Step | Hyp | Ref | Expression |
1 |
|
hyp h2 |
G -> nth n l2 = suc b |
2 |
|
hyp h1 |
G -> nth n l1 = suc a |
3 |
|
elall2 |
l1, l2 e. all2 R <-> len l1 = len l2 /\ A. _1 A. _2 A. _3 (nth _1 l1 = suc _2 -> nth _1 l2 = suc _3 -> _2, _3 e. R) |
4 |
|
hyp h |
G -> l1, l2 e. all2 R |
5 |
3, 4 |
sylib |
G -> len l1 = len l2 /\ A. _1 A. _2 A. _3 (nth _1 l1 = suc _2 -> nth _1 l2 = suc _3 -> _2, _3 e. R) |
6 |
5 |
anrd |
G -> A. _1 A. _2 A. _3 (nth _1 l1 = suc _2 -> nth _1 l2 = suc _3 -> _2, _3 e. R) |
7 |
|
id |
_1 = n -> _1 = n |
8 |
7 |
anwr |
G /\ _1 = n -> _1 = n |
9 |
8 |
anwl |
G /\ _1 = n /\ _2 = a -> _1 = n |
10 |
9 |
anwl |
G /\ _1 = n /\ _2 = a /\ _3 = b -> _1 = n |
11 |
|
eqidd |
G /\ _1 = n /\ _2 = a /\ _3 = b -> l1 = l1 |
12 |
10, 11 |
ntheqd |
G /\ _1 = n /\ _2 = a /\ _3 = b -> nth _1 l1 = nth n l1 |
13 |
|
id |
_2 = a -> _2 = a |
14 |
13 |
anwr |
G /\ _1 = n /\ _2 = a -> _2 = a |
15 |
14 |
anwl |
G /\ _1 = n /\ _2 = a /\ _3 = b -> _2 = a |
16 |
15 |
suceqd |
G /\ _1 = n /\ _2 = a /\ _3 = b -> suc _2 = suc a |
17 |
12, 16 |
eqeqd |
G /\ _1 = n /\ _2 = a /\ _3 = b -> (nth _1 l1 = suc _2 <-> nth n l1 = suc a) |
18 |
|
eqidd |
G /\ _1 = n /\ _2 = a /\ _3 = b -> l2 = l2 |
19 |
10, 18 |
ntheqd |
G /\ _1 = n /\ _2 = a /\ _3 = b -> nth _1 l2 = nth n l2 |
20 |
|
id |
_3 = b -> _3 = b |
21 |
20 |
anwr |
G /\ _1 = n /\ _2 = a /\ _3 = b -> _3 = b |
22 |
21 |
suceqd |
G /\ _1 = n /\ _2 = a /\ _3 = b -> suc _3 = suc b |
23 |
19, 22 |
eqeqd |
G /\ _1 = n /\ _2 = a /\ _3 = b -> (nth _1 l2 = suc _3 <-> nth n l2 = suc b) |
24 |
15, 21 |
preqd |
G /\ _1 = n /\ _2 = a /\ _3 = b -> _2, _3 = a, b |
25 |
|
eqsidd |
G /\ _1 = n /\ _2 = a /\ _3 = b -> R == R |
26 |
24, 25 |
eleqd |
G /\ _1 = n /\ _2 = a /\ _3 = b -> (_2, _3 e. R <-> a, b e. R) |
27 |
23, 26 |
imeqd |
G /\ _1 = n /\ _2 = a /\ _3 = b -> (nth _1 l2 = suc _3 -> _2, _3 e. R <-> nth n l2 = suc b -> a, b e. R) |
28 |
17, 27 |
imeqd |
G /\ _1 = n /\ _2 = a /\ _3 = b -> (nth _1 l1 = suc _2 -> nth _1 l2 = suc _3 -> _2, _3 e. R <-> nth n l1 = suc a -> nth n l2 = suc b -> a, b e. R) |
29 |
28 |
bi1d |
G /\ _1 = n /\ _2 = a /\ _3 = b -> (nth _1 l1 = suc _2 -> nth _1 l2 = suc _3 -> _2, _3 e. R) -> nth n l1 = suc a -> nth n l2 = suc b -> a, b e. R |
30 |
29 |
ealde |
G /\ _1 = n /\ _2 = a -> A. _3 (nth _1 l1 = suc _2 -> nth _1 l2 = suc _3 -> _2, _3 e. R) -> nth n l1 = suc a -> nth n l2 = suc b -> a, b e. R |
31 |
30 |
ealde |
G /\ _1 = n -> A. _2 A. _3 (nth _1 l1 = suc _2 -> nth _1 l2 = suc _3 -> _2, _3 e. R) -> nth n l1 = suc a -> nth n l2 = suc b -> a, b e. R |
32 |
31 |
ealde |
G -> A. _1 A. _2 A. _3 (nth _1 l1 = suc _2 -> nth _1 l2 = suc _3 -> _2, _3 e. R) -> nth n l1 = suc a -> nth n l2 = suc b -> a, b e. R |
33 |
6, 32 |
mpd |
G -> nth n l1 = suc a -> nth n l2 = suc b -> a, b e. R |
34 |
2, 33 |
mpd |
G -> nth n l2 = suc b -> a, b e. R |
35 |
1, 34 |
mpd |
G -> a, b e. R |