Metamath Proof Explorer


Theorem submul2

Description: Convert a subtraction to addition using multiplication by a negative. (Contributed by NM, 2-Feb-2007)

Ref Expression
Assertion submul2 ABCABC=A+BC

Proof

Step Hyp Ref Expression
1 mulneg2 BCBC=BC
2 1 adantl ABCBC=BC
3 2 oveq2d ABCA+BC=A+BC
4 mulcl BCBC
5 negsub ABCA+BC=ABC
6 4 5 sylan2 ABCA+BC=ABC
7 3 6 eqtr2d ABCABC=A+BC
8 7 3impb ABCABC=A+BC