Step |
Hyp |
Ref |
Expression |
1 |
|
2t1e2 |
|- ( 2 x. 1 ) = 2 |
2 |
|
df-2 |
|- 2 = ( 1 + 1 ) |
3 |
1 2
|
eqtri |
|- ( 2 x. 1 ) = ( 1 + 1 ) |
4 |
3
|
oveq2i |
|- ( ( 2 x. N ) + ( 2 x. 1 ) ) = ( ( 2 x. N ) + ( 1 + 1 ) ) |
5 |
|
nn0cn |
|- ( N e. NN0 -> N e. CC ) |
6 |
|
2cn |
|- 2 e. CC |
7 |
|
ax-1cn |
|- 1 e. CC |
8 |
|
adddi |
|- ( ( 2 e. CC /\ N e. CC /\ 1 e. CC ) -> ( 2 x. ( N + 1 ) ) = ( ( 2 x. N ) + ( 2 x. 1 ) ) ) |
9 |
6 7 8
|
mp3an13 |
|- ( N e. CC -> ( 2 x. ( N + 1 ) ) = ( ( 2 x. N ) + ( 2 x. 1 ) ) ) |
10 |
5 9
|
syl |
|- ( N e. NN0 -> ( 2 x. ( N + 1 ) ) = ( ( 2 x. N ) + ( 2 x. 1 ) ) ) |
11 |
|
2nn0 |
|- 2 e. NN0 |
12 |
|
nn0mulcl |
|- ( ( 2 e. NN0 /\ N e. NN0 ) -> ( 2 x. N ) e. NN0 ) |
13 |
11 12
|
mpan |
|- ( N e. NN0 -> ( 2 x. N ) e. NN0 ) |
14 |
13
|
nn0cnd |
|- ( N e. NN0 -> ( 2 x. N ) e. CC ) |
15 |
|
addass |
|- ( ( ( 2 x. N ) e. CC /\ 1 e. CC /\ 1 e. CC ) -> ( ( ( 2 x. N ) + 1 ) + 1 ) = ( ( 2 x. N ) + ( 1 + 1 ) ) ) |
16 |
7 7 15
|
mp3an23 |
|- ( ( 2 x. N ) e. CC -> ( ( ( 2 x. N ) + 1 ) + 1 ) = ( ( 2 x. N ) + ( 1 + 1 ) ) ) |
17 |
14 16
|
syl |
|- ( N e. NN0 -> ( ( ( 2 x. N ) + 1 ) + 1 ) = ( ( 2 x. N ) + ( 1 + 1 ) ) ) |
18 |
4 10 17
|
3eqtr4a |
|- ( N e. NN0 -> ( 2 x. ( N + 1 ) ) = ( ( ( 2 x. N ) + 1 ) + 1 ) ) |
19 |
18
|
oveq1d |
|- ( N e. NN0 -> ( ( 2 x. ( N + 1 ) ) _C ( N + 1 ) ) = ( ( ( ( 2 x. N ) + 1 ) + 1 ) _C ( N + 1 ) ) ) |
20 |
|
peano2nn0 |
|- ( ( 2 x. N ) e. NN0 -> ( ( 2 x. N ) + 1 ) e. NN0 ) |
21 |
13 20
|
syl |
|- ( N e. NN0 -> ( ( 2 x. N ) + 1 ) e. NN0 ) |
22 |
|
nn0p1nn |
|- ( N e. NN0 -> ( N + 1 ) e. NN ) |
23 |
22
|
nnzd |
|- ( N e. NN0 -> ( N + 1 ) e. ZZ ) |
24 |
|
bcpasc |
|- ( ( ( ( 2 x. N ) + 1 ) e. NN0 /\ ( N + 1 ) e. ZZ ) -> ( ( ( ( 2 x. N ) + 1 ) _C ( N + 1 ) ) + ( ( ( 2 x. N ) + 1 ) _C ( ( N + 1 ) - 1 ) ) ) = ( ( ( ( 2 x. N ) + 1 ) + 1 ) _C ( N + 1 ) ) ) |
25 |
21 23 24
|
syl2anc |
|- ( N e. NN0 -> ( ( ( ( 2 x. N ) + 1 ) _C ( N + 1 ) ) + ( ( ( 2 x. N ) + 1 ) _C ( ( N + 1 ) - 1 ) ) ) = ( ( ( ( 2 x. N ) + 1 ) + 1 ) _C ( N + 1 ) ) ) |
26 |
19 25
|
eqtr4d |
|- ( N e. NN0 -> ( ( 2 x. ( N + 1 ) ) _C ( N + 1 ) ) = ( ( ( ( 2 x. N ) + 1 ) _C ( N + 1 ) ) + ( ( ( 2 x. N ) + 1 ) _C ( ( N + 1 ) - 1 ) ) ) ) |
27 |
|
nn0z |
|- ( N e. NN0 -> N e. ZZ ) |
28 |
|
bccl |
