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Metamath Proof Explorer

Theorem sub2times

Description: Subtracting from a number, twice the number itself, gives negative the number. (Contributed by Glauco Siliprandi, 11-Dec-2019)

Ref Expression
Assertion sub2times 
|- ( A e. CC -> ( A - ( 2 x. A ) ) = -u A )

Proof

Step Hyp Ref Expression
1 2times 
 |-  ( A e. CC -> ( 2 x. A ) = ( A + A ) )
2 1 oveq2d 
 |-  ( A e. CC -> ( A - ( 2 x. A ) ) = ( A - ( A + A ) ) )
3 id 
 |-  ( A e. CC -> A e. CC )
4 3 3 addcld 
 |-  ( A e. CC -> ( A + A ) e. CC )
5 3 4 negsubd 
 |-  ( A e. CC -> ( A + -u ( A + A ) ) = ( A - ( A + A ) ) )
6 3 3 negdid 
 |-  ( A e. CC -> -u ( A + A ) = ( -u A + -u A ) )
7 6 oveq2d 
 |-  ( A e. CC -> ( A + -u ( A + A ) ) = ( A + ( -u A + -u A ) ) )
8 negcl 
 |-  ( A e. CC -> -u A e. CC )
9 3 8 8 addassd 
 |-  ( A e. CC -> ( ( A + -u A ) + -u A ) = ( A + ( -u A + -u A ) ) )
10 negid 
 |-  ( A e. CC -> ( A + -u A ) = 0 )
11 10 oveq1d 
 |-  ( A e. CC -> ( ( A + -u A ) + -u A ) = ( 0 + -u A ) )
12 8 addlidd 
 |-  ( A e. CC -> ( 0 + -u A ) = -u A )
13 11 12 eqtrd 
 |-  ( A e. CC -> ( ( A + -u A ) + -u A ) = -u A )
14 7 9 13 3eqtr2d 
 |-  ( A e. CC -> ( A + -u ( A + A ) ) = -u A )
15 2 5 14 3eqtr2d 
 |-  ( A e. CC -> ( A - ( 2 x. A ) ) = -u A )