Step |
Hyp |
Ref |
Expression |
1 |
|
6nn |
|- 6 e. NN |
2 |
1
|
nnzi |
|- 6 e. ZZ |
3 |
|
9nn |
|- 9 e. NN |
4 |
3
|
nnzi |
|- 9 e. ZZ |
5 |
|
neggcd |
|- ( ( 6 e. ZZ /\ 9 e. ZZ ) -> ( -u 6 gcd 9 ) = ( 6 gcd 9 ) ) |
6 |
2 4 5
|
mp2an |
|- ( -u 6 gcd 9 ) = ( 6 gcd 9 ) |
7 |
|
6cn |
|- 6 e. CC |
8 |
|
3cn |
|- 3 e. CC |
9 |
|
6p3e9 |
|- ( 6 + 3 ) = 9 |
10 |
7 8 9
|
addcomli |
|- ( 3 + 6 ) = 9 |
11 |
10
|
eqcomi |
|- 9 = ( 3 + 6 ) |
12 |
11
|
oveq2i |
|- ( 6 gcd 9 ) = ( 6 gcd ( 3 + 6 ) ) |
13 |
|
3z |
|- 3 e. ZZ |
14 |
|
gcdcom |
|- ( ( 6 e. ZZ /\ 3 e. ZZ ) -> ( 6 gcd 3 ) = ( 3 gcd 6 ) ) |
15 |
2 13 14
|
mp2an |
|- ( 6 gcd 3 ) = ( 3 gcd 6 ) |
16 |
|
3p3e6 |
|- ( 3 + 3 ) = 6 |
17 |
16
|
eqcomi |
|- 6 = ( 3 + 3 ) |
18 |
17
|
oveq2i |
|- ( 3 gcd 6 ) = ( 3 gcd ( 3 + 3 ) ) |
19 |
15 18
|
eqtri |
|- ( 6 gcd 3 ) = ( 3 gcd ( 3 + 3 ) ) |
20 |
|
gcdadd |
|- ( ( 6 e. ZZ /\ 3 e. ZZ ) -> ( 6 gcd 3 ) = ( 6 gcd ( 3 + 6 ) ) ) |
21 |
2 13 20
|
mp2an |
|- ( 6 gcd 3 ) = ( 6 gcd ( 3 + 6 ) ) |
22 |
|
gcdid |
|- ( 3 e. ZZ -> ( 3 gcd 3 ) = ( abs ` 3 ) ) |
23 |
13 22
|
ax-mp |
|- ( 3 gcd 3 ) = ( abs ` 3 ) |
24 |
|
gcdadd |
|- ( ( 3 e. ZZ /\ 3 e. ZZ ) -> ( 3 gcd 3 ) = ( 3 gcd ( 3 + 3 ) ) ) |
25 |
13 13 24
|
mp2an |
|- ( 3 gcd 3 ) = ( 3 gcd ( 3 + 3 ) ) |
26 |
|
3re |
|- 3 e. RR |
27 |
|
0re |
|- 0 e. RR |
28 |
|
3pos |
|- 0 < 3 |
29 |
27 26 28
|
ltleii |
|- 0 <_ 3 |
30 |
|
absid |
|- ( ( 3 e. RR /\ 0 <_ 3 ) -> ( abs ` 3 ) = 3 ) |
31 |
26 29 30
|
mp2an |
|- ( abs ` 3 ) = 3 |
32 |
23 25 31
|
3eqtr3i |
|- ( 3 gcd ( 3 + 3 ) ) = 3 |
33 |
19 21 32
|
3eqtr3i |
|- ( 6 gcd ( 3 + 6 ) ) = 3 |
34 |
12 33
|
eqtri |
|- ( 6 gcd 9 ) = 3 |
35 |
6 34
|
eqtri |
|- ( -u 6 gcd 9 ) = 3 |