Step |
Hyp |
Ref |
Expression |
1 |
|
elfzo0 |
|- ( K e. ( 0 ..^ N ) <-> ( K e. NN0 /\ N e. NN /\ K < N ) ) |
2 |
|
nnz |
|- ( N e. NN -> N e. ZZ ) |
3 |
2
|
adantr |
|- ( ( N e. NN /\ K e. NN0 ) -> N e. ZZ ) |
4 |
|
nn0z |
|- ( K e. NN0 -> K e. ZZ ) |
5 |
4
|
adantl |
|- ( ( N e. NN /\ K e. NN0 ) -> K e. ZZ ) |
6 |
3 5
|
zsubcld |
|- ( ( N e. NN /\ K e. NN0 ) -> ( N - K ) e. ZZ ) |
7 |
6
|
ancoms |
|- ( ( K e. NN0 /\ N e. NN ) -> ( N - K ) e. ZZ ) |
8 |
|
peano2zm |
|- ( ( N - K ) e. ZZ -> ( ( N - K ) - 1 ) e. ZZ ) |
9 |
7 8
|
syl |
|- ( ( K e. NN0 /\ N e. NN ) -> ( ( N - K ) - 1 ) e. ZZ ) |
10 |
9
|
3adant3 |
|- ( ( K e. NN0 /\ N e. NN /\ K < N ) -> ( ( N - K ) - 1 ) e. ZZ ) |
11 |
|
simp3 |
|- ( ( K e. NN0 /\ N e. NN /\ K < N ) -> K < N ) |
12 |
4 2
|
anim12i |
|- ( ( K e. NN0 /\ N e. NN ) -> ( K e. ZZ /\ N e. ZZ ) ) |
13 |
12
|
3adant3 |
|- ( ( K e. NN0 /\ N e. NN /\ K < N ) -> ( K e. ZZ /\ N e. ZZ ) ) |
14 |
|
znnsub |
|- ( ( K e. ZZ /\ N e. ZZ ) -> ( K < N <-> ( N - K ) e. NN ) ) |
15 |
13 14
|
syl |
|- ( ( K e. NN0 /\ N e. NN /\ K < N ) -> ( K < N <-> ( N - K ) e. NN ) ) |
16 |
11 15
|
mpbid |
|- ( ( K e. NN0 /\ N e. NN /\ K < N ) -> ( N - K ) e. NN ) |
17 |
|
nnm1ge0 |
|- ( ( N - K ) e. NN -> 0 <_ ( ( N - K ) - 1 ) ) |
18 |
16 17
|
syl |
|- ( ( K e. NN0 /\ N e. NN /\ K < N ) -> 0 <_ ( ( N - K ) - 1 ) ) |
19 |
|
elnn0z |
|- ( ( ( N - K ) - 1 ) e. NN0 <-> ( ( ( N - K ) - 1 ) e. ZZ /\ 0 <_ ( ( N - K ) - 1 ) ) ) |
20 |
10 18 19
|
sylanbrc |
|- ( ( K e. NN0 /\ N e. NN /\ K < N ) -> ( ( N - K ) - 1 ) e. NN0 ) |
21 |
|
simp2 |
|- ( ( K e. NN0 /\ N e. NN /\ K < N ) -> N e. NN ) |
22 |
|
nncn |
|- ( N e. NN -> N e. CC ) |
23 |
22
|
adantl |
|- ( ( K e. NN0 /\ N e. NN ) -> N e. CC ) |
24 |
|
nn0cn |
|- ( K e. NN0 -> K e. CC ) |
25 |
24
|
adantr |
|- ( ( K e. NN0 /\ N e. NN ) -> K e. CC ) |
26 |
|
1cnd |
|- ( ( K e. NN0 /\ N e. NN ) -> 1 e. CC ) |
27 |
23 25 26
|
subsub4d |
|- ( ( K e. NN0 /\ N e. NN ) -> ( ( N - K ) - 1 ) = ( N - ( K + 1 ) ) ) |
28 |
|
nn0p1gt0 |
|- ( K e. NN0 -> 0 < ( K + 1 ) ) |
29 |
28
|
adantr |
|- ( ( K e. NN0 /\ N e. NN ) -> 0 < ( K + 1 ) ) |
30 |
|
nn0re |
|- ( K e. NN0 -> K e. RR ) |
31 |
|
peano2re |
|- ( K e. RR -> ( K + 1 ) e. RR ) |
32 |
30 31
|
syl |
|- ( K e. NN0 -> ( K + 1 ) e. RR ) |
33 |
|
nnre |
|- ( N e. NN -> N e. RR ) |
34 |
|
ltsubpos |
|- ( ( ( K + 1 ) e. RR /\ N e. RR ) -> ( 0 < ( K + 1 ) <-> ( N - ( K + 1 ) ) < N ) ) |
35 |
32 33 34
|
syl2an |
|- ( ( K e. NN0 /\ N e. NN ) -> ( 0 < ( K + 1 ) <-> ( N - ( K + 1 ) ) < N ) ) |
36 |
29 35
|
mpbid |
|- ( ( K e. NN0 /\ N e. NN ) -> ( N - ( K + 1 ) ) < N ) |
37 |
27 36
|
eqbrtrd |
|- ( ( K e. NN0 /\ N e. NN ) -> ( ( N - K ) - 1 ) < N ) |
38 |
37
|
3adant3 |
|- ( ( K e. NN0 /\ N e. NN /\ K < N ) -> ( ( N - K ) - 1 ) < N ) |
39 |
|
elfzo0 |
|- ( ( ( N - K ) - 1 ) e. ( 0 ..^ N ) <-> ( ( ( N - K ) - 1 ) e. NN0 /\ N e. NN /\ ( ( N - K ) - 1 ) < N ) ) |
40 |
20 21 38 39
|
syl3anbrc |
|- ( ( K e. NN0 /\ N e. NN /\ K < N ) -> ( ( N - K ) - 1 ) e. ( 0 ..^ N ) ) |
41 |
1 40
|
sylbi |
|- ( K e. ( 0 ..^ N ) -> ( ( N - K ) - 1 ) e. ( 0 ..^ N ) ) |