd |
|- ( ( ph /\ n e. ( 0 ..^ M ) ) -> B e. dom ( ( S Dn ( ( S Dn F ) ` n ) ) ` ( ( N + ( M - ( n + 1 ) ) ) + 1 ) ) ) |
126 |
81 84 91 97 125
|
dvtaylp |
|- ( ( ph /\ n e. ( 0 ..^ M ) ) -> ( CC _D ( ( ( N + ( M - ( n + 1 ) ) ) + 1 ) ( S Tayl ( ( S Dn F ) ` n ) ) B ) ) = ( ( N + ( M - ( n + 1 ) ) ) ( S Tayl ( S _D ( ( S Dn F ) ` n ) ) ) B ) ) |
127 |
115
|
oveq1d |
|- ( ( ph /\ n e. ( 0 ..^ M ) ) -> ( ( ( N + ( M - ( n + 1 ) ) ) + 1 ) ( S Tayl ( ( S Dn F ) ` n ) ) B ) = ( ( N + ( M - n ) ) ( S Tayl ( ( S Dn F ) ` n ) ) B ) ) |
128 |
127
|
oveq2d |
|- ( ( ph /\ n e. ( 0 ..^ M ) ) -> ( CC _D ( ( ( N + ( M - ( n + 1 ) ) ) + 1 ) ( S Tayl ( ( S Dn F ) ` n ) ) B ) ) = ( CC _D ( ( N + ( M - n ) ) ( S Tayl ( ( S Dn F ) ` n ) ) B ) ) ) |
129 |
59
|
adantr |
|- ( ( ph /\ n e. ( 0 ..^ M ) ) -> S C_ CC ) |
130 |
|
dvnp1 |
|- ( ( S C_ CC /\ F e. ( CC ^pm S ) /\ n e. NN0 ) -> ( ( S Dn F ) ` ( n + 1 ) ) = ( S _D ( ( S Dn F ) ` n ) ) ) |
131 |
129 82 78 130
|
syl3anc |
|- ( ( ph /\ n e. ( 0 ..^ M ) ) -> ( ( S Dn F ) ` ( n + 1 ) ) = ( S _D ( ( S Dn F ) ` n ) ) ) |
132 |
131
|
oveq2d |
|- ( ( ph /\ n e. ( 0 ..^ M ) ) -> ( S Tayl ( ( S Dn F ) ` ( n + 1 ) ) ) = ( S Tayl ( S _D ( ( S Dn F ) ` n ) ) ) ) |
133 |
132
|
eqcomd |
|- ( ( ph /\ n e. ( 0 ..^ M ) ) -> ( S Tayl ( S _D ( ( S Dn F ) ` n ) ) ) = ( S Tayl ( ( S Dn F ) ` ( n + 1 ) ) ) ) |
134 |
133
|
oveqd |
|- ( ( ph /\ n e. ( 0 ..^ M ) ) -> ( ( N + ( M - ( n + 1 ) ) ) ( S Tayl ( S _D ( ( S Dn F ) ` n ) ) ) B ) = ( ( N + ( M - ( n + 1 ) ) ) ( S Tayl ( ( S Dn F ) ` ( n + 1 ) ) ) B ) ) |
135 |
126 128 134
|
3eqtr3rd |
|- ( ( ph /\ n e. ( 0 ..^ M ) ) -> ( ( N + ( M - ( n + 1 ) ) ) ( S Tayl ( ( S Dn F ) ` ( n + 1 ) ) ) B ) = ( CC _D ( ( N + ( M - n ) ) ( S Tayl ( ( S Dn F ) ` n ) ) B ) ) ) |
136 |
80 135
|
eqeq12d |
|- ( ( ph /\ n e. ( 0 ..^ M ) ) -> ( ( ( CC Dn ( ( N + M ) ( S Tayl F ) B ) ) ` ( n + 1 ) ) = ( ( N + ( M - ( n + 1 ) ) ) ( S Tayl ( ( S Dn F ) ` ( n + 1 ) ) ) B ) <-> ( CC _D ( ( CC Dn ( ( N + M ) ( S Tayl F ) B ) ) ` n ) ) = ( CC _D ( ( N + ( M - n ) ) ( S Tayl ( ( S Dn F ) ` n ) ) B ) ) ) ) |
137 |
73 136
|
syl5ibr |
|- ( ( ph /\ n e. ( 0 ..^ M ) ) -> ( ( ( CC Dn ( ( N + M ) ( S Tayl F ) B ) ) ` n ) = ( ( N + ( M - n ) ) ( S Tayl ( ( S Dn F ) ` n ) ) B ) -> ( ( CC Dn ( ( N + M ) ( S Tayl F ) B ) ) ` ( n + 1 ) ) = ( ( N + ( M - ( n + 1 ) ) ) ( S Tayl ( ( S Dn F ) ` ( n + 1 ) ) ) B ) ) ) |
138 |
137
|
expcom |
|- ( n e. ( 0 ..^ M ) -> ( ph -> ( ( ( CC Dn ( ( N + M ) ( S Tayl F ) B ) ) ` n ) = ( ( N + ( M - n ) ) ( S Tayl ( ( S Dn F ) ` n ) ) B ) -> ( ( CC Dn ( ( N + M ) ( S Tayl F ) B ) ) ` ( n + 1 ) ) = ( ( N + ( M - ( n + 1 ) ) ) ( S Tayl ( ( S Dn F ) ` ( n + 1 ) ) ) B ) ) ) ) |
139 |
138
|
a2d |
|- ( n e. ( 0 ..^ M ) -> ( ( ph -> ( ( CC Dn ( ( N + M ) ( S Tayl F ) B ) ) ` n ) = ( ( N + ( M - n ) ) ( S Tayl ( ( S Dn F ) ` n ) ) B ) ) -> ( ph -> ( ( CC Dn ( ( N + M ) ( S Tayl F ) B ) ) ` ( n + 1 ) ) = ( ( N + ( M - ( n + 1 ) ) ) ( S Tayl ( ( S Dn F ) ` ( n + 1 ) ) ) B ) ) ) ) |
140 |
19 28 37 46 72 139
|
fzind2 |
|- ( M e. ( 0 ... M ) -> ( ph -> ( ( CC Dn ( ( N + M ) ( S Tayl F ) B ) ) ` M ) = ( ( N + ( M - M ) ) ( S Tayl ( ( S Dn F ) ` M ) ) B ) ) ) |
141 |
10 140
|
mpcom |
|- ( ph -> ( ( CC Dn ( ( N + M ) ( S Tayl F ) B ) ) ` M ) = ( ( N + ( M - M ) ) ( S Tayl ( ( S Dn F ) ` M ) ) B ) ) |
142 |
66
|
subidd |
|- ( ph -> ( M - M ) = 0 ) |
143 |
142
|
oveq2d |
|- ( ph -> ( N + ( M - M ) ) = ( N + 0 ) ) |
144 |
106
|
addid1d |
|- ( ph -> ( N + 0 ) = N ) |
145 |
143 144
|
eqtrd |
|- ( ph -> ( N + ( M - M ) ) = N ) |
146 |
145
|
oveq1d |
|- ( ph -> ( ( N + ( M - M ) ) ( S Tayl ( ( S Dn F ) ` M ) ) B ) = ( N ( S Tayl ( ( S Dn F ) ` M ) ) B ) ) |
147 |
141 146
|
eqtrd |
|- ( ph -> ( ( CC Dn ( ( N + M ) ( S Tayl F ) B ) ) ` M ) = ( N ( S Tayl ( ( S Dn F ) ` M ) ) B ) ) |