Step |
Hyp |
Ref |
Expression |
1 |
|
elfzo0 |
|- ( A e. ( 0 ..^ N ) <-> ( A e. NN0 /\ N e. NN /\ A < N ) ) |
2 |
|
elfzo0 |
|- ( I e. ( 0 ..^ N ) <-> ( I e. NN0 /\ N e. NN /\ I < N ) ) |
3 |
|
nnre |
|- ( N e. NN -> N e. RR ) |
4 |
3
|
3ad2ant2 |
|- ( ( I e. NN0 /\ N e. NN /\ I < N ) -> N e. RR ) |
5 |
|
nn0re |
|- ( A e. NN0 -> A e. RR ) |
6 |
5
|
adantr |
|- ( ( A e. NN0 /\ A < N ) -> A e. RR ) |
7 |
|
resubcl |
|- ( ( N e. RR /\ A e. RR ) -> ( N - A ) e. RR ) |
8 |
4 6 7
|
syl2anr |
|- ( ( ( A e. NN0 /\ A < N ) /\ ( I e. NN0 /\ N e. NN /\ I < N ) ) -> ( N - A ) e. RR ) |
9 |
|
nn0re |
|- ( I e. NN0 -> I e. RR ) |
10 |
9
|
3ad2ant1 |
|- ( ( I e. NN0 /\ N e. NN /\ I < N ) -> I e. RR ) |
11 |
10
|
adantl |
|- ( ( ( A e. NN0 /\ A < N ) /\ ( I e. NN0 /\ N e. NN /\ I < N ) ) -> I e. RR ) |
12 |
|
lenlt |
|- ( ( ( N - A ) e. RR /\ I e. RR ) -> ( ( N - A ) <_ I <-> -. I < ( N - A ) ) ) |
13 |
12
|
bicomd |
|- ( ( ( N - A ) e. RR /\ I e. RR ) -> ( -. I < ( N - A ) <-> ( N - A ) <_ I ) ) |
14 |
8 11 13
|
syl2anc |
|- ( ( ( A e. NN0 /\ A < N ) /\ ( I e. NN0 /\ N e. NN /\ I < N ) ) -> ( -. I < ( N - A ) <-> ( N - A ) <_ I ) ) |
15 |
14
|
biimpa |
|- ( ( ( ( A e. NN0 /\ A < N ) /\ ( I e. NN0 /\ N e. NN /\ I < N ) ) /\ -. I < ( N - A ) ) -> ( N - A ) <_ I ) |
16 |
|
nnz |
|- ( N e. NN -> N e. ZZ ) |
17 |
16
|
3ad2ant2 |
|- ( ( I e. NN0 /\ N e. NN /\ I < N ) -> N e. ZZ ) |
18 |
|
nn0z |
|- ( A e. NN0 -> A e. ZZ ) |
19 |
18
|
adantr |
|- ( ( A e. NN0 /\ A < N ) -> A e. ZZ ) |
20 |
|
zsubcl |
|- ( ( N e. ZZ /\ A e. ZZ ) -> ( N - A ) e. ZZ ) |
21 |
17 19 20
|
syl2anr |
|- ( ( ( A e. NN0 /\ A < N ) /\ ( I e. NN0 /\ N e. NN /\ I < N ) ) -> ( N - A ) e. ZZ ) |
22 |
|
ltle |
|- ( ( A e. RR /\ N e. RR ) -> ( A < N -> A <_ N ) ) |
23 |
5 4 22
|
syl2an |
|- ( ( A e. NN0 /\ ( I e. NN0 /\ N e. NN /\ I < N ) ) -> ( A < N -> A <_ N ) ) |
24 |
23
|
impancom |
|- ( ( A e. NN0 /\ A < N ) -> ( ( I e. NN0 /\ N e. NN /\ I < N ) -> A <_ N ) ) |
25 |
24
|
imp |
|- ( ( ( A e. NN0 /\ A < N ) /\ ( I e. NN0 /\ N e. NN /\ I < N ) ) -> A <_ N ) |
26 |
|
subge0 |
|- ( ( N e. RR /\ A e. RR ) -> ( 0 <_ ( N - A ) <-> A <_ N ) ) |
27 |
4 6 26
|
syl2anr |
|- ( ( ( A e. NN0 /\ A < N ) /\ ( I e. NN0 /\ N e. NN /\ I < N ) ) -> ( 0 <_ ( N - A ) <-> A <_ N ) ) |
28 |
25 27
|
mpbird |
|- ( ( ( A e. NN0 /\ A < N ) /\ ( I e. NN0 /\ N e. NN /\ I < N ) ) -> 0 <_ ( N - A ) ) |