|- ( ( ( 2 x. N ) e. NN0 /\ N e. ZZ ) -> ( ( 2 x. N ) _C N ) e. NN0 ) |
29 |
13 27 28
|
syl2anc |
|- ( N e. NN0 -> ( ( 2 x. N ) _C N ) e. NN0 ) |
30 |
29
|
nn0cnd |
|- ( N e. NN0 -> ( ( 2 x. N ) _C N ) e. CC ) |
31 |
|
2cnd |
|- ( N e. NN0 -> 2 e. CC ) |
32 |
21
|
nn0red |
|- ( N e. NN0 -> ( ( 2 x. N ) + 1 ) e. RR ) |
33 |
32 22
|
nndivred |
|- ( N e. NN0 -> ( ( ( 2 x. N ) + 1 ) / ( N + 1 ) ) e. RR ) |
34 |
33
|
recnd |
|- ( N e. NN0 -> ( ( ( 2 x. N ) + 1 ) / ( N + 1 ) ) e. CC ) |
35 |
30 31 34
|
mul12d |
|- ( N e. NN0 -> ( ( ( 2 x. N ) _C N ) x. ( 2 x. ( ( ( 2 x. N ) + 1 ) / ( N + 1 ) ) ) ) = ( 2 x. ( ( ( 2 x. N ) _C N ) x. ( ( ( 2 x. N ) + 1 ) / ( N + 1 ) ) ) ) ) |
36 |
|
1cnd |
|- ( N e. NN0 -> 1 e. CC ) |
37 |
14 36 5
|
addsubd |
|- ( N e. NN0 -> ( ( ( 2 x. N ) + 1 ) - N ) = ( ( ( 2 x. N ) - N ) + 1 ) ) |
38 |
5
|
2timesd |
|- ( N e. NN0 -> ( 2 x. N ) = ( N + N ) ) |
39 |
5 5 38
|
mvrladdd |
|- ( N e. NN0 -> ( ( 2 x. N ) - N ) = N ) |
40 |
39
|
oveq1d |
|- ( N e. NN0 -> ( ( ( 2 x. N ) - N ) + 1 ) = ( N + 1 ) ) |
41 |
37 40
|
eqtr2d |
|- ( N e. NN0 -> ( N + 1 ) = ( ( ( 2 x. N ) + 1 ) - N ) ) |
42 |
41
|
oveq2d |
|- ( N e. NN0 -> ( ( ( 2 x. N ) + 1 ) / ( N + 1 ) ) = ( ( ( 2 x. N ) + 1 ) / ( ( ( 2 x. N ) + 1 ) - N ) ) ) |
43 |
42
|
oveq2d |
|- ( N e. NN0 -> ( ( ( 2 x. N ) _C N ) x. ( ( ( 2 x. N ) + 1 ) / ( N + 1 ) ) ) = ( ( ( 2 x. N ) _C N ) x. ( ( ( 2 x. N ) + 1 ) / ( ( ( 2 x. N ) + 1 ) - N ) ) ) ) |
44 |
|
fzctr |
|- ( N e. NN0 -> N e. ( 0 ... ( 2 x. N ) ) ) |
45 |
|
bcp1n |
|- ( N e. ( 0 ... ( 2 x. N ) ) -> ( ( ( 2 x. N ) + 1 ) _C N ) = ( ( ( 2 x. N ) _C N ) x. ( ( ( 2 x. N ) + 1 ) / ( ( ( 2 x. N ) + 1 ) - N ) ) ) ) |
46 |
44 45
|
syl |
|- ( N e. NN0 -> ( ( ( 2 x. N ) + 1 ) _C N ) = ( ( ( 2 x. N ) _C N ) x. ( ( ( 2 x. N ) + 1 ) / ( ( ( 2 x. N ) + 1 ) - N ) ) ) ) |
47 |
43 46
|
eqtr4d |
|- ( N e. NN0 -> ( ( ( 2 x. N ) _C N ) x. ( ( ( 2 x. N ) + 1 ) / ( N + 1 ) ) ) = ( ( ( 2 x. N ) + 1 ) _C N ) ) |
48 |
47
|
oveq2d |
|- ( N e. NN0 -> ( 2 x. ( ( ( 2 x. N ) _C N ) x. ( ( ( 2 x. N ) + 1 ) / ( N + 1 ) ) ) ) = ( 2 x. ( ( ( 2 x. N ) + 1 ) _C N ) ) ) |
49 |
35 48
|
eqtrd |
|- ( N e. NN0 -> ( ( ( 2 x. N ) _C N ) x. ( 2 x. ( ( ( 2 x. N ) + 1 ) / ( N + 1 ) ) ) ) = ( 2 x. ( ( ( 2 x. N ) + 1 ) _C N ) ) ) |
50 |
|
bccmpl |
