Metamath Proof Explorer


Theorem divsubdivd

Description: Subtraction of two ratios. (Contributed by Mario Carneiro, 27-May-2016)

Ref Expression
Hypotheses div1d.1
|- ( ph -> A e. CC )
divcld.2
|- ( ph -> B e. CC )
divmuld.3
|- ( ph -> C e. CC )
divmuldivd.4
|- ( ph -> D e. CC )
divmuldivd.5
|- ( ph -> B =/= 0 )
divmuldivd.6
|- ( ph -> D =/= 0 )
Assertion divsubdivd
|- ( ph -> ( ( A / B ) - ( C / D ) ) = ( ( ( A x. D ) - ( C x. B ) ) / ( B x. D ) ) )

Proof

Step Hyp Ref Expression
1 div1d.1
 |-  ( ph -> A e. CC )
2 divcld.2
 |-  ( ph -> B e. CC )
3 divmuld.3
 |-  ( ph -> C e. CC )
4 divmuldivd.4
 |-  ( ph -> D e. CC )
5 divmuldivd.5
 |-  ( ph -> B =/= 0 )
6 divmuldivd.6
 |-  ( ph -> D =/= 0 )
7 2 5 jca
 |-  ( ph -> ( B e. CC /\ B =/= 0 ) )
8 4 6 jca
 |-  ( ph -> ( D e. CC /\ D =/= 0 ) )
9 divsubdiv
 |-  ( ( ( A e. CC /\ C e. CC ) /\ ( ( B e. CC /\ B =/= 0 ) /\ ( D e. CC /\ D =/= 0 ) ) ) -> ( ( A / B ) - ( C / D ) ) = ( ( ( A x. D ) - ( C x. B ) ) / ( B x. D ) ) )
10 1 3 7 8 9 syl22anc
 |-  ( ph -> ( ( A / B ) - ( C / D ) ) = ( ( ( A x. D ) - ( C x. B ) ) / ( B x. D ) ) )