29 |
|
elnn0z |
|- ( ( N - A ) e. NN0 <-> ( ( N - A ) e. ZZ /\ 0 <_ ( N - A ) ) ) |
30 |
21 28 29
|
sylanbrc |
|- ( ( ( A e. NN0 /\ A < N ) /\ ( I e. NN0 /\ N e. NN /\ I < N ) ) -> ( N - A ) e. NN0 ) |
31 |
30
|
adantr |
|- ( ( ( ( A e. NN0 /\ A < N ) /\ ( I e. NN0 /\ N e. NN /\ I < N ) ) /\ -. I < ( N - A ) ) -> ( N - A ) e. NN0 ) |
32 |
|
simplr1 |
|- ( ( ( ( A e. NN0 /\ A < N ) /\ ( I e. NN0 /\ N e. NN /\ I < N ) ) /\ -. I < ( N - A ) ) -> I e. NN0 ) |
33 |
|
nn0sub |
|- ( ( ( N - A ) e. NN0 /\ I e. NN0 ) -> ( ( N - A ) <_ I <-> ( I - ( N - A ) ) e. NN0 ) ) |
34 |
31 32 33
|
syl2anc |
|- ( ( ( ( A e. NN0 /\ A < N ) /\ ( I e. NN0 /\ N e. NN /\ I < N ) ) /\ -. I < ( N - A ) ) -> ( ( N - A ) <_ I <-> ( I - ( N - A ) ) e. NN0 ) ) |
35 |
15 34
|
mpbid |
|- ( ( ( ( A e. NN0 /\ A < N ) /\ ( I e. NN0 /\ N e. NN /\ I < N ) ) /\ -. I < ( N - A ) ) -> ( I - ( N - A ) ) e. NN0 ) |
36 |
|
elnn0uz |
|- ( ( I - ( N - A ) ) e. NN0 <-> ( I - ( N - A ) ) e. ( ZZ>= ` 0 ) ) |
37 |
35 36
|
sylib |
|- ( ( ( ( A e. NN0 /\ A < N ) /\ ( I e. NN0 /\ N e. NN /\ I < N ) ) /\ -. I < ( N - A ) ) -> ( I - ( N - A ) ) e. ( ZZ>= ` 0 ) ) |
38 |
19
|
adantr |
|- ( ( ( A e. NN0 /\ A < N ) /\ ( I e. NN0 /\ N e. NN /\ I < N ) ) -> A e. ZZ ) |
39 |
38
|
adantr |
|- ( ( ( ( A e. NN0 /\ A < N ) /\ ( I e. NN0 /\ N e. NN /\ I < N ) ) /\ -. I < ( N - A ) ) -> A e. ZZ ) |
40 |
9
|
adantr |
|- ( ( I e. NN0 /\ N e. NN ) -> I e. RR ) |
41 |
40
|
adantl |
|- ( ( A e. NN0 /\ ( I e. NN0 /\ N e. NN ) ) -> I e. RR ) |
42 |
3
|
adantl |
|- ( ( I e. NN0 /\ N e. NN ) -> N e. RR ) |
43 |
42
|
adantl |
|- ( ( A e. NN0 /\ ( I e. NN0 /\ N e. NN ) ) -> N e. RR ) |
44 |
42 5 7
|
syl2anr |
|- ( ( A e. NN0 /\ ( I e. NN0 /\ N e. NN ) ) -> ( N - A ) e. RR ) |
45 |
41 43 44
|
ltsub1d |
|- ( ( A e. NN0 /\ ( I e. NN0 /\ N e. NN ) ) -> ( I < N <-> ( I - ( N - A ) ) < ( N - ( N - A ) ) ) ) |
46 |
|
nncn |
|- ( N e. NN -> N e. CC ) |
47 |
46
|
adantl |
|- ( ( I e. NN0 /\ N e. NN ) -> N e. CC ) |
48 |
|
nn0cn |
|- ( A e. NN0 -> A e. CC ) |
49 |
|
nncan |
|- ( ( N e. CC /\ A e. CC ) -> ( N - ( N - A ) ) = A ) |
50 |
47 48 49
|
syl2anr |
|- ( ( A e. NN0 /\ ( I e. NN0 /\ N e. NN ) ) -> ( N - ( N - A ) ) = A ) |
51 |
50
|
breq2d |
|- ( ( A e. NN0 /\ ( I e. NN0 /\ N e. NN ) ) -> ( ( I - ( N - A ) ) < ( N - ( N - A ) ) <-> ( I - ( N - A ) ) < A ) ) |
