Step |
Hyp |
Ref |
Expression |
1 |
|
2z |
|- 2 e. ZZ |
2 |
|
evenz |
|- ( n e. Even -> n e. ZZ ) |
3 |
|
zltp1le |
|- ( ( 2 e. ZZ /\ n e. ZZ ) -> ( 2 < n <-> ( 2 + 1 ) <_ n ) ) |
4 |
1 2 3
|
sylancr |
|- ( n e. Even -> ( 2 < n <-> ( 2 + 1 ) <_ n ) ) |
5 |
|
2p1e3 |
|- ( 2 + 1 ) = 3 |
6 |
5
|
breq1i |
|- ( ( 2 + 1 ) <_ n <-> 3 <_ n ) |
7 |
|
3re |
|- 3 e. RR |
8 |
7
|
a1i |
|- ( n e. Even -> 3 e. RR ) |
9 |
2
|
zred |
|- ( n e. Even -> n e. RR ) |
10 |
8 9
|
leloed |
|- ( n e. Even -> ( 3 <_ n <-> ( 3 < n \/ 3 = n ) ) ) |
11 |
|
3z |
|- 3 e. ZZ |
12 |
|
zltp1le |
|- ( ( 3 e. ZZ /\ n e. ZZ ) -> ( 3 < n <-> ( 3 + 1 ) <_ n ) ) |
13 |
11 2 12
|
sylancr |
|- ( n e. Even -> ( 3 < n <-> ( 3 + 1 ) <_ n ) ) |
14 |
|
3p1e4 |
|- ( 3 + 1 ) = 4 |
15 |
14
|
breq1i |
|- ( ( 3 + 1 ) <_ n <-> 4 <_ n ) |
16 |
|
4re |
|- 4 e. RR |
17 |
16
|
a1i |
|- ( n e. Even -> 4 e. RR ) |
18 |
17 9
|
leloed |
|- ( n e. Even -> ( 4 <_ n <-> ( 4 < n \/ 4 = n ) ) ) |
19 |
|
pm3.35 |
|- ( ( 4 < n /\ ( 4 < n -> n e. GoldbachEven ) ) -> n e. GoldbachEven ) |
20 |
|
isgbe |
|- ( n e. GoldbachEven <-> ( n e. Even /\ E. p e. Prime E. q e. Prime ( p e. Odd /\ q e. Odd /\ n = ( p + q ) ) ) ) |
21 |
|
simp3 |
|- ( ( p e. Odd /\ q e. Odd /\ n = ( p + q ) ) -> n = ( p + q ) ) |
22 |
21
|
a1i |
|- ( ( ( n e. Even /\ p e. Prime ) /\ q e. Prime ) -> ( ( p e. Odd /\ q e. Odd /\ n = ( p + q ) ) -> n = ( p + q ) ) ) |
23 |
22
|
reximdva |
|- ( ( n e. Even /\ p e. Prime ) -> ( E. q e. Prime ( p e. Odd /\ q e. Odd /\ n = ( p + q ) ) -> E. q e. Prime n = ( p + q ) ) ) |
24 |
23
|
reximdva |
|- ( n e. Even -> ( E. p e. Prime E. q e. Prime ( p e. Odd /\ q e. Odd /\ n = ( p + q ) ) -> E. p e. Prime E. q e. Prime n = ( p + q ) ) ) |
25 |
24
|
imp |
|- ( ( n e. Even /\ E. p e. Prime E. q e. Prime ( p e. Odd /\ q e. Odd /\ n = ( p + q ) ) ) -> E. p e. Prime E. q e. Prime n = ( p + q ) ) |
26 |
20 25
|
sylbi |
|- ( n e. GoldbachEven -> E. p e. Prime E. q e. Prime n = ( p + q ) ) |
27 |
26
|
a1d |
|- ( n e. GoldbachEven -> ( n e. Even -> E. p e. Prime E. q e. Prime n = ( p + q ) ) ) |
28 |
19 27
|
syl |
|- ( ( 4 < n /\ ( 4 < n -> n e. GoldbachEven ) ) -> ( n e. Even -> E. p e. Prime E. q e. Prime n = ( p + q ) ) ) |
29 |
28
|
ex |
|- ( 4 < n -> ( ( 4 < n -> n e. GoldbachEven ) -> ( n e. Even -> E. p e. Prime E. q e. Prime n = ( p + q ) ) ) ) |
