Step | Hyp | Ref | Expression |
1 |
|
psetsep |
E. a pset a == {y | y < n /\ y e. A} |
2 |
|
bian1a |
(y e. A -> y < n) -> (y < n /\ y e. A <-> y e. A) |
3 |
|
eleq1 |
x = y -> (x e. A <-> y e. A) |
4 |
|
lteq1 |
x = y -> (x < n <-> y < n) |
5 |
3, 4 |
imeqd |
x = y -> (x e. A -> x < n <-> y e. A -> y < n) |
6 |
5 |
eale |
A. x (x e. A -> x < n) -> y e. A -> y < n |
7 |
2, 6 |
syl |
A. x (x e. A -> x < n) -> (y < n /\ y e. A <-> y e. A) |
8 |
7 |
eqab1d |
A. x (x e. A -> x < n) -> {y | y < n /\ y e. A} == A |
9 |
8 |
eqseq2d |
A. x (x e. A -> x < n) -> (pset a == {y | y < n /\ y e. A} <-> pset a == A) |
10 |
9 |
exeqd |
A. x (x e. A -> x < n) -> (E. a pset a == {y | y < n /\ y e. A} <-> E. a pset a == A) |
11 |
1, 10 |
mpbii |
A. x (x e. A -> x < n) -> E. a pset a == A |
12 |
11 |
eex |
E. n A. x (x e. A -> x < n) -> E. a pset a == A |
13 |
|
lesucid |
fst a * suc x <= suc (fst a * suc x) |
14 |
|
letr |
suc x <= fst a * suc x -> fst a * suc x <= suc (fst a * suc x) -> suc x <= suc (fst a * suc x) |
15 |
13, 14 |
mpi |
suc x <= fst a * suc x -> suc x <= suc (fst a * suc x) |
16 |
|
leeq1 |
1 * suc x = suc x -> (1 * suc x <= fst a * suc x <-> suc x <= fst a * suc x) |
17 |
|
mul11 |
1 * suc x = suc x |
18 |
16, 17 |
ax_mp |
1 * suc x <= fst a * suc x <-> suc x <= fst a * suc x |
19 |
|
lemul1a |
1 <= fst a -> 1 * suc x <= fst a * suc x |
20 |
|
an3l |
1 <= fst a /\ 0 < snd a /\ A. y (0 < y /\ y <= x -> y || fst a) /\ suc (fst a * suc x) || snd a -> 1 <= fst a |
21 |
|
elpset |
x e. pset (fst a, snd a) <-> 0 < fst a /\ 0 < snd a /\ A. y (0 < y /\ y <= x -> y || fst a) /\ suc (fst a * suc x) || snd a |
22 |
|
pseteq |
fst a, snd a = a -> pset (fst a, snd a) == pset a |
23 |
|
fstsnd |
fst a, snd a = a |
24 |
22, 23 |
ax_mp |
pset (fst a, snd a) == pset a |
25 |
24 |
a1i |
pset a == A /\ n = snd a /\ x e. A -> pset (fst a, snd a) == pset a |
26 |
|
anll |
pset a == A /\ n = snd a /\ x e. A -> pset a == A |
27 |
25, 26 |
eqstrd |
pset a == A /\ n = snd a /\ x e. A -> pset (fst a, snd a) == A |
28 |
27 |
eleq2d |
pset a == A /\ n = snd a /\ x e. A -> (x e. pset (fst a, snd a) <-> x e. A) |
29 |
|
anr |
pset a == A /\ n = snd a /\ x e. A -> x e. A |
30 |
28, 29 |
mpbird |
pset a == A /\ n = snd a /\ x e. A -> x e. pset (fst a, snd a) |
31 |
21, 30 |
sylib |
pset a == A /\ n = snd a /\ x e. A -> 0 < fst a /\ 0 < snd a /\ A. y (0 < y /\ y <= x -> y || fst a) /\ suc (fst a * suc x) || snd a |
32 |
31 |
conv d1, lt |
pset a == A /\ n = snd a /\ x e. A -> 1 <= fst a /\ 0 < snd a /\ A. y (0 < y /\ y <= x -> y || fst a) /\ suc (fst a * suc x) || snd a |
33 |
20, 32 |
syl |
pset a == A /\ n = snd a /\ x e. A -> 1 <= fst a |
34 |
19, 33 |
syl |
pset a == A /\ n = snd a /\ x e. A -> 1 * suc x <= fst a * suc x |
35 |
18, 34 |
sylib |
pset a == A /\ n = snd a /\ x e. A -> suc x <= fst a * suc x |
36 |
15, 35 |
syl |
pset a == A /\ n = snd a /\ x e. A -> suc x <= suc (fst a * suc x) |
37 |
|
ltner |
0 < snd a -> snd a != 0 |
38 |
|
anllr |
0 < fst a /\ 0 < snd a /\ A. y (0 < y /\ y <= x -> y || fst a) /\ suc (fst a * suc x) || snd a -> 0 < snd a |
39 |
38, 31 |
syl |
pset a == A /\ n = snd a /\ x e. A -> 0 < snd a |
40 |
37, 39 |
syl |
pset a == A /\ n = snd a /\ x e. A -> snd a != 0 |
41 |
31 |
anrd |
pset a == A /\ n = snd a /\ x e. A -> suc (fst a * suc x) || snd a |
42 |
40, 41 |
dvdle |
pset a == A /\ n = snd a /\ x e. A -> suc (fst a * suc x) <= snd a |
43 |
|
eqler |
n = snd a -> snd a <= n |
44 |
|
anlr |
pset a == A /\ n = snd a /\ x e. A -> n = snd a |
45 |
43, 44 |
syl |
pset a == A /\ n = snd a /\ x e. A -> snd a <= n |
46 |
42, 45 |
letrd |
pset a == A /\ n = snd a /\ x e. A -> suc (fst a * suc x) <= n |
47 |
36, 46 |
letrd |
pset a == A /\ n = snd a /\ x e. A -> suc x <= n |
48 |
47 |
conv lt |
pset a == A /\ n = snd a /\ x e. A -> x < n |
49 |
48 |
ialda |
pset a == A /\ n = snd a -> A. x (x e. A -> x < n) |
50 |
49 |
iexde |
pset a == A -> E. n A. x (x e. A -> x < n) |
51 |
50 |
eex |
E. a pset a == A -> E. n A. x (x e. A -> x < n) |
52 |
12, 51 |
ibii |
E. n A. x (x e. A -> x < n) <-> E. a pset a == A |
53 |
52 |
conv finite |
finite A <-> E. a pset a == A |