Step |
Hyp |
Ref |
Expression |
1 |
|
olc |
|- ( R = 2 -> ( ( P = 2 \/ Q = 2 ) \/ R = 2 ) ) |
2 |
1
|
a1d |
|- ( R = 2 -> ( ( N e. Even /\ ( P e. Prime /\ Q e. Prime /\ R e. Prime ) /\ N = ( ( P + Q ) + R ) ) -> ( ( P = 2 \/ Q = 2 ) \/ R = 2 ) ) ) |
3 |
|
df-ne |
|- ( R =/= 2 <-> -. R = 2 ) |
4 |
|
eldifsn |
|- ( R e. ( Prime \ { 2 } ) <-> ( R e. Prime /\ R =/= 2 ) ) |
5 |
|
oddprmALTV |
|- ( R e. ( Prime \ { 2 } ) -> R e. Odd ) |
6 |
|
emoo |
|- ( ( N e. Even /\ R e. Odd ) -> ( N - R ) e. Odd ) |
7 |
6
|
expcom |
|- ( R e. Odd -> ( N e. Even -> ( N - R ) e. Odd ) ) |
8 |
5 7
|
syl |
|- ( R e. ( Prime \ { 2 } ) -> ( N e. Even -> ( N - R ) e. Odd ) ) |
9 |
4 8
|
sylbir |
|- ( ( R e. Prime /\ R =/= 2 ) -> ( N e. Even -> ( N - R ) e. Odd ) ) |
10 |
9
|
ex |
|- ( R e. Prime -> ( R =/= 2 -> ( N e. Even -> ( N - R ) e. Odd ) ) ) |
11 |
3 10
|
syl5bir |
|- ( R e. Prime -> ( -. R = 2 -> ( N e. Even -> ( N - R ) e. Odd ) ) ) |
12 |
11
|
com23 |
|- ( R e. Prime -> ( N e. Even -> ( -. R = 2 -> ( N - R ) e. Odd ) ) ) |
13 |
12
|
3ad2ant3 |
|- ( ( P e. Prime /\ Q e. Prime /\ R e. Prime ) -> ( N e. Even -> ( -. R = 2 -> ( N - R ) e. Odd ) ) ) |
14 |
13
|
impcom |
|- ( ( N e. Even /\ ( P e. Prime /\ Q e. Prime /\ R e. Prime ) ) -> ( -. R = 2 -> ( N - R ) e. Odd ) ) |
15 |
14
|
3adant3 |
|- ( ( N e. Even /\ ( P e. Prime /\ Q e. Prime /\ R e. Prime ) /\ N = ( ( P + Q ) + R ) ) -> ( -. R = 2 -> ( N - R ) e. Odd ) ) |
16 |
15
|
impcom |
|- ( ( -. R = 2 /\ ( N e. Even /\ ( P e. Prime /\ Q e. Prime /\ R e. Prime ) /\ N = ( ( P + Q ) + R ) ) ) -> ( N - R ) e. Odd ) |
17 |
|
3simpa |
|- ( ( P e. Prime /\ Q e. Prime /\ R e. Prime ) -> ( P e. Prime /\ Q e. Prime ) ) |
18 |
17
|
3ad2ant2 |
|- ( ( N e. Even /\ ( P e. Prime /\ Q e. Prime /\ R e. Prime ) /\ N = ( ( P + Q ) + R ) ) -> ( P e. Prime /\ Q e. Prime ) ) |
19 |
18
|
adantl |
|- ( ( -. R = 2 /\ ( N e. Even /\ ( P e. Prime /\ Q e. Prime /\ R e. Prime ) /\ N = ( ( P + Q ) + R ) ) ) -> ( P e. Prime /\ Q e. Prime ) ) |
20 |
|
eqcom |
|- ( N = ( ( P + Q ) + R ) <-> ( ( P + Q ) + R ) = N ) |
21 |
|
evenz |
|- ( N e. Even -> N e. ZZ ) |
22 |
21
|
zcnd |
|- ( N e. Even -> N e. CC ) |
23 |
22
|
adantr |
|- ( ( N e. Even /\ ( P e. Prime /\ Q e. Prime /\ R e. Prime ) ) -> N e. CC ) |
24 |
|
prmz |
|- ( R e. Prime -> R e. ZZ ) |
25 |
24
|
zcnd |
|- ( R e. Prime -> R e. CC ) |
26 |
25
|
3ad2ant3 |