30 |
29
|
com23 |
|- ( 4 < n -> ( n e. Even -> ( ( 4 < n -> n e. GoldbachEven ) -> E. p e. Prime E. q e. Prime n = ( p + q ) ) ) ) |
31 |
|
2prm |
|- 2 e. Prime |
32 |
|
2p2e4 |
|- ( 2 + 2 ) = 4 |
33 |
32
|
eqcomi |
|- 4 = ( 2 + 2 ) |
34 |
|
rspceov |
|- ( ( 2 e. Prime /\ 2 e. Prime /\ 4 = ( 2 + 2 ) ) -> E. p e. Prime E. q e. Prime 4 = ( p + q ) ) |
35 |
31 31 33 34
|
mp3an |
|- E. p e. Prime E. q e. Prime 4 = ( p + q ) |
36 |
|
eqeq1 |
|- ( 4 = n -> ( 4 = ( p + q ) <-> n = ( p + q ) ) ) |
37 |
36
|
2rexbidv |
|- ( 4 = n -> ( E. p e. Prime E. q e. Prime 4 = ( p + q ) <-> E. p e. Prime E. q e. Prime n = ( p + q ) ) ) |
38 |
35 37
|
mpbii |
|- ( 4 = n -> E. p e. Prime E. q e. Prime n = ( p + q ) ) |
39 |
38
|
a1d |
|- ( 4 = n -> ( ( 4 < n -> n e. GoldbachEven ) -> E. p e. Prime E. q e. Prime n = ( p + q ) ) ) |
40 |
39
|
a1d |
|- ( 4 = n -> ( n e. Even -> ( ( 4 < n -> n e. GoldbachEven ) -> E. p e. Prime E. q e. Prime n = ( p + q ) ) ) ) |
41 |
30 40
|
jaoi |
|- ( ( 4 < n \/ 4 = n ) -> ( n e. Even -> ( ( 4 < n -> n e. GoldbachEven ) -> E. p e. Prime E. q e. Prime n = ( p + q ) ) ) ) |
42 |
41
|
com12 |
|- ( n e. Even -> ( ( 4 < n \/ 4 = n ) -> ( ( 4 < n -> n e. GoldbachEven ) -> E. p e. Prime E. q e. Prime n = ( p + q ) ) ) ) |
43 |
18 42
|
sylbid |
|- ( n e. Even -> ( 4 <_ n -> ( ( 4 < n -> n e. GoldbachEven ) -> E. p e. Prime E. q e. Prime n = ( p + q ) ) ) ) |
44 |
15 43
|
syl5bi |
|- ( n e. Even -> ( ( 3 + 1 ) <_ n -> ( ( 4 < n -> n e. GoldbachEven ) -> E. p e. Prime E. q e. Prime n = ( p + q ) ) ) ) |
45 |
13 44
|
sylbid |
|- ( n e. Even -> ( 3 < n -> ( ( 4 < n -> n e. GoldbachEven ) -> E. p e. Prime E. q e. Prime n = ( p + q ) ) ) ) |
46 |
45
|
com12 |
|- ( 3 < n -> ( n e. Even -> ( ( 4 < n -> n e. GoldbachEven ) -> E. p e. Prime E. q e. Prime n = ( p + q ) ) ) ) |
47 |
|
3odd |
|- 3 e. Odd |
48 |
|
eleq1 |
|- ( 3 = n -> ( 3 e. Odd <-> n e. Odd ) ) |
49 |
47 48
|
mpbii |
|- ( 3 = n -> n e. Odd ) |
50 |
|
oddneven |
|- ( n e. Odd -> -. n e. Even ) |
51 |
49 50
|
syl |
|- ( 3 = n -> -. n e. Even ) |
52 |
51
|
pm2.21d |
|- ( 3 = n -> ( n e. Even -> ( ( 4 < n -> n e. GoldbachEven ) -> E. p e. Prime E. q e. Prime n = ( p + q ) ) ) ) |
53 |
46 52
|
jaoi |
|- ( ( 3 < n \/ 3 = n ) -> ( n e. Even -> ( ( 4 < n -> n e. GoldbachEven ) -> E. p e. Prime E. q e. Prime n = ( p + q ) ) ) ) |
