Description: Cancellation law for addition. Theorem I.1 of Apostol p. 18. (Contributed by Mario Carneiro, 27-May-2016)
Ref | Expression | ||
---|---|---|---|
Hypotheses | muld.1 | |- ( ph -> A e. CC ) |
|
addcomd.2 | |- ( ph -> B e. CC ) |
||
addcand.3 | |- ( ph -> C e. CC ) |
||
Assertion | addcand | |- ( ph -> ( ( A + B ) = ( A + C ) <-> B = C ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | muld.1 | |- ( ph -> A e. CC ) |
|
2 | addcomd.2 | |- ( ph -> B e. CC ) |
|
3 | addcand.3 | |- ( ph -> C e. CC ) |
|
4 | addcan | |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) = ( A + C ) <-> B = C ) ) |
|
5 | 1 2 3 4 | syl3anc | |- ( ph -> ( ( A + B ) = ( A + C ) <-> B = C ) ) |