Metamath Proof Explorer

Theorem divneg2

Description: Move negative sign inside of a division. (Contributed by Mario Carneiro, 15-Sep-2014)

Ref Expression
Assertion divneg2 
|- ( ( A e. CC /\ B e. CC /\ B =/= 0 ) -> -u ( A / B ) = ( A / -u B ) )

Proof

Step Hyp Ref Expression
1 divneg 
 |-  ( ( A e. CC /\ B e. CC /\ B =/= 0 ) -> -u ( A / B ) = ( -u A / B ) )
2 negcl 
 |-  ( A e. CC -> -u A e. CC )
3 div2neg 
 |-  ( ( -u A e. CC /\ B e. CC /\ B =/= 0 ) -> ( -u -u A / -u B ) = ( -u A / B ) )
4 2 3 syl3an1 
 |-  ( ( A e. CC /\ B e. CC /\ B =/= 0 ) -> ( -u -u A / -u B ) = ( -u A / B ) )
5 negneg 
 |-  ( A e. CC -> -u -u A = A )
6 5 3ad2ant1 
 |-  ( ( A e. CC /\ B e. CC /\ B =/= 0 ) -> -u -u A = A )
7 6 oveq1d 
 |-  ( ( A e. CC /\ B e. CC /\ B =/= 0 ) -> ( -u -u A / -u B ) = ( A / -u B ) )
8 1 4 7 3eqtr2d 
 |-  ( ( A e. CC /\ B e. CC /\ B =/= 0 ) -> -u ( A / B ) = ( A / -u B ) )