54 |
53
|
com12 |
|- ( n e. Even -> ( ( 3 < n \/ 3 = n ) -> ( ( 4 < n -> n e. GoldbachEven ) -> E. p e. Prime E. q e. Prime n = ( p + q ) ) ) ) |
55 |
10 54
|
sylbid |
|- ( n e. Even -> ( 3 <_ n -> ( ( 4 < n -> n e. GoldbachEven ) -> E. p e. Prime E. q e. Prime n = ( p + q ) ) ) ) |
56 |
6 55
|
syl5bi |
|- ( n e. Even -> ( ( 2 + 1 ) <_ n -> ( ( 4 < n -> n e. GoldbachEven ) -> E. p e. Prime E. q e. Prime n = ( p + q ) ) ) ) |
57 |
4 56
|
sylbid |
|- ( n e. Even -> ( 2 < n -> ( ( 4 < n -> n e. GoldbachEven ) -> E. p e. Prime E. q e. Prime n = ( p + q ) ) ) ) |
58 |
57
|
com23 |
|- ( n e. Even -> ( ( 4 < n -> n e. GoldbachEven ) -> ( 2 < n -> E. p e. Prime E. q e. Prime n = ( p + q ) ) ) ) |
59 |
|
2lt4 |
|- 2 < 4 |
60 |
|
2re |
|- 2 e. RR |
61 |
60
|
a1i |
|- ( n e. Even -> 2 e. RR ) |
62 |
|
lttr |
|- ( ( 2 e. RR /\ 4 e. RR /\ n e. RR ) -> ( ( 2 < 4 /\ 4 < n ) -> 2 < n ) ) |
63 |
61 17 9 62
|
syl3anc |
|- ( n e. Even -> ( ( 2 < 4 /\ 4 < n ) -> 2 < n ) ) |
64 |
59 63
|
mpani |
|- ( n e. Even -> ( 4 < n -> 2 < n ) ) |
65 |
64
|
imp |
|- ( ( n e. Even /\ 4 < n ) -> 2 < n ) |
66 |
|
simpll |
|- ( ( ( n e. Even /\ 4 < n ) /\ E. p e. Prime E. q e. Prime n = ( p + q ) ) -> n e. Even ) |
67 |
|
simpr |
|- ( ( ( n e. Even /\ 4 < n ) /\ p e. Prime ) -> p e. Prime ) |
68 |
67
|
anim1i |
|- ( ( ( ( n e. Even /\ 4 < n ) /\ p e. Prime ) /\ q e. Prime ) -> ( p e. Prime /\ q e. Prime ) ) |
69 |
68
|
adantr |
|- ( ( ( ( ( n e. Even /\ 4 < n ) /\ p e. Prime ) /\ q e. Prime ) /\ n = ( p + q ) ) -> ( p e. Prime /\ q e. Prime ) ) |
70 |
|
simpll |
|- ( ( ( ( n e. Even /\ 4 < n ) /\ p e. Prime ) /\ q e. Prime ) -> ( n e. Even /\ 4 < n ) ) |
71 |
70
|
anim1i |
|- ( ( ( ( ( n e. Even /\ 4 < n ) /\ p e. Prime ) /\ q e. Prime ) /\ n = ( p + q ) ) -> ( ( n e. Even /\ 4 < n ) /\ n = ( p + q ) ) ) |
72 |
|
df-3an |
|- ( ( n e. Even /\ 4 < n /\ n = ( p + q ) ) <-> ( ( n e. Even /\ 4 < n ) /\ n = ( p + q ) ) ) |
73 |
71 72
|
sylibr |
|- ( ( ( ( ( n e. Even /\ 4 < n ) /\ p e. Prime ) /\ q e. Prime ) /\ n = ( p + q ) ) -> ( n e. Even /\ 4 < n /\ n = ( p + q ) ) ) |
74 |
|
sbgoldbaltlem2 |
|- ( ( p e. Prime /\ q e. Prime ) -> ( ( n e. Even /\ 4 < n /\ n = ( p + q ) ) -> ( p e. Odd /\ q e. Odd ) ) ) |
75 |
69 73 74
|
sylc |
