Metamath Proof Explorer

Theorem subsub4

Description: Law for double subtraction. (Contributed by NM, 19-Aug-2005) (Revised by Mario Carneiro, 27-May-2016)

Ref Expression
Assertion subsub4 
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A - B ) - C ) = ( A - ( B + C ) ) )

Proof

Step Hyp Ref Expression
1 nppcan2 
 |-  ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A - ( B + C ) ) + C ) = ( A - B ) )
2 simp1 
 |-  ( ( A e. CC /\ B e. CC /\ C e. CC ) -> A e. CC )
3 simp2 
 |-  ( ( A e. CC /\ B e. CC /\ C e. CC ) -> B e. CC )
4 subcl 
 |-  ( ( A e. CC /\ B e. CC ) -> ( A - B ) e. CC )
5 2 3 4 syl2anc 
 |-  ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A - B ) e. CC )
6 simp3 
 |-  ( ( A e. CC /\ B e. CC /\ C e. CC ) -> C e. CC )
7 3 6 addcld 
 |-  ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( B + C ) e. CC )
8 subcl 
 |-  ( ( A e. CC /\ ( B + C ) e. CC ) -> ( A - ( B + C ) ) e. CC )
9 2 7 8 syl2anc 
 |-  ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A - ( B + C ) ) e. CC )
10 subadd2 
 |-  ( ( ( A - B ) e. CC /\ C e. CC /\ ( A - ( B + C ) ) e. CC ) -> ( ( ( A - B ) - C ) = ( A - ( B + C ) ) <-> ( ( A - ( B + C ) ) + C ) = ( A - B ) ) )
11 5 6 9 10 syl3anc 
 |-  ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( ( A - B ) - C ) = ( A - ( B + C ) ) <-> ( ( A - ( B + C ) ) + C ) = ( A - B ) ) )
12 1 11 mpbird 
 |-  ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A - B ) - C ) = ( A - ( B + C ) ) )