|- ( ( ( ( 2 x. N ) + 1 ) e. NN0 /\ ( N + 1 ) e. ZZ ) -> ( ( ( 2 x. N ) + 1 ) _C ( N + 1 ) ) = ( ( ( 2 x. N ) + 1 ) _C ( ( ( 2 x. N ) + 1 ) - ( N + 1 ) ) ) ) |
51 |
21 23 50
|
syl2anc |
|- ( N e. NN0 -> ( ( ( 2 x. N ) + 1 ) _C ( N + 1 ) ) = ( ( ( 2 x. N ) + 1 ) _C ( ( ( 2 x. N ) + 1 ) - ( N + 1 ) ) ) ) |
52 |
22
|
nncnd |
|- ( N e. NN0 -> ( N + 1 ) e. CC ) |
53 |
38
|
oveq1d |
|- ( N e. NN0 -> ( ( 2 x. N ) + 1 ) = ( ( N + N ) + 1 ) ) |
54 |
5 5 36
|
addassd |
|- ( N e. NN0 -> ( ( N + N ) + 1 ) = ( N + ( N + 1 ) ) ) |
55 |
53 54
|
eqtrd |
|- ( N e. NN0 -> ( ( 2 x. N ) + 1 ) = ( N + ( N + 1 ) ) ) |
56 |
5 52 55
|
mvrraddd |
|- ( N e. NN0 -> ( ( ( 2 x. N ) + 1 ) - ( N + 1 ) ) = N ) |
57 |
56
|
oveq2d |
|- ( N e. NN0 -> ( ( ( 2 x. N ) + 1 ) _C ( ( ( 2 x. N ) + 1 ) - ( N + 1 ) ) ) = ( ( ( 2 x. N ) + 1 ) _C N ) ) |
58 |
51 57
|
eqtrd |
|- ( N e. NN0 -> ( ( ( 2 x. N ) + 1 ) _C ( N + 1 ) ) = ( ( ( 2 x. N ) + 1 ) _C N ) ) |
59 |
|
pncan |
|- ( ( N e. CC /\ 1 e. CC ) -> ( ( N + 1 ) - 1 ) = N ) |
60 |
5 7 59
|
sylancl |
|- ( N e. NN0 -> ( ( N + 1 ) - 1 ) = N ) |
61 |
60
|
oveq2d |
|- ( N e. NN0 -> ( ( ( 2 x. N ) + 1 ) _C ( ( N + 1 ) - 1 ) ) = ( ( ( 2 x. N ) + 1 ) _C N ) ) |
62 |
58 61
|
oveq12d |
|- ( N e. NN0 -> ( ( ( ( 2 x. N ) + 1 ) _C ( N + 1 ) ) + ( ( ( 2 x. N ) + 1 ) _C ( ( N + 1 ) - 1 ) ) ) = ( ( ( ( 2 x. N ) + 1 ) _C N ) + ( ( ( 2 x. N ) + 1 ) _C N ) ) ) |
63 |
|
bccl |
|- ( ( ( ( 2 x. N ) + 1 ) e. NN0 /\ N e. ZZ ) -> ( ( ( 2 x. N ) + 1 ) _C N ) e. NN0 ) |
64 |
21 27 63
|
syl2anc |
|- ( N e. NN0 -> ( ( ( 2 x. N ) + 1 ) _C N ) e. NN0 ) |
65 |
64
|
nn0cnd |
|- ( N e. NN0 -> ( ( ( 2 x. N ) + 1 ) _C N ) e. CC ) |
66 |
65
|
2timesd |
|- ( N e. NN0 -> ( 2 x. ( ( ( 2 x. N ) + 1 ) _C N ) ) = ( ( ( ( 2 x. N ) + 1 ) _C N ) + ( ( ( 2 x. N ) + 1 ) _C N ) ) ) |
67 |
62 66
|
eqtr4d |
|- ( N e. NN0 -> ( ( ( ( 2 x. N ) + 1 ) _C ( N + 1 ) ) + ( ( ( 2 x. N ) + 1 ) _C ( ( N + 1 ) - 1 ) ) ) = ( 2 x. ( ( ( 2 x. N ) + 1 ) _C N ) ) ) |
68 |
49 67
|
eqtr4d |
|- ( N e. NN0 -> ( ( ( 2 x. N ) _C N ) x. ( 2 x. ( ( ( 2 x. N ) + 1 ) / ( N + 1 ) ) ) ) = ( ( ( ( 2 x. N ) + 1 ) _C ( N + 1 ) ) + ( ( ( 2 x. N ) + 1 ) _C ( ( N + 1 ) - 1 ) ) ) ) |
69 |
26 68
|
eqtr4d |
|- ( N e. NN0 -> ( ( 2 x. ( N + 1 ) ) _C ( N + 1 ) ) = ( ( ( 2 x. N ) _C N ) x. ( 2 x. ( ( ( 2 x. N ) + 1 ) / ( N + 1 ) ) ) ) ) |