|- ( ( P e. Prime /\ Q e. Prime /\ R e. Prime ) -> R e. CC ) |
27 |
26
|
adantl |
|- ( ( N e. Even /\ ( P e. Prime /\ Q e. Prime /\ R e. Prime ) ) -> R e. CC ) |
28 |
|
prmz |
|- ( P e. Prime -> P e. ZZ ) |
29 |
|
prmz |
|- ( Q e. Prime -> Q e. ZZ ) |
30 |
|
zaddcl |
|- ( ( P e. ZZ /\ Q e. ZZ ) -> ( P + Q ) e. ZZ ) |
31 |
28 29 30
|
syl2an |
|- ( ( P e. Prime /\ Q e. Prime ) -> ( P + Q ) e. ZZ ) |
32 |
31
|
zcnd |
|- ( ( P e. Prime /\ Q e. Prime ) -> ( P + Q ) e. CC ) |
33 |
32
|
3adant3 |
|- ( ( P e. Prime /\ Q e. Prime /\ R e. Prime ) -> ( P + Q ) e. CC ) |
34 |
33
|
adantl |
|- ( ( N e. Even /\ ( P e. Prime /\ Q e. Prime /\ R e. Prime ) ) -> ( P + Q ) e. CC ) |
35 |
23 27 34
|
subadd2d |
|- ( ( N e. Even /\ ( P e. Prime /\ Q e. Prime /\ R e. Prime ) ) -> ( ( N - R ) = ( P + Q ) <-> ( ( P + Q ) + R ) = N ) ) |
36 |
35
|
biimprd |
|- ( ( N e. Even /\ ( P e. Prime /\ Q e. Prime /\ R e. Prime ) ) -> ( ( ( P + Q ) + R ) = N -> ( N - R ) = ( P + Q ) ) ) |
37 |
20 36
|
syl5bi |
|- ( ( N e. Even /\ ( P e. Prime /\ Q e. Prime /\ R e. Prime ) ) -> ( N = ( ( P + Q ) + R ) -> ( N - R ) = ( P + Q ) ) ) |
38 |
37
|
3impia |
|- ( ( N e. Even /\ ( P e. Prime /\ Q e. Prime /\ R e. Prime ) /\ N = ( ( P + Q ) + R ) ) -> ( N - R ) = ( P + Q ) ) |
39 |
38
|
adantl |
|- ( ( -. R = 2 /\ ( N e. Even /\ ( P e. Prime /\ Q e. Prime /\ R e. Prime ) /\ N = ( ( P + Q ) + R ) ) ) -> ( N - R ) = ( P + Q ) ) |
40 |
|
odd2prm2 |
|- ( ( ( N - R ) e. Odd /\ ( P e. Prime /\ Q e. Prime ) /\ ( N - R ) = ( P + Q ) ) -> ( P = 2 \/ Q = 2 ) ) |
41 |
16 19 39 40
|
syl3anc |
|- ( ( -. R = 2 /\ ( N e. Even /\ ( P e. Prime /\ Q e. Prime /\ R e. Prime ) /\ N = ( ( P + Q ) + R ) ) ) -> ( P = 2 \/ Q = 2 ) ) |
42 |
41
|
orcd |
|- ( ( -. R = 2 /\ ( N e. Even /\ ( P e. Prime /\ Q e. Prime /\ R e. Prime ) /\ N = ( ( P + Q ) + R ) ) ) -> ( ( P = 2 \/ Q = 2 ) \/ R = 2 ) ) |
43 |
42
|
ex |
|- ( -. R = 2 -> ( ( N e. Even /\ ( P e. Prime /\ Q e. Prime /\ R e. Prime ) /\ N = ( ( P + Q ) + R ) ) -> ( ( P = 2 \/ Q = 2 ) \/ R = 2 ) ) ) |
44 |
2 43
|
pm2.61i |
|- ( ( N e. Even /\ ( P e. Prime /\ Q e. Prime /\ R e. Prime ) /\ N = ( ( P + Q ) + R ) ) -> ( ( P = 2 \/ Q = 2 ) \/ R = 2 ) ) |
45 |
|
df-3or |
|- ( ( P = 2 \/ Q = 2 \/ R = 2 ) <-> ( ( P = 2 \/ Q = 2 ) \/ R = 2 ) ) |
46 |
44 45
|
sylibr |
|- ( ( N e. Even /\ ( P e. Prime /\ Q e. Prime /\ R e. Prime ) /\ N = ( ( P + Q ) + R ) ) -> ( P = 2 \/ Q = 2 \/ R = 2 ) ) |