Step |
Hyp |
Ref |
Expression |
1 |
|
nn0cn |
|- ( N e. NN0 -> N e. CC ) |
2 |
|
1cnd |
|- ( N e. NN0 -> 1 e. CC ) |
3 |
1 2 2
|
subsub4d |
|- ( N e. NN0 -> ( ( N - 1 ) - 1 ) = ( N - ( 1 + 1 ) ) ) |
4 |
|
1p1e2 |
|- ( 1 + 1 ) = 2 |
5 |
4
|
oveq2i |
|- ( N - ( 1 + 1 ) ) = ( N - 2 ) |
6 |
3 5
|
eqtr2di |
|- ( N e. NN0 -> ( N - 2 ) = ( ( N - 1 ) - 1 ) ) |
7 |
6
|
3ad2ant1 |
|- ( ( N e. NN0 /\ N =/= 0 /\ N =/= 1 ) -> ( N - 2 ) = ( ( N - 1 ) - 1 ) ) |
8 |
|
3simpa |
|- ( ( N e. NN0 /\ N =/= 0 /\ N =/= 1 ) -> ( N e. NN0 /\ N =/= 0 ) ) |
9 |
|
elnnne0 |
|- ( N e. NN <-> ( N e. NN0 /\ N =/= 0 ) ) |
10 |
8 9
|
sylibr |
|- ( ( N e. NN0 /\ N =/= 0 /\ N =/= 1 ) -> N e. NN ) |
11 |
|
nnm1nn0 |
|- ( N e. NN -> ( N - 1 ) e. NN0 ) |
12 |
10 11
|
syl |
|- ( ( N e. NN0 /\ N =/= 0 /\ N =/= 1 ) -> ( N - 1 ) e. NN0 ) |
13 |
1 2
|
subeq0ad |
|- ( N e. NN0 -> ( ( N - 1 ) = 0 <-> N = 1 ) ) |
14 |
13
|
biimpd |
|- ( N e. NN0 -> ( ( N - 1 ) = 0 -> N = 1 ) ) |
15 |
14
|
necon3d |
|- ( N e. NN0 -> ( N =/= 1 -> ( N - 1 ) =/= 0 ) ) |
16 |
15
|
imp |
|- ( ( N e. NN0 /\ N =/= 1 ) -> ( N - 1 ) =/= 0 ) |
17 |
16
|
3adant2 |
|- ( ( N e. NN0 /\ N =/= 0 /\ N =/= 1 ) -> ( N - 1 ) =/= 0 ) |
18 |
|
elnnne0 |
|- ( ( N - 1 ) e. NN <-> ( ( N - 1 ) e. NN0 /\ ( N - 1 ) =/= 0 ) ) |
19 |
12 17 18
|
sylanbrc |
|- ( ( N e. NN0 /\ N =/= 0 /\ N =/= 1 ) -> ( N - 1 ) e. NN ) |
20 |
|
nnm1nn0 |
|- ( ( N - 1 ) e. NN -> ( ( N - 1 ) - 1 ) e. NN0 ) |
21 |
19 20
|
syl |
|- ( ( N e. NN0 /\ N =/= 0 /\ N =/= 1 ) -> ( ( N - 1 ) - 1 ) e. NN0 ) |
22 |
7 21
|
eqeltrd |
|- ( ( N e. NN0 /\ N =/= 0 /\ N =/= 1 ) -> ( N - 2 ) e. NN0 ) |
23 |
|
2nn0 |
|- 2 e. NN0 |
24 |
23
|
jctl |
|- ( N e. NN0 -> ( 2 e. NN0 /\ N e. NN0 ) ) |
25 |
24
|
3ad2ant1 |
|- ( ( N e. NN0 /\ N =/= 0 /\ N =/= 1 ) -> ( 2 e. NN0 /\ N e. NN0 ) ) |
26 |
|
nn0sub |
|- ( ( 2 e. NN0 /\ N e. NN0 ) -> ( 2 <_ N <-> ( N - 2 ) e. NN0 ) ) |
27 |
25 26
|
syl |
|- ( ( N e. NN0 /\ N =/= 0 /\ N =/= 1 ) -> ( 2 <_ N <-> ( N - 2 ) e. NN0 ) ) |
28 |
22 27
|
mpbird |
|- ( ( N e. NN0 /\ N =/= 0 /\ N =/= 1 ) -> 2 <_ N ) |