Step |
Hyp |
Ref |
Expression |
1 |
|
gcdcom |
|- ( ( B e. ZZ /\ C e. ZZ ) -> ( B gcd C ) = ( C gcd B ) ) |
2 |
1
|
3adant1 |
|- ( ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) -> ( B gcd C ) = ( C gcd B ) ) |
3 |
2
|
oveq2d |
|- ( ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) -> ( A gcd ( B gcd C ) ) = ( A gcd ( C gcd B ) ) ) |
4 |
|
gcdass |
|- ( ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) -> ( ( A gcd B ) gcd C ) = ( A gcd ( B gcd C ) ) ) |
5 |
|
gcdass |
|- ( ( A e. ZZ /\ C e. ZZ /\ B e. ZZ ) -> ( ( A gcd C ) gcd B ) = ( A gcd ( C gcd B ) ) ) |
6 |
5
|
3com23 |
|- ( ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) -> ( ( A gcd C ) gcd B ) = ( A gcd ( C gcd B ) ) ) |
7 |
3 4 6
|
3eqtr4d |
|- ( ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) -> ( ( A gcd B ) gcd C ) = ( ( A gcd C ) gcd B ) ) |