|- ( ( ( ( ( n e. Even /\ 4 < n ) /\ p e. Prime ) /\ q e. Prime ) /\ n = ( p + q ) ) -> ( p e. Odd /\ q e. Odd ) ) |
76 |
|
simpr |
|- ( ( ( ( ( n e. Even /\ 4 < n ) /\ p e. Prime ) /\ q e. Prime ) /\ n = ( p + q ) ) -> n = ( p + q ) ) |
77 |
|
df-3an |
|- ( ( p e. Odd /\ q e. Odd /\ n = ( p + q ) ) <-> ( ( p e. Odd /\ q e. Odd ) /\ n = ( p + q ) ) ) |
78 |
75 76 77
|
sylanbrc |
|- ( ( ( ( ( n e. Even /\ 4 < n ) /\ p e. Prime ) /\ q e. Prime ) /\ n = ( p + q ) ) -> ( p e. Odd /\ q e. Odd /\ n = ( p + q ) ) ) |
79 |
78
|
ex |
|- ( ( ( ( n e. Even /\ 4 < n ) /\ p e. Prime ) /\ q e. Prime ) -> ( n = ( p + q ) -> ( p e. Odd /\ q e. Odd /\ n = ( p + q ) ) ) ) |
80 |
79
|
reximdva |
|- ( ( ( n e. Even /\ 4 < n ) /\ p e. Prime ) -> ( E. q e. Prime n = ( p + q ) -> E. q e. Prime ( p e. Odd /\ q e. Odd /\ n = ( p + q ) ) ) ) |
81 |
80
|
reximdva |
|- ( ( n e. Even /\ 4 < n ) -> ( E. p e. Prime E. q e. Prime n = ( p + q ) -> E. p e. Prime E. q e. Prime ( p e. Odd /\ q e. Odd /\ n = ( p + q ) ) ) ) |
82 |
81
|
imp |
|- ( ( ( n e. Even /\ 4 < n ) /\ E. p e. Prime E. q e. Prime n = ( p + q ) ) -> E. p e. Prime E. q e. Prime ( p e. Odd /\ q e. Odd /\ n = ( p + q ) ) ) |
83 |
66 82
|
jca |
|- ( ( ( n e. Even /\ 4 < n ) /\ E. p e. Prime E. q e. Prime n = ( p + q ) ) -> ( n e. Even /\ E. p e. Prime E. q e. Prime ( p e. Odd /\ q e. Odd /\ n = ( p + q ) ) ) ) |
84 |
83
|
ex |
|- ( ( n e. Even /\ 4 < n ) -> ( E. p e. Prime E. q e. Prime n = ( p + q ) -> ( n e. Even /\ E. p e. Prime E. q e. Prime ( p e. Odd /\ q e. Odd /\ n = ( p + q ) ) ) ) ) |
85 |
84 20
|
syl6ibr |
|- ( ( n e. Even /\ 4 < n ) -> ( E. p e. Prime E. q e. Prime n = ( p + q ) -> n e. GoldbachEven ) ) |
86 |
65 85
|
embantd |
|- ( ( n e. Even /\ 4 < n ) -> ( ( 2 < n -> E. p e. Prime E. q e. Prime n = ( p + q ) ) -> n e. GoldbachEven ) ) |
87 |
86
|
ex |
|- ( n e. Even -> ( 4 < n -> ( ( 2 < n -> E. p e. Prime E. q e. Prime n = ( p + q ) ) -> n e. GoldbachEven ) ) ) |
88 |
87
|
com23 |
|- ( n e. Even -> ( ( 2 < n -> E. p e. Prime E. q e. Prime n = ( p + q ) ) -> ( 4 < n -> n e. GoldbachEven ) ) ) |
89 |
58 88
|
impbid |
|- ( n e. Even -> ( ( 4 < n -> n e. GoldbachEven ) <-> ( 2 < n -> E. p e. Prime E. q e. Prime n = ( p + q ) ) ) ) |
90 |
89
|
ralbiia |
|- ( A. n e. Even ( 4 < n -> n e. GoldbachEven ) <-> A. n e. Even ( 2 < n -> E. p e. Prime E. q e. Prime n = ( p + q ) ) ) |