52 |
51
|
biimpd |
|- ( ( A e. NN0 /\ ( I e. NN0 /\ N e. NN ) ) -> ( ( I - ( N - A ) ) < ( N - ( N - A ) ) -> ( I - ( N - A ) ) < A ) ) |
53 |
45 52
|
sylbid |
|- ( ( A e. NN0 /\ ( I e. NN0 /\ N e. NN ) ) -> ( I < N -> ( I - ( N - A ) ) < A ) ) |
54 |
53
|
ex |
|- ( A e. NN0 -> ( ( I e. NN0 /\ N e. NN ) -> ( I < N -> ( I - ( N - A ) ) < A ) ) ) |
55 |
54
|
adantr |
|- ( ( A e. NN0 /\ A < N ) -> ( ( I e. NN0 /\ N e. NN ) -> ( I < N -> ( I - ( N - A ) ) < A ) ) ) |
56 |
55
|
com3l |
|- ( ( I e. NN0 /\ N e. NN ) -> ( I < N -> ( ( A e. NN0 /\ A < N ) -> ( I - ( N - A ) ) < A ) ) ) |
57 |
56
|
3impia |
|- ( ( I e. NN0 /\ N e. NN /\ I < N ) -> ( ( A e. NN0 /\ A < N ) -> ( I - ( N - A ) ) < A ) ) |
58 |
57
|
impcom |
|- ( ( ( A e. NN0 /\ A < N ) /\ ( I e. NN0 /\ N e. NN /\ I < N ) ) -> ( I - ( N - A ) ) < A ) |
59 |
58
|
adantr |
|- ( ( ( ( A e. NN0 /\ A < N ) /\ ( I e. NN0 /\ N e. NN /\ I < N ) ) /\ -. I < ( N - A ) ) -> ( I - ( N - A ) ) < A ) |
60 |
37 39 59
|
3jca |
|- ( ( ( ( A e. NN0 /\ A < N ) /\ ( I e. NN0 /\ N e. NN /\ I < N ) ) /\ -. I < ( N - A ) ) -> ( ( I - ( N - A ) ) e. ( ZZ>= ` 0 ) /\ A e. ZZ /\ ( I - ( N - A ) ) < A ) ) |
61 |
60
|
exp31 |
|- ( ( A e. NN0 /\ A < N ) -> ( ( I e. NN0 /\ N e. NN /\ I < N ) -> ( -. I < ( N - A ) -> ( ( I - ( N - A ) ) e. ( ZZ>= ` 0 ) /\ A e. ZZ /\ ( I - ( N - A ) ) < A ) ) ) ) |
62 |
2 61
|
syl5bi |
|- ( ( A e. NN0 /\ A < N ) -> ( I e. ( 0 ..^ N ) -> ( -. I < ( N - A ) -> ( ( I - ( N - A ) ) e. ( ZZ>= ` 0 ) /\ A e. ZZ /\ ( I - ( N - A ) ) < A ) ) ) ) |
63 |
62
|
3adant2 |
|- ( ( A e. NN0 /\ N e. NN /\ A < N ) -> ( I e. ( 0 ..^ N ) -> ( -. I < ( N - A ) -> ( ( I - ( N - A ) ) e. ( ZZ>= ` 0 ) /\ A e. ZZ /\ ( I - ( N - A ) ) < A ) ) ) ) |
64 |
1 63
|
sylbi |
|- ( A e. ( 0 ..^ N ) -> ( I e. ( 0 ..^ N ) -> ( -. I < ( N - A ) -> ( ( I - ( N - A ) ) e. ( ZZ>= ` 0 ) /\ A e. ZZ /\ ( I - ( N - A ) ) < A ) ) ) ) |
65 |
64
|
3imp |
|- ( ( A e. ( 0 ..^ N ) /\ I e. ( 0 ..^ N ) /\ -. I < ( N - A ) ) -> ( ( I - ( N - A ) ) e. ( ZZ>= ` 0 ) /\ A e. ZZ /\ ( I - ( N - A ) ) < A ) ) |
66 |
|
elfzo2 |
|- ( ( I - ( N - A ) ) e. ( 0 ..^ A ) <-> ( ( I - ( N - A ) ) e. ( ZZ>= ` 0 ) /\ A e. ZZ /\ ( I - ( N - A ) ) < A ) ) |
67 |
65 66
|
sylibr |
|- ( ( A e. ( 0 ..^ N ) /\ I e. ( 0 ..^ N ) /\ -. I < ( N - A ) ) -> ( I - ( N - A ) ) e. ( 0 ..^ A ) ) |