Metamath Proof Explorer


Theorem divdivdivd

Description: Division of two ratios. Theorem I.15 of Apostol p. 18. (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 )
divdivdivd.7
|- ( ph -> C =/= 0 )
Assertion divdivdivd
|- ( ph -> ( ( A / B ) / ( C / D ) ) = ( ( A x. D ) / ( B x. C ) ) )

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 divdivdivd.7
 |-  ( ph -> C =/= 0 )
8 2 5 jca
 |-  ( ph -> ( B e. CC /\ B =/= 0 ) )
9 3 7 jca
 |-  ( ph -> ( C e. CC /\ C =/= 0 ) )
10 4 6 jca
 |-  ( ph -> ( D e. CC /\ D =/= 0 ) )
11 divdivdiv
 |-  ( ( ( A e. CC /\ ( B e. CC /\ B =/= 0 ) ) /\ ( ( C e. CC /\ C =/= 0 ) /\ ( D e. CC /\ D =/= 0 ) ) ) -> ( ( A / B ) / ( C / D ) ) = ( ( A x. D ) / ( B x. C ) ) )
12 1 8 9 10 11 syl22anc
 |-  ( ph -> ( ( A / B ) / ( C / D ) ) = ( ( A x. D ) / ( B x. C ) ) )